queueing/0000755000175000017500000000000013635377242012274 5ustar morenomorenoqueueing/DESCRIPTION0000644000175000017500000000170613635377222014004 0ustar morenomorenoName: queueing Version: 1.2.7 Date: 2020-03-21 Author: Moreno Marzolla Maintainer: Moreno Marzolla Title: Octave package for Queueing Networks and Markov chains analysis Description: The queueing package provides functions for queueing networks and Markov chains analysis. This package can be used to compute steady-state performance measures for open, closed and mixed networks with single or multiple job classes. Mean Value Analysis (MVA), convolution, and various bounding techniques are implemented. Furthermore, several transient and steady-state performance measures for Markov chains can be computed, such as state occupancy probabilities, mean time to absorption, time-averaged sojourn times and so forth. Discrete- and continuous-time Markov chains are supported. Categories: Misc Depends: octave (>= 4.0.0) Autoload: no License: GPLv3+ Url: http://www.moreno.marzolla.name/software/queueing/ queueing/COPYING0000644000175000017500000010451313570046274013327 0ustar morenomoreno GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. 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But first, please read . queueing/inst/0000755000175000017500000000000013635377242013251 5ustar morenomorenoqueueing/inst/qnmix.m0000644000175000017500000002120713570046274014561 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmix (@var{lambda}, @var{N}, @var{S}, @var{V}, @var{m}) ## ## @cindex Mean Value Analysys (MVA) ## @cindex mixed network ## ## Mean Value Analysis for mixed queueing networks. The network ## consists of @math{K} service centers (single-server or delay ## centers) and @math{C} independent customer chains. Both open and ## closed chains are possible. @var{lambda} is the vector of per-chain ## arrival rates (open classes); @var{N} is the vector of populations ## for closed chains. ## ## Class switching is @strong{not} allowed. Each customer class ## @emph{must} correspond to an independent chain. ## ## If the network is made of open or closed classes only, then this ## function calls @code{qnom} or @code{qncmmva} respectively, and ## prints a warning message. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda}(c) ## @itemx @var{N}(c) ## For each customer chain @math{c}: ## ## @itemize ## ## @item if @math{c} is a closed chain, then @code{@var{N}(c)>0} is the ## number of class @math{c} requests and @code{@var{lambda}(c)} must be ## zero; ## ## @item If @math{c} is an open chain, ## @code{@var{lambda}(c)>0} is the arrival rate of class @math{c} ## requests and @code{@var{N}(c)} must be zero; ## ## @end itemize ## ## @noindent In other words, for each class @math{c} the following must hold: ## ## @example ## (@var{lambda}(c)>0 && @var{N}(c)==0) || (@var{lambda}(c)==0 && @var{N}(c)>0) ## @end example ## ## @item @var{S}(c,k) ## mean class @math{c} service time at center @math{k}, ## @code{@var{S}(c,k) @geq{} 0}. For FCFS nodes, service times must be ## class-independent. ## ## @item @var{V}(c,k) ## average number of visits of class @math{c} customers to center ## @math{k} (@code{@var{V}(c,k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. Only single-server ## (@code{@var{m}(k)==1}) or IS (Infinite Server) nodes ## (@code{@var{m}(k)<1}) are supported. If omitted, each center is ## assumed to be of type @math{M/M/1}-FCFS. Queueing discipline for ## single-server nodes can be FCFS, PS or LCFS-PR. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(c,k) ## class @math{c} utilization at center @math{k}. ## ## @item @var{R}(c,k) ## class @math{c} response time at center @math{k}. ## ## @item @var{Q}(c,k) ## average number of class @math{c} requests at center @math{k}. ## ## @item @var{X}(c,k) ## class @math{c} throughput at center @math{k}. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. ## Sevcik, @cite{Quantitative System Performance: Computer System ## Analysis Using Queueing Network Models}, Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 7.4.3 ("Mixed Model Solution Techniques"). ## Note that in this function we compute the mean response time @math{R} ## instead of the mean residence time as in the reference. ## ## @item ## Herb Schwetman, @cite{Implementing the Mean Value Algorithm for the ## Solution of Queueing Network Models}, Technical Report ## @uref{http://docs.lib.purdue.edu/cstech/286/, CSD-TR-355}, Department ## of Computer Sciences, Purdue University, revised Feb 15, 1982. ## @end itemize ## ## @seealso{qncmmva, qncm} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnmix( lambda, N, S, V, m ) if ( nargin < 4 || nargin > 5 ) print_usage(); endif isvector(lambda) || ... error( "lambda must be a vector" ); lambda = lambda(:)'; isvector(N) || ... error( "N must be a vector" ); N = N(:)'; size_equal(lambda,N) || ... error( "lambda and N must be of equal length" ); ( !any( lambda>0 & N>0 ) ) || ... error("A class cannot be open and closed at the same time. Check lambda and N" ); ( all( lambda>0 | N>0 ) ) || ... error( "A class cannot be neither open nor closed. Check lambda and N" ); size_equal(S,V) || ... error( "S and V must have the same size" ); C = length(lambda); # number of classes K = columns(S); # number of service centers rows(S) == C || ... error( "S must have %d rows", C ); if ( nargin < 5 ) m = ones(1,K); else isvector( m ) || ... error( "m must be a vector" ); m = m(:)'; size_equal(lambda,m) || ... error( "lambda and m must be of equal length" ); endif all( m<=1 ) || ... error( "This function supports single-server and delay centers only. Check m" ); if ( !any(lambda>0) ) warning( "qnmix(): There are no open classes. Using qncmmva()" ); [U R Q X] = qncmmva( N, S, V, m ); return; endif if ( !any(N>0) ) warning( "qnmix(): There are no closed classes. Using qnom()" ); [U R Q X] = qnom( lambda, S, V, m ); return; endif D = S.*V; # service demands op = find( lambda>0 ); # indexes of open networks cl = find( N>0 ); # indexes of closed networks ## Initialize results U = R = Q = X = zeros(C,K); U(op,:) = diag( lambda(op) )* D(op,:); # U(c,:) = lambda(c)*D(c,:); Uo = sum(U,1); # Total utilization for open classes on service center k ## Build closed model to solve Ncl = N(cl); Scl = S; for c=cl Scl(c,:) = Scl(c,:) ./ (1-Uo); endfor Scl = Scl(cl,:); # select only rows for closed classes Vcl = V(cl,:); [Ucl Rcl Qcl Xcl] = qncmmva(Ncl, Scl, Vcl, m ); ## Results for closed classes X(cl,:) = Xcl; Q(cl,:) = Qcl; R(cl,:) = Rcl; U(cl,:) = X(cl,:) .* D(cl,:); ## Results for open classes Qc = sum(Q(cl,:),1); i_single=find(m==1); i_multi=find(m<1); for c=op R(c,i_single) = S(c,i_single).*(1+Qc) ./ (1-Uo); # This is the Response time, _not_ the residence time R(c,i_multi) = S(c,i_multi); endfor Q(op,:) = (diag(lambda(op))*V(op,:)).*R(op,:); # Q(c,k) = lambda(c)*V(c,k)*R(c,k) ## The following is needed to avoid division by zero idx = false(size(X)); idx(op,:) = ( S(op,:)>0 & V(op,:)>0 ); X(idx) = U(idx) ./ S(idx); endfunction %!test %! lambda = [1 0 0]; %! N = [1 1 1]; %! S = V = [1 1 1; 1 1 1; 1 1 1]; %! fail( "qnmix( lambda, N, S, V)", "same time"); %! N = [0 0 1]; %! fail( "qnmix( lambda, N, S, V)", "open nor closed" ); %! N = [0 1 2]; %! m = [ 1 1 2 ]; %! fail( "qnmix( lambda, N, S, V, m)", "single-server and delay" ); %! S = V = [1 1 1; 1 1 1]; %! fail( "qnmix( lambda, N, S, V)", "rows" ); %!test %! # Example p. 148 Zahorjan et al. %! lambda = [1 1/2 0 0]; %! N = [0 0 1 1]; %! V = [1 1; 1 1; 1 1; 1 1]; %! S = [1/4 1/6; 1/2 1; 1/2 1; 1 4/3]; %! [U R Q X] = qnmix(lambda, N, S, V ); %! assert( Q(3,1), 4/19, 1e-4 ); %! assert( Q(3,2), 15/19, 1e-4 ); %! assert( Q(4,1), 5/19, 1e-4 ); %! assert( Q(4,2), 14/19, 1e-4 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! # Example 8.6 p. 345 Bolch et al. %! lambda = [0.5 0.25 0 0]; %! N = [0 0 1 1]; %! V = [2 1; 2.5 1.5; 1 0.5; 1 0.4]; %! S = [0.4 0.6; 0.8 1.6; 0.3 0.5; 0.5 0.8]; %! [U R Q X] = qnmix( lambda, N, S, V ); %! assert( U([1 2],:), [0.4 0.3; 0.5 0.6], 1e-3 ); %! assert( R([3 4],:), [4.829 6.951; 7.727 11.636], 1e-3 ); %! assert( Q([3 4],:), [0.582 0.418; 0.624 0.376], 1e-3 ); %! assert( Q([1 2],:), [8.822 5.383; 11.028 10.766], 1e-3 ); #FIXME %! assert( R([1 2],:), [8.822 10.766; 17.645 28.710], 1e-3 ); %! assert( X(3,1)/V(3,1), 0.120, 1e-3 ); %! assert( X(4,1)/V(4,1), 0.081, 1e-3 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! ## example figure 10 p. 26 Schwetman, "Implementing the Mean Value %! ## Analysis for the Solution of Queueing Network Models", Technical %! ## Report CSD-TR-355, feb 15, 1982, Purdue University. %! S = [.25 0; .25 .10]; %! V = [1 0; 1 1]; %! lambda = [1 0]; %! N = [0 3]; %! [U R Q X] = qnmix( lambda, N, S, V ); %! assert( U(1,1), .25, 1e-3 ); %! assert( X(1,1), 1.0, 1e-3 ); %! assert( [R(1,1) R(2,1) R(2,2)], [1.201 0.885 0.135], 1e-3 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law queueing/inst/qnom.m0000644000175000017500000002621613570046274014404 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{P}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{P}, @var{m}) ## ## @cindex open network, multiple classes ## @cindex multiclass network, open ## ## Exact analysis of open, multiple-class BCMP networks. The network can ## be made of @emph{single-server} queueing centers (FCFS, LCFS-PR or ## PS) or delay centers (IS). This function assumes a network with ## @math{K} service centers and @math{C} customer classes. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda}(c) ## If this function is invoked as @code{qnom(lambda, S, V, @dots{})}, ## then @code{@var{lambda}(c)} is the external arrival rate of class ## @math{c} customers (@code{@var{lambda}(c) @geq{} 0}). If this ## function is invoked as @code{qnom(lambda, S, P, @dots{})}, then ## @code{@var{lambda}(c,k)} is the external arrival rate of class ## @math{c} customers at center @math{k} (@code{@var{lambda}(c,k) ## @geq{} 0}). ## ## @item @var{S}(c,k) ## mean service time of class @math{c} customers on the service center ## @math{k} (@code{@var{S}(c,k)>0}). For FCFS nodes, mean service ## times must be class-independent. ## ## @item @var{V}(c,k) ## visit ratio of class @math{c} customers to service center @math{k} ## (@code{@var{V}(c,k) @geq{} 0 }). @strong{If you pass this argument, ## class switching is not allowed} ## ## @item @var{P}(r,i,s,j) ## probability that a class @math{r} job completing service at center ## @math{i} is routed to center @math{j} as a class @math{s} job. ## @strong{If you pass argument @var{P}, class switching is allowed}; ## however, all servers must be fixed-rate or infinite-server nodes ## (@code{@var{m}(k) @leq{} 1} for all @math{k}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. If @code{@var{m}(k) < 1}, ## enter @math{k} is a delay center (IS); otherwise it is a regular ## queueing center with @code{@var{m}(k)} servers. Default is ## @code{@var{m}(k) = 1} for all @math{k}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(c,k) ## If @math{k} is a queueing center, then @code{@var{U}(c,k)} is the ## class @math{c} utilization of center @math{k}. If @math{k} is an IS ## node, then @code{@var{U}(c,k)} is the class @math{c} @emph{traffic ## intensity} defined as @code{@var{X}(c,k)*@var{S}(c,k)}. ## ## @item @var{R}(c,k) ## class @math{c} response time at center @math{k}. The system ## response time for class @math{c} requests can be computed as ## @code{dot(@var{R}, @var{V}, 2)}. ## ## @item @var{Q}(c,k) ## average number of class @math{c} requests at center @math{k}. The ## average number of class @math{c} requests in the system @var{Qc} ## can be computed as @code{Qc = sum(@var{Q}, 2)} ## ## @item @var{X}(c,k) ## class @math{c} throughput at center @math{k}. ## ## @end table ## ## @strong{NOTES} ## ## If the function call specifies the visit ratios @var{V}, ## class switching is @strong{not} allowed. If the function call ## specifies the routing probability matrix @var{P}, then class ## switching @strong{is} allowed; however, all nodes are ## restricted to be fixed rate servers or delay centers: ## multiple-server and general load-dependent centers are not ## supported. Note that the meaning of parameter @var{lambda} is ## different from one case to the other (see below). ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. ## Sevcik, @cite{Quantitative System Performance: Computer System ## Analysis Using Queueing Network Models}, Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 7.4.1 ("Open Model Solution Techniques"). ## @end itemize ## ## @seealso{qnopen,qnos,qnomvisits} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnom( varargin ) if ( nargin < 2 || nargin > 4 ) print_usage(); endif if ( nargin == 2 || ndims(varargin{3}) == 2 ) [err lambda S V m] = qnomchkparam( varargin{:} ); else lambda = varargin{1}; ( ndims(lambda) == 2 && all( lambda(:) >= 0 ) ) || ... error( "lambda must be >= 0" ); [C,K] = size(lambda); S = varargin{2}; ( ndims(S) == 2 && size(S) == [C,K] ) || ... error( "S size mismatch (should be [%d,%d])", C, K ); P = varargin{3}; ( ndims(P) == 4 && size(P) == [C,K,C,K] ) || ... error( "P size mismatch (should be %dx%dx%dx%d)",C,K,C,K ); V = qnomvisits(P,lambda); if ( nargin < 4 ) m = ones(1,K); else m = varargin{4}; isvector(m) || ... error( "m must be a vector" ); m = m(:)'; # make m a row vector length(m) == K || ... error( "m size mismatch (should be %d, is %d)", K, length(m) ); all(m<=1) || ... error( "IF you use parameter P, m must be <= 1"); endif lambda = sum(lambda,2); # lambda(c) is the overall class c arrival rate endif [C K] = size(S); U = R = Q = X = zeros(C,K); ## NOTE; Zahorjan et al. define the class c throughput at center k as ## X(c,k) = lambda(c) * V(c,k). However, this assumes a definition of ## V(c,k) that is different from what is returned by the qnomvisits() ## function. The queueing package defines V(c,k) as the class c visit ## _ratio_ at center k (see the documentation of the queueing package ## to see the formal definition of V(c,k) as the solution of a linear ## system of equations), while Zahorjan et al. define V(c,k) as the ## _number of visits_ at center k. If you want to try the examples ## on Zahorjan with this function, you need to scale V(c,k) ## as lambda / lambda(c) * V(c,k). X = sum(lambda)*V; # X(c,k) = lambda*V(c,k); ## If there are M/M/k servers with k>=1, compute the maximum ## processing capacity m(m<1) = -1; # avoid division by zero in next line rho = X .* S * diag( 1 ./ m ); # rho(c,k) = X(c,k) * S(x,k) / m(k) [Umax kmax] = max( sum(rho,1) ); (Umax < 1) || ... error( "Processing capacity exceeded at center %d", kmax ); ## Compute utilizations (for IS nodes compute also response time and ## queue lenghts) for k=1:K for c=1:C if ( m(k) > 1 ) # M/M/m-FCFS [U(c,k)] = qsmmm( X(c,k), 1/S(c,k), m(k) ); elseif ( m(k) == 1 ) # M/M/1 or -/G/1-PS [U(c,k)] = qsmm1( X(c,k), 1/S(c,k) ); else # -/G/inf [U(c,k) R(c,k) Q(c,k)] = qsmminf( X(c,k), 1/S(c,k) ); endif endfor endfor assert( sum(U,1) < 1 ); # sanity check ## Adjust response times and queue lengths for FCFS queues k_fcfs = find(m>=1); for c=1:C Q(c,k_fcfs) = U(c,k_fcfs) ./ ( 1 - sum(U(:,k_fcfs),1) ); R(c,k_fcfs) = Q(c,k_fcfs) ./ X(c,k_fcfs); # Use Little's law endfor endfunction %!test %! fail( "qnom([1 1], [.9; 1.0])", "exceeded at center 1"); %! fail( "qnom([1 1], [0.9 .9; 0.9 1.0])", "exceeded at center 2"); %! #qnom([1 1], [.9; 1.0],[],2); # should not fail, M/M/2-FCFS %! #qnom([1 1], [.9; 1.0],[],-1); # should not fail, -/G/1-PS %! fail( "qnom(1./[2 3], [1.9 1.9 0.9; 2.9 3.0 2.9])", "exceeded at center 2"); %! #qnom(1./[2 3], [1 1.9 0.9; 0.3 3.0 1.5],[],[1 2 1]); # should not fail %!test %! V = [1 1; 1 1]; %! S = [1 3; 2 4]; %! lambda = [3/19 2/19]; %! [U R Q X] = qnom(lambda, S, diag( lambda / sum(lambda) ) * V ); %! assert( U(1,1), 3/19, 1e-6 ); %! assert( U(2,1), 4/19, 1e-6 ); %! assert( R(1,1), 19/12, 1e-6 ); %! assert( R(1,2), 57/2, 1e-6 ); %! assert( Q(1,1), .25, 1e-6 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! # example p. 138 Zahorjan et al. %! V = [ 10 9; 5 4]; %! S = [ 1/10 1/3; 2/5 1]; %! lambda = [3/19 2/19]; %! [U R Q X] = qnom(lambda, S, diag( lambda / sum(lambda) ) * V ); %! assert( X(1,1), 1.58, 1e-2 ); %! assert( U(1,1), .158, 1e-3 ); %! assert( R(1,1), .158, 1e-3 ); # modified from the original example, as the reference above considers R as the residence time, not the response time %! assert( Q(1,1), .25, 1e-2 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! # example 7.7 p. 304 Bolch et al. Please note that the book uses the %! # notation P(i,r,j,s) (i,j are service centers, r,s are job %! # classes) while the queueing package uses P(r,i,s,j) %! P = zeros(2,3,2,3); %! lambda = S = zeros(2,3); %! P(1,1,1,2) = 0.4; %! P(1,1,1,3) = 0.3; %! P(1,2,1,1) = 0.6; %! P(1,2,1,3) = 0.4; %! P(1,3,1,1) = 0.5; %! P(1,3,1,2) = 0.5; %! P(2,1,2,2) = 0.3; %! P(2,1,2,3) = 0.6; %! P(2,2,2,1) = 0.7; %! P(2,2,2,3) = 0.3; %! P(2,3,2,1) = 0.4; %! P(2,3,2,2) = 0.6; %! S(1,1) = 1/8; %! S(1,2) = 1/12; %! S(1,3) = 1/16; %! S(2,1) = 1/24; %! S(2,2) = 1/32; %! S(2,3) = 1/36; %! lambda(1,1) = lambda(2,1) = 1; %! V = qnomvisits(P,lambda); %! assert( V, [ 3.333 2.292 1.917; 10 8.049 8.415] ./ 2, 1e-3); %! [U R Q X] = qnom(sum(lambda,2), S, V); %! assert( sum(U,1), [0.833 0.442 0.354], 1e-3 ); %! # Note: the value of K_22 (corresponding to Q(2,2)) reported in the book %! # is 0.5. However, hand computation using the exact same formulas %! # from the book produces a different value, 0.451 %! assert( Q, [2.5 0.342 0.186; 2.5 0.451 0.362], 1e-3 ); ## Check that the results of qnom_nocs and qnom_cs are the same ## for multiclass networks WITHOUT class switching. %!test %! P = zeros(2,2,2,2); %! P(1,1,1,2) = 0.8; P(1,2,1,1) = 1; %! P(2,1,2,2) = 0.9; P(2,2,2,1) = 1; %! S = zeros(2,2); %! S(1,1) = 1.5; S(1,2) = 1.2; %! S(2,1) = 0.8; S(2,2) = 2.5; %! lambda = zeros(2,2); %! lambda(1,1) = 1/20; %! lambda(2,1) = 1/30; %! [U1 R1 Q1 X1] = qnom(lambda, S, P); # qnom_cs %! [U2 R2 Q2 X2] = qnom(sum(lambda,2), S, qnomvisits(P,lambda)); # qnom_nocs %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); %! assert( X1, X2, 1e-5 ); %!demo %! P = zeros(2,2,2,2); %! lambda = zeros(2,2); %! S = zeros(2,2); %! P(1,1,2,1) = P(1,2,2,1) = 0.2; %! P(1,1,2,2) = P(2,2,2,2) = 0.8; %! S(1,1) = S(1,2) = 0.1; %! S(2,1) = S(2,2) = 0.05; %! rr = 1:100; %! Xk = zeros(2,length(rr)); %! for r=rr %! lambda(1,1) = lambda(1,2) = 1/r; %! [U R Q X] = qnom(lambda,S,P); %! Xk(:,r) = sum(X,1)'; %! endfor %! plot(rr,Xk(1,:),";Server 1;","linewidth",2, ... %! rr,Xk(2,:),";Server 2;","linewidth",2); %! legend("boxoff"); %! xlabel("Class 1 interarrival time"); %! ylabel("Throughput"); queueing/inst/dtmcbd.m0000644000175000017500000000715513570046274014670 0ustar morenomoreno## Copyright (C) 2012, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{P} =} dtmcbd (@var{b}, @var{d}) ## ## @cindex Markov chain, discrete time ## @cindex DTMC ## @cindex discrete time Markov chain ## @cindex birth-death process, DTMC ## ## Returns the transition probability matrix @math{P} for a discrete ## birth-death process over state space @math{@{1, @dots{}, N@}}. ## For each @math{i=1, @dots{}, (N-1)}, ## @code{@var{b}(i)} is the transition probability from state ## @math{i} to @math{(i+1)}, and @code{@var{d}(i)} is the transition ## probability from state @math{(i+1)} to @math{i}. ## ## Matrix @math{\bf P} is defined as: ## ## @tex ## $$ \pmatrix{ (1-\lambda_1) & \lambda_1 & & & & \cr ## \mu_1 & (1 - \mu_1 - \lambda_2) & \lambda_2 & & \cr ## & \mu_2 & (1 - \mu_2 - \lambda_3) & \lambda_3 & & \cr ## \cr ## & & \ddots & \ddots & \ddots & & \cr ## \cr ## & & & \mu_{N-2} & (1 - \mu_{N-2}-\lambda_{N-1}) & \lambda_{N-1} \cr ## & & & & \mu_{N-1} & (1-\mu_{N-1}) } ## $$ ## @end tex ## @ifnottex ## @example ## @group ## / \ ## | 1-b(1) b(1) | ## | d(1) (1-d(1)-b(2)) b(2) | ## | d(2) (1-d(2)-b(3)) b(3) | ## | | ## | ... ... ... | ## | | ## | d(N-2) (1-d(N-2)-b(N-1)) b(N-1) | ## | d(N-1) 1-d(N-1) | ## \ / ## @end group ## @end example ## @end ifnottex ## ## @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and ## death probabilities, respectively. ## ## @seealso{ctmcbd} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function P = dtmcbd( b, d ) if ( nargin != 2 ) print_usage(); endif ( isvector( b ) && isvector( d ) ) || ... error( "birht and death must be vectors" ); b = b(:); # make b a column vector d = d(:); # make d a column vector size_equal( b, d ) || ... error( "birth and death vectors must have the same length" ); all( b >= 0 ) || ... error( "birth probabilities must be >= 0" ); all( d >= 0 ) || ... error( "death probabilities must be >= 0" ); all( ([b; 0] + [0; d]) <= 1 ) || ... error( "d(i)+b(i+1) must be <= 1"); P = diag( b, 1 ) + diag( d, -1 ); P += diag( 1-sum(P,2) ); endfunction %!test %! birth = [.5 .5 .3]; %! death = [.6 .2 .3]; %! fail("dtmcbd(birth,death)","must be"); %!demo %! birth = [ .2 .3 .4 ]; %! death = [ .1 .2 .3 ]; %! P = dtmcbd( birth, death ); %! disp(P) queueing/inst/dtmcmtta.m0000644000175000017500000001711313612113034015225 0ustar morenomoreno## Copyright (C) 2012, 2014, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{t} @var{N} @var{B}] =} dtmcmtta (@var{P}) ## @deftypefnx {Function File} {[@var{t} @var{N} @var{B}] =} dtmcmtta (@var{P}, @var{p0}) ## ## @cindex mean time to absorption, DTMC ## @cindex absorption probabilities, DTMC ## @cindex fundamental matrix ## @cindex DTMC ## @cindex discrete time Markov chain ## @cindex Markov chain, discrete time ## ## Compute the expected number of steps before absorption for a ## DTMC with state space @math{@{1, @dots{}, N@}} ## and transition probability matrix @var{P}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## @math{N \times N} transition probability matrix. ## @code{@var{P}(i,j)} is the transition probability from state ## @math{i} to state @math{j}. ## ## @item @var{p0}(i) ## Initial state occupancy probabilities (vector of length @math{N}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{t} ## @itemx @var{t}(i) ## When called with a single argument, @var{t} is a vector of length ## @math{N} such that @code{@var{t}(i)} is the expected number of steps ## before being absorbed in any absorbing state, starting from state ## @math{i}; if @math{i} is absorbing, @code{@var{t}(i) = 0}. When ## called with two arguments, @var{t} is a scalar, and represents the ## expected number of steps before absorption, starting from the initial ## state occupancy probability @var{p0}. ## ## @item @var{N}(i) ## @itemx @var{N}(i,j) ## When called with a single argument, @var{N} is the @math{N \times N} ## fundamental matrix for @var{P}. @code{@var{N}(i,j)} is the expected ## number of visits to transient state @var{j} before absorption, if the ## system started in transient state @var{i}. The initial state is counted ## if @math{i = j}. When called with two arguments, @var{N} is a vector ## of length @math{N} such that @code{@var{N}(j)} is the expected number ## of visits to transient state @var{j} before absorption, given initial ## state occupancy probability @var{P0}. ## ## @item @var{B}(i) ## @itemx @var{B}(i,j) ## When called with a single argument, @var{B} is a @math{N \times N} ## matrix where @code{@var{B}(i,j)} is the probability of being ## absorbed in state @math{j}, starting from transient state @math{i}; ## if @math{j} is not absorbing, @code{@var{B}(i,j) = 0}; if @math{i} ## is absorbing, @code{@var{B}(i,i) = 1} and @code{@var{B}(i,j) = 0} ## for all @math{i \neq j}. When called with two arguments, @var{B} is ## a vector of length @math{N} where @code{@var{B}(j)} is the ## probability of being absorbed in state @var{j}, given initial state ## occupancy probabilities @var{p0}. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item Grinstead, Charles M.; Snell, J. Laurie (July ## 1997). @cite{Introduction to Probability}, Ch. 11: Markov ## Chains. American Mathematical Society. ISBN 978-0821807491. ## @end itemize ## ## @seealso{ctmcmtta} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [t N B] = dtmcmtta( P, p0 ) persistent epsilon = 10*eps; if ( nargin < 1 || nargin > 2 ) print_usage(); endif [K err] = dtmcchkP(P); (K>0) || ... error(err); if ( nargin == 2 ) ( isvector(p0) && length(p0) == K && all(p0>=0) && abs(sum(p0)-1.0). ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosaba (@var{lambda}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosaba (@var{lambda}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosaba (@var{lambda}, @var{S}, @var{V}, @var{m}) ## ## @cindex bounds, asymptotic ## @cindex open network ## ## Compute Asymptotic Bounds for open, single-class networks with @math{K} service centers. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate of requests (scalar, @code{@var{lambda} @geq{} 0}). ## ## @item @var{D}(k) ## service demand at center @math{k}. ## (vector of length @math{K}, @code{@var{D}(k) @geq{} 0}). ## ## @item @var{S}(k) ## mean service time at center @math{k}. ## (vector of length @math{K}, @code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## mean number of visits to center @math{k}. ## (vector of length @math{K}, @code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. ## This function only supports @math{M/M/1} queues, therefore ## @var{m} must be @code{ones(size(S))}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl} ## @item @var{Xu} ## Lower and upper bounds on the system throughput. @var{Xl} is ## always set to @math{0} since there can be no lower bound on the ## throughput of open networks (scalar). ## ## @item @var{Rl} ## @item @var{Ru} ## Lower and upper bounds on the system response time. @var{Ru} ## is always set to @code{+inf} since there can be no upper bound on the ## throughput of open networks (scalar). ## ## @end table ## ## @seealso{qnomaba} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [X_lower X_upper R_lower R_upper] = qnosaba( varargin ) if ( nargin < 2 || nargin > 4 ) print_usage(); endif [err lambda S V m] = qnoschkparam( varargin{:} ); isempty(err) || error(err); all(m==1) || ... error("this function supports M/M/1 servers only"); D = S .* V; X_lower = 0; X_upper = 1/max(D); R_lower = sum(D); R_upper = +inf; endfunction %!test %! fail( "qnosaba( 0.1, [] )", "vector" ); %! fail( "qnosaba( 0.1, [0 -1])", "nonnegative" ); %! fail( "qnosaba( 0, [1 2] )", "lambda" ); %! fail( "qnosaba( -1, [1 2])", "lambda" ); %! fail( "qnosaba( 1, [1 2 3], [1 2] )", "incompatible size"); %! fail( "qnosaba( 1, [1 2 3], [-1 2 3] )", "nonnegative"); %!test %! [Xl Xu Rl Ru] = qnosaba( 1, [1 1] ); %! assert( Xl, 0 ); %! assert( Ru, +inf ); queueing/inst/qncmvisits.m0000644000175000017500000002223213603663713015624 0ustar morenomoreno## Copyright (C) 2012, 2016, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{V} @var{ch}] =} qncmvisits (@var{P}) ## @deftypefnx {Function File} {[@var{V} @var{ch}] =} qncmvisits (@var{P}, @var{r}) ## ## Compute the average number of visits for the nodes of a closed multiclass network with @math{K} service centers and @math{C} customer classes. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(r,i,s,j) ## probability that a ## class @math{r} request which completed service at center @math{i} is ## routed to center @math{j} as a class @math{s} request. Class switching ## is allowed. ## ## @item @var{r}(c) ## index of class @math{c} reference station, ## @math{r(c) \in @{1, @dots{}, K@}}, @math{1 @leq{} c @leq{} C}. ## The class @math{c} visit count to server @code{@var{r}(c)} ## (@code{@var{V}(c,r(c))}) is conventionally set to 1. The reference ## station serves two purposes: (i) its throughput is assumed to be the ## system throughput, and (ii) a job returning to the reference station ## is assumed to have completed one cycle. Default is to consider ## station 1 as the reference station for all classes. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{V}(c,i) ## number of visits of class @math{c} requests at center @math{i}. ## ## @item @var{ch}(c) ## chain number that class @math{c} belongs ## to. Different classes can belong to the same chain. Chains are ## numbered sequentially starting from 1 (@math{1, 2, @dots{}}). The ## total number of chains is @code{max(@var{ch})}. ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [V chains] = qncmvisits( P, r ) if ( nargin < 1 || nargin > 2 ) print_usage(); endif ndims(P) == 4 || ... error("P must be a 4-dimensional matrix"); [C, K, C2, K2] = size( P ); (K == K2 && C == C2) || ... error( "P must be a C*K*C*K matrix"); if ( nargin < 2) r = ones(1,C); else isvector(r) && length(r) == C || ... error("r must be a vector with %d elements",C); all( r>=1 && r<=K ) || ... error("elements in r must be in the range [1, %d]",K); r = r(:)'; endif ## solve the traffic equations: V(s,j) = sum_r sum_i V(r,i) * ## P(r,i,s,j), for all s,j V(s,r(s)) = 1 for all s. A = reshape(P,[K*C K*C])-eye(K*C); b = zeros(1,K*C); CH = __scc(reshape(P,[C*K C*K])>0); nCH = max(CH); # number of chains CH = reshape(CH,C,K); # CH(c,k) is the chain that class c at center k belongs to chains = zeros(1,C); for k=1:K for c=1:C if ( chains(c) == 0 ) chains(c) = CH(c,k); else ( CH(c,k) == 0 || chains(c) == CH(c,k) ) || ... error("Class %d belongs to different chains",c); endif endfor endfor constraints = zeros(1,nCH); # constraint(cc) = 1 iff we set a constraint for a class belonging to chain cc; we only set one constraint per chain for c=1:C cc = CH(c,r(c)); if ( cc == 0 || constraints(cc) == 0 ) ii = sub2ind([C K],c,r(c)); A(:,ii) = 0; A(ii,ii) = 1; if ( cc > 0 ) ## if r(c) is not an isolated node constraints(cc) = 1; b(ii) = 1; endif endif endfor V = reshape(b/A, C, K); ## Make sure that no negative values appear (sometimes, numerical ## errors produce tiny negative values instead of zeros) V = max(0,V); endfunction %!test %! %! ## Closed, multiclass network %! %! C = 2; K = 3; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = 1; %! P(1,2,1,1) = 1; %! P(2,1,2,3) = 1; %! P(2,3,2,1) = 1; %! V = qncmvisits(P); %! for c=1:C %! for k=1:K %! assert(V(c,k), sum(sum(V .* P(:,:,c,k))), 1e-5); %! endfor %! endfor %!test %! %! ## Test multiclass network. Example from Schwetman (figure 7, page 9 of %! ## http://docs.lib.purdue.edu/cstech/259/ %! ## "Testing network-of-queues software, technical report CSD-TR 330, %! ## Purdue University). %! %! C = 2; K = 4; %! P = zeros(C,K,C,K); %! # class 1 routing %! P(1,1,1,1) = .05; %! P(1,1,1,2) = .45; %! P(1,1,1,3) = .5; %! P(1,2,1,1) = 1; %! P(1,3,1,1) = 1; %! # class 2 routing %! P(2,1,2,1) = .01; %! P(2,1,2,3) = .5; %! P(2,1,2,4) = .49; %! P(2,3,2,1) = 1; %! P(2,4,2,1) = 1; %! V = qncmvisits(P); %! for c=1:C %! for i=1:K %! assert(V(c,i), sum(sum(V .* P(:,:,c,i))), 1e-5); %! endfor %! endfor %!test %! %! ## Network with class switching. %! ## This is the example in figure 9 of %! ## Schwetman, "Implementing the Mean Value Analysis %! ## Algorithm fort the solution of Queueing Network Models", Technical %! ## Report CSD-TR-355, Department of Computer Science, Purdue Univrsity, %! ## Feb 15, 1982, http://docs.lib.purdue.edu/cstech/286/ %! %! C = 2; K = 3; %! S = [.01 .07 .10; ... %! .05 0.7 .10 ]; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = .7; %! P(1,1,1,3) = .2; %! P(1,1,2,1) = .1; %! P(2,1,2,2) = .3; %! P(2,1,2,3) = .5; %! P(2,1,1,1) = .2; %! P(1,2,1,1) = P(1,3,1,1) = 1; %! P(2,2,2,1) = P(2,3,2,1) = 1; %! N = [3 0]; %! V = qncmvisits(P); %! VV = [10 7 2; 5 1.5 2.5]; # result given in Schwetman; our function computes something different, but that's ok since visit counts are actually ratios %! assert( V ./ repmat(V(:,1),1,K), VV ./ repmat(VV(:,1),1,K), 1e-5 ); %!test %! %! ## two disjoint classes: must produce two disjoing chains %! %! C = 2; K = 3; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = 1; %! P(1,2,1,1) = 1; %! P(2,1,2,3) = 1; %! P(2,3,2,1) = 1; %! [nc r] = qncmvisits(P); %! assert( r(1) != r(2) ); %!test %! %! ## two classes, one chain %! %! C = 2; K = 3; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = .5; %! P(1,2,2,1) = 1; %! P(2,1,2,3) = .5; %! P(2,3,1,1) = 1; %! [nc r] = qncmvisits(P); %! assert( r(1) == r(2) ); %!test %! %! ## a "Moebius strip". Note that this configuration is invalid, and %! ## therefore our algorithm must raise an error. This is because this %! ## network has two chains, but both chains contain both classes %! %! C = 2; K = 2; %! P = zeros(C,K,C,K); %! P(1,1,2,2) = 1; %! P(2,2,1,1) = 1; %! P(2,1,1,2) = 1; %! P(1,2,2,1) = 1; %! fail( "qncmvisits(P)", "different"); %!test %! %! ## Network with two classes representing independent chains. %! ## This is example in figure 8 of %! ## Schwetman, "Implementing the Mean Value Analysis %! ## Algorithm fort the solution of Queueing Network Models", Technical %! ## Report CSD-TR-355, Department of Computer Science, Purdue Univrsity, %! ## Feb 15, 1982, http://docs.lib.purdue.edu/cstech/286/ %! %! C = 2; K = 2; %! P = zeros(C,K,C,K); %! P(1,1,1,3) = P(1,3,1,1) = 1; %! P(2,2,2,3) = P(2,3,2,2) = 1; %! V = qncmvisits(P,[1,2]); %! assert( V, [1 0 1; 0 1 1], 1e-5 ); %!test %! C = 2; %! K = 3; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = 1; %! P(1,2,1,3) = 1; %! P(1,3,2,2) = 1; %! P(2,2,1,1) = 1; %! [V ch] = qncmvisits(P); %! assert( ch, [1 1] ); ## The following transition probability matrix is not well formed: note ## that there is an outgoing transition from center 1, class 1 but not ## incoming transition. %!test %! C = 2; %! K = 3; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = 1; %! P(1,2,1,3) = 1; %! P(1,3,2,2) = 1; %! P(2,2,2,1) = 1; %! P(2,1,1,2) = 1; %! [V ch] = qncmvisits(P); %! assert( ch, [1 1] ); ## compute strongly connected components using Kosaraju's algorithm, ## which requires two DFS visits. A better solution would be to use ## Tarjan's algorithm. ## ## In this implementation, an isolated node without self loops will NOT ## belong to any SCC. Although this is not formally correct from the ## graph theoretic point of view, it is necessary to compute chains ## correctly. function s = __scc(G) assert(issquare(G)); N = rows(G); GF = (G>0); GB = (G'>0); s = zeros(N,1); c=1; for n=1:N if (s(n) == 0) fw = __dfs(GF,n); bw = __dfs(GB,n); r = (fw & bw); if (any(r)) s(r) = c++; endif endif endfor endfunction ## Executes a dfs visit on graph G, starting from source node s function v = __dfs(G, s) assert( issquare(G) ); N = rows(G); v = stack = zeros(1,N); ## v(i) == 1 iff node i has been visited q = 1; # first empty slot in queue stack(q++) = s; ## Note: node s is NOT marked as visited; it will be marked as visited ## only if we visit it again. This is necessary to ensure that ## isolated nodes without self loops will not belong to any SCC. while( q>1 ) n = stack(--q); ## explore neighbors of n: all f in G(n,:) such that v(f) == 0 ## The following instruction is equivalent to: ## for f=find(G(n,:)) ## if ( v(f) == 0 ) for f = find ( G(n,:) & (v==0) ) stack(q++) = f; v(f) = 1; endfor endwhile endfunction queueing/inst/qncspb.m0000644000175000017500000001167013570046274014716 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{D} ) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{S}, @var{V} ) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{S}, @var{V}, @var{m} ) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z} ) ## ## @cindex bounds, PB ## @cindex PB bounds ## @cindex closed network, single class ## ## Compute PB Bounds (C. H. Hsieh and S. Lam, 1987) for single-class, ## closed networks with @math{K} service centers. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{} ## number of requests in the system (scalar, @code{@var{N} > 0}). ## ## @item @var{D}(k) ## service demand of service center @math{k} (@code{@var{D}(k) @geq{} 0}). ## ## @item @var{S}(k) ## mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## visit ratio to center @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. This function only supports ## @math{M/M/1} queues, therefore @var{m} must be ## @code{ones(size(S))}. ## ## @item @var{Z} ## external delay (think time, @code{@var{Z} @geq{} 0}). Default 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl} ## @itemx @var{Xu} ## Lower and upper bounds on the system throughput. ## ## @item @var{Rl} ## @itemx @var{Ru} ## Lower and upper bounds on the system response time. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## C. H. Hsieh and S. Lam, @cite{Two classes of performance bounds for ## closed queueing networks}, Performance Evaluation, Vol. 7 Issue 1, ## pp. 3--30, February 1987, DOI ## @uref{http://dx.doi.org/10.1016/0166-5316(87)90054-X, ## 10.1016/0166-5316(87)90054-X}. Also available as ## @uref{ftp://ftp.cs.utexas.edu/pub/techreports/tr85-09.pdf, Technical ## Report TR-85-09}, Department of Computer Science, University of Texas ## at Austin, June 1985 ## @end itemize ## ## This function implements the non-iterative variant described in G. ## Casale, R. R. Muntz, G. Serazzi, @cite{Geometric Bounds: a ## Non-Iterative Analysis Technique for Closed Queueing Networks}, IEEE ## Transactions on Computers, 57(6):780-794, June 2008. ## ## @seealso{qncsaba, qbcsbsb, qncsgb} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [X_lower X_upper R_lower R_upper] = qncspb( varargin ) if ( nargin < 2 || nargin > 5 ) print_usage(); endif [err N S V m Z] = qncschkparam( varargin{:} ); isempty(err) || error(err); ( N>0 ) || ... error("N must be positive"); all(m==1) || ... error("qncspb only supports single server nodes"); D = S .* V; D_tot = sum(D); X_max = 1/max(D); X_min = 0; X_lower = N/( Z + D_tot + ... ( sum( D .^ N * (N-1-Z*X_min) ) / sum( D .^ (N-1) ) ) ); X_upper = N/( Z + D_tot + ... ( sum( D .^ 2 * (N-1-Z*X_max) ) / sum( D ) ) ); X_upper = min( X_upper, X_max ); # cap X upper bound to 1/max(D) R_lower = N/X_upper-Z; R_upper = N/X_lower-Z; endfunction %!test %! fail( "qncspb( 1, [] )", "vector" ); %! fail( "qncspb( 1, [0 -1])", "nonnegative" ); %! fail( "qncspb( 0, [1 2] )", "positive" ); %! fail( "qncspb( -1, [1 2])", "nonnegative" ); %! fail( "qncspb( 1, [1 2], [1,1], [2, 2])", "single server" ); %! fail( "qncspb( 1, [1 2], [1,1], [1, 1], -1)", "nonnegative" ); %!# shared test function %!function test_pb( D, expected, Z=0 ) %! for i=1:rows(expected) %! N = expected(i,1); %! [X_lower X_upper] = qncspb(N,D,ones(size(D)),ones(size(D)),Z); %! X_exp_lower = expected(i,2); %! X_exp_upper = expected(i,3); %! assert( [N X_lower X_upper], [N X_exp_lower X_exp_upper], 1e-4 ) %! endfor %!test %! # table IV %! D = [ 0.1 0.1 0.09 0.08 ]; %! # N X_lower X_upper %! expected = [ 2 4.3174 4.3174; ... %! 5 6.6600 6.7297; ... %! 10 8.0219 8.2700; ... %! 20 8.8672 9.3387; ... %! 80 9.6736 10.000 ]; %! test_pb(D, expected); queueing/inst/qnopen.m0000644000175000017500000000427113570046274014727 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnopen (@var{lambda}, @var{S}, @var{V}, @dots{}) ## ## @cindex open network ## ## Compute utilization, response time, average number of requests in the ## system, and throughput for open queueing networks. If @var{lambda} is ## a scalar, the network is considered a single-class QN and is solved ## using @code{qnopensingle}. If @var{lambda} is a vector, the network ## is considered as a multiclass QN and solved using @code{qnopenmulti}. ## ## @seealso{qnos, qnom} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnopen( lambda, S, V, varargin ) if ( nargin < 3 ) print_usage(); endif if ( isscalar(lambda) ) [U R Q X] = qnos(lambda, S, V, varargin{:}); else [U R Q X] = qnom(lambda, S, V, varargin{:}); endif endfunction %!test %! # Example 34.1 p. 572 %! lambda = 3; %! V = [16 7 8]; %! S = [0.01 0.02 0.03]; %! [U R Q X] = qnopen( lambda, S, V ); %! assert( R, [0.0192 0.0345 0.107], 1e-2 ); %! assert( U, [0.48 0.42 0.72], 1e-2 ); %!test %! V = [1 1; 1 1]; %! S = [1 3; 2 4]; %! lambda = [3/19 2/19]; %! [U R Q] = qnopen(lambda, S, diag( lambda / sum(lambda) ) * V); %! assert( U(1,1), 3/19, 1e-6 ); %! assert( U(2,1), 4/19, 1e-6 ); %! assert( R(1,1), 19/12, 1e-6 ); %! assert( R(1,2), 57/2, 1e-6 ); %! assert( Q(1,1), .25, 1e-6 ); queueing/inst/qsmg1.m0000644000175000017500000000532013570046274014453 0ustar morenomoreno## Copyright (C) 2009 Dmitry Kolesnikov ## Copyright (C) 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmg1 (@var{lambda}, @var{xavg}, @var{x2nd}) ## ## @cindex @math{M/G/1} system ## ## Compute utilization, response time, average number of requests and ## throughput for a @math{M/G/1} system. The service time distribution ## is described by its mean @var{xavg}, and by its second moment ## @var{x2nd}. The computations are based on results from L. Kleinrock, ## @cite{Queuing Systems}, Wiley, Vol 2, and Pollaczek-Khinchine formula. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate ## ## @item @var{xavg} ## Average service time ## ## @item @var{x2nd} ## Second moment of service time distribution ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Service center utilization ## ## @item @var{R} ## Service center response time ## ## @item @var{Q} ## Average number of requests in the system ## ## @item @var{X} ## Service center throughput ## ## @item @var{p0} ## Probability that there is not any request at system ## ## @end table ## ## @var{lambda}, @var{xavg}, @var{t2nd} can be vectors of the ## same size. In this case, the results will be vectors as well. ## ## @seealso{qsmh1} ## ## @end deftypefn ## Author: Dmitry Kolesnikov function [U R Q X p0] = qsmg1(lambda, xavg, x2nd) if ( nargin != 3 ) print_usage(); endif ## bring the parameters to a common size [ err lambda xavg x2nd ] = common_size( lambda, xavg, x2nd ); if ( err ) error( "parameters are of incompatible size" ); endif mu = 1 ./ xavg; rho = lambda ./ mu; #coefficient of variation Cx = (x2nd .- xavg .* xavg) ./ (xavg .* xavg); #PK mean formula(s) Q = rho .+ rho .* rho .* (1 .+ Cx) ./ (2 .* (1 .- rho)); R = xavg .+ xavg .* rho .* (1 .+ Cx) ./ (2 .* (1 .- rho)); p0 = exp(-rho); #General Results #utilization U = rho; X = lambda; endfunction queueing/inst/ctmc.m0000644000175000017500000002216613631666131014356 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{p} =} ctmc (@var{Q}) ## @deftypefnx {Function File} {@var{p} =} ctmc (@var{Q}, @var{t}, @var{p0}) ## ## @cindex Markov chain, continuous time ## @cindex continuous time Markov chain ## @cindex Markov chain, state occupancy probabilities ## @cindex stationary probabilities ## @cindex CTMC ## ## Compute stationary or transient state occupancy probabilities for a continuous-time Markov chain. ## ## With a single argument, compute the stationary state occupancy ## probabilities @math{@var{p}(1), @dots{}, @var{p}(N)} for a ## continuous-time Markov chain with finite state space @math{@{1, @dots{}, ## N@}} and @math{N \times N} infinitesimal generator matrix @var{Q}. ## With three arguments, compute the state occupancy probabilities ## @math{@var{p}(1), @dots{}, @var{p}(N)} that the system is in state @math{i} ## at time @var{t}, given initial state occupancy probabilities ## @math{@var{p0}(1), @dots{}, @var{p0}(N)} at time 0. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{Q}(i,j) ## Infinitesimal generator matrix. @var{Q} is a @math{N \times N} square ## matrix where @code{@var{Q}(i,j)} is the transition rate from state ## @math{i} to state @math{j}, for @math{1 @leq{} i \neq j @leq{} N}. ## @var{Q} must satisfy the property that @math{\sum_{j=1}^N Q_{i, j} = ## 0} ## ## @item @var{t} ## Time at which to compute the transient probability (@math{t @geq{} ## 0}). If omitted, the function computes the steady state occupancy ## probability vector. ## ## @item @var{p0}(i) ## probability that the system is in state @math{i} at time 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{p}(i) ## If this function is invoked with a single argument, @code{@var{p}(i)} ## is the steady-state probability that the system is in state @math{i}, ## @math{i = 1, @dots{}, N}. If this function is invoked with three ## arguments, @code{@var{p}(i)} is the probability that the system is in ## state @math{i} at time @var{t}, given the initial occupancy ## probabilities @var{p0}(1), @dots{}, @var{p0}(N). ## ## @end table ## ## @seealso{dtmc} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function q = ctmc( Q, t, p0 ) persistent epsilon = 10*eps; if ( nargin != 1 && nargin != 3 ) print_usage(); endif [N err] = ctmcchkQ(Q); ( N>0 ) || ... error(err); if ( nargin == 1 ) # steady-state analysis ## non zero columns nonzero=find( any(abs(Q)>epsilon,1 ) ); if ( length(nonzero) == 0 ) error( "Q is the zero matrix" ); endif normcol = nonzero(1); # normalization condition column ## force probability of unvisited states to zero for i=find( all(abs(Q)=0 ) || ... error("t must be a scalar >= 0"); ( isvector(p0) && length(p0) == N && all(p0>=0) && abs(sum(p0)-1.0). ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsmva (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsmva (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsmva (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## ## @cindex Mean Value Analysys (MVA) ## @cindex closed network, single class ## @cindex normalization constant ## ## Analyze closed, single class queueing networks using the exact Mean Value Analysis (MVA) algorithm. ## ## The following queueing disciplines are supported: FCFS, LCFS-PR, PS ## and IS (Infinite Server). This function supports fixed-rate service ## centers or multiple server nodes. For general load-dependent service ## centers, use the function @code{qncsmvald} instead. ## ## Additionally, the normalization constant @math{G(n)}, @math{n=0, ## @dots{}, N} is computed; @math{G(n)} can be used in conjunction with ## the BCMP theorem to compute steady-state probabilities. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## Population size (number of requests in the system, @code{@var{N} @geq{} 0}). ## If @code{@var{N} == 0}, this function returns ## @code{@var{U} = @var{R} = @var{Q} = @var{X} = 0} ## ## @item @var{S}(k) ## mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## average number of visits to service center @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{Z} ## External delay for customers (@code{@var{Z} @geq{} 0}). Default is 0. ## ## @item @var{m}(k) ## number of servers at center @math{k} (if @var{m} is a scalar, all ## centers have that number of servers). If @code{@var{m}(k) < 1}, ## center @math{k} is a delay center (IS); otherwise it is a regular ## queueing center (FCFS, LCFS-PR or PS) with @code{@var{m}(k)} ## servers. Default is @code{@var{m}(k) = 1} for all @math{k} (each ## service center has a single server). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## If @math{k} is a FCFS, LCFS-PR or PS node (@code{@var{m}(k) @geq{} ## 1}), then @code{@var{U}(k)} is the utilization of center @math{k}, ## @math{0 @leq{} U(k) @leq{} 1}. If @math{k} is an IS node ## (@code{@var{m}(k) < 1}), then @code{@var{U}(k)} is the @emph{traffic ## intensity} defined as @code{@var{X}(k)*@var{S}(k)}. In this case the ## value of @code{@var{U}(k)} may be greater than one. ## ## @item @var{R}(k) ## center @math{k} response time. The @emph{Residence Time} at center ## @math{k} is @code{@var{R}(k) * @var{V}(k)}. The system response ## time @var{Rsys} can be computed either as @code{@var{Rsys} = ## @var{N}/@var{Xsys} - Z} or as @code{@var{Rsys} = ## dot(@var{R},@var{V})} ## ## @item @var{Q}(k) ## average number of requests at center @math{k}. The number of ## requests in the system can be computed either as ## @code{sum(@var{Q})}, or using the formula ## @code{@var{N}-@var{Xsys}*@var{Z}}. ## ## @item @var{X}(k) ## center @math{K} throughput. The system throughput @var{Xsys} can be ## computed as @code{@var{Xsys} = @var{X}(1) / @var{V}(1)} ## ## @item @var{G}(n) ## Normalization constants. @code{@var{G}(n+1)} contains the value of ## the normalization constant @math{G(n)}, @math{n=0, @dots{}, N} as ## array indexes in Octave start from 1. @math{G(n)} can be used in ## conjunction with the BCMP theorem to compute steady-state ## probabilities. ## ## @end table ## ## @strong{NOTES} ## ## In presence of load-dependent servers (i.e., if @code{@var{m}(k)>1} ## for some @math{k}), the MVA algorithm is known to be numerically ## unstable. Generally, this issue manifests itself as negative values ## for the response times or utilizations. This is not a problem of ## the @code{queueing} toolbox, but of the MVA algorithm, and has ## currently no known solution. This function prints a warning if ## numerical problems are detected; the warning can be disabled with ## the command @code{warning("off", "qn:numerical-instability")}. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed ## Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, April ## 1980, pp. 313--322. @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} ## @end itemize ## ## This implementation is described in R. Jain , @cite{The Art of ## Computer Systems Performance Analysis}, Wiley, 1991, p. 577. ## Multi-server nodes are treated according to G. Bolch, S. Greiner, ## H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: ## Modeling and Performance Evaluation with Computer Science ## Applications}, Wiley, 1998, Section 8.2.1, "Single Class Queueing ## Networks". ## ## @seealso{qncsmvald,qncscmva} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X G] = qncsmva( varargin ) if ( nargin < 3 || nargin > 5 ) print_usage(); endif [err N S V m Z] = qncschkparam( varargin{:} ); isempty(err) || error(err); K = length(S); # Number of servers U = R = Q = X = zeros( 1, K ); G = zeros(1,N+1); G(1) = 1; if ( N == 0 ) # Trivial case of empty population: just return all zeros return; endif i_single = find( m==1 ); i_multi = find( m>1 ); i_delay = find( m<1 ); ## Initialize results if ( length(i_multi)>0 ) p = zeros( K, max(m)+1 ); # p(i,j+1) is the probability that there are j jobs at server i p(:,1) = 1; endif X_s = 0; # System throughput ## Main MVA loop, iterates over the population size for n=1:N R(i_single) = S(i_single) .* (1 + Q(i_single)); for i=i_multi # I cannot easily vectorize this j=0:m(i)-2; R(i) = S(i) / m(i) * (1+Q(i)+dot( m(i)-j-1, p( i, 1+j ) ) ); endfor R(i_delay) = S(i_delay); R_s = dot( V, R ); # System response time X_s = n / ( Z + R_s ); # System Throughput Q = X_s * ( V .* R ); G(1+n) = G(n) / X_s; ## prepare for next iteration lambda_i = V * X_s; # lambda_i(i) is the node i throughput for i=i_multi j=1:m(i)-1; # range p(i, j+1) = lambda_i(i) .* S(i) ./ min( j,m(i) ) .* p(i,j); p(i,1) = 1 - 1/m(i) * ... (V(i)*S(i)*X_s + dot( m(i)-j, p(i,j+1)) ); endfor endfor X = X_s * V; # Service centers throughput U(i_single) = X(i_single) .* S(i_single); U(i_delay) = X(i_delay) .* S(i_delay); U(i_multi) = X(i_multi) .* S(i_multi) ./ m(i_multi); if ( any(U<0) || any(R<0) ) warning("qn:numerical-instability", "Numerical instability detected. Type 'help qncsmva' for details"); endif endfunction #{ ## This function is slightly faster (and more compact) than the above ## when all servers are single-server or delay centers. Improvements are ## quite small (10%-15% faster, depends on the network size), so at the ## moment it is commented out. function [U R Q X G] = __qncsmva_fast( N, S, V, m, Z ) U = R = Q = X = zeros( 1, length(S) ); X_s = 0; # System throughput G = zeros(1,N+1); G(1) = 1; ## Main MVA loop for n=1:N R = S .* (1+Q.*(m==1)); R_s = dot( V, R ); # System response time X_s = n / ( Z + R_s ); # System Throughput Q = X_s * ( V .* R ); G(1+n) = G(n) / X_s; endfor X = X_s * V; # Service centers throughput U = X .* S; endfunction #} %!test %! fail( "qncsmva()", "Invalid" ); %! fail( "qncsmva( 10, [1 2], [1 2 3] )", "incompatible size" ); %! fail( "qncsmva( 10, [-1 1], [1 1] )", "nonnegative" ); %! fail( "qncsmva( 10.3, [-1 1], [1 1] )", "integer" ); %! fail( "qncsmva( -0.3, [-1 1], [1 1] )", "nonnegative" ); ## Check if networks with only one type of server are handled correctly %!test %! qncsmva(1,1,1,1); %! qncsmva(1,1,1,-1); %! qncsmva(1,1,1,2); %! qncsmva(1,[1 1],[1 1],[-1 -1]); %! qncsmva(1,[1 1],[1 1],[1 1]); %! qncsmva(1,[1 1],[1 1],[2 2]); ## Check degenerate case of N==0 (LI case) %!test %! N = 0; %! S = [1 2 3 4]; %! V = [1 1 1 4]; %! [U R Q X] = qncsmva(N, S, V); %! assert( U, 0*S ); %! assert( R, 0*S ); %! assert( Q, 0*S ); %! assert( X, 0*S ); ## Check degenerate case of N==0 (LD case) %!test %! N = 0; %! S = [1 2 3 4]; %! V = [1 1 1 4]; %! m = [2 3 4 5]; %! [U R Q X] = qncsmva(N, S, V, m); %! assert( U, 0*S ); %! assert( R, 0*S ); %! assert( Q, 0*S ); %! assert( X, 0*S ); %!test %! # Exsample 3.42 p. 577 Jain %! S = [ 0.125 0.3 0.2 ]'; %! V = [ 16 10 5 ]; %! N = 20; %! m = ones(1,3)'; %! Z = 4; %! [U R Q X] = qncsmva(N,S,V,m,Z); %! assert( R, [ .373 4.854 .300 ], 1e-3 ); %! assert( Q, [ 1.991 16.177 0.500 ], 1e-3 ); %! assert( all( U>=0 ) ); %! assert( all( U<=1 ) ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! # Exsample 3.42 p. 577 Jain %! S = [ 0.125 0.3 0.2 ]; %! V = [ 16 10 5 ]; %! N = 20; %! m = ones(1,3); %! Z = 4; %! [U R Q X] = qncsmva(N,S,V,m,Z); %! assert( R, [ .373 4.854 .300 ], 1e-3 ); %! assert( Q, [ 1.991 16.177 0.500 ], 1e-3 ); %! assert( all( U>=0 ) ); %! assert( all( U<=1 ) ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! # Example 8.4 p. 333 Bolch et al. %! S = [ .5 .6 .8 1 ]; %! N = 3; %! m = [2 1 1 -1]; %! V = [ 1 .5 .5 1 ]; %! [U R Q X] = qncsmva(N,S,V,m); %! assert( Q, [ 0.624 0.473 0.686 1.217 ], 1e-3 ); %! assert( X, [ 1.218 0.609 0.609 1.218 ], 1e-3 ); %! assert( all(U >= 0 ) ); %! assert( all(U( m>0 ) <= 1 ) ); %! assert( Q, R.*X, 1e-5 ); # Little's Law %!test %! # Example 8.3 p. 331 Bolch et al. %! # This is a single-class network, which however nothing else than %! # a special case of multiclass network %! S = [ 0.02 0.2 0.4 0.6 ]; %! K = 6; %! V = [ 1 0.4 0.2 0.1 ]; %! [U R Q X] = qncsmva(K, S, V); %! assert( U, [ 0.198 0.794 0.794 0.595 ], 1e-3 ); %! assert( R, [ 0.025 0.570 1.140 1.244 ], 1e-3 ); %! assert( Q, [ 0.244 2.261 2.261 1.234 ], 1e-3 ); %! assert( X, [ 9.920 3.968 1.984 0.992 ], 1e-3 ); %!test %! # Check bound analysis %! N = 10; # max population %! for n=1:N %! S = [1 0.8 1.2 0.5]; %! V = [1 2 2 1]; %! [U R Q X] = qncsmva(n, S, V); %! Xs = X(1)/V(1); %! Rs = dot(R,V); %! # Compare with balanced system bounds %! [Xlbsb Xubsb Rlbsb Rubsb] = qncsbsb( n, S .* V ); %! assert( Xlbsb<=Xs ); %! assert( Xubsb>=Xs ); %! assert( Rlbsb<=Rs ); %! assert( Rubsb>=Rs ); %! # Compare with asymptotic bounds %! [Xlab Xuab Rlab Ruab] = qncsaba( n, S .* V ); %! assert( Xlab<=Xs ); %! assert( Xuab>=Xs ); %! assert( Rlab<=Rs ); %! assert( Ruab>=Rs ); %! endfor %!demo %! S = [ 0.125 0.3 0.2 ]; %! V = [ 16 10 5 ]; %! N = 20; %! m = ones(1,3); %! Z = 4; %! [U R Q X] = qncsmva(N,S,V,m,Z); %! X_s = X(1)/V(1); # System throughput %! R_s = dot(R,V); # System response time %! printf("\t Util Qlen RespT Tput\n"); %! printf("\t-------- -------- -------- --------\n"); %! for k=1:length(S) %! printf("Dev%d\t%8.4f %8.4f %8.4f %8.4f\n", k, U(k), Q(k), R(k), X(k) ); %! endfor %! printf("\nSystem\t %8.4f %8.4f %8.4f\n\n", N-X_s*Z, R_s, X_s ); queueing/inst/dtmcexps.m0000644000175000017500000001023113570046274015247 0ustar morenomoreno## Copyright (C) 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{L} =} dtmcexps (@var{P}, @var{n}, @var{p0}) ## @deftypefnx {Function File} {@var{L} =} dtmcexps (@var{P}, @var{p0}) ## ## @cindex expected sojourn times, DTMC ## @cindex DTMC ## @cindex discrete time Markov chain ## @cindex Markov chain, discrete time ## ## Compute the expected number of visits to each state during the first ## @var{n} transitions, or until abrosption. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## @math{N \times N} transition matrix. @code{@var{P}(i,j)} is the ## transition probability from state @math{i} to state @math{j}. ## ## @item @var{n} ## Number of steps during which the expected number of visits are ## computed (@math{@var{n} @geq{} 0}). If @code{@var{n}=0}, returns ## @var{p0}. If @code{@var{n} > 0}, returns the expected number of ## visits after exactly @var{n} transitions. ## ## @item @var{p0}(i) ## Initial state occupancy probabilities; @code{@var{p0}(i)} is ## the probability that the system is in state @math{i} at step 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{L}(i) ## When called with two arguments, @code{@var{L}(i)} is the expected ## number of visits to state @math{i} before absorption. When ## called with three arguments, @code{@var{L}(i)} is the expected number ## of visits to state @math{i} during the first @var{n} transitions. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item Grinstead, Charles M.; Snell, J. Laurie (July ## 1997). @cite{Introduction to Probability}, Ch. 11: Markov ## Chains. American Mathematical Society. ISBN 978-0821807491. ## @end itemize ## ## @seealso{ctmcexps} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function L = dtmcexps ( P, varargin ) persistent epsilon = 10*eps; if ( nargin < 2 || nargin > 3 ) print_usage(); endif [K err] = dtmcchkP(P); (K>0) || ... error(err); if ( nargin == 2 ) p0 = varargin{1}; else n = varargin{1}; p0 = varargin{2}; endif ( isvector(p0) && length(p0) == K && all(p0>=0) && abs(sum(p0)-1.0)=0 || ... error("n must be >= 0"); n = fix(n); L = zeros(sizeof(p0)); ## It is know that ## ## I + P + P^2 + P^3 + ... + P^n = (I-P)^-1 * (I-P^(n+1)) ## ## and therefore we could succintly write ## ## L = p0*inv(eye(K)-P)*(eye(K)-P^(n+1)); ## ## Unfortunatly, the method above is numerically unstable (at least ## for small values of n), so we use the crude approach below. PP = p0; L = zeros(1,K); for p=0:n L += PP; PP *= P; endfor else ## identify transient states tr = find(diag(P) < 1); k = length(tr); # number of transient states if ( k == K ) error("There are no absorbing states"); endif N = zeros(size(P)); tmpN = inv(eye(k) - P(tr,tr)); # matrix N = (I-Q)^-1 N(tr,tr) = tmpN; L = p0*N; endif endfunction %!test %! P = dtmcbd([1 1 1 1], [0 0 0 0]); %! L = dtmcexps(P,[1 0 0 0 0]); %! t = dtmcmtta(P,[1 0 0 0 0]); %! assert( L, [1 1 1 1 0] ); %! assert( sum(L), t ); %!test %! P = dtmcbd(linspace(0.1,0.4,5),linspace(0.4,0.1,5)); %! p0 = [1 0 0 0 0 0]; %! L = dtmcexps(P,0,p0); %! assert( L, p0 ); queueing/inst/qncsmvablo.m0000644000175000017500000001404213570046274015571 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvablo (@var{N}, @var{S}, @var{M}, @var{P} ) ## ## @cindex queueing network with blocking ## @cindex blocking queueing network ## @cindex closed network, finite capacity ## @cindex MVABLO ## ## Approximate MVA algorithm for closed queueing networks with blocking. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## number of requests in the system. @var{N} must be strictly greater ## than zero, and less than the overall network capacity: @code{0 < ## @var{N} < sum(@var{M})}. ## ## @item @var{S}(k) ## average service time on server @math{k} (@code{@var{S}(k) > 0}). ## ## @item @var{M}(k) ## capacity of center @math{k}. The capacity is the maximum number of requests in a service ## center, including the request in service (@code{@var{M}(k) @geq{} 1}). ## ## @item @var{P}(i,j) ## probability that a request which completes ## service at server @math{i} will be transferred to server @math{j}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## center @math{k} utilization. ## ## @item @var{R}(k) ## average response time of service center @math{k}. ## ## @item @var{Q}(k) ## average number of requests in service center @math{k} (including ## the request in service). ## ## @item @var{X}(k) ## center @math{k} throughput. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Ian F. Akyildiz, @cite{Mean Value Analysis for Blocking Queueing ## Networks}, IEEE Transactions on Software Engineering, vol. 14, n. 2, ## april 1988, pp. 418--428. @uref{http://dx.doi.org/10.1109/32.4663, 10.1109/32.4663} ## @end itemize ## ## @seealso{qnopen, qnclosed} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncsmvablo( K, S, M, P ) ## Note that we use "K" instead of "N" as the number of requests in ## order to be compliant with the paper by Akyildiz describing this ## algorithm. if ( nargin != 4 ) print_usage(); endif ( isscalar(K) && K > 0 ) || ... error( "K must be a positive integer" ); isvector(S) && all(S>0) || ... error ("S must be a vector > 0"); S = S(:)'; # make S a row vector N = length(S); ( isvector(M) && length(M) == N ) || ... error( "M must be a vector with %d elements", N ); all( M >= 1) || ... error( "M must be >= 1"); M = M(:)'; # make M a row vector (K < sum(M)) || ... error( "The population size K=%d exceeds the total system capacity %d", K, sum(M) ); [na err] = dtmcchkP(P); ( na>0 ) || ... error( err ); rows(P) == N || ... error("The number of rows of P must be equal to the length of S"); ## Note: in this implementation we make use of the same notation found ## in Akyildiz's paper cited in the REFERENCES above, with the minor ## exception of using 'v' instead of 'e' as the visit count vector. ## k_bar(i) is the average number of jobs in the i-th server, lambda ## is the network throughput, t_bar(i) is the mean residence time ## (time spent in queue and in service) for requests in the i-th ## service center. ## Initialization k_bar_m1 = zeros(1,N); # k_bar(k-1) BT = zeros(1,N); z = ones(1,N); lambda = 0; ## Computation of the visit counts v = qncsvisits(P); D = S .* v; # Service demand ## Main loop for k=1:K do ## t_bar_i(k) = S(i) *(z_i(k) + k_bar_i(k-1))+BT_i(k) t_bar = S .* ( z + k_bar_m1 ) + BT; lambda = k / dot(v,t_bar); k_bar = t_bar .* v * lambda; if ( any(k_bar>M) ) i = find( k_bar > M, 1 ); z(i) = 0; BT = BT + S(i) * ( v .* P(:,i)' ) / v(i); endif until( all(k_bar<=M) ); k_bar_m1 = k_bar; endfor R = t_bar; X = v * lambda; # Throughputs ## w_bar = t_bar - S - BT; # mean waiting time U = X .* S; Q = X .* R; endfunction %!test %! fail( "qncsmvablo( 10, [1 1], [4 5], [0 1; 1 0] )", "capacity"); %! fail( "qncsmvablo( 6, [1 1], [4 5], [0 1; 1 1] )", "stochastic"); %! fail( "qncsmvablo( 5, [1 1 1], [1 1], [0 1; 1 1] )", "3 elements"); %!test %! # This is the example on section v) p. 422 of the reference paper %! M = [12 10 14]; %! P = [0 1 0; 0 0 1; 1 0 0]; %! S = [1/1 1/2 1/3]; %! K = 27; %! [U R Q X]=qncsmvablo( K, S, M, P ); %! assert( R, [11.80 1.66 14.4], 1e-2 ); %!test %! # This is example 2, i) and ii) p. 424 of the reference paper %! M = [4 5 5]; %! S = [1.5 2 1]; %! P = [0 1 0; 0 0 1; 1 0 0]; %! K = 10; %! [U R Q X]=qncsmvablo( K, S, M, P ); %! assert( R, [6.925 8.061 4.185], 1e-3 ); %! K = 12; %! [U R Q X]=qncsmvablo( K, S, M, P ); %! assert( R, [7.967 9.019 8.011], 1e-3 ); %!test %! # This is example 3, i) and ii) p. 424 of the reference paper %! M = [8 7 6]; %! S = [0.2 1.2 1.4]; %! P = [ 0 0.5 0.5; 1 0 0; 1 0 0 ]; %! K = 10; %! [U R Q X] = qncsmvablo( K, S, M, P ); %! assert( R, [1.674 5.007 7.639], 1e-3 ); %! K = 12; %! [U R Q X] = qncsmvablo( K, S, M, P ); %! assert( R, [2.166 5.372 6.567], 1e-3 ); %!test %! # Network which never blocks, central server model %! M = [50 50 50]; %! S = [1 1/0.8 1/0.4]; %! P = [0 0.7 0.3; 1 0 0; 1 0 0]; %! K = 40; %! [U1 R1 Q1] = qncsmvablo( K, S, M, P ); %! V = qncsvisits(P); %! [U2 R2 Q2] = qncsmva( K, S, V ); %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); queueing/inst/qncscmva.m0000644000175000017500000001732013570046274015241 0ustar morenomoreno## Copyright (C) 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncscmva (@var{N}, @var{S}, @var{Sld}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncscmva (@var{N}, @var{S}, @var{Sld}, @var{V}, @var{Z}) ## ## @cindex conditional MVA (CMVA) ## @cindex Mean Value Analysis, conditional (CMVA) ## @cindex closed network, single class ## @cindex CMVA ## ## Conditional MVA (CMVA) algorithm, a numerically stable variant of ## MVA. This function supports a network of @math{M @geq{} 1} service ## centers and a single delay center. Servers @math{1, @dots{}, (M-1)} ## are load-independent; server @math{M} is load-dependent. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## Number of requests in the system, @code{@var{N} @geq{} 0}. If ## @code{@var{N} == 0}, this function returns @code{@var{U} = @var{R} ## = @var{Q} = @var{X} = 0} ## ## @item @var{S}(k) ## mean service time on server @math{k = 1, @dots{}, (M-1)} ## (@code{@var{S}(k) > 0}). If there are no fixed-rate servers, then ## @code{S = []} ## ## @item @var{Sld}(n) ## inverse service rate at server @math{M} (the load-dependent server) ## when there are @math{n} requests, @math{n=1, @dots{}, N}. ## @code{@var{Sld}(n) = } @math{1 / \mu(n)}. ## ## @item @var{V}(k) ## average number of visits to service center @math{k=1, @dots{}, M}, ## where @code{@var{V}(k) @geq{} 0}. @code{@var{V}(1:M-1)} are the ## visit rates to the fixed rate servers; @code{@var{V}(M)} is the ## visit rate to the load dependent server. ## ## @item @var{Z} ## External delay for customers (@code{@var{Z} @geq{} 0}). Default is 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## center @math{k} utilization (@math{k=1, @dots{}, M}) ## ## @item @var{R}(k) ## response time of center @math{k} (@math{k=1, @dots{}, M}). The ## system response time @var{Rsys} can be computed as @code{@var{Rsys} ## = @var{N}/@var{Xsys} - Z} ## ## @item @var{Q}(k) ## average number of requests at center @math{k} (@math{k=1, @dots{}, M}). ## ## @item @var{X}(k) ## center @math{k} throughput (@math{k=1, @dots{}, M}). ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Casale. @cite{A note on stable flow-equivalent aggregation in ## closed networks}. Queueing Syst. Theory Appl., 60:193–-202, December ## 2008, @uref{http://dx.doi.org/10.1007/s11134-008-9093-6, 10.1007/s11134-008-9093-6} ## @end itemize ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncscmva( N, S, Sld, V, Z ) ## This is a numerically stable implementation of the MVA algorithm, ## described in G. Casale, A note on stable flow-equivalent aggregation in ## closed networks. Queueing Syst. Theory Appl., 60:193–-202, December ## 2008, http://dx.doi.org/10.1007/s11134-008-9093-6 if ( nargin < 4 || nargin > 5 ) print_usage(); endif isscalar(N) && N >= 0 || ... error("N must be a positive scalar"); (isempty(S) || isvector(S)) || ... error("S must be a vector"); S = S(:)'; # make S a row vector M = length(S)+1; # total number of service centers (excluding the delay center) isvector(Sld) && length(Sld) == N && all(Sld>=0) || ... error("Sld must be a vector with %d elements >= 0", N); Sld = Sld(:)'; # Make Sld a row vector isvector(V) && length(V) == M && all(V>=0) || ... error("V must be a vector with %d elements", M); V = V(:)'; # Make V a row vector ## The reference paper assumes queue M (LD center) as reference. ## Therefore, we need to rescale V V(M) > 0 || ... error("V(M) must be >0"); V = V / V(M); if ( nargin == 5 ) isscalar(Z) && Z>=0 || ... error("Z must be nonnegative"); else Z = 0; endif if ( N == 0 ) U = R = Q = X = zeros(1,M); return; endif ## Di(1+k) = service demand of server k=0,1,...,M-1 (server 0 is the delay center) Di = zeros(1,M); Di(1) = Z; Di(2:M) = S .* V(1:M-1); ## DM(n,t), n=1, ..., N, t=1, ..., N DM = zeros(N,N); mu = 1./Sld; # rate function ## Ri(1+k,:,:) = response time of server k=0,1,...,M Ri = zeros(1+M,N,N); ## Qi(k,1+n,t) = queue length of server k=1,...,M, n=0,1,...,N, t=1,...,N+1 Qi = zeros(M,1+N,N+1); Xs = zeros(N,N); # Xs = system throughput ## Main MVA loop for n=1:N for t=1:(N-n+1) if ( n==1 ) DM(n,t) = 1/mu(t); else # n>=2 DM(n,t) = Xs(n-1,t)/Xs(n-1,t+1)*DM(n-1,t); endif Ri(1+0,n,t) = Di(1+0); i=1:M-1; Ri(1+i,n,t) = Di(1+i).*(1+Qi(i,1+n-1,t))'; Ri(1+M,n,t) = DM(n,t)*(1+Qi(M,1+n-1,t+1)); Xs(n,t) = n/sum(Ri(:,n,t)); i=1:M-1; Qi(i,1+n,t) = Di(1+i) .* Xs(n,t) .* (1+Qi(i,1+n-1,t))'; Qi(M,1+n,t) = DM(n,t) * Xs(n,t) * (1+Qi(M,1+n-1,t+1)); endfor endfor X = Xs(N,1).*V; Q = Qi(1:M,1+N,1)'; ## Note that the result R is the *response time*, while the value ## computed by the reference paper is the *residence time*. The ## response time is equal to the residence time divided by the visit ## ratios. FIXME: This will choke if the visit ratio is zero for some server k R = Ri(2:M+1,N,1)' ./ V; U = [Di(2:M) DM(N,1)] .* X ./ V; endfunction %!test %! N=5; %! S = [1 0.3 0.8 0.9]; %! V = [1 1 1 1]; %! [U1 R1 Q1 X1] = qncscmva( N, S(1:3), repmat(S(4),1,N), V ); %! [U2 R2 Q2 X2] = qncsmva(N, S, V); %! assert( X1, X2, 1e-5 ); %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); %!test %! N=5; %! S = [1 1 1 1 1; ... %! 1 1 1 1 1; ... %! 1 1 1 1 1; ... %! 1 1/2 1/3 1/4 1/5]; %! V = [1 1 1 1]; %! [U1 R1 Q1 X1] = qncscmva( N, S(1:3,1), S(4,:), V ); %! [U2 R2 Q2 X2] = qncsmvald(N, S, V); %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); %! assert( X1, X2, 1e-5 ); %!test %! N=5; %! S = [1 1 1 1 1; ... %! 1 1 1 1 1; ... %! 1 1 1 1 1; ... %! 1 1/2 1/3 1/4 1/5]; %! V = [1 2 1 1]; %! Z = 3; %! [U1 R1 Q1 X1] = qncscmva( N, S(1:3,1), S(4,:), V, Z ); %! [U2 R2 Q2 X2] = qncsmvald(N, S, V, Z); %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); %! assert( X1, X2, 1e-5 ); %!demo %! maxN = 90; # Max population size %! Rmva = Rconv = Rcmva = zeros(1,maxN); # Results %! S = 4; Z = 10; m = 8; %! old = warning("query","qn:numerical-instability"); %! warning("off","qn:numerical-instability"); %! for N=1:maxN %! [U R] = qncsmva(N,S,1,m,Z); # Use MVA %! Rmva(N) = R(1); %! [U R] = qncsconv(N,[S Z],[1 1],[m -1]); # Use Convolution %! Rconv(N) = R(1); %! if ( N > m ) %! Scmva = S ./ min(1:N,m); %! else %! Scmva = S ./ (1:N); %! endif %! [U R] = qncscmva(N,[],Scmva,1,Z); # Use CMVA %! Rcmva(N) = R(1); %! endfor %! warning(old.state,"qn:numerical-instability"); %! plot(1:maxN, Rmva, ";MVA;", ... %! 1:maxN, Rconv, ";Convolution;", ... %! 1:maxN, Rcmva, ";CNVA;", "linewidth",2); %! xlabel("Population size (N)"); %! ylabel("Response Time"); %! ax=axis(); ax(3) = 0; ax(4) = 40; axis(ax); %! legend("location","northwest"); legend("boxoff"); queueing/inst/qsmh1.m0000644000175000017500000000607713570046274014466 0ustar morenomoreno## Copyright (C) 2009 Dmitry Kolesnikov ## Copyright (C) 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmh1 (@var{lambda}, @var{mu}, @var{alpha}) ## ## @cindex @math{M/H_m/1} system ## ## Compute utilization, response time, average number of requests and ## throughput for a @math{M/H_m/1} system. In this system, the customer ## service times have hyper-exponential distribution: ## ## @tex ## $$ B(x) = \sum_{j=1}^m \alpha_j(1-e^{-\mu_j x}),\quad x>0 $$ ## @end tex ## ## @ifnottex ## @example ## @group ## ___ m ## \ ## B(x) = > alpha(j) * (1-exp(-mu(j)*x)) x>0 ## /__ ## j=1 ## @end group ## @end example ## @end ifnottex ## ## where @math{\alpha_j} is the probability that the request is served ## at phase @math{j}, in which case the average service rate is ## @math{\mu_j}. After completing service at phase @math{j}, for ## some @math{j}, the request exits the system. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate ## ## @item @var{mu} ## @code{@var{mu}(j)} is the phase @math{j} service rate. The total ## number of phases @math{m} is @code{length(@var{mu})}. ## ## @item @var{alpha} ## @code{@var{alpha}(j)} is the probability that a request ## is served at phase @math{j}. @var{alpha} must have the same size ## as @var{mu}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Service center utilization ## ## @item @var{R} ## Service center response time ## ## @item @var{Q} ## Average number of requests in the system ## ## @item @var{X} ## Service center throughput ## ## @end table ## ## @end deftypefn ## Author: Dmitry Kolesnikov function [U R Q X p0] = qsmh1(lambda, mu, alpha) if ( nargin != 3 ) print_usage(); endif if ( size(mu) != size(alpha) ) error( "parameters are of incompatible size" ); endif [n c] = size(mu); if (!is_scalar(lambda) && (n != length(lambda)) ) error( "parameters are of incompatible size" ); endif for i=1:n avg = sum( alpha(i,:) .* (1 ./ mu(i,:)) ); m2nd = sum( alpha(i,:) .* (1 ./ (mu(i,:) .* mu(i,:))) ); if (is_scalar(lambda)) xavg = avg; x2nd = m2nd; else xavg(i) = avg; x2nd(i) = m2nd; endif endfor [U R Q X p0] = qsmg1(lambda, xavg, x2nd); endfunction queueing/inst/ctmctaexps.m0000644000175000017500000001172013570046274015577 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{M} =} ctmctaexps (@var{Q}, @var{t}, @var{p0}) ## @deftypefnx {Function File} {@var{M} =} ctmctaexps (@var{Q}, @var{p0}) ## ## @cindex Markov chain, continuous time ## @cindex time-alveraged sojourn time, CTMC ## @cindex continuous time Markov chain ## @cindex CTMC ## ## Compute the @emph{time-averaged sojourn time} @code{@var{M}(i)}, ## defined as the fraction of the time interval @math{[0,t]} (or until ## absorption) spent in state @math{i}, assuming that the state ## occupancy probabilities at time 0 are @var{p}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{Q}(i,j) ## Infinitesimal generator matrix. @code{@var{Q}(i,j)} is the transition ## rate from state @math{i} to state @math{j}, ## @math{1 @leq{} i,j @leq{} N}, @math{i \neq j}. The ## matrix @var{Q} must also satisfy the condition @math{\sum_{j=1}^N Q_{i,j} = 0} ## ## @item @var{t} ## Time. If omitted, the results are computed until absorption. ## ## @item @var{p0}(i) ## initial state occupancy probabilities. @code{@var{p0}(i)} is the ## probability that the system is in state @math{i} at time 0, @math{i ## = 1, @dots{}, N} ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{M}(i) ## When called with three arguments, @code{@var{M}(i)} is the expected ## fraction of the interval @math{[0,t]} spent in state @math{i} ## assuming that the state occupancy probability at time zero is ## @var{p}. When called with two arguments, @code{@var{M}(i)} is the ## expected fraction of time until absorption spent in state @math{i}; ## in this case the mean time to absorption is @code{sum(@var{M})}. ## ## @end table ## ## @seealso{ctmcexps} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function M = ctmctaexps( Q, varargin ) persistent epsilon = 10*eps; if ( nargin < 2 || nargin > 3 ) print_usage(); endif L = ctmcexps(Q,varargin{:}); M = L ./ repmat(sum(L,2),1,columns(L)); endfunction %!test %! Q = [ 0 0.1 0 0; ... %! 0.9 0 0.1 0; ... %! 0 0.9 0 0.1; ... %! 0 0 0 0 ]; %! Q -= diag( sum(Q,2) ); %! M = ctmctaexps(Q, [1 0 0 0]); %! assert( sum(M), 1, 10*eps ); %!demo %! lambda = 0.5; %! N = 4; %! birth = lambda*linspace(1,N-1,N-1); %! death = zeros(1,N-1); %! Q = diag(birth,1)+diag(death,-1); %! Q -= diag(sum(Q,2)); %! t = linspace(1e-5,30,100); %! p = zeros(1,N); p(1)=1; %! M = zeros(length(t),N); %! for i=1:length(t) %! M(i,:) = ctmctaexps(Q,t(i),p); %! endfor %! clf; %! plot(t, M(:,1), ";State 1;", "linewidth", 2, ... %! t, M(:,2), ";State 2;", "linewidth", 2, ... %! t, M(:,3), ";State 3;", "linewidth", 2, ... %! t, M(:,4), ";State 4 (absorbing);", "linewidth", 2 ); %! legend("location","east"); legend("boxoff"); %! xlabel("Time"); %! ylabel("Time-averaged Expected sojourn time"); ## This example is from: David I. Heimann, Nitin Mittal, Kishor S. Trivedi, ## "Availability and Reliability Modeling for Computer Systems", sep 1989, ## section 2.5 %!demo %! sec = 1; %! min = sec*60; %! hour = 60*min; %! day = 24*hour; %! %! # state space enumeration {2, RC, RB, 1, 0} %! a = 1/(10*min); # 1/a = duration of reboot (10 min) %! b = 1/(30*sec); # 1/b = reconfiguration time (30 sec) %! g = 1/(5000*hour); # 1/g = processor MTTF (5000 hours) %! d = 1/(4*hour); # 1/d = processor MTTR (4 hours) %! c = 0.9; # coverage %! Q = [ -2*g 2*c*g 2*(1-c)*g 0 0; ... %! 0 -b 0 b 0; ... %! 0 0 -a a 0; ... %! d 0 0 -(g+d) g; ... %! 0 0 0 d -d]; %! p = ctmc(Q); %! printf("System availability: %f\n",p(1)+p(4)); %! TT = linspace(0,1*day,101); %! PP = ctmctaexps(Q,TT,[1 0 0 0 0]); %! A = At = Abart = zeros(size(TT)); %! A(:) = p(1) + p(4); # steady-state availability %! for n=1:length(TT) %! t = TT(n); %! p = ctmc(Q,t,[1 0 0 0 0]); %! At(n) = p(1) + p(4); # instantaneous availability %! Abart(n) = PP(n,1) + PP(n,4); # interval base availability %! endfor %! clf; %! semilogy(TT,A,";Steady-state;", ... %! TT,At,";Instantaneous;", ... %! TT,Abart,";Interval base;"); %! ax = axis(); %! ax(3) = 1-1e-5; %! axis(ax); %! legend("boxoff"); queueing/inst/qnmarkov.m0000644000175000017500000002756713570046274015302 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{lambda}, @var{S}, @var{C}, @var{P}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{lambda}, @var{S}, @var{C}, @var{P}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{N}, @var{S}, @var{C}, @var{P}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{N}, @var{S}, @var{C}, @var{P}, @var{m}) ## ## @cindex closed network, single class ## @cindex open network, single class ## @cindex closed network, finite capacity ## @cindex blocking queueing network ## @cindex RS blocking ## ## Compute utilization, response time, average queue length and ## throughput for open or closed queueing networks with finite capacity ## and a single class of requests. ## Blocking type is Repetitive-Service (RS). This function explicitly ## generates and solve the underlying Markov chain, and thus might ## require a large amount of memory. ## ## More specifically, networks which can me analyzed by this ## function have the following properties: ## ## @itemize @bullet ## ## @item There exists only a single class of customers. ## ## @item The network has @math{K} service centers. Center ## @math{k \in @{1, @dots{}, K@}} ## has @math{m_k > 0} servers, and has a total (finite) capacity of ## @math{C_k \geq m_k} which includes both buffer space and servers. ## The buffer space at service center @math{k} is therefore ## @math{C_k - m_k}. ## ## @item The network can be open, with external arrival rate to ## center @math{k} equal to ## @math{\lambda_k}, or closed with fixed ## population size @math{N}. For closed networks, the population size ## @math{N} must be strictly less than the network capacity: ## @math{N < \sum_{k=1}^K C_k}. ## ## @item Average service times are load-independent. ## ## @item @math{P_{i, j}} is the probability that requests completing ## execution at center @math{i} are transferred to ## center @math{j}, @math{i \neq j}. For open networks, a request may leave the system ## from any node @math{i} with probability @math{1-\sum_{j=1}^K P_{i, j}}. ## ## @item Blocking type is Repetitive-Service (RS). Service ## center @math{j} is @emph{saturated} if the number of requests is equal ## to its capacity @math{C_j}. Under the RS blocking discipline, ## a request completing service at center @math{i} which is being ## transferred to a saturated server @math{j} is put back at the end of ## the queue of @math{i} and will receive service again. Center @math{i} ## then processes the next request in queue. External arrivals to a ## saturated servers are dropped. ## ## @end itemize ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda}(k) ## @itemx @var{N} ## If the first argument is a vector @var{lambda}, it is considered to be ## the external arrival rate @code{@var{lambda}(k) @geq{} 0} to service center ## @math{k} of an open network. If the first argument is a scalar, it is ## considered as the population size @var{N} of a closed network; in this case ## @var{N} must be strictly ## less than the network capacity: @code{@var{N} < sum(@var{C})}. ## ## @item @var{S}(k) ## average service time at service center @math{k} ## ## @item @var{C}(k) ## capacity of service center @math{k}. The capacity includes both ## the buffer and server space @code{@var{m}(k)}. Thus the buffer space is ## @code{@var{C}(k)-@var{m}(k)}. ## ## @item @var{P}(i,j) ## transition probability from service center ## @math{i} to service center @math{j}. ## ## @item @var{m}(k) ## number of servers at service center ## @math{k}. Note that @code{@var{m}(k) @geq{} @var{C}(k)} for each @var{k}. ## If @var{m} is omitted, all service centers are assumed to have a ## single server (@code{@var{m}(k) = 1} for all @math{k}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## center @math{k} utilization. ## ## @item @var{R}(k) ## response time on service center @math{k}. ## ## @item @var{Q}(k) ## average number of customers in the ## service center @math{k}, @emph{including} the request in service. ## ## @item @var{X}(k) ## throughput of service center @math{k}. ## ## @end table ## ## @strong{NOTES} ## ## The space complexity of this implementation is @math{O(\prod_{k=1}^K (C_k + 1)^2)}. The time complexity is dominated by ## the time needed to solve a linear system with @math{\prod_{k=1}^K (C_k + 1)} unknowns. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnmarkov( x, S, C, P, m ) if ( nargin < 4 || nargin > 5 ) print_usage(); endif isvector(S) || error( "S must be a vector" ); K = length(S); # number of service centers if ( nargin < 5 ) m = ones(1,K); else size_equal(m,S) || error( "m must have the same langth as S" ); endif ( [K,K] == size(P) && all( all(P>=0)) && all(sum(P,2) <= 1)) || ... error( "P must be SxS and nonnegative" ); if ( isscalar(x) ) is_open = false; N = x; # closed network ( N < sum(C) ) || ... error( "The population size exceeds the network capacity" ); all( abs(sum(P,2)-1) < 1000*eps ) || ... error( "P for closed networks cannot have exit nodes" ); else is_open = true; lambda = x; # open network size_equal(lambda, S ) || ... error( "lambda must have the same langth as S" ); endif ( all(m > 0) && all(m <= C) ) || ... error( "Capacities C must be greater or equal than m" ); q_size = prod( C+1 ); # number of states of the system ## The infinitesimal generator matrix Q_markovmight be sparse, so it ## would be appropriate to represent it as a sparse matrix. In this ## case the UMFPACK library must be installed to solve the @math{{\bf ## A}x=b} sparse system. Since I'm not sure everyone built Octave with ## sparse matrix support, I leave Q_markov as a full matrix here. Q_markov = zeros( q_size, q_size ); cur_state = zeros(1, K); if ( is_open ) valid_populations = linspace(0, sum(C), sum(C)+1); else valid_populations = [ N ]; endif ## exit_prob(i) is the probability that a job leaves the system from node i exit_prob = 1 - sum(P,2); for n=valid_populations pop_mix = qncmpopmix( n, C ); for cur_state=pop_mix' # for each feasible configuration with n customers cur_idx = sub2cell( C, cur_state ); for i=1:K if ( is_open ) # for open networks only ## handle new external arrival to center i if ( lambda(i) > 0 && cur_state(i) < C(i) ) next_state = cur_state; next_state(i) += 1; next_idx = sub2cell( C, next_state ); Q_markov( cur_idx, next_idx ) = lambda(i); endif ## handle requests that leave the system from center i ## exit_prob = 1-sum(P(i,:)); if ( exit_prob(i) > 0 && cur_state(i) > 0 ) next_state = cur_state; next_state(i) -= 1; next_idx = sub2cell( C, next_state ); Q_markov( cur_idx, next_idx ) = min(m(i), cur_state(i))*exit_prob(i)/S(i); endif endif # end open networks only ## for both open and closed networks for j=[1:(i-1) (i+1):K] ## check whether a job can move from server i to server j!=i if ( cur_state(i) > 0 && cur_state(j) < C(j) && P(i,j) > 0 ) next_state = cur_state; next_state(i) -= 1; next_state(j) += 1; next_idx = sub2cell( C, next_state ); Q_markov( cur_idx, next_idx ) = min(m(i), cur_state(i))*P(i,j)/S(i); endif endfor endfor endfor endfor ##spy( Q_markov ); ## complete the diagonal elements of the matrix Q d = sum(Q_markov,2); Q_markov -= diag( d ); ## Solve the ctmc prob = ctmc( Q_markov ); ## Compute the average queue length p = zeros(K, max(C)+1); # p(k,i+1) = prob that there are i requests at service center k for n=valid_populations pop_mix = qncmpopmix( n, C ); for cur_state=pop_mix' cur_idx = sub2cell( C, cur_state ); for k=1:K i=cur_state(k); p(k,i+1) += prob(cur_idx); endfor endfor endfor ## We can now compute all the performance measures U = R = Q = X = zeros(1,K); for k=1:K j = [0:m(k)-1]; U(k) = 1 - sum( ( m(k) - j ) ./ m(k) .* p(k,1+j) ); ##X(k) = U(k)/S(k); j = [1:C(k)]; Q(k) = sum( j .* p(k,1+j) ); ##R(k) = Q(k)/X(k); endfor X = U./S; R = Q./X; endfunction %!test %! S = [5 2.5]; %! P = [0 1; 1 0]; %! C = [3 3]; %! m = [1 1]; %! [U R Q X] = qnmarkov( 3, S, C, P, m ); %! assert( U, [0.9333 0.4667], 1e-4 ); %! assert( X, [0.1867 0.1867], 1e-4 ); %! assert( R, [12.1429 3.9286], 1e-4 ); ## Example 7.5 p. 292 Bolch et al. %!test %! S = [1/0.8 1/0.6 1/0.4]; %! P = [0.6 0.3 0.1; 0.2 0.3 0.5; 0.4 0.1 0.5]; %! C = [3 3 3]; %! [U R Q X] = qnmarkov( 3, S, C, P ); %! assert( U, [0.543 0.386 0.797], 1e-3 ); %! assert( Q, [0.873 0.541 1.585], 1e-3 ); ## Example 10.19, p. 551 Bolch et al. %!xtest %! S = [2 0.9]; %! C = [7 5]; %! P = [0 1; 1 0]; %! [U R Q X] = qnmarkov( 10, S, C, P ); %! assert( Q, [6.73 3.27], 1e-3 ); ## Example 8.1 p. 317 Bolch et al. %!test %! S = [1/0.8 1/0.6 1/0.4]; %! P = [(1-0.667-0.2) 0.667 0.2; 1 0 0; 1 0 0]; %! m = [2 3 1]; %! C = [3 3 3]; %! [U R Q X] = qnmarkov( 3, S, C, P, m ); %! assert( U, [0.590 0.350 0.473], 1e-3 ); %! assert( Q(1:2), [1.290 1.050], 1e-3 ); ## This is a simple test of an open QN with fixed capacity queues. There ## are two service centers, S1 and S2. C(1) = 2 and C(2) = 1. Transition ## probability from S1 to S2 is 1. Transition probability from S2 to S1 ## is p. %!test %! p = 0.5; # transition prob. from S2 to S1 %! mu = [1 2]; # Service rates %! C = [2 1]; # Capacities %! lambda = [0.5 0]; # arrival rate at service center 1 %! %! PP = [ 0 1; p 0 ]; %! [U R Q X] = qnmarkov( lambda, 1./mu, C, PP ); %! ## Now we generate explicitly the infinitesimal generator matrix %! ## of the underlying MC. %! ## 00 01 10 11 20 21 %! QQ = [ 0 0 lambda(1) 0 0 0; ... ## 00 %! mu(2)*(1-p) 0 mu(2)*p lambda(1) 0 0; ... ## 01 %! 0 mu(1) 0 0 lambda(1) 0; ... ## 10 %! 0 0 mu(2)*(1-p) 0 mu(2)*p lambda(1); ... ## 11 %! 0 0 0 mu(1) 0 0; ... ## 20 %! 0 0 0 0 mu(2)*(1-p) 0 ]; ## 21 %! ## Complete matrix %! sum_el = sum(QQ,2); %! QQ -= diag(sum_el); %! q = ctmc(QQ); %! ## Compare results %! assert( U(1), 1-sum(q([1, 2])), 1e-5 ); %! assert( U(2), 1-sum(q([1,3,5])), 1e-5 ); ## This is a closed network with fixed-capacity queues. The population ## size N is such that blocking never occurs, so this model can be ## analyzed using the conventional MVA algorithm. MVA and qnmarkov() ## must produce the same results. %!test %! P = [0 0.5 0.5; 1 0 0; 1 0 0]; %! C = [6 6 6]; %! S = [1 0.8 1.8]; %! N = 6; %! [U1 R1 Q1 X1] = qnclosed( N, S, qncsvisits(P) ); %! [U2 R2 Q2 X2] = qnmarkov( N, S, C, P ); %! assert( U1, U2, 1e-6 ); %! assert( R1, R2, 1e-6 ); %! assert( Q1, Q2, 1e-6 ); %! assert( X1, X2, 1e-6 ); ## return a linear index corresponding to index idx on a ## multidimensional vector of dimension(s) dim function i = sub2cell( dim, idx ) idx_cell = num2cell( idx+1 ); i = sub2ind( dim+1, idx_cell{:} ); endfunction queueing/inst/ctmcfpt.m0000644000175000017500000000620513570046274015066 0ustar morenomoreno## Copyright (C) 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{M} =} ctmcfpt (@var{Q}) ## @deftypefnx {Function File} {@var{m} =} ctmcfpt (@var{Q}, @var{i}, @var{j}) ## ## @cindex first passage times, CTMC ## @cindex CTMC ## @cindex continuous time Markov chain ## @cindex Markov chain, continuous time ## ## Compute mean first passage times for an irreducible continuous-time ## Markov chain. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{Q}(i,j) ## Infinitesimal generator matrix. @var{Q} is a @math{N \times N} ## square matrix where @code{@var{Q}(i,j)} is the transition rate from ## state @math{i} to state @math{j}, for @math{1 @leq{} i, j @leq{} N}, ## @math{i \neq j}. Transition rates must be nonnegative, and ## @math{\sum_{j=1}^N Q_{i,j} = 0} ## ## @item @var{i} ## Initial state. ## ## @item @var{j} ## Destination state. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{M}(i,j) ## average time before state ## @var{j} is visited for the first time, starting from state @var{i}. ## We let @code{@var{M}(i,i) = 0}. ## ## @item m ## @var{m} is the average time before state @var{j} is visited for the first ## time, starting from state @var{i}. ## ## @end table ## ## @seealso{ctmcmtta} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function result = ctmcfpt( Q, i, j ) persistent epsilon = 10*eps; if ( nargin != 1 && nargin != 3 ) print_usage(); endif [N err] = ctmcchkQ(Q); (N>0) || ... error(err); if ( nargin == 1 ) result = zeros(N,N); for j=1:N QQ = Q; QQ(j,:) = 0; # make state j absorbing for i=1:N p0 = zeros(1,N); p0(i) = 1; result(i,j) = ctmcmtta(QQ,p0); endfor endfor else (isscalar(i) && i>=1 && j<=N) || error("i must be an integer in the range 1..%d", N); (isvector(j) && all(j>=1) && all(j<=N)) || error("j must be an integer or vector with elements in 1..%d", N); j = j(:)'; # make j a row vector Q(j,:) = 0; # make state(s) j absorbing p0 = zeros(1,N); p0(i) = 1; result = ctmcmtta(Q,p0); endif endfunction %!demo %! Q = [ -1.0 0.9 0.1; ... %! 0.1 -1.0 0.9; ... %! 0.9 0.1 -1.0 ]; %! M = ctmcfpt(Q) %! m = ctmcfpt(Q,1,3) %!test %! N = 10; %! Q = reshape(1:N^2,N,N); %! Q(1:N+1:end) = 0; %! Q -= diag(sum(Q,2)); %! M = ctmcfpt(Q); %! assert( all(diag(M) < 10*eps) ); queueing/inst/engset.m0000644000175000017500000000740513570046274014716 0ustar morenomoreno## Copyright (C) 2014, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{B} =} engset (@var{A}, @var{m}, @var{n}) ## ## @cindex Engset loss formula ## ## Evaluate the Engset loss formula. ## ## The Engset formula computes the blocking probability ## @math{P_b(A,m,n)} for a system with a finite population of @math{n} ## users, @math{m} identical servers, no queue, individual service ## rate @math{\mu}, individual arrival rate @math{\lambda} (i.e., the ## time until a user tries to request service is exponentially ## distributed with mean @math{1/\lambda}), and offered load ## @math{A=\lambda/\mu}. ## ## @tex ## @math{P_b(A, m, n)} is defined for @math{n > m} as: ## ## $$ ## P_b(A, m, n) = {{\displaystyle{A^m {n \choose m}}} \over {\displaystyle{\sum_{k=0}^m A^k {n \choose k}}}} ## $$ ## ## and is 0 if @math{n @leq{} m}. ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{A} ## Offered load, defined as @math{A = \lambda / \mu} where ## @math{\lambda} is the mean arrival rate and @math{\mu} the mean ## service rate of each individual server (real, @math{A > 0}). ## ## @item @var{m} ## Number of identical servers (integer, @math{m @geq{} 1}). Default @math{m = 1} ## ## @item @var{n} ## Number of requests (integer, @math{n @geq{} 1}). Default @math{n = 1} ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{B} ## The value @math{P_b(A, m, n)} ## ## @end table ## ## @var{A}, @var{m} or @math{n} can be vectors, and in this case, the ## results will be vectors as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Zeng, @cite{Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 ## @end itemize ## ## @seealso{erlangb, erlangc} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function P = engset(A, m, n) if ( nargin < 1 || nargin > 3 ) print_usage(); endif ( isnumeric(A) && all( A(:) > 0 ) ) || error("A must be positive"); if ( nargin < 2 ) m = 1; else ( isnumeric(m) && all( fix(m(:)) == m(:)) && all( m(:) > 0 ) ) || error("m must be a positive integer"); endif if ( nargin < 3 ) n = 1; else ( isnumeric(n) && all( fix(n(:)) == n(:)) && all( n(:) > 0 ) ) || error("n must be a positive integer"); endif [err A m n] = common_size(A, m, n); if ( err ) error("parameters are not of common size"); endif P = arrayfun( @__engset_compute, A, m, n); endfunction ## Compute P_b(A,m,n) recursively function P = __engset_compute(A, m, n) if ( m >= n ) P = 0.0; else P = 1.0; for i = 1:m P=(A*(n-i)*P)/(i+A*(n-i)*P); endfor endif endfunction %!test %! fail("erlangb(1, -1)", "positive"); %! fail("erlangb(-1, 1)", "positive"); %! fail("erlangb(1, 0)", "positive"); %! fail("erlangb(0, 1)", "positive"); %! fail("erlangb('foo',1)", "positive"); %! fail("erlangb(1,'bar')", "positive"); %! fail("erlangb([1 1],[1 1 1])","common size"); queueing/inst/qnomvisits.m0000644000175000017500000001076213570046274015645 0ustar morenomoreno## Copyright (C) 2012, 2013, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{V} =} qnomvisits (@var{P}, @var{lambda}) ## ## Compute the visit ratios to the service centers of an open multiclass network with @math{K} service centers and @math{C} customer classes. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(r,i,s,j) ## probability that a class @math{r} request which completed service at center @math{i} is ## routed to center @math{j} as a class @math{s} request. Class switching ## is supported. ## ## @item @var{lambda}(r,i) ## external arrival rate of class @math{r} requests to center @math{i}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{V}(r,i) ## visit ratio of class @math{r} requests at center @math{i}. ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function V = qnomvisits( P, lambda ) if ( nargin != 2 ) print_usage(); endif ndims(P) == 4 || ... error("P must be a 4-dimensional matrix"); [C, K, C2, K2] = size( P ); (K == K2 && C == C2) || ... error( "P must be a [%d,%d,%d,%d] matrix", C, K, C, K); ( ndims(lambda) == 2 && [C,K] == size(lambda) ) || ... error( "lambda must be a %d x %d matrix", C, K ); all(lambda(:)>=0) || ... error(" lambda contains negative values" ); ## solve the traffic equations: V(s,j) = lambda(s,j) / lambda + sum_r ## sum_i V(r,i) * P(r,i,s,j), for all s,j where lambda is defined as ## sum_r sum_i lambda(r,i) A = eye(K*C) - reshape(P,[K*C K*C]); b = reshape(lambda / sum(lambda(:)), [1,K*C]); V = reshape(b/A, [C, K]); ## Make sure that no negative values appear (sometimes, numerical ## errors produce tiny negative values instead of zeros) V = max(0,V); endfunction %!test %! fail( "qnomvisits( zeros(3,3,3), [1 1 1] )", "matrix"); %!test %! C = 2; K = 4; %! P = zeros(C,K,C,K); %! # class 1 routing %! P(1,1,1,1) = .05; %! P(1,1,1,2) = .45; %! P(1,1,1,3) = .5; %! P(1,2,1,1) = 0.1; %! P(1,3,1,1) = 0.2; %! # class 2 routing %! P(2,1,2,1) = .01; %! P(2,1,2,3) = .5; %! P(2,1,2,4) = .49; %! P(2,3,2,1) = 0.2; %! P(2,4,2,1) = 0.16; %! lambda = [0.1 0 0 0.1 ; 0 0 0.2 0.1]; %! lambda_sum = sum(lambda(:)); %! V = qnomvisits(P, lambda); %! assert( all(V(:)>=0) ); %! for i=1:K %! for c=1:C %! assert(V(c,i), lambda(c,i) / lambda_sum + sum(sum(V .* P(:,:,c,i))), 1e-5); %! endfor %! endfor %!test %! # example 7.7 p. 304 Bolch et al. %! # Note that the book uses a slightly different notation than %! # what we use here. Specifically, the book defines the routing %! # probabilities as P(i,r,j,s) (i,j are service centers, r,s are job %! # classes) while the queueing package uses P(r,i,s,j). %! # A more serious problem arises in the definition of external arrivals. %! # The computation of V(r,i) as given in the book (called e_ir %! # in Eq 7.14) is performed in terms of P_{0, js}, defined as %! # "the probability in an open network that a job from outside the network %! # enters the jth node as a job of the sth class" (p. 267). This is %! # compliant with eq. 7.12 where the external class r arrival rate at center %! # i is computed as \lambda * P_{0,ir}. However, example 7.7 wrongly %! # defines P_{0,11} = P_{0,12} = 1, instead of P_{0,11} = P_{0,12} = 0.5 %! # Therefore the resulting visit ratios they obtain must be divided by two. %! P = zeros(2,3,2,3); %! lambda = S = zeros(2,3); %! P(1,1,1,2) = 0.4; %! P(1,1,1,3) = 0.3; %! P(1,2,1,1) = 0.6; %! P(1,2,1,3) = 0.4; %! P(1,3,1,1) = 0.5; %! P(1,3,1,2) = 0.5; %! P(2,1,2,2) = 0.3; %! P(2,1,2,3) = 0.6; %! P(2,2,2,1) = 0.7; %! P(2,2,2,3) = 0.3; %! P(2,3,2,1) = 0.4; %! P(2,3,2,2) = 0.6; %! lambda(1,1) = lambda(2,1) = 1; %! V = qnomvisits(P,lambda); %! assert( V, [ 3.333 2.292 1.917; 10 8.049 8.415] ./ 2, 1e-3); queueing/inst/qsmm1.m0000644000175000017500000001076113614611357014465 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2018, 2019, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmm1 (@var{lambda}, @var{mu}) ## @deftypefnx {Function File} {@var{pk} =} qsmm1 (@var{lambda}, @var{mu}, @var{k}) ## ## @cindex @math{M/M/1} system ## ## Compute utilization, response time, average number of requests and throughput for a @math{M/M/1} queue. ## ## @tex ## The steady-state probability @math{\pi_k} that there are @math{k} ## jobs in the system, @math{k \geq 0}, can be computed as: ## ## $$ ## \pi_k = (1-\rho)\rho^k ## $$ ## ## where @math{\rho = \lambda/\mu} is the server utilization. ## ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate (@code{@var{lambda} @geq{} 0}). ## ## @item @var{mu} ## Service rate (@code{@var{mu} > @var{lambda}}). ## ## @item @var{k} ## Number of requests in the system (@code{@var{k} @geq{} 0}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Server utilization ## ## @item @var{R} ## Server response time ## ## @item @var{Q} ## Average number of requests in the system ## ## @item @var{X} ## Server throughput. If the system is ergodic (@code{@var{mu} > ## @var{lambda}}), we always have @code{@var{X} = @var{lambda}} ## ## @item @var{p0} ## Steady-state probability that there are no requests in the system. ## ## @item @var{pk} ## Steady-state probability that there are @var{k} requests in the system. ## (including the one being served). ## ## @end table ## ## If this function is called with less than three input parameters, ## @var{lambda} and @var{mu} can be vectors of the same size. In this ## case, the results will be vectors as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks ## and Markov Chains: Modeling and Performance Evaluation with Computer ## Science Applications}, Wiley, 1998, Section 6.3 ## @end itemize ## ## @seealso{qsmmm, qsmminf, qsmmmk} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U_or_pk R Q X p0] = qsmm1( lambda, mu, k ) if ( nargin < 2 || nargin > 3 ) print_usage(); endif ( isvector(lambda) && isvector(mu) ) || ... error( "lambda and mu must be vectors" ); [ err lambda mu ] = common_size( lambda, mu ); if ( err ) error( "parameters are of incompatible size" ); endif lambda = lambda(:)'; mu = mu(:)'; all( lambda >= 0 ) || error( "lambda must be >= 0" ); rho = lambda ./ mu; all( rho < 1 ) || error( "Processing capacity exceeded" ); if (nargin == 2) U_or_pk = rho; # utilization p0 = 1-rho; Q = rho ./ (1-rho); R = 1 ./ ( mu .* (1-rho) ); X = lambda; else (length(lambda) == 1) || error("lambda must be a scalar if this function is called with three arguments"); isvector(k) || error("k must be a vector"); all(k>=0) || error("k must be >= 0"); k = k(:)'; # make k a row vector U_or_pk = (1 - rho).*rho.^k; endif endfunction %!test %! fail( "qsmm1(10,5)", "capacity exceeded" ); %! fail( "qsmm1(1,1)", "capacity exceeded" ); %! fail( "qsmm1([2 2], [1 1 1])", "incompatible size"); %!test %! [U R Q X P0] = qsmm1(0, 1); %! assert( U, 0 ); %! assert( R, 1 ); %! assert( Q, 0 ); %! assert( X, 0 ); %! assert( P0, 1 ); %!test %! [U R Q X P0] = qsmm1(0.2, 1.0); %! pk = qsmm1(0.2, 1.0, 0); %! assert(P0, pk); %!demo %! ## Given a M/M/1 queue, compute the steady-state probability pk %! ## of having k requests in the systen. %! lambda = 0.2; %! mu = 0.25; %! k = 0:10; %! pk = qsmm1(lambda, mu, k); %! plot(k, pk, "-o", "linewidth", 2); %! xlabel("N. of requests (k)"); %! ylabel("p_k"); %! title(sprintf("M/M/1 system, \\lambda = %g, \\mu = %g", lambda, mu)); queueing/inst/qsmmmk.m0000644000175000017500000001747513612113034014730 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pK}] =} qsmmmk (@var{lambda}, @var{mu}, @var{m}, @var{K}) ## @deftypefnx {Function File} {@var{pn} =} qsmmmk (@var{lambda}, @var{mu}, @var{m}, @var{K}, @var{n}) ## ## @cindex @math{M/M/m/K} system ## ## Compute utilization, response time, average number of requests and ## throughput for a @math{M/M/m/K} finite capacity system. In a ## @math{M/M/m/K} system there are @math{m \geq 1} identical service centers ## sharing a fixed-capacity queue. At any time, at most @math{K @geq{} m} requests can be in the system, including those being served. The maximum queue length ## is @math{K-m}. This function generates and ## solves the underlying CTMC. ## ## @tex ## ## The steady-state probability @math{\pi_n} that there are @math{n} ## jobs in the system, @math{0 @leq{} n @leq{} K}, is: ## ## $$ ## \pi_n = \cases{ \displaystyle{{\rho^n \over n!} \pi_0} & if $0 \leq n \leq m$;\cr\cr ## \displaystyle{{\rho^m \over m!} \left( \rho \over m \right)^{n-m} \pi_0} & if $m < n \leq K$\cr} ## $$ ## ## where @math{\rho = \lambda/\mu} is the offered load. The probability ## @math{\pi_0} that the system is empty can be computed by considering ## that all probabilities must sum to one: @math{\sum_{k=0}^K \pi_k = 1}, ## that gives: ## ## $$ ## \pi_0 = \left[ \sum_{k=0}^m {\rho^k \over k!} + {\rho^m \over m!} \sum_{k=m+1}^K \left( {\rho \over m}\right)^{k-m} \right]^{-1} ## $$ ## ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate (@code{@var{lambda}>0}) ## ## @item @var{mu} ## Service rate (@code{@var{mu}>0}) ## ## @item @var{m} ## Number of servers (@code{@var{m} @geq{} 1}) ## ## @item @var{K} ## Maximum number of requests allowed in the system, ## including those being served (@code{@var{K} @geq{} @var{m}}) ## ## @item @var{n} ## Number of requests in the (@code{0 @leq{} @var{n} @leq{} K}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Service center utilization ## ## @item @var{R} ## Service center response time ## ## @item @var{Q} ## Average number of requests in the system ## ## @item @var{X} ## Service center throughput ## ## @item @var{p0} ## Steady-state probability that there are no requests in the system. ## ## @item @var{pK} ## Steady-state probability that there are @var{K} requests in the system ## (i.e., probability that the system is full). ## ## @item @var{pn} ## Steady-state probability that there are @var{n} requests in the system ## (including those being served). ## ## @end table ## ## If this function is called with less than five arguments, ## @var{lambda}, @var{mu}, @var{m} and @var{K} can be either scalars, or ## vectors of the same size. In this case, the results will be vectors ## as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks ## and Markov Chains: Modeling and Performance Evaluation with Computer ## Science Applications}, Wiley, 1998, Section 6.6 ## @end itemize ## ## @seealso{qsmm1,qsmminf,qsmmm} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U_or_pn R Q X p0 pK] = qsmmmk( lambda, mu, m, K, n ) if ( nargin < 4 || nargin > 5 ) print_usage(); endif ( isvector(lambda) && isvector(mu) && isvector(m) && isvector(K) ) || ... error( "lambda, mu, m, K must be vectors" ); lambda = lambda(:)'; # make lambda a row vector mu = mu(:)'; # make mu a row vector m = m(:)'; # make m a row vector K = K(:)'; # make K a row vector [err lambda mu m K] = common_size( lambda, mu, m, K ); if ( err ) error( "Parameters are not of common size" ); endif all( K>0 ) || ... error( "k must be strictly positive" ); all( m>0 ) && all( m <= K ) || ... error( "m must be in the range 1:k" ); all( lambda>0 ) && all( mu>0 ) || ... error( "lambda and mu must be >0" ); if (nargin < 5) U_or_pn = R = Q = X = p0 = pK = 0*lambda; for i=1:length(lambda) ## Build and solve the birth-death process describing the M/M/m/k system birth_rate = lambda(i)*ones(1,K(i)); death_rate = [ linspace(1,m(i),m(i))*mu(i) ones(1,K(i)-m(i))*m(i)*mu(i) ]; p = ctmc(ctmcbd(birth_rate, death_rate)); p0(i) = p(1); pK(i) = p(1+K(i)); j = [1:K(i)]; Q(i) = dot( p(1+j),j ); endfor ## Compute other performance measures X = lambda.*(1-pK); U_or_pn = X ./ (m .* mu ); R = Q ./ X; else (length(lambda) == 1) || error("lambda must be a scalar if this function is called with five arguments"); isvector(n) || error("n must be a vector"); (all(n >= 0) && all(n <= K)) || error("n must be >= 0 and <= K"); n = n(:)'; birth_rate = lambda*ones(1,K); death_rate = [ linspace(1,m,m)*mu ones(1,K-m)*m*mu ]; p = ctmc(ctmcbd(birth_rate, death_rate)); U_or_pn = p(1+n); endif endfunction %!test %! lambda = mu = m = 1; %! k = 10; %! [U R Q X p0] = qsmmmk(lambda,mu,m,k); %! assert( Q, k/2, 1e-7 ); %! assert( U, 1-p0, 1e-7 ); %!test %! lambda = [1 0.8 2 9.2 0.01]; %! mu = lambda + 0.17; %! k = 12; %! [U1 R1 Q1 X1] = qsmm1k(lambda,mu,k); %! [U2 R2 Q2 X2] = qsmmmk(lambda,mu,1,k); %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); %! assert( X1, X2, 1e-5 ); %! #assert( [U1 R1 Q1 X1], [U2 R2 Q2 X2], 1e-5 ); %!test %! lambda = 0.9; %! mu = 0.75; %! k = 10; %! [U1 R1 Q1 X1 p01] = qsmmmk(lambda,mu,1,k); %! [U2 R2 Q2 X2 p02] = qsmm1k(lambda,mu,k); %! assert( [U1 R1 Q1 X1 p01], [U2 R2 Q2 X2 p02], 1e-5 ); %!test %! lambda = 0.8; %! mu = 0.85; %! m = 3; %! k = 5; %! [U1 R1 Q1 X1 p0] = qsmmmk( lambda, mu, m, k ); %! birth = lambda*ones(1,k); %! death = [ mu*linspace(1,m,m) mu*m*ones(1,k-m) ]; %! q = ctmc(ctmcbd( birth, death )); %! U2 = dot( q, min( 0:k, m )/m ); %! assert( U1, U2, 1e-4 ); %! Q2 = dot( [0:k], q ); %! assert( Q1, Q2, 1e-4 ); %! assert( p0, q(1), 1e-4 ); %!test %! # This test comes from an example I found on the web %! lambda = 40; %! mu = 30; %! m = 3; %! k = 7; %! [U R Q X p0] = qsmmmk( lambda, mu, m, k ); %! assert( p0, 0.255037, 1e-6 ); %! assert( R, 0.036517, 1e-6 ); %!test %! # This test comes from an example I found on the web %! lambda = 50; %! mu = 10; %! m = 4; %! k = 6; %! [U R Q X p0 pk] = qsmmmk( lambda, mu, m, k ); %! assert( pk, 0.293543, 1e-6 ); %!test %! # This test comes from an example I found on the web %! lambda = 3; %! mu = 2; %! m = 2; %! k = 5; %! [U R Q X p0 pk] = qsmmmk( lambda, mu, m, k ); %! assert( p0, 0.179334, 1e-6 ); %! assert( pk, 0.085113, 1e-6 ); %! assert( Q, 2.00595, 1e-5 ); %! assert( R-1/mu, 0.230857, 1e-6 ); # waiting time in the queue %!demo %! ## Given a M/M/m/K queue, compute the steady-state probability pn %! ## of having n jobs in the systen. %! lambda = 0.2; %! mu = 0.25; %! m = 5; %! K = 20; %! n = 0:10; %! pn = qsmmmk(lambda, mu, m, K, n); %! plot(n, pn, "-o", "linewidth", 2); %! xlabel("N. of jobs (n)"); %! ylabel("P_n"); %! title(sprintf("M/M/%d/%d system, \\lambda = %g, \\mu = %g", m, K, lambda, mu)); queueing/inst/dtmctaexps.m0000644000175000017500000000533213570046274015602 0ustar morenomoreno## Copyright (C) 2012, 2014, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{M} =} dtmctaexps (@var{P}, @var{n}, @var{p0}) ## @deftypefnx {Function File} {@var{M} =} dtmctaexps (@var{P}, @var{p0}) ## ## @cindex time-alveraged sojourn time, DTMC ## @cindex discrete time Markov chain ## @cindex Markov chain, discrete time ## @cindex DTMC ## ## Compute the @emph{time-averaged sojourn times} @code{@var{M}(i)}, ## defined as the fraction of time spent in state @math{i} during the ## first @math{n} transitions (or until absorption), assuming that the ## state occupancy probabilities at time 0 are @var{p0}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## @math{N \times N} transition probability matrix. ## ## @item @var{n} ## Number of transitions during which the time-averaged expected sojourn times ## are computed (scalar, @math{@var{n} @geq{} 0}). if @math{@var{n} = 0}, ## returns @var{p0}. ## ## @item @var{p0}(i) ## Initial state occupancy probabilities (vector of length @math{N}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{M}(i) ## If this function is called with three arguments, @code{@var{M}(i)} is ## the expected fraction of steps @math{@{0, @dots{}, n@}} spent in ## state @math{i}, assuming that the state occupancy probabilities at ## time zero are @var{p0}. If this function is called with two ## arguments, @code{@var{M}(i)} is the expected fraction of steps spent ## in state @math{i} until absorption. @var{M} is a vector of length ## @math{N}. ## ## @end table ## ## @seealso{dtmcexps} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function M = dtmctaexps( P, varargin ) persistent epsilon = 10*eps; if ( nargin < 2 || nargin > 3 ) print_usage(); endif L = dtmcexps(P,varargin{:}); M = L ./ sum(L); endfunction %!test %! P = dtmcbd([1 1 1 1], [0 0 0 0]); %! p0 = [1 0 0 0 0]; %! L = dtmctaexps(P,p0); %! assert( L, [.25 .25 .25 .25 0], 10*eps ); queueing/inst/ctmcbd.m0000644000175000017500000000705613570046274014667 0ustar morenomoreno## Copyright (C) 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{Q} =} ctmcbd (@var{b}, @var{d}) ## ## @cindex Markov chain, continuous time ## @cindex continuous time Markov chain ## @cindex CTMC ## @cindex birth-death process, CTMC ## ## Returns the infinitesimal generator matrix @math{Q} for a ## continuous birth-death process over the finite state space ## @math{@{1, @dots{}, N@}}. For each @math{i=1, @dots{}, (N-1)}, ## @code{@var{b}(i)} is the transition rate from state @math{i} to ## state @math{(i+1)}, and @code{@var{d}(i)} is the transition rate from state ## @math{(i+1)} to state @math{i}. ## ## Matrix @math{\bf Q} is therefore defined as: ## ## @tex ## $$ \pmatrix{ -\lambda_1 & \lambda_1 & & & & \cr ## \mu_1 & -(\mu_1 + \lambda_2) & \lambda_2 & & \cr ## & \mu_2 & -(\mu_2 + \lambda_3) & \lambda_3 & & \cr ## \cr ## & & \ddots & \ddots & \ddots & & \cr ## \cr ## & & & \mu_{N-2} & -(\mu_{N-2}+\lambda_{N-1}) & \lambda_{N-1} \cr ## & & & & \mu_{N-1} & -\mu_{N-1} } ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## / \ ## | -b(1) b(1) | ## | d(1) -(d(1)+b(2)) b(2) | ## | d(2) -(d(2)+b(3)) b(3) | ## | | ## | ... ... ... | ## | | ## | d(N-2) -(d(N-2)+b(N-1)) b(N-1) | ## | d(N-1) -d(N-1) | ## \ / ## @end group ## @end example ## @end ifnottex ## ## @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and ## death rates, respectively. ## ## @seealso{dtmcbd} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function Q = ctmcbd( birth, death ) if ( nargin != 2 ) print_usage(); endif ( isvector( birth ) && isvector( death ) ) || ... error( "birth and death must be vectors" ); birth = birth(:); # make birth a column vector death = death(:); # make death a column vector size_equal( birth, death ) || ... error( "birth and death rates must have the same length" ); all( birth >= 0 ) || ... error( "birth rates must be >= 0" ); all( death >= 0 ) || ... error( "death rates must be >= 0" ); ## builds the infinitesimal generator matrix Q = diag( birth, 1 ) + diag( death, -1 ); Q -= diag( sum(Q,2) ); endfunction %!test %! birth = [ 1 1 1 ]; %! death = [ 2 2 2 ]; %! Q = ctmcbd( birth, death ); %! assert( ctmc(Q), [ 8/15 4/15 2/15 1/15 ], 1e-5 ); queueing/inst/qncmnpop.m0000644000175000017500000000562313570046274015264 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{H} =} qncmnpop (@var{N}) ## ## @cindex population mix ## @cindex closed network, multiple classes ## ## Given a network with @math{C} customer classes, this function ## computes the number of @math{k}-mixes @code{@var{H}(r,k)} that can ## be constructed by the multiclass MVA algorithm by allocating ## @math{k} customers to the first @math{r} classes. ## @xref{doc-qncmpopmix} for the definition of @math{k}-mix. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N}(c) ## number of class-@math{c} requests in the system. The total number ## of requests in the network is @code{sum(@var{N})}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{H}(r,k) ## is the number of @math{k} mixes that can be constructed allocating ## @math{k} customers to the first @math{r} classes. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item Zahorjan, J. and Wong, E. @cite{The solution of separable queueing ## network models using mean value analysis}. SIGMETRICS ## Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI ## @uref{http://doi.acm.org/10.1145/1010629.805477, 10.1145/1010629.805477} ## @end itemize ## ## @seealso{qncmmva,qncmpopmix} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function H = qncmnpop( N ) (isvector(N) && all( N > 0 ) ) || ... error( "N must be a vector of strictly positive integers" ); N = N(:)'; # make N a row vector Ns = sum(N); R = length(N); ## In the implementation below we initialize the variable ## @code{TOTAL_POP} variable to 0 instead of @code{@var{N}(1)} as in ## the paper. Moreover, we increment @code{TOTAL_POP} ## @code{@var{N}(r)} at each iteration, instead of ## @code{@var{N}(r-1)} as in the paper. total_pop = N(1); H = zeros(R, Ns+1); H(1,1:N(1)+1) = 1; for r=2:R total_pop += N(r); for n=0:total_pop range = max(0,n-N(r)) : n; H(r,n+1) = sum( H(r-1, range+1 ) ); endfor endfor endfunction %!test %! H = qncmnpop( [1 2 2] ); %! assert( H, [1 1 0 0 0 0; 1 2 2 1 0 0; 1 3 5 5 3 1] ); queueing/inst/qncsmvaap.m0000644000175000017500000001760413570046274015424 0ustar morenomoreno## Copyright (C) 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}, @var{iter_max}) ## ## @cindex Mean Value Analysys (MVA), approximate ## @cindex MVA, approximate ## @cindex approximate MVA ## @cindex closed network, single class ## @cindex closed network, approximate analysis ## ## Analyze closed, single class queueing networks using the Approximate ## Mean Value Analysis (MVA) algorithm. This function is based on ## approximating the number of customers seen at center @math{k} when a ## new request arrives as @math{Q_k(N) \times (N-1)/N}. This function ## only handles single-server and delay centers; if your network ## contains general load-dependent service centers, use the function ## @code{qncsmvald} instead. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## Population size (number of requests in the system, @code{@var{N} > 0}). ## ## @item @var{S}(k) ## mean service time on server @math{k} ## (@code{@var{S}(k)>0}). ## ## @item @var{V}(k) ## average number of visits to service center ## @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k} ## (if @var{m} is a scalar, all centers have that number of servers). If ## @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); if ## @code{@var{m}(k) == 1}, center @math{k} is a regular queueing ## center (FCFS, LCFS-PR or PS) with one server (default). This function ## does not support multiple server nodes (@code{@var{m}(k) > 1}). ## ## @item @var{Z} ## External delay for customers (@code{@var{Z} @geq{} 0}). Default is 0. ## ## @item @var{tol} ## Stopping tolerance. The algorithm stops when the maximum relative ## difference between the new and old value of the queue lengths ## @var{Q} becomes less than the tolerance. Default is @math{10^{-5}}. ## ## @item @var{iter_max} ## Maximum number of iterations (@code{@var{iter_max}>0}. ## The function aborts if convergenge is not reached within the maximum ## number of iterations. Default is 100. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## If @math{k} is a FCFS, LCFS-PR or PS node (@code{@var{m}(k) == 1}), ## then @code{@var{U}(k)} is the utilization of center @math{k}. If ## @math{k} is an IS node (@code{@var{m}(k) < 1}), then ## @code{@var{U}(k)} is the @emph{traffic intensity} defined as ## @code{@var{X}(k)*@var{S}(k)}. ## ## @item @var{R}(k) ## response time at center @math{k}. ## The system response time @var{Rsys} ## can be computed as @code{@var{Rsys} = @var{N}/@var{Xsys} - Z} ## ## @item @var{Q}(k) ## average number of requests at center @math{k}. The number of ## requests in the system can be computed either as ## @code{sum(@var{Q})}, or using the formula ## @code{@var{N}-@var{Xsys}*@var{Z}}. ## ## @item @var{X}(k) ## center @math{k} throughput. The system throughput @var{Xsys} can be ## computed as @code{@var{Xsys} = @var{X}(1) / @var{V}(1)} ## ## @end table ## ## @strong{REFERENCES} ## ## This implementation is based on Edward D. Lazowska, John Zahorjan, ## G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System ## Performance: Computer System Analysis Using Queueing Network Models}, ## Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 6.4.2.2 ("Approximate Solution Techniques"). ## ## @seealso{qncsmva,qncsmvald} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncsmvaap( N, S, V, m, Z, tol, iter_max ) if ( nargin < 3 || nargin > 7 ) print_usage(); endif isscalar(N) && N >= 0 || ... error( "N must be >= 0" ); isvector(S) || ... error( "S must be a vector" ); isvector(V) || ... error( "V must be a vector" ); S = S(:)'; # make S a row vector V = V(:)'; # make V a row vector K = length(S); # Number of servers if ( nargin < 4 ) m = ones(1,K); else isvector(m) || ... error( "m must be a vector" ); m = m(:)'; # make m a row vector endif [err S V m] = common_size(S, V, m); (err == 0) || ... error( "S, V and m are of incompatible size" ); all(S>=0) || ... error( "S must be a vector >= 0" ); all(V>=0) || ... error( "V must be a vector >= 0" ); all(m<=1) || ... error( "Vector m must be <= 1 (this function supports IS and single-server nodes only)" ); if ( nargin < 5 ) Z = 0; else (isscalar(Z) && Z >= 0) || ... error( "Z must be >= 0" ); endif if ( nargin < 6 ) tol = 1e-5; else ( isscalar(tol) && tol>0 ) || ... error("tol must be a positive scalar"); endif if ( nargin < 7 ) iter_max = 100; else ( isscalar(iter_max) && iter_max > 0 ) || ... error("iter_max must be a positive integer"); endif U = R = Q = X = zeros( 1, K ); ## Trivial case of empty population: just return all zeros if ( N == 0 ) return; endif Q = N/K * ones(1,K); # initialize queue lengths iter = 0; do iter++; Qold = Q; A = (N-1)/N * Q; R = S.*(1+A.*(m==1)); Rs = dot(V,R); Xs = N/(Z+Rs); Q = Xs*(V.*R); err = norm((Q-Qold)./Qold, "inf"); until (err < tol || iter>iter_max); if ( iter > iter_max ) warning( "qncsmvaap(): Convergence not reached after %d iterations", iter_max ); endif X = Xs * V; U = X .* S; endfunction %!test %! fail( "qncsmvaap()", "Invalid" ); %! fail( "qncsmvaap( 10, [1 2], [1 2 3] )", "S, V and m" ); %! fail( "qncsmvaap( 10, [-1 1], [1 1] )", ">= 0" ); %! fail( "qncsmvaap( 10, [1 2], [1 2], [1 2] )", "supports"); %! fail( "qncsmvaap( 10, [1 2], [1 2], [1 1], 0, -1)", "tol"); %!test %! # Example p. 117 Lazowska et al. %! S = [0.605 2.1 1.35]; %! V = [1 1 1]; %! N = 3; %! Z = 15; %! m = 1; %! [U R Q X] = qncsmvaap(N, S, V, m, Z); %! Rs = dot(V,R); %! Xs = N/(Z+Rs); %! assert( Q, [0.0973 0.4021 0.2359], 1e-3 ); %! assert( Xs, 0.1510, 1e-3 ); %! assert( Rs, 4.87, 1e-3 ); %!demo %! S = [ 0.125 0.3 0.2 ]; %! V = [ 16 10 5 ]; %! N = 30; %! m = ones(1,3); %! Z = 4; %! Xmva = Xapp = Rmva = Rapp = zeros(1,N); %! for n=1:N %! [U R Q X] = qncsmva(n,S,V,m,Z); %! Xmva(n) = X(1)/V(1); %! Rmva(n) = dot(R,V); %! [U R Q X] = qncsmvaap(n,S,V,m,Z); %! Xapp(n) = X(1)/V(1); %! Rapp(n) = dot(R,V); %! endfor %! subplot(2,1,1); %! plot(1:N, Xmva, ";Exact;", "linewidth", 2, 1:N, Xapp, "x;Approximate;", "markersize", 7); %! legend("location","southeast"); legend("boxoff"); %! ylabel("Throughput X(n)"); %! subplot(2,1,2); %! plot(1:N, Rmva, ";Exact;", "linewidth", 2, 1:N, Rapp, "x;Approximate;", "markersize", 7); %! legend("location","southeast"); legend("boxoff"); %! ylabel("Response Time R(n)"); %! xlabel("Number of Requests n"); queueing/inst/qncmmva.m0000644000175000017500000007460413570046274015100 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S} ) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{P}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{P}, @var{r}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{P}, @var{r}, @var{m}) ## ## @cindex Mean Value Analysys (MVA) ## @cindex closed network, multiple classes ## @cindex multiclass network, closed ## ## Compute steady-state performance measures for closed, multiclass ## queueing networks using the Mean Value Analysys (MVA) algorithm. ## ## Queueing policies at service centers can be any of the following: ## ## @table @strong ## ## @item FCFS ## (First-Come-First-Served) customers are served in order of arrival; ## multiple servers are allowed. For this kind of queueing discipline, ## average service times must be class-independent. ## ## @item PS ## (Processor Sharing) customers are served in parallel by a single ## server, each customer receiving an equal share of the service rate. ## ## @item LCFS-PR ## (Last-Come-First-Served, Preemptive Resume) customers are served in ## reverse order of arrival by a single server and the last arrival ## preempts the customer in service who will later resume service at the ## point of interruption. ## ## @item IS ## (Infinite Server) customers are delayed independently of other ## customers at the service center (there is effectively an infinite ## number of servers). ## ## @end table ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N}(c) ## number of class @math{c} requests; @code{@var{N}(c) @geq{} 0}. If ## class @math{c} has no requests (@code{@var{N}(c) == 0}), then for ## all @var{k}, this function returns ## @code{@var{U}(c,k) = @var{R}(c,k) = @var{Q}(c,k) = @var{X}(c,k) = 0} ## ## @item @var{S}(c,k) ## mean service time for class @math{c} requests at center @math{k} ## (@code{@var{S}(c,k) @geq{} 0}). If the service time at center ## @math{k} is class-dependent, then center @math{k} is assumed ## to be of type @math{-/G/1}--PS (Processor Sharing). If center ## @math{k} is a FCFS node (@code{@var{m}(k)>1}), then the service ## times @strong{must} be class-independent, i.e., all classes ## @strong{must} have the same service time. ## ## @item @var{V}(c,k) ## average number of visits of class @math{c} requests at ## center @math{k}; @code{@var{V}(c,k) @geq{} 0}, default is 1. ## @strong{If you pass this argument, class switching is not allowed} ## ## @item @var{P}(r,i,s,j) ## probability that a class @math{r} request completing service at center ## @math{i} is routed to center @math{j} as a class @math{s} request; the ## reference stations for each class are specified with the paramter ## @var{r}. @strong{If you pass argument @var{P}, class switching is ## allowed}; however, you can not specify any external delay (i.e., ## @var{Z} must be zero) and all servers must be fixed-rate or ## infinite-server nodes (@code{@var{m}(k) @leq{} 1} for all ## @math{k}). ## ## @item @var{r}(c) ## reference station for class @math{c}. If omitted, station 1 is the ## reference station for all classes. See @command{qncmvisits}. ## ## @item @var{m}(k) ## If @code{@var{m}(k)<1}, then center @math{k} is assumed to be a delay ## center (IS node @math{-/G/\infty}). If @code{@var{m}(k)==1}, then ## service center @math{k} is a regular queueing center ## (@math{M/M/1}--FCFS, @math{-/G/1}--LCFS-PR or @math{-/G/1}--PS). ## Finally, if @code{@var{m}(k)>1}, center @math{k} is a ## @math{M/M/m}--FCFS center with @code{@var{m}(k)} identical servers. ## Default is @code{@var{m}(k)=1} for each @math{k}. ## ## @item @var{Z}(c) ## class @math{c} external delay (think time); @code{@var{Z}(c) @geq{} ## 0}. Default is 0. This parameter can not be used if you pass a ## routing matrix as the second parameter of @code{qncmmva}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(c,k) ## If @math{k} is a FCFS, LCFS-PR or PS node (@code{@var{m}(k) @geq{} ## 1}), then @code{@var{U}(c,k)} is the class @math{c} utilization at ## center @math{k}, @math{0 @leq{} U(c,k) @leq{} 1}. If @math{k} is an ## IS node, then @code{@var{U}(c,k)} is the class @math{c} @emph{traffic ## intensity} at center @math{k}, defined as @code{@var{U}(c,k) = ## @var{X}(c,k)*@var{S}(c,k)}. In this case the value of ## @code{@var{U}(c,k)} may be greater than one. ## ## @item @var{R}(c,k) ## class @math{c} response time at center @math{k}. The class @math{c} ## @emph{residence time} at center @math{k} is @code{@var{R}(c,k) * ## @var{C}(c,k)}. The total class @math{c} system response time is ## @code{dot(@var{R}, @var{V}, 2)}. ## ## @item @var{Q}(c,k) ## average number of class @math{c} requests at center @math{k}. The ## total number of requests at center @math{k} is ## @code{sum(@var{Q}(:,k))}. The total number of class @math{c} ## requests in the system is @code{sum(@var{Q}(c,:))}. ## ## @item @var{X}(c,k) ## class @math{c} throughput at center @math{k}. The class @math{c} ## throughput can be computed as @code{@var{X}(c,1) / @var{V}(c,1)}. ## ## @end table ## ## @strong{NOTES} ## ## If the function call specifies the visit ratios @var{V}, then class ## switching is @strong{not} allowed. If the function call specifies ## the routing probability matrix @var{P}, then class switching ## @strong{is} allowed; however, in this case all nodes are restricted ## to be fixed rate servers or delay centers: multiple-server and ## general load-dependent centers are not supported. ## ## In presence of load-dependent servers (e.g., if @code{@var{m}(i)>1} ## for some @math{i}), the MVA algorithm is known to be numerically ## unstable. Generally this problem shows up as negative values for the ## computed response times or utilizations. This is not a problem with the ## @code{queueing} package, but with the MVA algorithm; ## as such, there is no known workaround at the moment (aoart from using a ## different solution technique, if available). This function prints a ## warning if it detects numerical problems; you can disable the warning ## with the command @code{warning("off", "qn:numerical-instability")}. ## ## Given a network with @math{K} service centers, @math{C} job classes ## and population vector @math{{\bf N}=\left[N_1, @dots{}, N_C\right]}, the MVA ## algorithm requires space @math{O(C \prod_i (N_i + 1))}. The time ## complexity is @math{O(CK\prod_i (N_i + 1))}. This implementation is ## slightly more space-efficient (see details in the code). While the ## space requirement can be mitigated by using some optimizations, the ## time complexity can not. If you need to analyze large closed networks ## you should consider the @command{qncmmvaap} function, which implements ## the approximate MVA algorithm. Note however that @command{qncmmvaap} ## will only provide approximate results. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed ## Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, April ## 1980, pp. 313--322. @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} ## @end itemize ## ## This implementation is based on G. Bolch, S. Greiner, H. de Meer and ## K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and ## Performance Evaluation with Computer Science Applications}, Wiley, ## 1998 and Edward D. Lazowska, John Zahorjan, G. Scott Graham, and ## Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer ## System Analysis Using Queueing Network Models}, Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 7.4.2.1 ("Exact Solution Techniques"). ## ## @seealso{qnclosed, qncmmvaapprox, qncmvisits} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncmmva( varargin ) if ( nargin < 2 || nargin > 5 ) print_usage(); endif if ( nargin == 2 || ndims(varargin{3}) == 2 ) [err N S V m Z] = qncmchkparam( varargin{:} ); isempty(err) || error(err); [U R Q X] = __qncmmva_nocs( N, S, V, m, Z ); else [U R Q X] = __qncmmva_cs( varargin{:} ); endif endfunction ############################################################################## ## Analyze closed, multiclass QNs with class switching function [U R Q X] = __qncmmva_cs( N, S, P, r, m ) if ( nargin < 3 || nargin > 5 ) print_usage(); endif isvector(N) && all( N>=0 ) || ... error( "N must be >=0" ); N = N(:)'; # make N a row vector C = length(N); ## Number of classes ( ndims(S) == 2 ) || ... error( "S must be a matrix" ); K = columns(S); ## Number of service centers size(S) == [C,K] || ... error( "S size mismatch (is %dx%d, should be %dx%d)", rows(S), columns(S), C, K ); ndims(P) == 4 && size(P) == [C,K,C,K] || ... error( "P size mismatch (should be %dx%dx%dx%d)",C,K,C,K ); if ( nargin < 4 ) r = ones(1,C); # reference station else if (isscalar(r)) r = r*ones(1,C); endif ( isvector(r) && length(r) == C ) || ... error("r must be a vector with %d elements", C); r = r(:)'; all( r>=1 && r<=K ) || ... error("elements of r are out of range [1,%d]",K); endif if ( nargin < 5 ) m = ones(1,K); else isvector(m) || ... error( "m must be a vector" ); m = m(:)'; # make m a row vector length(m) == K || ... error( "m size mismatch (should be %d, is %d)", K, length(m) ); endif ## Check consistency of parameters all(S(:) >= 0) || ... error( "S must be >= 0" ); all( any(S>0,2) ) || ... error( "S must contain at least a value >0 for each row" ); all(P(:)>= 0) || ... error( "P must be >=0" ); U = R = Q = X = zeros(C,K); ## 1. Compute visit counts [V ch] = qncmvisits(P,r); ## 2. Identify chains nch = max(ch); ## 3. Compute visit counts for the equivalent network Vstar = zeros(nch,K); for q=1:nch r = (ch == q); Vstar(q,:) = sum(V(r,:),1); endfor ## 4. Compute proportionality constants alpha = zeros(C,K); for r=1:C for k=find( Vstar(ch(r),:) > 0 ) alpha(r,k) = V(r,k) / Vstar( ch(r), k ); endfor endfor ## 5. Compute service times Sstar = zeros(nch,K); for q=1:nch r = (ch==q); Sstar(q,:) = dot( alpha(r,:), S(r,:), 1 ); endfor ## 6. Compute populations of superclasses Nstar = zeros(1,nch); for q=1:nch r = (ch == q); Nstar(q) = sum( N(r) ); endfor ## 7. Solve the equivalent network [Ustar Rstar Qstar Xstar Qnm1] = __qncmmva_nocs( Nstar, Sstar, Vstar, m, zeros(size(Nstar)) ); ## 8. Compute solutions of the original network for r=1:C for k=1:K R(r,k) = S(r,k) * (1 + Qnm1(ch(r),k)*(m(k)==1)); X(r,k) = alpha(r,k) * Xstar(ch(r),k); Q(r,k) = X(r,k) * R(r,k); U(r,k) = S(r,k) * X(r,k) / max(1,m(k)); endfor endfor ## 9. Check for numerical instability if ( any(U(:)<0) || any(R(:)<0) ) warning("qn:numerical-instability", "Numerical instability detected. Type 'help qncmmva' for details"); endif endfunction ############################################################################## ## Analyze closed, multiclass QNs WITHOUT class switching ## ## This implementation is based on: ## ## Herb Schwetman, "Implementing the Mean Value Algorithm for the ## Solution of Queueing Network Models", technical report OSD-TR-355, ## dept. of Computer Science, Purdue University, feb. 1982. ## function [U R Q X Qnm1] = __qncmmva_nocs( N, S, V, m, Z ) assert( nargin == 5 ); all( any(S>0,2) ) || ... error( "S must contain at least a value >0 for each row" ); [C K] = size(S); ## ensure that the service times for multiserver nodes ## are class-independent for k=find(m>1) all( S(:,k) == S(1,k) ) || ... error( "Service times for FCFS node %d are not class-independent", k ); endfor ## Initialize results R = zeros( C, K ); X = zeros( 1, C ); D = S .* V; ## The multiclass MVA algorithm requires to store the queue lengths Q( ## _n_, k ) at center k where the population vector is _n_. The space ## required would be K*prod(N+1), but this can be reduced by ## considering that, at each iteration of the main MVA loop, the total ## number of requests is n; therefore it is sufficient to consider the ## first (C-1) components of vector _n_ to uniquely identify the cell ## containing Q( _n_, k ). See Schwetman for a better explanation. bufsize = prod((N+1)(1:end-1)); Q_next = Q = zeros( bufsize,K ); p = cell(1,K); for k=find(m>1) ## p{i}(j+1,k+1) is the probability to have j jobs at node i ## where the network is in state k p{k} = zeros( m(k)+1,bufsize ); p{k}(1,__getidx(N,0*N)) = 1; endfor Qnm1= zeros(C,K); ## Qnm1(c,k) is the number of requests in center k, provided that the population size is N-1_c (N is the total population vector). This value is needed by __qncmmva_cs. Qnm1 is only filled for M/M/1 or PS centers. The values are not computed for PS nodes dd = zeros(1,C); for c=find(N>0) h = zeros(1,C); h(c) = 1; dd(c) = __getidx(N,h)-1; endfor for n=1:sum(N) ## MVA iteration for population size n n_bar = zeros(1, C); const = min(n, N); mp = 0; while ( all(n_bar(C) <= const(C)) ) ## Fill the current configuration (algorithm 3b, p. 10, Schwetman) x=n-mp; i=1; while ( x>0 && i<=C ) n_bar(i) = min(x,const(i)); x -= n_bar(i); mp += n_bar(i); i += 1; endwhile idx = __getidx( N, n_bar ); R = S; ## Compute response time for LI servers k=find(m==1); for c=find(n_bar>0) ## idx-dd(c) is the index of element n_bar - 1_c R(c,k) = S(c,k).*(1 + Q( idx-dd(c), k ) ); Qnm1(c,k) = Q( idx-dd(c), k); ## for FCFS nodes with class-dependent service times, ## it is possible to use the following approximation ## (p. 469 Bolch et al.) ## ## R(c,k) = S(c,k) + sum( S(:,k) * Q(idx(:), k) ); ## R(c,k) = S(c,k) + sum( S(:,k) .* Q(idx, k) .* V(:,k) ) / sum(V(:,k)); endfor ## Compute response time for LD servers for k=find(m>1) j=0:m(k)-2; # range for c=find(n_bar > 0 ) R(c,k) = S(c,k)/m(k)*(1 + Q( idx-dd(c), k ) + ... dot(m(k)-j-1,p{k}(j+1,idx-dd(c)) ) ); endfor endfor X = n_bar ./ ( Z .+ dot(R,V,2)' ); # X(c) = N(c) / ( Z(c) + sum_k R(c,k) * V(c,k) ) ## Q_k = sum_c X(c) * R(c,k) * V(c,k) Q_next( idx, : ) = (X * (R .* V))'; ## Update marginal probabilities for LD servers for k=find(m>1) j = 1:m(k)-1; s = zeros(size(j)); for r=find(n_bar>0) # FIXME: vectorize this s+=(D(r,k)*X(r)*p{k}(j,idx-dd(r)))'; endfor p{k}(j+1,idx) = s./j; p{k}(1,idx) = 1-1/m(k)*(dot( D(:,k),X ) + ... dot( m(k)-j, p{k}(j+1,idx) ) ); endfor #{ ## The following "if" is in the paper, but it makes the algorithm ## incorrect. Therefore, it is commented out. if ( n_bar(C) == N(C) ) break; endif #} ## Advance to next feasible configuration (Algorithm 3c, p. 10 Schwetman) i = 1; sw = true; while sw if ( ( mp==n || n_bar(i)==const(i)) && ( i14.323 %! assert( U, [14.323 0 .358; 0 4.707 .157], 1e-3 ); %! # FIXME: I replaced 14.3->14.323 %! assert( X, [14.323 0 14.323; 0 .314 .314 ], 1e-3 ); %! # FIXME: I replaced 14.3->14.323 %! assert( Q, [14.323 0 .677; 0 4.707 .293 ], 1e-3 ); %! assert( R, [1 0 .047; 0 15 .934 ], 1e-3 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law ## Example 9.5 p. 337, Bolch et al. %!test %! S = [ 0.2 0.4 1; 0.2 0.6 2 ]; %! V = [ 1 0.6 0.4; 1 0.3 0.7 ]; %! N = [ 2 1 ]; %! m = [ 2 1 -1 ]; %! [U R Q X] = qncmmva(N,S,V,m); %! assert( Q, [ 0.428 0.726 0.845; 0.108 0.158 0.734 ], 1e-3 ); %! assert( X(1,1), 2.113, 1e-3 ); # CHECK %! assert( X(2,1), 0.524, 1e-3 ); # CHECK %! assert( all(U(:)<=1) ); %! assert( Q, R.*X, 1e-5 ); # Little's Law ## Multiclass network with two classes; however, class 2 has 0 requests. ## Therefore, we check that the results for class 1 are the same as those ## computed by the single-class MVA %!test %! C = 2; # two classes %! K = 4; # four servers %! S = V = zeros(C,K); %! S(1,:) = linspace(1,2,K); %! S(2,:) = linspace(2,3,K); %! V(1,:) = linspace(4,1,K); %! V(2,:) = linspace(6,3,K); %! N = [10 0]; # class 2 has no customers %! [U1 R1 Q1 X1] = qncmmva(N,S,V); %! [U2 R2 Q2 X2] = qncsmva(N(1),S(1,:),V(1,:)); %! assert( U1(1,:), U2, 1e-5 ); %! assert( R1(1,:), R2, 1e-5 ); %! assert( Q1(1,:), Q2, 1e-5 ); %! assert( X1(1,:), X2, 1e-5 ); ## This is example 5(b) page 7 of ## http://docs.lib.purdue.edu/cstech/259/ ## "Testing network-of-queues software", technical report CSD-TR 330, %!test %! Z = [1 15]; %! V = [1; 1]; %! S = [.025; .5]; %! N = [15; 5]; %! [U R Q X] = qncmmva(N, S, V, 1, Z); %! assert( U, [.358; .157], 1e-3 ); %! assert( Q, [.677; .293], 1e-3 ); %! assert( X, [14.323; .314], 1e-3 ); ## NOTE: X(1,1) = 14.3 in Schwetman %! assert( R, [.047; .934], 1e-3 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law ## This is example of Figure 6, page 9 of ## http://docs.lib.purdue.edu/cstech/259/ ## "Testing network-of-queues software", technical report CSD-TR 330, %!test %! C = 2; %! K = 6; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = P(2,1,2,2) = 1; %! P(1,2,1,3) = P(1,2,1,4) = P(1,2,1,5) = P(1,2,1,6) = .25; %! P(2,2,2,3) = P(2,2,2,4) = P(2,2,2,5) = P(2,2,2,6) = .25; %! P(1,3,1,1) = P(1,4,1,1) = P(1,5,1,1) = P(1,6,1,1) = .9; %! P(1,3,1,2) = P(1,4,1,2) = P(1,5,1,2) = P(1,6,1,2) = .1; %! P(2,3,2,1) = P(2,4,2,1) = P(2,5,2,1) = P(2,6,2,1) = .05; %! P(2,3,2,2) = P(2,4,2,2) = P(2,5,2,2) = P(2,6,2,2) = .95; %! N = [40 4]; %! S = [ 5.0 .010 .035 .035 .035 .035; ... %! 10.0 .100 .035 .035 .035 .035 ]; %! V = qncmvisits(P); %! [U R Q X] = qncmmva(N, S, V, [-1 1 1 1 1 1]); %! # FIXME: The results below were computed with JMVA; the numbers %! # in the paper are different (wrong?!?)!! %! assert( U, [39.457941 0.087684 0.076724 0.076724 0.076724 0.076724; ... %! 2.772704 0.554541 0.048522 0.048522 0.048522 0.048522 ], 1e-5 ); %! assert( R.*V, [5 0.024363 0.011081 0.011081 0.011081 0.011081; ... %! 10 3.636155 0.197549 0.197549 0.197549 0.197549 ], 1e-5 ); %! assert( Q(:,1), [39.457941 2.772704]', 1e-5 ); %! assert( Q(:,2), [0.192262 1.008198]', 1e-5 ); %! assert( Q(:,3), [0.087449 0.054775]', 1e-5 ); %! assert( Q(:,4), Q(:,5), 1e-5 ); %! assert( Q(:,5), Q(:,6), 1e-5 ); %! assert( X(:,1), [7.891588 0.277270]', 1e-5 ); %! assert( X(:,2), [8.768431 5.545407]', 1e-5 ); %! assert( X(:,3), [2.192108 1.386352]', 1e-5 ); %! assert( X(:,4), X(:,5), 1e-5 ); %! assert( X(:,5), X(:,6), 1e-5 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law ## If there is no class switching, we must get the same results as ## the plain application of multiclass MVA %!test %! C = 2; # two classes %! K = 4; # four servers %! C = 2; K = 4; %! P = zeros(C,K,C,K); %! S = zeros(C,K); %! %! # Routing %! %! # class 1 routing %! P(1,1,1,1) = .05; %! P(1,1,1,2) = .45; %! P(1,1,1,3) = .5; %! P(1,2,1,1) = 1; %! P(1,3,1,1) = 1; %! # class 2 routing %! P(2,1,2,1) = .01; %! P(2,1,2,3) = .5; %! P(2,1,2,4) = .49; %! P(2,3,2,1) = 1; %! P(2,4,2,1) = 1; %! %! # Compute visits %! %! V = qncmvisits(P); %! %! # Define population and service times %! %! N = [3 2]; %! S = [0.01 0.09 0.10 0.08; ... %! 0.05 0.09 0.10 0.08]; %! [U1 R1 Q1 X1] = qncmmva(N,S,V); # this invokes __qncmmva_nocs %! [U2 R2 Q2 X2] = qncmmva(N,S,P); # this invokes __qncmmva_cs %! assert( U2, U1, 1e-5 ); %! assert( R2, R1, 1e-5 ); %! assert( Q2, Q1, 1e-5 ); %! assert( X2, X1, 1e-5 ); ## Example from table 5, p. 23, Herb Schwetman, "Implementing the Mean ## Value Algorith for the Solution of Queueing Network Models", ## Technical Report CSD-TR-355, Department of Computer Sciences, Purdue ## University, feb 15, 1982. %!test %! S = [1 0 .025; 0 15 .5]; %! V = [1 0 1; 0 1 1]; %! N = [15 5]; %! m = [-1 -1 1]; %! [U R Q X] = qncmmva(N,S,V,m); %! assert( U, [14.323 0 .358; 0 4.707 .157], 1e-3 ); %! assert( R, [1.0 0 .047; 0 15 .934], 1e-3 ); %! assert( Q, [14.323 0 .677; 0 4.707 .293], 1e-3 ); %! assert( X, [14.323 0 14.323; 0 .314 .314], 1e-3 ); ## Same test as above, but using routing probabilities instead of ## visits. Also, reordered the nodes such that server 1 is the PS node ## labeled "Sys 3" in the example; server 2 is the IS labeled "APL1" and ## server e is the IS labeled "IMS2" %!test %! S = [.025 1 15; .5 1 15 ]; %! P = zeros(2,3,2,3); %! P(1,1,1,2) = P(1,2,1,1) = 1; %! P(2,1,2,3) = P(2,3,2,1) = 1; %! N = [15 5]; %! m = [1 -1 -1]; %! r = [1 1]; # reference station is station 1 %! [U R Q X] = qncmmva(N,S,P,r,m); %! # FIXME: I replaced 14.3->14.323 %! assert( U, [0.358 14.323 0; 0.156 0 4.707], 1e-3 ); %! # FIXME: I replaced 14.3->14.323 %! assert( X, [14.323 14.3230 0; .314 0 .314 ], 1e-3 ); %! # FIXME: I replaced 14.3->14.323 %! assert( Q, [.677 14.323 0; .293 0 4.707], 1e-3 ); %! assert( R, [.047 1 15.0; .934 1 15.0], 1e-3 ); %! assert( Q, R.*X, 1e-5 ); # Little's Law ## Example figure 9 Herb Schwetman "Implementing the Mean ## Value Algorith for the Solution of Queueing Network Models", ## Technical Report CSD-TR-355, Department of Computer Sciences, Purdue ## University, feb 15, 1982. %!test %! C = 2; K = 3; %! S = [.01 .07 .10; ... %! .05 .07 .10 ]; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = .7; %! P(1,1,1,3) = .2; %! P(1,1,2,1) = .1; %! P(2,1,2,2) = .3; %! P(2,1,2,3) = .5; %! P(2,1,1,1) = .2; %! P(1,2,1,1) = P(2,2,2,1) = 1; %! P(1,3,1,1) = P(2,3,2,1) = 1; %! N = [3 0]; %! [U R Q X] = qncmmva(N, S, P); %! assert( R, [.015 .133 .163; .073 .133 .163], 1e-3 ); %! assert( X, [12.609 8.826 2.522; 6.304 1.891 3.152], 1e-3 ); %! assert( Q, [.185 1.175 .412; .462 .252 .515], 1e-3 ); %! assert( U, [.126 .618 .252; .315 .132 .315], 1e-3 ); ## Example from Schwetman (figure 7, page 9 of ## http://docs.lib.purdue.edu/cstech/259/ ## "Testing network-of-queues software, technical report CSD-TR 330, ## Purdue University). Note that the results for that network (table 9 ## of the reference above) seems to be wrong. The "correct" results ## below have been computed using the multiclass MVA implementation of ## JMT (http://jmt.sourceforge.net/) %!test %! V = [ 1.00 0.45 0.50 0.00; ... %! 1.00 0.00 0.50 0.49 ]; %! N = [3 2]; %! S = [0.01 0.09 0.10 0.08; ... %! 0.05 0.09 0.10 0.08]; %! [U R Q X] = qncmmva(N, S, V); %! assert( U, [ 0.1215 0.4921 0.6075 0.0000; ... %! 0.3433 0.0000 0.3433 0.2691 ], 1e-4 ); %! assert( Q, [ 0.2131 0.7539 2.0328 0.0000; ... %! 0.5011 0.0000 1.1839 0.3149 ], 1e-4 ); %! assert( R.*V, [0.0175 0.0620 0.1672 0.0000; ... %! 0.0729 0.0000 0.1724 0.0458 ], 1e-4 ); %! assert( X, [12.1517 5.4682 6.0758 0.0000; ... %! 6.8669 0.0000 3.4334 3.3648 ], 1e-4 ); ## The following test case was used in a bug report from F. Paolieri. It ## consists of a three-class network with four nodes. There is just ## one job for each class n; that job visits node 1 and node 1+n. ## Singe service times at nodes 2:4 are all the same, and service time ## at node 1 is class-independent, we expect that all performance ## measures for nodes 2:4 are equal. %!test %! N = [1 1 1]; %! S = [0.20000 0.02000 0.00000 0.00000; %! 0.20000 0.00000 0.02000 0.00000; %! 0.20000 0.00000 0.00000 0.02000]; %! V = [1 1 0 0; %! 1 0 1 0; %! 1 0 0 1]; %! [U R Q X] = qncmmva(N,S,V); %! assert( Q(1,2), Q(2,3), 1e-5); %! assert( Q(2,3), Q(3,4), 1e-5); %! assert( abs(max(Q(:,1)) - min(Q(:,1))) < 1e-5 ); ## This example is from G. Casale and G. Serazzi. Quantitative system ## evaluation with java modeling tools. In Proceedings of the second ## joint WOSP/SIPEW international conference on Performance engineering, ## ICPE '11, pages 449-454, New York, NY, USA, 2011. ACM %!demo %! Ntot = 100; # total population size %! b = linspace(0.1,0.9,10); # fractions of class-1 requests %! S = [20 80 31 14 23 12; ... %! 90 30 33 20 14 7]; %! V = ones(size(S)); %! X1 = X1 = XX = zeros(size(b)); %! R1 = R2 = RR = zeros(size(b)); %! for i=1:length(b) %! N = [fix(b(i)*Ntot) Ntot-fix(b(i)*Ntot)]; %! # printf("[%3d %3d]\n", N(1), N(2) ); %! [U R Q X] = qncmmva( N, S, V ); %! X1(i) = X(1,1) / V(1,1); %! X2(i) = X(2,1) / V(2,1); %! XX(i) = X1(i) + X2(i); %! R1(i) = dot(R(1,:), V(1,:)); %! R2(i) = dot(R(2,:), V(2,:)); %! RR(i) = Ntot / XX(i); %! endfor %! subplot(2,1,1); %! plot(b, X1, ";Class 1;", "linewidth", 2, ... %! b, X2, ";Class 2;", "linewidth", 2, ... %! b, XX, ";System;", "linewidth", 2 ); %! legend("location","south"); legend("boxoff"); %! ylabel("Throughput"); %! subplot(2,1,2); %! plot(b, R1, ";Class 1;", "linewidth", 2, ... %! b, R2, ";Class 2;", "linewidth", 2, ... %! b, RR, ";System;", "linewidth", 2 ); %! legend("location","south"); legend("boxoff"); %! xlabel("Population mix \\beta for Class 1"); %! ylabel("Resp. Time"); %!demo %! S = [1 0 .025; 0 15 .5]; %! P = zeros(2,3,2,3); %! P(1,1,1,3) = P(1,3,1,1) = 1; %! P(2,2,2,3) = P(2,3,2,2) = 1; %! V = qncmvisits(P,[3 3]); # reference station is station 3 %! N = [15 5]; %! m = [-1 -1 1]; %! [U R Q X] = qncmmva(N,S,V,m) ## Example shown on Figure 9: Herb Schwetman, "Implementing the Mean ## Value Algorith for the Solution of Queueing Network Models", ## Technical Report CSD-TR-355, Department of Computer Sciences, Purdue ## University, feb 15, 1982. %!demo %! C = 2; K = 3; %! S = [.01 .07 .10; ... %! .05 .07 .10 ]; %! P = zeros(C,K,C,K); %! P(1,1,1,2) = .7; P(1,1,1,3) = .2; P(1,1,2,1) = .1; %! P(2,1,2,2) = .3; P(2,1,2,3) = .5; P(2,1,1,1) = .2; %! P(1,2,1,1) = P(2,2,2,1) = 1; %! P(1,3,1,1) = P(2,3,2,1) = 1; %! N = [3 0]; %! [U R Q X] = qncmmva(N, S, P) queueing/inst/qnos.m0000644000175000017500000001447513570046274014416 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnos (@var{lambda}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnos (@var{lambda}, @var{S}, @var{V}, @var{m}) ## ## @cindex open network, single class ## @cindex BCMP network ## ## Analyze open, single class BCMP queueing networks with @math{K} service centers. ## ## This function works for a subset of BCMP single-class open networks ## satisfying the following properties: ## ## @itemize ## ## @item The allowed service disciplines at network nodes are: FCFS, ## PS, LCFS-PR, IS (infinite server); ## ## @item Service times are exponentially distributed and ## load-independent; ## ## @item Center @math{k} can consist of @code{@var{m}(k) @geq{} 1} ## identical servers. ## ## @item Routing is load-independent ## ## @end itemize ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Overall external arrival rate (@code{@var{lambda}>0}). ## ## @item @var{S}(k) ## average service time at center @math{k} (@code{@var{S}(k)>0}). ## ## @item @var{V}(k) ## average number of visits to center @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{i}. If @code{@var{m}(k) < 1}, ## enter @math{k} is a delay center (IS); otherwise it is a regular ## queueing center with @code{@var{m}(k)} servers. Default is ## @code{@var{m}(k) = 1} for all @math{k}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## If @math{k} is a queueing center, ## @code{@var{U}(k)} is the utilization of center @math{k}. ## If @math{k} is an IS node, then @code{@var{U}(k)} is the ## @emph{traffic intensity} defined as @code{@var{X}(k)*@var{S}(k)}. ## ## @item @var{R}(k) ## center @math{k} average response time. ## ## @item @var{Q}(k) ## average number of requests at center @math{k}. ## ## @item @var{X}(k) ## center @math{k} throughput. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks ## and Markov Chains: Modeling and Performance Evaluation with Computer ## Science Applications}, Wiley, 1998 ## @end itemize ## ## @seealso{qnopen,qnclosed,qnosvisits} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnos( varargin ) if ( nargin < 3 || nargin > 4 ) print_usage(); endif [err lambda S V m] = qnoschkparam( varargin {:} ); isempty(err) || error(err); all(S>0) || ... error( "S must be positive" ); ## If there are M/M/k servers with k>=1, compute the maximum ## processing capacity m(m<1) = -1; # avoids division by zero in next line [Umax kmax] = max( lambda * S .* V ./ m ); (Umax < 1) || ... error( "Processing capacity exceeded at center %d", kmax ); l = lambda*V; # arrival rates i = find( m == 1 ); # single station queueing centers if numel(i) [U(i) R(i) Q(i) X(i)] = qsmm1( l(i), 1./S(i) ); endif i = find( m<1 ); # delay centers if numel(i) [U(i) R(i) Q(i) X(i)] = qsmminf( l(i), 1./S(i) ); endif i = find( m>1 ); # multiple stations queueing centers if numel(i) [U(i) R(i) Q(i) X(i)] = qsmmm( l(i), 1./S(i), m(i) ); endif endfunction %!test %! lambda = 0; %! S = [1 1 1]; %! V = [1 1 1]; %! fail( "qnos(lambda,S,V)","lambda must be"); %! lambda = 1; %! S = [1 0 1]; %! fail( "qnos(lambda,S,V)","S must be"); %! S = [1 1 1]; %! m = [1 1]; %! fail( "qnos(lambda,S,V,m)","incompatible size"); %! V = [1 1 1 1]; %! fail( "qnos(lambda,S,V)","incompatible size"); %! fail( "qnos(1.0, [0.9 1.2], [1 1])", "exceeded at center 2"); %! fail( "qnos(1.0, [0.9 2.0], [1 1], [1 2])", "exceeded at center 2"); %! qnos(1.0, [0.9 1.9], [1 1], [1 2]); # should not fail %! qnos(1.0, [0.9 1.9], [1 1], [1 0]); # should not fail %! qnos(1.0, [1.9 1.9], [1 1], [0 0]); # should not fail %! qnos(1.0, [1.9 1.9], [1 1], [2 2]); # should not fail %!test %! # Example 34.1 p. 572 Bolch et al. %! lambda = 3; %! V = [16 7 8]; %! S = [0.01 0.02 0.03]; %! [U R Q X] = qnos( lambda, S, V ); %! assert( R, [0.0192 0.0345 0.107], 1e-2 ); %! assert( U, [0.48 0.42 0.72], 1e-2 ); %! assert( Q, R.*X, 1e-5 ); # check Little's Law %!test %! # Example p. 113, Lazowska et al. %! V = [121 70 50]; %! S = [0.005 0.03 0.027]; %! lambda=0.3; %! [U R Q X] = qnos( lambda, S, V ); %! assert( U(1), 0.182, 1e-3 ); %! assert( X(1), 36.3, 1e-2 ); %! assert( Q(1), 0.222, 1e-3 ); %! assert( Q, R.*X, 1e-5 ); # check Little's Law %!test %! lambda=[1]; %! P=[0]; %! V=qnosvisits(P,lambda); %! S=[0.25]; %! [U1 R1 Q1 X1]=qnos(sum(lambda),S,V); %! [U2 R2 Q2 X2]=qsmm1(lambda(1),1/S(1)); %! assert( U1, U2, 1e-5 ); %! assert( R1, R2, 1e-5 ); %! assert( Q1, Q2, 1e-5 ); %! assert( X1, X2, 1e-5 ); ## Check if processing capacity is properly accounted for %!test %! lambda = 1.1; %! V = 1; %! m = [2]; %! S = [1]; %! [U1 R1 Q1 X1] = qnos(lambda,S,V,m); %! m = [-1]; %! lambda = 90.0; %! [U1 R1 Q1 X1] = qnos(lambda,S,V,m); %!demo %! lambda = 3; %! V = [16 7 8]; %! S = [0.01 0.02 0.03]; %! [U R Q X] = qnos( lambda, S, V ); %! R_s = dot(R,V) # System response time %! N = sum(Q) # Average number in system %!test %! # Example 7.4 p. 287 Bolch et al. %! S = [ 0.04 0.03 0.06 0.05 ]; %! P = [ 0 0.5 0.5 0; 1 0 0 0; 0.6 0 0 0; 1 0 0 0 ]; %! lambda = [0 0 0 4]; %! V=qnosvisits(P,lambda); %! k = [ 3 2 4 1 ]; %! [U R Q X] = qnos( sum(lambda), S, V ); %! assert( X, [20 10 10 4], 1e-4 ); %! assert( U, [0.8 0.3 0.6 0.2], 1e-2 ); %! assert( R, [0.2 0.043 0.15 0.0625], 1e-3 ); %! assert( Q, [4, 0.429 1.5 0.25], 1e-3 ); queueing/inst/ctmcchkQ.m0000644000175000017500000000465113570046274015166 0ustar morenomoreno## Copyright (C) 2012 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{result} @var{err}] =} ctmcchkQ (@var{Q}) ## ## @cindex Markov chain, continuous time ## ## If @var{Q} is a valid infinitesimal generator matrix, return ## the size (number of rows or columns) of @var{Q}. If @var{Q} is not ## an infinitesimal generator matrix, set @var{result} to zero, and ## @var{err} to an appropriate error string. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [result err] = ctmcchkQ( Q ) persistent epsilon = 100*eps; if ( nargin != 1 ) print_usage(); endif result = 0; err = ""; if ( !issquare(Q) ) err = "P is not a square matrix"; return; endif if (any(Q(~logical(eye(size(Q))))<0) || ... # there is any negative non-diagonal element norm( sum(Q,2), "inf" ) > epsilon ) err = "Q is not an infinitesimal generator matrix"; return; endif result = rows(Q); endfunction %!test %! Q = [0]; %! [result err] = ctmcchkQ(Q); %! assert( result, 1 ); %! assert( err, "" ); %!test %! N = 10; %! Q = ctmcbd(rand(1,N-1),rand(1,N-1)); %! [result err] = ctmcchkQ(Q); %! assert( result, N ); %! assert( err, "" ); %!test %! Q = [1 2 3; 4 5 6]; %! [result err] = ctmcchkQ(Q); %! assert( result, 0 ); %! assert( index(err, "square") > 0 ); %!test %! N = 10; %! Q = ctmcbd(rand(1,N-1),rand(1,N-1)); %! Q(2,1) = -1; %! [result err] = ctmcchkQ(Q); %! assert( result, 0 ); %! assert( index(err, "infinitesimal") > 0 ); %!test %! N = 10; %! Q = ctmcbd(linspace(1,N-1,N-1),linspace(1,N-1,N-1)); %! Q(1,1) += 7; %! [result err] = ctmcchkQ(Q); %! assert( result, 0 ); %! assert( index(err, "infinitesimal") > 0 ); queueing/inst/qnclosed.m0000644000175000017500000000733413570046274015242 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnclosed (@var{N}, @var{S}, @var{V}, @dots{}) ## ## @cindex closed network, single class ## @cindex closed network, multiple classes ## ## This function computes steady-state performance measures of closed ## queueing networks using the Mean Value Analysis (MVA) algorithm. The ## qneneing network is allowed to contain fixed-capacity centers, delay ## centers or general load-dependent centers. Multiple request ## classes are supported. ## ## This function dispatches the computation to one of ## @code{qncsemva}, @code{qncsmvald} or @code{qncmmva}. ## ## @itemize ## ## @item If @var{N} is a scalar, the network is assumed to have a single ## class of requests; in this case, the exact MVA algorithm is used to ## analyze the network. If @var{S} is a vector, then @code{@var{S}(k)} ## is the average service time of center @math{k}, and this function ## calls @code{qncsmva} which supports load-independent ## service centers. If @var{S} is a matrix, @code{@var{S}(k,i)} is the ## average service time at center @math{k} when @math{i=1, @dots{}, N} ## jobs are present; in this case, the network is analyzed with the ## @code{qncmmvald} function. ## ## @item If @var{N} is a vector, the network is assumed to have multiple ## classes of requests, and is analyzed using the exact multiclass ## MVA algorithm as implemented in the @code{qncmmva} function. ## ## @end itemize ## ## @seealso{qncsmva, qncsmvald, qncmmva} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnclosed( N, S, V, varargin ) if ( nargin < 3 ) print_usage(); endif if ( isscalar(N) ) if ( isvector(S) ) [U R Q X] = qncsmva( N, S, V, varargin{:} ); else [U R Q X] = qncsmvald( N, S, V, varargin{:} ); endif else [U R Q X] = qncmmva( N, S, V, varargin{:} ); endif endfunction %!demo %! P = [0 0.3 0.7; 1 0 0; 1 0 0]; # Transition probability matrix %! S = [1 0.6 0.2]; # Average service times %! m = ones(size(S)); # All centers are single-server %! Z = 2; # External delay %! N = 15; # Maximum population to consider %! V = qncsvisits(P); # Compute number of visits %! X_bsb_lower = X_bsb_upper = X_ab_lower = X_ab_upper = X_mva = zeros(1,N); %! for n=1:N %! [X_bsb_lower(n) X_bsb_upper(n)] = qncsbsb(n, S, V, m, Z); %! [X_ab_lower(n) X_ab_upper(n)] = qncsaba(n, S, V, m, Z); %! [U R Q X] = qnclosed( n, S, V, m, Z ); %! X_mva(n) = X(1)/V(1); %! endfor %! close all; %! plot(1:N, X_ab_lower,"g;Asymptotic Bounds;", ... %! 1:N, X_bsb_lower,"k;Balanced System Bounds;", ... %! 1:N, X_mva,"b;MVA;", "linewidth", 2, ... %! 1:N, X_bsb_upper,"k", 1:N, X_ab_upper,"g" ); %! axis([1,N,0,1]); legend("location","southeast"); legend("boxoff"); %! xlabel("Number of Requests n"); ylabel("System Throughput X(n)"); queueing/inst/qncmpopmix.m0000644000175000017500000001161413570046274015621 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {pop_mix =} qncmpopmix (@var{k}, @var{N}) ## ## @cindex population mix ## @cindex closed network, multiple classes ## ## Return the set of population mixes for a closed multiclass queueing ## network with exactly @var{k} customers. Specifically, given a ## closed multiclass QN with @math{C} customer classes, where there ## are @code{@var{N}(c)} class @math{c} requests, @math{c = 1, @dots{}, C} ## a @math{k}-mix @math{M} is a vector of length @math{C} with the following ## properties: ## ## @itemize ## @item @math{0 @leq{} M_c @leq{} @var{N}(c)} for all @math{c = 1, @dots{}, C}; ## @item @math{\sum_{c=1}^C M_c = k} ## @end itemize ## ## In other words, a @math{k}-mix is an allocation of @math{k} ## requests to @math{C} classes such that the number of requests ## assigned to class @math{c} does not exceed the maximum value ## @code{@var{N}(c)}. ## ## @var{pop_mix} is a matrix with @math{C} columns, such ## that each row represents a valid mix. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{k} ## Size of the requested mix (scalar, @code{@var{k} @geq{} 0}). ## ## @item @var{N}(c) ## number of class @math{c} requests (@code{@var{k} @leq{} sum(@var{N})}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{pop_mix}(i,c) ## number of class @math{c} requests in the @math{i}-th population ## mix. The number of mixes is @code{rows(@var{pop_mix})}. ## ## @end table ## ## If you are interested in the number of @math{k}-mixes only, you can ## use the funcion @code{qnmvapop}. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Herb Schwetman, @cite{Implementing the Mean Value Algorithm for the ## Solution of Queueing Network Models}, Technical Report ## @uref{http://docs.lib.purdue.edu/cstech/286/, 80-355}, Department of Computer ## Sciences, Purdue University, revised February 15, 1982. ## @end itemize ## ## The slightly different problem of enumerating all tuples @math{k_1, ## @dots{}, k_N} such that @math{\sum_i k_i = k} and @math{k_i ## @geq{} 0}, for a given @math{k @geq{} 0} has been described in ## S. Santini, @cite{Computing the Indices for a Complex Summation}, ## unpublished report, available at ## @url{http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf} ## ## @seealso{qncmnpop} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function pop_mix = qncmpopmix( k, population ) if ( nargin != 2 ) print_usage(); endif isvector( population ) && all( population>=0 ) || ... error( "N must be an array >=0" ); R = length(population); # number of classes ( isscalar(k) && k >= 0 && k <= sum(population) ) || ... error( "valid range for k is [0, %d]", sum(population)); N = zeros(1, R); const = min(k, population); mp = 0; pop_mix = []; # Init result while ( N(R) <= const(R) ) x=k-mp; ## Fill the current configuration i=1; while ( x>0 && i<=R ) N(i) = min(x,const(i)); x = x-N(i); mp = mp+N(i); i = i+1; endwhile ## here the configuration is filled. add it to the set of mixes assert( sum(N), k ); pop_mix = [pop_mix; N]; ## FIXME: pop_mix is continuously resized ## advance to the next feasible configuration i = 1; sw = true; while sw if ( ( mp==k || N(i)==const(i)) && ( i. ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{r} @var{err}] =} dtmcchkP (@var{P}) ## ## @cindex Markov chain, discrete time ## @cindex DTMC ## @cindex discrete time Markov chain ## ## Check whether @var{P} is a valid transition probability matrix. ## ## If @var{P} is valid, @var{r} is the size (number of rows or columns) ## of @var{P}. If @var{P} is not a transition probability matrix, ## @var{r} is set to zero, and @var{err} to an appropriate error string. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [result err] = dtmcchkP( P ) if ( nargin != 1 ) print_usage(); endif result = 0; err = ""; if ( !issquare(P) ) err = "P is not a square matrix"; return; endif if ( any(P(:)<-eps) || norm( sum(P,2) - 1, "inf" ) > columns(P)*eps ) err = "P is not a stochastic matrix"; return; endif result = rows(P); endfunction %!test %! [r err] = dtmcchkP( [1 1 1; 1 1 1] ); %! assert( r, 0 ); %! assert( index(err, "square") > 0 ); %!test %! [r err] = dtmcchkP( [1 0 0; 0 0.5 0; 0 0 0] ); %! assert( r, 0 ); %! assert( index(err, "stochastic") > 0 ); %!test %! P = [0 1; 1 0]; %! assert( dtmcchkP(P), 2 ); %!test %! P = dtmcbd( linspace(0.1,0.4,10), linspace(0.4,0.1,10) ); %! assert( dtmcchkP(P), rows(P) ); %!test %! N = 1000; %! P = reshape( 1:N^2, N, N ); %! P(1:N+1:end) = 0; %! P = P ./ repmat(sum(P,2),1,N); %! assert( dtmcchkP(P), N );queueing/inst/erlangc.m0000644000175000017500000000635413570046274015046 0ustar morenomoreno## Copyright (C) 2014, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{C} =} erlangc (@var{A}, @var{m}) ## ## @cindex Erlang-C formula ## ## Compute the steady-state probability of delay in the Erlang delay model. ## ## The Erlang-C formula @math{E_C(A, m)} gives the probability that an ## open queueing system with @math{m} identical servers, infinite ## wating space, arrival rate @math{\lambda}, individual service rate ## @math{\mu} and offered load @math{A = \lambda / \mu} has all the ## servers busy. This is the waiting probability in an ## @math{M/M/m/\infty} system with @math{m} servers and an infinite ## queue. ## ## @tex ## @math{E_C(A, m)} is defined as: ## ## $$ ## E_C(A, m) = \displaystyle{ {A^m \over m!} {1 \over 1-\rho} \left( \sum_{k=0}^{m-1} {A^k \over k!} + {A^m \over m!} {1 \over 1 - \rho} \right) ^{-1}} ## $$ ## ## where @math{\rho = A / m = \lambda / (m \mu)}. ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{A} ## Offered load. @math{A = \lambda / \mu} where ## @math{\lambda} is the mean arrival rate and @math{\mu} the mean ## service rate of each individual server (real, @math{0 < A < m}). ## ## @item @var{m} ## Number of identical servers (integer, @math{m @geq{} 1}). ## Default @math{m = 1} ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{B} ## The value @math{E_C(A, m)} ## ## @end table ## ## @var{A} or @var{m} can be vectors, and in this case, the results will ## be vectors as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Zeng, @cite{Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 ## @end itemize ## ## @seealso{erlangb,engset,qsmmm} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function C = erlangc(A, m) if ( nargin < 1 || nargin > 2 ) print_usage(); endif ( isnumeric(A) && all(A(:) > 0) ) || error("A must be positive"); if ( nargin == 1 ) m = 1; else ( isnumeric(m) && all( fix(m(:)) == m(:)) && all( m(:) > 0 ) ) || error("m must be a positive integer"); endif [err A, m] = common_size(A, m); if ( err ) error("parameters are not of common size"); endif all( A(:) < m(:) ) || error("A must be < m"); rho = A ./ m; B = erlangb(A, m); C = B ./ (1 - rho .* (1 - B)); endfunction %!test %! fail("erlangc('foo',1)", "positive"); %! fail("erlangc(1,'bar')", "positive"); queueing/inst/qncsmvald.m0000644000175000017500000001542513570046274015422 0ustar morenomoreno## Copyright (C) 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvald (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvald (@var{N}, @var{S}, @var{V}, @var{Z}) ## ## @cindex Mean Value Analysys (MVA) ## @cindex MVA ## @cindex closed network, single class ## @cindex load-dependent service center ## ## Mean Value Analysis algorithm for closed, single class queueing ## networks with @math{K} service centers and load-dependent service ## times. This function supports FCFS, LCFS-PR, PS and IS nodes. For ## networks with only fixed-rate centers and multiple-server ## nodes, the function @code{qncsmva} is more efficient. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## Population size (number of requests in the system, @code{@var{N} @geq{} 0}). ## If @code{@var{N} == 0}, this function returns @code{@var{U} = @var{R} = @var{Q} = @var{X} = 0} ## ## @item @var{S}(k,n) ## mean service time at center @math{k} ## where there are @math{n} requests, @math{1 @leq{} n ## @leq{} N}. @code{@var{S}(k,n)} @math{= 1 / \mu_{k}(n)}, ## where @math{\mu_{k}(n)} is the service rate of center @math{k} ## when there are @math{n} requests. ## ## @item @var{V}(k) ## average number of visits to service center @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{Z} ## external delay ("think time", @code{@var{Z} @geq{} 0}); default 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## utilization of service center @math{k}. The ## utilization is defined as the probability that service center ## @math{k} is not empty, that is, @math{U_k = 1-\pi_k(0)} where ## @math{\pi_k(0)} is the steady-state probability that there are 0 ## jobs at service center @math{k}. ## ## @item @var{R}(k) ## response time on service center @math{k}. ## ## @item @var{Q}(k) ## average number of requests in service center @math{k}. ## ## @item @var{X}(k) ## throughput of service center @math{k}. ## ## @end table ## ## @strong{NOTES} ## ## In presence of load-dependent servers, the MVA algorithm is known ## to be numerically unstable. Generally this problem manifests itself ## as negative response times or utilization. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed ## Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, ## April 1980, pp. 313--322. @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} ## @end itemize ## ## This implementation is described in G. Bolch, S. Greiner, H. de Meer ## and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling ## and Performance Evaluation with Computer Science Applications}, Wiley, ## 1998, Section 8.2.4.1, ``Networks with Load-Dependent Service: Closed ## Networks''. ## ## @seealso{qncsmva} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncsmvald( N, S, V, Z ) if ( nargin < 3 || nargin > 4 ) print_usage(); endif isvector(V) && all(V>=0) || ... error( "V must be a vector >= 0" ); V = V(:)'; # make V a row vector K = length(V); # Number of servers isscalar(N) && N >= 0 || ... error( "N must be >= 0" ); ( ismatrix(S) && rows(S) == K && columns(S) >= N ) || ... error( "S size mismatch: is %dx%d, should be %dx%d", rows(S), columns(S), K, N ); all(S(:)>=0) || ... error( "S must be >= 0" ); if ( nargin < 4 ) Z = 0; else isscalar(Z) && Z>=0 || ... error( "Z must be >= 0" ); endif ## Initialize results p = zeros( K, N+1 ); # p(k,i+1) is the probability that there are i jobs at server k, given that the network population is j p(:,1) = 1; U = R = Q = X = zeros( 1, K ); X_s = 0; # System throughput ## Main MVA loop, iterates over the population size for n=1:N # over population size j=1:n; ## for i=1:K ## R(i) = sum( j.*S(i,j).*p(i,j) ); ## endfor R = sum( repmat(j,K,1).*S(:,1:n).*p(:,1:n), 2)'; R_s = dot( V, R ); # System response time X_s = n / (Z+R_s); # System Throughput ## G_N = G_Nm1 / X_s; G_Nm1 = G_N; ## prepare for next iteration for i=1:K p(i, 1+j) = X_s * S(i,j) .* p(i,j) * V(i); p(i, 1) = 1-sum(p(i,1+j)); endfor endfor Q = X_s * ( V .* R ); U = 1-p(:,1)'; # Service centers utilization X = X_s * V; # Service centers throughput endfunction ## Check degenerate case of N==0 (general LD case) %!test %! N = 0; %! S = [1 2; 3 4; 5 6; 7 8]; %! V = [1 1 1 4]; %! [U R Q X] = qncsmvald(N, S, V); %! assert( U, 0*V ); %! assert( R, 0*V ); %! assert( Q, 0*V ); %! assert( X, 0*V ); %!test %! # Exsample 3.42 p. 577 Jain %! V = [ 16 10 5 ]; %! N = 20; %! S = [ 0.125 0.3 0.2 ]; %! Sld = repmat( S', 1, N ); %! Z = 4; %! [U1 R1 Q1 X1] = qncsmvald(N,Sld,V,Z); %! [U2 R2 Q2 X2] = qncsmva(N,S,V,ones(1,3),Z); %! assert( U1, U2, 1e-3 ); %! assert( R1, R2, 1e-3 ); %! assert( Q1, Q2, 1e-3 ); %! assert( X1, X2, 1e-3 ); %!test %! # Example 8.7 p. 349 Bolch et al. %! N = 3; %! Sld = 1 ./ [ 2 4 4; ... %! 1.667 1.667 1.667; ... %! 1.25 1.25 1.25; ... %! 1 2 3 ]; %! V = [ 1 .5 .5 1 ]; %! [U R Q X] = qncsmvald(N,Sld,V); %! assert( Q, [0.624 0.473 0.686 1.217], 1e-3 ); %! assert( R, [0.512 0.776 1.127 1], 1e-3 ); %! assert( all( U<=1) ); %!test %! # Example 8.4 p. 333 Bolch et al. %! N = 3; %! S = [ .5 .6 .8 1 ]; %! m = [2 1 1 N]; %! Sld = zeros(3,N); %! Sld(1,:) = .5 ./ [1 2 2]; %! Sld(2,:) = [.6 .6 .6]; %! Sld(3,:) = [.8 .8 .8]; %! Sld(4,:) = 1 ./ [1 2 3]; %! V = [ 1 .5 .5 1 ]; %! [U1 R1 Q1 X1] = qncsmvald(N,Sld,V); %! [U2 R2 Q2 X2] = qncsmva(N,S,V,m); %! ## Note that qncsmvald computes the utilization in a different %! ## way as qncsmva; in fact, qncsmva knows that service %! ## center i has m(i)>1 servers, but qncsmvald does not. Thus, %! ## utilizations for multiple-server nodes cannot be compared %! assert( U1([2,3]), U2([2,3]), 1e-3 ); %! assert( R1, R2, 1e-3 ); %! assert( Q1, Q2, 1e-3 ); %! assert( X1, X2, 1e-3 ); queueing/inst/qsmm1k.m0000644000175000017500000001303513572245572014642 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pK}] =} qsmm1k (@var{lambda}, @var{mu}, @var{K}) ## @deftypefnx {Function File} {@var{pn} =} qsmm1k (@var{lambda}, @var{mu}, @var{K}, @var{n}) ## ## @cindex @math{M/M/1/K} system ## ## Compute utilization, response time, average number of requests and ## throughput for a @math{M/M/1/K} finite capacity system. ## ## In a @math{M/M/1/K} queue there is a single server and a queue with ## finite capacity: the maximum number of requests in the system ## (including the request being served) is @math{K}, and the maximum ## queue length is therefore @math{K-1}. ## ## @tex ## The steady-state probability @math{\pi_n} that there are @math{n} ## jobs in the system, @math{0 @leq{} n @leq{} K}, is: ## ## $$ ## \pi_n = {(1-a)a^n \over 1-a^{K+1}} ## $$ ## ## where @math{a = \lambda/\mu}. ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate (@code{@var{lambda}>0}). ## ## @item @var{mu} ## Service rate (@code{@var{mu}>0}). ## ## @item @var{K} ## Maximum number of requests allowed in the system (@code{@var{K} @geq{} 1}). ## ## @item @var{n} ## Number of requests in the (@code{0 @leq{} @var{n} @leq{} K}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Service center utilization, which is defined as @code{@var{U} = 1-@var{p0}} ## ## @item @var{R} ## Service center response time ## ## @item @var{Q} ## Average number of requests in the system ## ## @item @var{X} ## Service center throughput ## ## @item @var{p0} ## Steady-state probability that there are no requests in the system ## ## @item @var{pK} ## Steady-state probability that there are @math{K} requests in the system ## (i.e., that the system is full) ## ## @item @var{pn} ## Steady-state probability that there are @math{n} requests in the system ## (including the one being served). ## ## @end table ## ## If this function is called with less than four arguments, ## @var{lambda}, @var{mu} and @var{K} can be vectors of the ## same size. In this case, the results will be vectors as well. ## ## @seealso{qsmm1,qsmminf,qsmmm} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U_or_pn R Q X p0 pK] = qsmm1k( lambda, mu, K, n ) if ( nargin < 3 || nargin > 4 ) print_usage(); endif ( isvector(lambda) && isvector(mu) && isvector(K) ) || ... error( "lambda, mu, K must be vectors of the same size" ); [err lambda mu K] = common_size( lambda, mu, K ); if ( err ) error( "Parameters are not of common size" ); endif all( K>0 ) || ... error( "K must be >0" ); ( all( lambda>0 ) && all( mu>0 ) ) || ... error( "lambda and mu must be >0" ); a = lambda./mu; if (nargin < 4) U_or_pn = R = Q = X = p0 = pK = 0*lambda; ## persistent tol = 1e-7; ## if a!=1 ## i = find( abs(a-1)>tol ); i = find( a != 1 ); p0(i) = (1-a(i))./(1-a(i).^(K(i)+1)); pK(i) = (1-a(i)).*(a(i).^K(i))./(1-a(i).^(K(i)+1)); Q(i) = a(i)./(1-a(i)) - (K(i)+1)./(1-a(i).^(K(i)+1)).*(a(i).^(K(i)+1)); ## if a==1 ## i = find( abs(a-1)<=tol ); i = find( a == 1 ); p0(i) = pK(i) = 1./(K(i)+1); Q(i) = K(i)/2; ## Compute other performance measures U_or_pn = 1-p0; X = lambda.*(1-pK); R = Q ./ X; else (length(lambda) == 1) || error("lambda must be a scalar if this function is called with four arguments"); isvector(n) || error("n must be a vector"); (all(n >= 0) && all(n <= K)) || error("n must be >= 0 and <= K"); n = n(:)'; if (a != 1) # we know that a must be a scalar U_or_pn = ((1 - a) * a.^n) ./ (1 - a^(K+1)); else U_or_pn = 1/(K+1) * ones(size(n)); endif endif endfunction %!test %! lambda = mu = 1; %! K = 10; %! [U R Q X p0] = qsmm1k(lambda,mu,K); %! assert( Q, K/2, 1e-7 ); %! assert( U, 1-p0, 1e-7 ); %!test %! lambda = 1; %! mu = 1.2; %! K = 10; %! [U R Q X p0 pK] = qsmm1k(lambda, mu, K); %! prob = qsmm1k(lambda, mu, K, 0:K); %! assert( p0, prob(1), 1e-7 ); %! assert( pK, prob(K+1), 1e-7 ); %!test %! # Compare the result with the equivalent Markov chain %! lambda = 0.8; %! mu = 0.8; %! K = 10; %! [U1 R1 Q1 X1] = qsmm1k( lambda, mu, K ); %! birth = lambda*ones(1,K); %! death = mu*ones(1,K); %! q = ctmc(ctmcbd( birth, death )); %! U2 = 1-q(1); %! Q2 = dot( [0:K], q ); %! assert( U1, U2, 1e-4 ); %! assert( Q1, Q2, 1e-4 ); %!demo %! ## Given a M/M/1/K queue, compute the steady-state probability pk %! ## of having n requests in the systen. %! lambda = 0.2; %! mu = 0.25; %! K = 10; %! n = 0:10; %! pn = qsmm1k(lambda, mu, K, n); %! plot(n, pn, "-o", "linewidth", 2); %! xlabel("N. of requests (n)"); %! ylabel("p_n"); %! title(sprintf("M/M/1/%d system, \\lambda = %g, \\mu = %g", K, lambda, mu)); queueing/inst/qncsconvld.m0000644000175000017500000001747713570046274015615 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsconvld (@var{N}, @var{S}, @var{V}) ## ## @cindex closed network ## @cindex normalization constant ## @cindex convolution algorithm ## @cindex load-dependent service center ## ## Convolution algorithm for product-form, single-class closed ## queueing networks with @math{K} general load-dependent service ## centers. ## ## This function computes steady-state performance measures for ## single-class, closed networks with load-dependent service centers ## using the convolution algorithm; the normalization constants are also ## computed. The normalization constants are returned as vector ## @code{@var{G}=[@var{G}(1), @dots{}, @var{G}(N+1)]} where ## @code{@var{G}(i+1)} is the value of @math{G(i)}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## Number of requests in the system (@code{@var{N}>0}). ## ## @item @var{S}(k,n) ## mean service time at center @math{k} where there are @math{n} ## requests, @math{1 @leq{} n @leq{} N}. @code{@var{S}(k,n)} @math{= 1 / \mu_{k,n}}, where @math{\mu_{k,n}} is the service rate of center ## @math{k} when there are @math{n} requests. ## ## @item @var{V}(k) ## visit count of service center @math{k} ## (@code{@var{V}(k) @geq{} 0}). The length of @var{V} is the number of ## servers @math{K} in the network. ## ## @end table ## ## @strong{OUTPUT} ## ## @table @code ## ## @item @var{U}(k) ## center @math{k} utilization. ## ## @item @var{R}(k) ## average response time at center @math{k}. ## ## @item @var{Q}(k) ## average number of requests in center @math{k}. ## ## @item @var{X}(k) ## center @math{k} throughput. ## ## @item @var{G}(n) ## Normalization constants (vector). @code{@var{G}(n+1)} ## corresponds to @math{G(n)}, as array indexes in Octave start ## from 1. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Herb Schwetman, @cite{Some Computational Aspects of Queueing Network ## Models}, Technical Report ## @uref{http://docs.lib.purdue.edu/cstech/285/, CSD-TR-354}, Department ## of Computer Sciences, Purdue University, February 1981 (revised). ## ## @item ## M. Reiser, H. Kobayashi, @cite{On The Convolution Algorithm for ## Separable Queueing Networks}, In Proceedings of the 1976 ACM ## SIGMETRICS Conference on Computer Performance Modeling Measurement and ## Evaluation (Cambridge, Massachusetts, United States, March 29--31, ## 1976). SIGMETRICS '76. ACM, New York, NY, ## pp. 109--117. @uref{http://doi.acm.org/10.1145/800200.806187, 10.1145/800200.806187} ## @end itemize ## ## This implementation is based on G. Bolch, S. Greiner, H. de Meer and ## K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and ## Performance Evaluation with Computer Science Applications}, Wiley, ## 1998, pp. 313--317. Function @command{qncsconvld} is slightly ## different from the version described in Bolch et al. because it ## supports general load-dependent centers (while the version in the book ## does not). The modification is in the definition of function ## @code{F()} in @command{qncsconvld} which has been made similar to ## function @math{f_i} defined in Schwetman, @cite{Some Computational ## Aspects of Queueing Network Models}. ## ## @seealso{qncsconv} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X G] = qncsconvld( N, S, V ) if ( nargin != 3 ) print_usage(); endif ( isscalar(N) && N>0 ) || ... error( "N must be a positive scalar" ); K = N; # To be compliant with the reference, we denote K as the population size ( isvector(V) && all(V>=0) ) || ... error( "V must be a vector >=0" ); V = V(:)'; # Make V a row vector N = length(V); # Number of service centers if ( isnumeric(S) ) ( rows(S) == N && columns(S) == K) || ... error( sprintf("S size mismatch: is %dx%d, should be %dx%d", rows(S), columns(S),K,N ) ); all(S(:)>=0) || ... error( "S must be >=0" ); endif ## Initialization G_n = G_nm1 = zeros(1,K+1); G_n(1) = 1; F_n = zeros(N,K+1); F_n(:,1) = 1; for k=1:K G_n(k+1) = F_n(1,k+1) = F(1,k,V,S); endfor ## Main convolution loop for n=2:N G_nm1 = G_n; for k=2:K+1 F_n(n,k) = F(n,k-1,V,S); endfor G_n = conv( F_n(n,:), G_nm1(:) )(1:K+1); endfor ## Done computation of G(n,k). G = G_n; G = G(:)'; # ensure G is a row vector ## Computes performance measures X = V*G(K)/G(K+1); Q = U = zeros(1,N); for i=1:N G_N_i = zeros(1,K+1); G_N_i(1) = 1; for k=1:K j=1:k; G_N_i(k+1) = G(k+1)-dot( F_n(i,j+1), G_N_i(k-j+1) ); endfor k=0:K; p_i(k+1) = F_n(i,k+1)./G(K+1).*G_N_i(K-k+1); Q(i) = dot( k, p_i( k+1 ) ); U(i) = 1-p_i(1); endfor R = Q ./ X; endfunction %!test %! K=3; %! S = [ 1 1 1; 1 1 1 ]; %! V = [ 1 .667 .2 ]; %! fail( "qncsconvld(K,S,V)", "size mismatch" ); %!test %! # Example 8.1 p. 318 Bolch et al. %! K=3; %! S = [ 1/0.8 ./ [1 2 2]; %! 1/0.6 ./ [1 2 3]; %! 1/0.4 ./ [1 1 1] ]; %! V = [ 1 .667 .2 ]; %! [U R Q X G] = qncsconvld( K, S, V ); %! assert( G, [1 2.861 4.218 4.465], 5e-3 ); %! assert( X, [0.945 0.630 0.189], 1e-3 ); %! assert( Q, [1.290 1.050 0.660], 1e-3 ); %! assert( R, [1.366 1.667 3.496], 1e-3 ); %!test %! # Example 8.3 p. 331 Bolch et al. %! # compare results of convolution with those of mva %! K = 6; %! S = [ 0.02 0.2 0.4 0.6 ]; %! V = [ 1 0.4 0.2 0.1 ]; %! [U_mva R_mva Q_mva X_mva] = qncsmva(K, S, V); %! [U_con R_con Q_con X_con G] = qncsconvld(K, repmat(S',1,K), V ); %! assert( U_mva, U_con, 1e-5 ); %! assert( R_mva, R_con, 1e-5 ); %! assert( Q_mva, Q_con, 1e-5 ); %! assert( X_mva, X_con, 1e-5 ); %!test %! # Compare the results of convolution to those of mva %! S = [ 0.02 0.2 0.4 0.6 ]; %! K = 6; %! V = [ 1 0.4 0.2 0.1 ]; %! m = [ 1 5 2 1 ]; %! [U_mva R_mva Q_mva X_mva] = qncsmva(K, S, V); %! [U_con R_con Q_con X_con G] = qncsconvld(K, repmat(S',1,K), V); %! assert( U_mva, U_con, 1e-5 ); %! assert( R_mva, R_con, 1e-5 ); %! assert( Q_mva, Q_con, 1e-5 ); %! assert( X_mva, X_con, 1e-5 ); %!function r = S_function(k,n) %! M = [ 1/0.8 ./ [1 2 2]; %! 1/0.6 ./ [1 2 3]; %! 1/0.4 ./ [1 1 1] ]; %! r = M(k,n); %!test %! # Example 8.1 p. 318 Bolch et al. %! K=3; %! V = [ 1 .667 .2 ]; %! [U R Q X G] = qncsconvld( K, @S_function, V ); %! assert( G, [1 2.861 4.218 4.465], 5e-3 ); %! assert( X, [0.945 0.630 0.189], 1e-3 ); %! assert( Q, [1.290 1.050 0.660], 1e-3 ); %! assert( R, [1.366 1.667 3.496], 1e-3 ); ## result = F(i,j,v,S) ## ## Helper fuction to compute a generalization of equation F(i,j) as ## defined in Eq 7.61 p. 289 of Bolch, Greiner, de Meer, Trivedi ## "Queueing Networks and Markov Chains: Modeling and Performance ## Evaluation with Computer Science Applications", Wiley, 1998. This ## generalization is taken from Schwetman, "Some Computational Aspects ## of Queueing Network Models", Technical Report CSD-TR 354, Dept. of ## CS, Purdue University, Dec 1980 (see definition of f_i(n) on p. 7). function result = F(i,j,v,S) k_i = j; if ( k_i == 0 ) result = 1; else result = v(i)^k_i * prod(S(i,1:k_i)); endif endfunction queueing/inst/dtmc.m0000644000175000017500000001244013570046274014353 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{p} =} dtmc (@var{P}) ## @deftypefnx {Function File} {@var{p} =} dtmc (@var{P}, @var{n}, @var{p0}) ## ## @cindex Markov chain, discrete time ## @cindex discrete time Markov chain ## @cindex DTMC ## @cindex Markov chain, stationary probabilities ## @cindex Markov chain, transient probabilities ## ## Compute stationary or transient state occupancy probabilities for a discrete-time Markov chain. ## ## With a single argument, compute the stationary state occupancy ## probabilities @code{@var{p}(1), @dots{}, @var{p}(N)} for a ## discrete-time Markov chain with finite state space @math{@{1, @dots{}, ## N@}} and with @math{N \times N} transition matrix ## @var{P}. With three arguments, compute the transient state occupancy ## probabilities @code{@var{p}(1), @dots{}, @var{p}(N)} that the system is in ## state @math{i} after @var{n} steps, given initial occupancy ## probabilities @var{p0}(1), @dots{}, @var{p0}(N). ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## transition probabilities from state @math{i} to state @math{j}. ## @var{P} must be an @math{N \times N} irreducible stochastic matrix, ## meaning that the sum of each row must be 1 (@math{\sum_{j=1}^N ## P_{i, j} = 1}), and the rank of @var{P} must be @math{N}. ## ## @item @var{n} ## Number of transitions after which state occupancy probabilities are computed ## (scalar, @math{n @geq{} 0}) ## ## @item @var{p0}(i) ## probability that at step 0 the system is in state @math{i} (vector ## of length @math{N}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{p}(i) ## If this function is called with a single argument, @code{@var{p}(i)} ## is the steady-state probability that the system is in state @math{i}. ## If this function is called with three arguments, @code{@var{p}(i)} ## is the probability that the system is in state @math{i} ## after @var{n} transitions, given the probabilities ## @code{@var{p0}(i)} that the initial state is @math{i}. ## ## @end table ## ## @seealso{ctmc} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function p = dtmc( P, n, p0 ) if ( nargin != 1 && nargin != 3 ) print_usage(); endif [N err] = dtmcchkP(P); ( N>0 ) || ... error( err ); if ( nargin == 1 ) # steady-state analysis A = P-eye(N); A(:,N) = 1; # add normalization condition rank( A ) == N || ... warning( "dtmc(): P is reducible" ); b = [ zeros(1,N-1) 1 ]; p = b/A; else # transient analysis ( isscalar(n) && n>=0 ) || ... error( "n must be >=0" ); ( isvector(p0) && length(p0) == N && all(p0>=0) && abs(sum(p0)-1.0). ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{r} @var{s}] =} dtmcisir (@var{P}) ## ## @cindex Markov chain, discrete time ## @cindex discrete time Markov chain ## @cindex DTMC ## @cindex irreducible Markov chain ## ## Check if @var{P} is irreducible, and identify Strongly Connected ## Components (SCC) in the transition graph of the DTMC with transition ## matrix @var{P}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## transition probability from state @math{i} to state @math{j}. ## @var{P} must be an @math{N \times N} stochastic matrix. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{r} ## 1 if @var{P} is irreducible, 0 otherwise (scalar) ## ## @item @var{s}(i) ## strongly connected component (SCC) that state @math{i} belongs to ## (vector of length @math{N}). SCCs are numbered @math{1, 2, @dots{}}. ## The number of SCCs is @code{max(s)}. If the graph is ## strongly connected, then there is a single SCC and the predicate ## @code{all(s == 1)} evaluates to true ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [r s] = dtmcisir( P ) if ( nargin != 1 ) print_usage(); endif [N err] = dtmcchkP(P); if ( N == 0 ) error(err); endif s = scc(P); r = (max(s) == 1); endfunction %!test %! P = [0 .5 0; 0 0 0]; %! fail( "dtmcisir(P)" ); %!test %! P = [0 1 0; 0 .5 .5; 0 1 0]; %! [r s] = dtmcisir(P); %! assert( r == 0 ); %! assert( max(s), 2 ); %! assert( min(s), 1 ); %!test %! P = [.5 .5 0; .2 .3 .5; 0 .2 .8]; %! [r s] = dtmcisir(P); %! assert( r == 1 ); %! assert( max(s), 1 ); %! assert( min(s), 1 ); queueing/inst/ctmcisir.m0000644000175000017500000000471413570046274015246 0ustar morenomoreno## Copyright (C) 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{r} @var{s}] =} ctmcisir (@var{P}) ## ## @cindex Markov chain, continuous time ## @cindex continuous time Markov chain ## @cindex CTMC ## @cindex irreducible Markov chain ## ## Check if @var{Q} is irreducible, and identify Strongly Connected ## Components (SCC) in the transition graph of the DTMC with infinitesimal ## generator matrix @var{Q}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{Q}(i,j) ## Infinitesimal generator matrix. @var{Q} is a @math{N \times N} square ## matrix where @code{@var{Q}(i,j)} is the transition rate from state ## @math{i} to state @math{j}, for @math{1 @leq{} i, j @leq{} N}, ## @math{i \neq j}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{r} ## 1 if @var{Q} is irreducible, 0 otherwise. ## ## @item @var{s}(i) ## strongly connected component (SCC) that state @math{i} belongs to. ## SCCs are numbered @math{1, 2, @dots{}}. If the graph is strongly ## connected, then there is a single SCC and the predicate @code{all(s == 1)} ## evaluates to true. ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [r s] = ctmcisir( Q ) if ( nargin != 1 ) print_usage(); endif [N err] = ctmcchkQ(Q); ( N > 0 ) || ... error(err); s = scc(Q); r = (max(s) == 1); endfunction %!test %! Q = [-.5 .5 0; 1 0 0]; %! fail( "ctmcisir(Q)" ); %!test %! Q = [-1 1 0; .5 -.5 0; 0 0 0]; %! [r s] = ctmcisir(Q); %! assert( r == 0 ); %! assert( max(s), 2 ); %! assert( min(s), 1 ); %!test %! Q = [-.5 .5 0; .2 -.7 .5; .2 0 -.2]; %! [r s] = ctmcisir(Q); %! assert( r == 1 ); %! assert( max(s), 1 ); %! assert( min(s), 1 ); queueing/inst/qncsvisits.m0000644000175000017500000000707213603663754015644 0ustar morenomoreno## Copyright (C) 2012, 2016, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{V} =} qncsvisits (@var{P}) ## @deftypefnx {Function File} {@var{V} =} qncsvisits (@var{P}, @var{r}) ## ## Compute the mean number of visits to the service centers of a ## single class, closed network with @math{K} service centers. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## probability that a request which completed service at center ## @math{i} is routed to center @math{j} (@math{K \times K} matrix). ## For closed networks it must hold that @code{sum(@var{P},2)==1}. The ## routing graph must be strongly connected, meaning that each node ## must be reachable from every other node. ## ## @item @var{r} ## Index of the reference station, @math{r \in @{1, @dots{}, K@}}; ## Default @code{@var{r}=1}. The traffic equations are solved by ## imposing the condition @code{@var{V}(r) = 1}. A request returning to ## the reference station completes its activity cycle. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{V}(k) ## average number of visits to service center @math{k}, assuming ## @math{r} as the reference station. ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function V = qncsvisits( P, r ) if ( nargin < 1 || nargin > 2 ) print_usage(); endif issquare(P) || ... error("P must be a square matrix"); [res err] = dtmcchkP(P); (res>0) || ... error( "invalid transition probability matrix P" ); K = rows(P); if ( nargin < 2 ) r = 1; else isscalar(r) || ... error("r must be a scalar"); (r>=1 && r<=K) || ... error("r must be an integer in the range [1, %d]",K); endif V = zeros(size(P)); A = P-eye(K); b = zeros(1,K); A(:,r) = 0; A(r,r) = 1; b(r) = 1; V = b/A; ## Make sure that no negative values appear (sometimes, numerical ## errors produce tiny negative values instead of zeros) V = max(0,V); endfunction %!test %! P = [-1 0; 0 0]; %! fail( "qncsvisits(P)", "invalid" ); %! P = [1 0; 0.5 0]; %! fail( "qncsvisits(P)", "invalid" ); %! P = [1 2 3; 1 2 3]; %! fail( "qncsvisits(P)", "square" ); %! P = [0 1; 1 0]; %! fail( "qncsvisits(P,0)", "range" ); %! fail( "qncsvisits(P,3)", "range" ); %!test %! %! ## Closed, single class network %! %! P = [0 0.3 0.7; 1 0 0; 1 0 0]; %! V = qncsvisits(P); %! assert( V*P,V,1e-5 ); %! assert( V, [1 0.3 0.7], 1e-5 ); %!test %! %! ## Test tolerance of the qncsvisits() function. %! ## This test builds transition probability matrices and tries %! ## to compute the visit counts on them. %! %! for k=[5, 10, 20, 50] %! P = reshape(1:k^2, k, k); %! P = P ./ repmat(sum(P,2),1,k); %! V = qncsvisits(P); %! assert( V*P, V, 1e-5 ); %! endfor %!demo %! P = [0 0.3 0.7; ... %! 1 0 0 ; ... %! 1 0 0 ]; %! V = qncsvisits(P) queueing/inst/qnosbsb.m0000644000175000017500000000575013570046274015101 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosbsb (@var{lambda}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosbsb (@var{lambda}, @var{S}, @var{V}) ## ## @cindex bounds, balanced system ## @cindex open network ## ## Compute Balanced System Bounds for single-class, open networks with ## @math{K} service centers. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## overall arrival rate to the system (scalar, @code{@var{lambda} @geq{} 0}). ## ## @item @var{D}(k) ## service demand at center @math{k} (@code{@var{D}(k) @geq{} 0}). ## ## @item @var{S}(k) ## service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## mean number of visits at center @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. This function only supports ## @math{M/M/1} queues, therefore @var{m} must be ## @code{ones(size(S))}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl} ## @item @var{Xu} ## Lower and upper bounds on the system throughput. @var{Xl} is always ## set to @math{0}, since there can be no lower bound on open ## networks throughput. ## ## @item @var{Rl} ## @itemx @var{Ru} ## Lower and upper bounds on the system response time. ## ## @end table ## ## @seealso{qnosaba} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [X_lower X_upper R_lower R_upper] = qnosbsb( varargin ) if ( nargin < 2 || nargin > 4 ) print_usage(); endif [err lambda S V m] = qnoschkparam( varargin{:} ); isempty(err) || error(err); all(m==1) || ... error("this function supports M/M/1 servers only"); D = S .* V; D_max = max(D); D_tot = sum(D); D_ave = mean(D_tot); X_upper = 1/D_max; X_lower = 0; R_lower = D_tot / (1-lambda*D_ave); R_upper = D_tot / (1-lambda*D_max); endfunction %!test %! fail( "qnosbsb( 0.1, [] )", "vector" ); %! fail( "qnosbsb( 0.1, [0 -1])", "nonnegative" ); %! fail( "qnosbsb( 0, [1 2] )", "lambda" ); %! fail( "qnosbsb( -1, [1 2])", "lambda" ); %!test %! [Xl Xu Rl Ru] = qnosbsb(0.1,[1 2 3]); %! assert( Xl, 0 ); queueing/inst/qncsaba.m0000644000175000017500000001020613570046274015032 0ustar morenomoreno## Copyright (C) 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## ## @cindex bounds, asymptotic ## @cindex asymptotic bounds ## @cindex closed network, single class ## ## Compute Asymptotic Bounds for the system throughput and response ## time of closed, single-class networks with @math{K} service ## centers. ## ## Single-server and infinite-server nodes are supported. ## Multiple-server nodes and general load-dependent servers are not ## supported. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## number of requests in the system (scalar, @code{@var{N}>0}). ## ## @item @var{D}(k) ## service demand at center @math{k} ## (@code{@var{D}(k) @geq{} 0}). ## ## @item @var{S}(k) ## mean service time at center @math{k} ## (@code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## average number of visits to center ## @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k} ## (if @var{m} is a scalar, all centers have that number of servers). If ## @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); ## if @code{@var{m}(k) = 1}, center @math{k} is a M/M/1-FCFS server. ## This function does not support multiple-server nodes. Default ## is 1. ## ## @item @var{Z} ## External delay (scalar, @code{@var{Z} @geq{} 0}). Default is 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl} ## @itemx @var{Xu} ## Lower and upper bounds on the system throughput. ## ## @item @var{Rl} ## @itemx @var{Ru} ## Lower and upper bounds on the system response time. ## ## @end table ## ## @seealso{qncmaba} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [Xl Xu Rl Ru] = qncsaba( varargin ) #N, S, V, m, Z ) if (nargin<2 || nargin>5) print_usage(); endif [err N S V m Z] = qncschkparam( varargin{:} ); isempty(err) || error(err); all(m<=1) || ... error( "multiple server nodes are not supported" ); D = S.*V; Dtot_single = sum(D(m==1)); # total demand at single-server nodes Dtot_delay = sum(D(m<1)); # total demand at IS nodes Dtot = sum(D); # total demand Dmax = max(D); # max demand Xl = N/(N*Dtot_single + Dtot_delay + Z); Xu = min( N/(Dtot+Z), 1/Dmax ); Rl = max( Dtot, N*Dmax - Z ); Ru = N*Dtot_single + Dtot_delay; endfunction %!test %! fail("qncsaba(-1,0)", "N must be"); %! fail("qncsaba(1,[])", "nonempty"); %! fail("qncsaba(1,[-1 2])", "nonnegative"); %! fail("qncsaba(1,[1 2],[1 2 3])", "incompatible size"); %! fail("qncsaba(1,[1 2 3],[1 2 -1])", "nonnegative"); %! fail("qncsaba(1,[1 2 3],[1 2 3],[1 2])", "incompatible size"); %! fail("qncsaba(1,[1 2 3],[1 2 3],[1 2 1])", "not supported"); %! fail("qncsaba(1,[1 2 3],[1 2 3],[1 1 1],-1)", "nonnegative"); %! fail("qncsaba(1,[1 2 3],[1 2 3],1,[0 0])", "scalar"); ## Example 9.6 p. 913 Bolch et al. %!test %! N = 20; %! S = [ 4.6*2 8 ]; %! Z = 120; %! [X_l X_u R_l R_u] = qncsaba(N, S, ones(size(S)), ones(size(S)), Z); %! assert( [X_u R_l], [0.109 64], 1e-3 ); queueing/inst/qncsbsb.m0000644000175000017500000000777613570046274015077 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## ## @cindex bounds, balanced system ## @cindex closed network, single class ## @cindex balanced system bounds ## ## Compute Balanced System Bounds on system throughput and response time for closed, single-class networks with @math{K} service centers. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## number of requests in the system (scalar, @code{@var{N} @geq{} 0}). ## ## @item @var{D}(k) ## service demand at center @math{k} (@code{@var{D}(k) @geq{} 0}). ## ## @item @var{S}(k) ## mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## average number of visits to center @math{k} (@code{@var{V}(k) ## @geq{} 0}). Default is 1. ## ## @item @var{m}(k) ## number of servers at center @math{k}. This function supports ## @code{@var{m}(k) = 1} only (single-eserver FCFS nodes); this ## parameter is only for compatibility with @code{qncsaba}. Default is ## 1. ## ## @item @var{Z} ## External delay (@code{@var{Z} @geq{} 0}). Default is 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl} ## @itemx @var{Xu} ## Lower and upper bound on the system throughput. ## ## @item @var{Rl} ## @itemx @var{Ru} ## Lower and upper bound on the system response time. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth ## C. Sevcik, @cite{Quantitative System Performance: Computer System ## Analysis Using Queueing Network Models}, Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 5.4 ("Balanced Systems Bounds"). ## @end itemize ## ## @seealso{qncmbsb} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [Xl Xu Rl Ru] = qncsbsb( varargin ) if (nargin<2 || nargin>5) print_usage(); endif [err N S V m Z] = qncschkparam( varargin{:} ); isempty(err) || error(err); all(m==1) || ... error( "this function supports M/M/1 servers only" ); D = S .* V; D_max = max(D); D_tot = sum(D); D_ave = mean(D); Xl = N/(D_tot+Z+( (N-1)*D_max )/( 1+Z/(N*D_tot) ) ); Xu = min( 1/D_max, N/( D_tot+Z+( (N-1)*D_ave )/(1+Z/D_tot) ) ); Rl = max( N*D_max-Z, D_tot+( (N-1)*D_ave )/( 1+Z/D_tot) ); Ru = D_tot + ( (N-1)*D_max )/( 1+Z/(N*D_tot) ); endfunction %!test %! fail("qncsbsb(-1,0)", "N must be"); %! fail("qncsbsb(1,[])", "nonempty"); %! fail("qncsbsb(1,[-1 2])", "nonnegative"); %! fail("qncsbsb(1,[1 2],[1 2 3])", "incompatible size"); %! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 2])", "incompatible size"); %! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 2 1])", "M/M/1 servers"); %! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 1 1],-1)", "nonnegative"); %! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 1 1],[0 0])", "scalar"); queueing/inst/erlangb.m0000644000175000017500000000741513570046274015044 0ustar morenomoreno## Copyright (C) 2014, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{B} =} erlangb (@var{A}, @var{m}) ## ## @cindex Erlang-B formula ## ## Compute the steady-state blocking probability in the Erlang loss model. ## ## The Erlang-B formula @math{E_B(A, m)} gives the probability that an ## open system with @math{m} identical servers, arrival rate ## @math{\lambda}, individual service rate @math{\mu} and offered load ## @math{A = \lambda / \mu} has all servers busy. This corresponds to ## the rejection probability of an @math{M/M/m/0} system with @math{m} ## servers and no queue. ## ## @tex ## @math{E_B(A, m)} is defined as: ## ## $$ ## E_B(A, m) = \displaystyle{{A^m \over m!} \left( \sum_{k=0}^m {A^k \over k!} \right) ^{-1}} ## $$ ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{A} ## Offered load, defined as @math{A = \lambda / \mu} where ## @math{\lambda} is the mean arrival rate and @math{\mu} the mean ## service rate of each individual server (real, @math{A > 0}). ## ## @item @var{m} ## Number of identical servers (integer, @math{m @geq{} 1}). Default @math{m = 1} ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{B} ## The value @math{E_B(A, m)} ## ## @end table ## ## @var{A} or @var{m} can be vectors, and in this case, the results will ## be vectors as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Zeng, @cite{Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 ## @end itemize ## ## @seealso{erlangc,engset,qsmmm} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function B = erlangb(A, m) if ( nargin < 1 || nargin > 2 ) print_usage(); endif ( isnumeric(A) && all( A(:) > 0 ) ) || error("A must be positive"); if ( nargin == 1 ) m = 1; else ( isnumeric(m) && all( fix(m(:)) == m(:)) && all( m(:) > 0 ) ) || error("m must be a positive integer"); endif [err A m] = common_size(A, m); if ( err ) error("parameters are not of common size"); endif B = arrayfun( @__erlangb_compute, A, m); endfunction ## Compute E_B(A,m) recursively, as described in: ## ## Guoping Zeng, Two common properties of the erlang-B function, ## erlang-C function, and Engset blocking function, Mathematical and ## Computer Modelling Volume 37, Issues 12–13, June 2003, Pages ## 1287–1296 http://dx.doi.org/10.1016/S0895-7177(03)90040-9 ## ## To improve numerical stability, the recursion is based on the inverse ## 1 / E_B rather than E_B itself. function B = __erlangb_compute(A, m) Binv = 1.0; for k=1:m Binv = 1.0 + k/A*Binv; endfor B = 1.0 / Binv; endfunction %!test %! fail("erlangb(1, -1)", "positive"); %! fail("erlangb(-1, 1)", "positive"); %! fail("erlangb(1, 0)", "positive"); %! fail("erlangb(0, 1)", "positive"); %! fail("erlangb('foo',1)", "positive"); %! fail("erlangb(1,'bar')", "positive"); %! fail("erlangb([1 1],[1 1 1])","common size"); queueing/inst/qnsolve.m0000644000175000017500000005532113570046274015120 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"closed"}, @var{N}, @var{QQ}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"closed"}, @var{N}, @var{QQ}, @var{V}, @var{Z}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"open"}, @var{lambda}, @var{QQ}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"mixed"}, @var{lambda}, @var{N}, @var{QQ}, @var{V}) ## ## High-level function for analyzing QN models. ## ## @itemize ## ## @item For @strong{closed} networks, the following server types are ## supported: @math{M/M/m}--FCFS, @math{-/G/\infty}, @math{-/G/1}--LCFS-PR, ## @math{-/G/1}--PS and load-dependent variants. ## ## @item For @strong{open} networks, the following server types are supported: ## @math{M/M/m}--FCFS, @math{-/G/\infty} and @math{-/G/1}--PS. General ## load-dependent nodes are @emph{not} supported. Multiclass open networks ## do not support multiple server @math{M/M/m} nodes, but only ## single server @math{M/M/1}--FCFS. ## ## @item For @strong{mixed} networks, the following server types are supported: ## @math{M/M/1}--FCFS, @math{-/G/\infty} and @math{-/G/1}--PS. General ## load-dependent nodes are @emph{not} supported. ## ## @end itemize ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## @itemx @var{N}(c) ## Number of requests in the system for closed networks. For ## single-class networks, @var{N} must be a scalar. For multiclass ## networks, @code{@var{N}(c)} is the population size of closed class ## @math{c}. ## ## @item @var{lambda} ## @itemx @var{lambda}(c) ## External arrival rate (scalar) for open networks. For single-class ## networks, @var{lambda} must be a scalar. For multiclass networks, ## @code{@var{lambda}(c)} is the class @math{c} overall arrival rate. ## ## @item @var{QQ}@{i@} ## List of queues in the network. This must be a cell array ## with @math{N} elements, such that @code{@var{QQ}@{i@}} is ## a struct produced by the @code{qnmknode} function. ## ## @item @var{Z} ## External delay ("think time") for closed networks. Default 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(k) ## If @math{k} is a FCFS node, then @code{@var{U}(k)} is the utilization ## of service center @math{k}. If @math{k} is an IS node, then ## @code{@var{U}(k)} is the @emph{traffic intensity} defined as ## @code{@var{X}(k)*@var{S}(k)}. ## ## @item @var{R}(k) ## average response time of service center @math{k}. ## ## @item @var{Q}(k) ## average number of customers in service center @math{k}. ## ## @item @var{X}(k) ## throughput of service center @math{k}. ## ## @end table ## ## Note that for multiclass networks, the computed results are per-class ## utilization, response time, number of customers and throughput: ## @code{@var{U}(c,k)}, @code{@var{R}(c,k)}, @code{@var{Q}(c,k)}, ## @code{@var{X}(c,k)}. ## ## String literals are case-insensitive, so @var{"closed"}, @var{"Closed"} ## and @var{"CLoSEd"} are all equivalent. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qnsolve( network_type, varargin ) if ( nargin < 2 ) print_usage(); endif ischar(network_type) || ... error("First parameter must be a string"); network_type = tolower(network_type); if ( strcmp(network_type, "open" ) ) [U R Q X] = __qnsolve_open( varargin{:} ); elseif ( strcmp(network_type, "closed" ) ) [U R Q X] = __qnsolve_closed( varargin{:} ); elseif (strcmp(network_type, "mixed" ) ) [U R Q X] = __qnsolve_mixed( varargin{:} ); else error( "Invalid network type %s: must be one of \"open\", \"closed\" or \"mixed\"", network_type ); endif endfunction ############################################################################## ## Dispatcher function for open networks function [U R Q X] = __qnsolve_open( lambda, varargin ) if ( isscalar(lambda) ) [U R Q X] = __qnsolve_open_single( lambda, varargin{:} ); else [U R Q X] = __qnsolve_open_multi( lambda, varargin{:} ); endif endfunction ############################################################################## ## Worker function for open, single class networks function [U R Q X] = __qnsolve_open_single( lambda, QQ, V ) if ( nargin != 3 ) print_usage(); endif ( isscalar(lambda) && (lambda>0) ) || ... error( "lambda must be a scalar > 0" ); iscell(QQ) || ... error( "QQ must be a cell array" ); N = length(QQ); ( isvector(V) && length(V) == N ) || ... error( "V must be a vector of length %d", N ); V = V(:); # make V a row vector all(V>=0) || ... error( "V must be >= 0" ); ## Initialize vectors S = zeros(size(V)); m = ones(size(V)); for i=1:N QQ{i}.c == 1 || ... error( "Multiclass networks are not supported by this function" ); S(i) = QQ{i}.S; if __is_li(QQ{i}) ; # nothing to do elseif __is_multi(QQ{i}) m(i) = QQ{i}.m; elseif __is_is(QQ{i}) m(i) = -1; else error( "Unsupported type \"%s\" for node %d", QQ{i}.node, i ); endif endfor [U R Q X] = qnos( lambda, S, V, m ); __prettyprint( 0, lambda, QQ, V, U, R, Q, X ); endfunction ############################################################################## ## Worker function for open, multiclass networks function [U R Q X] = __qnsolve_open_multi( lambda, QQ, V ) if ( nargin != 3 ) print_usage(); endif isvector(lambda) && all(lambda > 0) || ... error( "lambda must be a vector >0" ); lambda = lambda(:)'; # make lambda a row vector iscell(QQ) || ... error( "QQ must be a cell array" ); C = length(lambda); K = length(QQ); [C,K] == size(V) || ... error( "V size mismatch" ); all( all( V>= 0 ) ) || ... error( "V must be >= 0 " ); S = zeros(C,K); m = ones(1,K); for i=1:K QQ{i}.c == C || ... error( "Wrong number of classes for center %d (is %d, should be %d)", i, QQ{i}.c, C ); S(:,i) = QQ{i}.S(:); if __is_li(QQ{i}) ; # nothing to do elseif __is_is(QQ{i}) m(i) = -1; else error( "Unsupported type \"%s\" for node %d", QQ{i}.node, i ); endif endfor [U R Q X] = qnom( lambda, S, V, m ); __prettyprint( 0, lambda, QQ, V, U, R, Q, X ); endfunction ############################################################################## ## Dispatcher function for closed networks function [U R Q X] = __qnsolve_closed( N, varargin ) if ( isscalar(N) ) [U R Q X] = __qnsolve_closed_single( N, varargin{:} ); else [U R Q X] = __qnsolve_closed_multi( N, varargin{:} ); endif endfunction ############################################################################## ## Worker function for closed, single-class networks function [U R Q X] = __qnsolve_closed_single( N, QQ, V, Z ) if ( nargin < 3 || nargin > 4 ) error(); endif isscalar(N) || ... error( "Multiclass networks are not supported by this function" ); iscell(QQ) || ... error( "QQ must be a cell array" ); if ( nargin < 4 ) Z = 0; else isscalar(Z) && Z >= 0 || ... error( "Z must be >= 0" ); endif K = length(QQ); ( isvector(V) && length(V) == K ) || ... error( "V must be a vector of length %d", K ); found_ld = false; for k=1:K if ( __is_ld(QQ{k}) ) found_ld = true; break; endif endfor if ( found_ld ) S = zeros(K, N); for k=1:K ( QQ{k}.c == 1 ) || ... error( "Multiclass networks are not supported by this function" ); if __is_li(QQ{k}) S(k,:) = QQ{k}.S; elseif __is_multi(QQ{k}) S(k,:) = QQ{k}.S ./ min(1:N,QQ{k}.m); elseif __is_is(QQ{k}) S(k,:) = QQ{k}.S ./ (1:N); elseif __is_ld(QQ{k}) S(k,:) = QQ{k}.S; else error( "Unsupported type \"%s\" for node %d", QQ{k}.node, k ); endif endfor [U R Q X] = qncsmvald(N, S, V, Z); else S = zeros(1,K); m = ones(1,K); for k=1:K ( QQ{k}.c == 1 ) || ... error( "Multiclass networks are not supported by this function" ); S(k) = QQ{k}.S; if __is_li(QQ{k}) # nothing to do elseif __is_multi(QQ{k}) m(k) = QQ{k}.m; elseif __is_is(QQ{k}) m(k) = -1; else error( "Unsupported type \"%s\" for node %d", QQ{k}.node, k ); endif endfor [U R Q X] = qncsmva(N, S, V, m, Z); endif __prettyprint( N, 0, QQ, V, U, R, Q, X ); endfunction ############################################################################## ## Worker function for closed, multi-class networks function [U R Q X] = __qnsolve_closed_multi( N, QQ, V, Z ) if ( nargin < 3 || nargin > 4 ) print_usage(); endif isvector(N) && all( N>0 ) || ... error( "N must be >0" ); iscell(QQ) || ... error( "QQ must be a cell array" ); C = length(N); ## Number of classes K = length(QQ); ## Number of service centers size(V) == [C,K] || ... error( "V size mismatch" ); if ( nargin < 4 ) Z = zeros(1,C); else isvector(Z) && length(Z) == C || ... error( "Z size mismatch" ); endif ## Check consistence of parameters all( all( V >= 0 ) ) || ... error( "V must be >=0" ); ## Initialize vectors i_single = i_multi = i_delay = i_ld = []; S = zeros(C,K); for i=1:K ( QQ{i}.c == C ) || ... error( "Service center %d has wrong number of classes (is %d, should be %d)", i, QQ{i}.c, C ); if __is_li(QQ{i}) i_single = [i_single i]; ( !strcmpi( QQ{i}.node, "m/m/m-fcfs" ) || all( QQ{i}.S(1) == QQ{i}.S )) || ... error( "Service times at FIFO node %d are not class-independent", i ); elseif __is_multi(QQ{i}) i_multi = [i_multi i]; elseif __is_is(QQ{i}) i_delay = [i_delay i]; elseif __is_ld(QQ{i}) columns( QQ{i}.S ) == sum(N) || ... error( "Load-dependent center %d has insufficient data (is %d, should be %d", i, columns(QQ{i}.S), sum(N) ); i_ld = [i_ld i]; else error( "Unknown or unsupported type \"%s\" for node %d", QQ{i}.node, i ); endif endfor ## Initialize results U = R = zeros( C, K ); X = zeros( 1, C ); Q_next = Q = sparse( prod(N+1),K ); p = cell(1,K); for k=i_multi ## p{i}(j+1,k+1) is the probability to have j jobs at node i ## where the network is in state k p{k} = zeros( QQ{k}.m+1,prod(N+1) ); p{k}(1,__get_idx( N, 0*N )) = 1; endfor for k=i_ld ## p{i}(j+1,k+1) is the probability to have j jobs at node i ## where the network is in state k p{k} = zeros( columns(QQ{k}.S )+1, prod(N+1) ); p{k}(1,__get_idx( N, 0*N )) = 1; endfor ## Main loop for n=1:sum(N) feasible_set = qncmpopmix( n, N ); for nn=1:rows(feasible_set) n_bar = feasible_set(nn,:); for c=1:C if ( n_bar(c) > 0 ) ## single server nodes for k=i_single n_bar_c = __minusonec(n_bar,c); idx = __get_idx( N, n_bar_c ); R(c,k) = QQ{k}.S(c)*(1 + Q( idx, k ) ); ## for FCFS nodes with class-dependent service times, ## it is possible to use the following approximation ## (p. 469 Bolch et al.) ## ## R(c,k) = S(c,k) + sum( S(:,k) * Q(idx(:), k) ); endfor ## multi server nodes for k=i_multi n_bar_c = __minusonec(n_bar,c); idx = __get_idx( N, n_bar_c ); j=0:QQ{k}.m-2; # range R(c,k) = QQ{k}.S(c)/QQ{k}.m*(1 + Q( idx, k ) + ... dot(QQ{k}.m-j-1,p{k}(j+1,idx) ) ); endfor ## General load-dependent nodes for k=i_ld n_bar_c = __minusonec(n_bar,c); idx = __get_idx( N, n_bar_c ); j=1:sum(n_bar); # range R(c,k) = sum( j .* QQ{k}.S(c,j) .* p{k}(j,idx)' ); endfor endif ## delay centers for k=i_delay R(c,k) = QQ{k}.S(c); endfor endfor # c X = n_bar ./ ( Z .+ dot(R,V,2)' ); # X(c) = N(c) / ( Z(c) + sum_k R(c,k) * V(c,k) ) idx = __get_idx( N, n_bar ); ## Q_k = sum_c X(c) * R(c,k) for k=1:K Q_next( idx, k ) = dot( X, R(:,k) .* V(:,k) ); endfor ## Adjust probabilities for multiple server nodes for k=i_multi s=0; # it is actually a vector j=1:QQ{k}.m-1; for r=find(n_bar>0) # I don't know how to vectorize this ii = __minusonec(n_bar,r); s+=QQ{k}.S(r)*V(r,k)*X(r)*p{k}(j,__get_idx(N,ii)); endfor p{k}(j+1,idx) = s./j; p{k}(1,idx) = 1-1/QQ{k}.m*(sum( QQ{k}.S(:) .* V(:,k) .* X(:) ) + ... dot( QQ{k}.m-j, p{k}(j+1,idx) ) ); endfor ## Adjust probabilities for general load-dependent server nodes for k=i_ld s=0; # it is actually a vector j=1:sum(n_bar); for r=find(n_bar>0) ii = __minusonec(n_bar,r); s+=QQ{k}.S(r,sum(n_bar))*V(r,k)*X(r)*p{k}(j,__get_idx(N,ii)); endfor p{k}(j+1,idx) = s; p{k}(1,idx) = 1-sum(p{k}(1+j,idx)); endfor endfor Q = Q_next; Q_next = sparse( prod(N+1), K ); endfor for k=1:K if __is_ld(QQ{k}) U(:,k) = 1-p{k}(1, __get_idx(N,N)); else U(:,k) = X(:) .* QQ{k}.S(:) .* V(:,k); # U(c,k) = X(c)*D(c,k) endif endfor Q = (diag(X)*R).*V; # dmult(X,R).*V; X = diag(X)*V; # dmult(X,V); endfunction ############################################################################## ## Compute the linear index corresponding to vector i from a population ## of N. function idx = __get_idx( N, i ) i_cell = num2cell( i+1 ); idx = sub2ind( N+1, i_cell{:} ); endfunction ############################################################################## ## Given an input vector n, returns an output vector r which is equal to ## n except that the element at the c-th position is decreased by one: ## r(c) = n(c)-1. Warning: no check is made on the parameters function r = __minusonec( n, c ) r = n; r(c) -= 1; endfunction ############################################################################## ## Worker function for mixed networks. This function delegates to qnmix function [U R Q X] = __qnsolve_mixed( lambda, N, QQ, V ) if ( nargin != 4 ) print_usage(); endif ( isvector(lambda) && isvector(N) && size_equal(lambda,N) ) || ... error( "lambda and N must be vectors of the same size" ); ( iscell(QQ) && length(QQ) == length(lambda) ) || ... error( "QQ size mismatch (is %d, should be %d)", length(QQ), length(lambda) ); C = length(lambda); # number of classes K = length(QQ); # number of service centers S = zeros(C,K); m = ones(1,K); ## fill S matrix for k=1:K if __is_ld(QQ{k}) error( "General load-dependent service center %d is not supported", k ); elseif __is_is(QQ{k}) m(k) = -1; else m(k) = QQ{k}.m; endif S(:,k) = QQ{k}.S; endfor [U R Q X] = qnmix( lambda, N, S, V, m ); __prettyprint( N, lambda, QQ, V, U, R, Q, X ); endfunction ############################################################################## ## return true iff Q is an infinite server (IS) node function result = __is_is( Q ) result = strcmp(Q.node, "-/g/inf" ); endfunction ############################################################################## ## return true iff Q is a multi-server FIFO node function result = __is_multi( Q ) result = (strcmp(Q.node, "m/m/m-fcfs") && Q.m>1); endfunction ############################################################################## ## return true iff Q is a single-server, load-dependent node function result = __is_ld( Q ) result = ( (strcmp(Q.node, "m/m/m-fcfs") || ... strcmp(Q.node, "-/g/1-lcfs-pr") || ... strcmp(Q.node, "-/g/1-ps" ) ) && ... columns( Q.S ) > 1 ); endfunction ############################################################################## ## return ture iff Q is a single-server, load-independent node function result = __is_li( Q ) result = ((Q.m==1) && (1 == columns( Q.S )) && !strcmp( Q.node, "-/g/inf" ) ); endfunction ############################################################################## ## This function is used to "pretty-print" a solved network. Used for ## debugging function __prettyprint( N, lambda, QQ, V, U, R, Q, X ) return; ## immediately return [errorcode, N, lambda] = common_size( N, lambda ); if ( errorcode) error( "N and lambda are of incompatible size" ); endif ( isvector(N) && isvector(lambda) && size_equal(lambda,N) ) || ... error( "N and lambda must be vector of the same length" ); C = length(N); K = length(QQ); # number of service centers [C,K] == size(V) || ... error( "V size mismatch" ); [C,K] == size(U) || ... error( "U size mismatch" ); [C,K] == size(R) || ... error( "R size mismatch" ); [C,K] == size(Q) || ... error( "Q size mismatch" ); [C,K] == size(X) || ... error( "X size mismatch" ); for c=1:C printf("\n"); printf("=== CLASS %d ===\n", c ); if ( N(c)>0 ) printf("Type: CLOSED\nPopulation: %d\n", N(c)) else printf("Type: OPEN\nRequests arrival rate: %6.2f\n", lambda(c)) endif printf("\n"); printf("+---+---------------+---+------+------+------+------+------+------+\n"); printf("| i | Node type | m | S(i) | V(i) | U(i) | R(i) | Q(i) | X(i) |\n"); printf("+---+---------------+---+------+------+------+------+------+------+\n"); for i=1:K if ( isscalar(QQ{i}.S(c)) ) serv = sprintf("%6.2f",QQ{i}.S(c)); else serv = "LD"; endif printf("|%3d|%-33s| | | | |\n", i, QQ{i}.comment); printf("| |%15s|%3d|%6s|%6.2f|%6.4f|%6.2f|%6.2f|%6.2f|\n", QQ{i}.node, QQ{i}.m, serv, V(c,i), U(c,i), R(c,i), Q(c,i), X(c,i) ); endfor printf("+---+---------------+---+------+------+------+------+------+------+\n"); printf("| THIS CLASS STATISTICS | ---- |%6.2f|%6.2f|%6.2f|\n", dot(R(c,:),V(c,:)), sum(Q(c,:)), X(c,1)/V(c,1) ); printf("+---+---------------+---+------+------+------+------+------+------+\n\n"); endfor endfunction %!test %! # Example 8.7 p. 349 Bolch et al. %! N = 3; %! Q1 = qnmknode( "m/m/m-fcfs", .5, 2 ); %! Q2 = qnmknode( "m/m/m-fcfs", 1/1.667 ); %! Q3 = qnmknode( "m/m/m-fcfs", 1/1.25 ); %! Q4 = qnmknode( "m/m/m-fcfs", 1./[1 2 3] ); %! V = [ 1 .5 .5 1 ]; %! [U R Q X] = qnsolve("closed",N, { Q1, Q2, Q3, Q4 }, V); %! assert( Q, [0.624 0.473 0.686 1.217], 1e-3 ); %! assert( R, [0.512 0.776 1.127 1], 1e-3 ); %!test %! # Example 8.7 p. 349 Bolch et al. %! N = 3; %! Q1 = qnmknode( "m/m/m-fcfs", 1/2, 2 ); %! Q2 = qnmknode( "m/m/m-fcfs", 1/1.667 ); %! Q3 = qnmknode( "m/m/m-fcfs", 1/1.25 ); %! Q4 = qnmknode( "-/g/inf", 1 ); %! V = [ 1 .5 .5 1 ]; %! [U R Q X] = qnsolve("closed",N, { Q1, Q2, Q3, Q4 }, V); %! assert( Q, [0.624 0.473 0.686 1.217], 1e-3 ); %! assert( R, [0.512 0.776 1.127 1], 1e-3 ); %!test %! # Example 8.4 p. 333 Bolch et al. %! N = 3; %! Q1 = qnmknode( "m/m/m-fcfs", .5, 2 ); %! Q2 = qnmknode( "m/m/m-fcfs", .6 ); %! Q3 = qnmknode( "m/m/m-fcfs", .8 ); %! Q4 = qnmknode( "-/g/inf", 1 ); %! V = [ 1 .5 .5 1 ]; %! [U R Q X] = qnsolve("closed",N, { Q1, Q2, Q3, Q4 }, V); %! assert( U(1:3), [.304 .365 .487], 1e-3 ); %! assert( X, [1.218 0.609 0.609 1.218], 1e-3 ); %!test %! # Same as above, with center 1 replaced with a load-dependent service center %! N = 3; %! Q1 = qnmknode( "m/m/m-fcfs", [.5 .25 .25] ); %! Q2 = qnmknode( "m/m/m-fcfs", .6 ); %! Q3 = qnmknode( "m/m/m-fcfs", .8 ); %! Q4 = qnmknode( "m/m/m-fcfs", [1 1/2 1/3] ); %! V = [ 1 .5 .5 1 ]; %! [U R Q X] = qnsolve("closed",N, { Q1, Q2, Q3, Q4 }, V); %! assert( U(2:3), [.365 .487], 1e-3 ); ## NOTE: Utilization U(1) is computed differently from M/M/m nodes and load-dependent M/M/1 nodes %! assert( X, [1.218 0.609 0.609 1.218], 1e-3 ); %!test %! # Example 7.4 p. 287 Bolch et al. %! QQ = { qnmknode( "m/m/m-fcfs", 0.04 ), ... %! qnmknode( "m/m/m-fcfs", 0.03 ), ... %! qnmknode( "m/m/m-fcfs", 0.06 ), ... %! qnmknode( "m/m/m-fcfs", 0.05 ) }; %! P = [ 0 0.5 0.5 0; 1 0 0 0; 0.6 0 0 0; 1 0 0 0 ]; %! lambda = [0 0 0 4]; %! [U R Q X] = qnsolve("open", sum(lambda), QQ, qnosvisits(P,lambda) ); %! assert( X, [20 10 10 4], 1e-4 ); %! assert( U, [0.8 0.3 0.6 0.2], 1e-2 ); %! assert( R, [0.2 0.043 0.15 0.0625], 1e-3 ); %! assert( Q, [4, 0.429 1.5 0.25], 1e-3 ); %!test %! V = [1 1; 1 1]; %! Q1 = qnmknode( "m/m/m-fcfs", [1;2] ); %! Q2 = qnmknode( "m/m/m-fcfs", [3;4] ); %! lambda = [3/19 2/19]; %! [U R Q] = qnsolve("open", lambda, { Q1, Q2 }, diag( lambda / sum(lambda) ) * V); %! assert( U(1,1), 3/19, 1e-6 ); %! assert( U(2,1), 4/19, 1e-6 ); %! assert( R(1,1), 19/12, 1e-6 ); %! assert( R(1,2), 57/2, 1e-6 ); %! assert( Q(1,1), .25, 1e-6 ); ## Example 9.5 p. 337, Bolch et al. %!test %! QQ = { qnmknode( "m/m/m-fcfs", [0.2; 0.2], 2 ), ... %! qnmknode( "-/g/1-ps", [0.4; 0.6] ), ... %! qnmknode( "-/g/inf", [1; 2] ) }; %! V = [ 1 0.6 0.4; 1 0.3 0.7 ]; %! N = [ 2 1 ]; %! [U R Q X] = qnsolve( "closed", N, QQ, V ); %! assert( Q, [ 0.428 0.726 0.845; 0.108 0.158 0.734 ], 1e-3 ); %! assert( X(1,1), 2.113, 1e-3 ); # CHECK %! assert( X(2,1), 0.524, 1e-3 ); # CHECK %! assert( all( all(U(:,[1,2])<=1) ) ); ## Same as above, but with general load-dependent centers %!test %! QQ = { qnmknode( "m/m/m-fcfs", [0.2 0.1 0.1; 0.2 0.1 0.1] ), ... %! qnmknode( "-/g/1-ps", [0.4; 0.6] ), ... %! qnmknode( "-/g/inf", [1; 2] ) }; %! V = [ 1 0.6 0.4; 1 0.3 0.7 ]; %! N = [ 2 1 ]; %! [U R Q X] = qnsolve( "closed", N, QQ, V ); %! assert( Q, [ 0.428 0.726 0.845; 0.108 0.158 0.734 ], 1e-3 ); %! assert( X(1,1), 2.113, 1e-3 ); # CHECK %! assert( X(2,1), 0.524, 1e-3 ); # CHECK %! assert( all( all(U(:,[1,2])<=1) ) ); %!test %! # example p. 26 Schwetman %! QQ = { qnmknode( "m/m/m-fcfs", [.25; .25] ), %! qnmknode( "-/g/1-ps", [0; .1] ) }; %! V = [1 0; 1 1]; %! lambda = [1 0]; %! N = [0 3]; %! [U R Q X] = qnsolve( "mixed", lambda, N, QQ, V ); %! assert( U(1,1), .25, 1e-3 ); %! assert( X(1,1), 1.0, 1e-3 ); %! assert( [R(1,1) R(2,1) R(2,2)], [1.201 0.885 0.135], 1e-3 ); %!demo %! QQ = { qnmknode( "m/m/m-fcfs", [0.2 0.1 0.1; 0.2 0.1 0.1] ), ... %! qnmknode( "-/g/1-ps", [0.4; 0.6] ), ... %! qnmknode( "-/g/inf", [1; 2] ) }; %! V = [ 1 0.6 0.4; ... %! 1 0.3 0.7 ]; %! N = [ 2 1 ]; %! [U R Q X] = qnsolve( "closed", N, QQ, V ); queueing/inst/qnomaba.m0000644000175000017500000000670413570046274015050 0ustar morenomoreno## Copyright (C) 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnomaba (@var{lambda}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Rl}] =} qnomaba (@var{lambda}, @var{S}, @var{V}) ## ## @cindex bounds, asymptotic ## @cindex open network ## @cindex multiclass network, open ## ## Compute Asymptotic Bounds for open, multiclass networks with @math{K} ## service centers and @math{C} customer classes. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda}(c) ## class @math{c} arrival rate to the system (vector of length ## @math{C}, @code{@var{lambda}(c) > 0}). ## ## @item @var{D}(c, k) ## class @math{c} service demand at center @math{k} (@math{C \times K} ## matrix, @code{@var{D}(c, k) @geq{} 0}). ## ## @item @var{S}(c, k) ## mean service time of class @math{c} requests at center @math{k} ## (@math{C \times K} matrix, @code{@var{S}(c, k) @geq{} 0}). ## ## @item @var{V}(c, k) ## mean number of visits of class @math{c} requests at center @math{k} ## (@math{C \times K} matrix, @code{@var{V}(c, k) @geq{} 0}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl}(c) ## @item @var{Xu}(c) ## lower and upper bounds of class @math{c} throughput. ## @code{@var{Xl}(c)} is always @math{0} since there can be no lower ## bound on the throughput of open networks (vector of length ## @math{C}). ## ## @item @var{Rl}(c) ## @item @var{Ru}(c) ## lower and upper bounds of class @math{c} response time. ## @code{@var{Ru}(c)} is always @code{+inf} since there can be no ## upper bound on the response time of open networks (vector of length ## @math{C}). ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [X_lower X_upper R_lower R_upper] = qnomaba( lambda, S, V ) if ( nargin < 2 || nargin > 3 ) print_usage(); endif (isvector(lambda) && length(lambda)>0) || ... error( "lambda must be a nonempty vector" ); all(lambda > 0) || ... error( "lambda must contain positive values" ); lambda = lambda(:)'; C = length(lambda); ( ismatrix(S) && rows(S)==C ) || ... error( "S/D must be a matrix >=0 with %d rows", C ); all(S(:)>=0) || ... error( "S/D must contain nonnegative values" ); K = columns(S); if ( nargin < 3 ) V = ones(size(S)); else ( ismatrix(V) && size_equal(S,V) ) || ... error( "V must be a %d x %d matrix", C, K); all(V(:)>=0) || ... error( "V must contain nonnegative values" ); endif D = S.*V; X_lower = zeros(1,C); X_upper = 1./max(D,[],2)'; R_lower = sum(D,2)'; R_upper = +inf(1,C); endfunction %!test %! fail( "qnomaba( [1 1], [1 1 1; 1 1 1; 1 1 1] )", "2 rows" ); queueing/inst/qncsconv.m0000644000175000017500000001724613570046274015267 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsconv (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsconv (@var{N}, @var{S}, @var{V}, @var{m}) ## ## @cindex closed network, single class ## @cindex normalization constant ## @cindex convolution algorithm ## ## Analyze product-form, single class closed networks with @math{K} service centers using the convolution algorithm. ## ## Load-independent service centers, multiple servers (@math{M/M/m} ## queues) and IS nodes are supported. For general load-dependent ## service centers, use @code{qncsconvld} instead. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## Number of requests in the system (@code{@var{N}>0}). ## ## @item @var{S}(k) ## average service time on center @math{k} (@code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## visit count of service center @math{k} (@code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. If @code{@var{m}(k) < 1}, ## center @math{k} is a delay center (IS); if @code{@var{m}(k) @geq{} ## 1}, center @math{k} it is a regular @math{M/M/m} queueing center ## with @code{@var{m}(k)} identical servers. Default is ## @code{@var{m}(k) = 1} for all @math{k}. ## ## @end table ## ## @strong{OUTPUT} ## ## @table @code ## ## @item @var{U}(k) ## center @math{k} utilization. ## For IS nodes, @code{@var{U}(k)} is the @emph{traffic intensity} ## @code{@var{X}(k) * @var{S}(k)}. ## ## @item @var{R}(k) ## average response time of center @math{k}. ## ## @item @var{Q}(k) ## average number of customers at center @math{k}. ## ## @item @var{X}(k) ## throughput of center @math{k}. ## ## @item @var{G}(n) ## Vector of normalization constants. @code{@var{G}(n+1)} contains the value of ## the normalization constant with @math{n} requests ## @math{G(n)}, @math{n=0, @dots{}, N}. ## ## @end table ## ## @strong{NOTE} ## ## For a network with @math{K} service centers and @math{N} requests, ## this implementation of the convolution algorithm has time and space ## complexity @math{O(NK)}. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Jeffrey P. Buzen, @cite{Computational Algorithms for Closed Queueing ## Networks with Exponential Servers}, Communications of the ACM, volume ## 16, number 9, September 1973, ## pp. 527--531. @uref{http://doi.acm.org/10.1145/362342.362345, 10.1145/362342.362345} ## @end itemize ## ## This implementation is based on G. Bolch, S. Greiner, H. de Meer and ## K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and ## Performance Evaluation with Computer Science Applications}, Wiley, ## 1998, pp. 313--317. ## ## @seealso{qncsconvld} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X G] = qncsconv( varargin ) if ( nargin < 3 || nargin > 4 ) print_usage(); endif ## To be compliant with the reference, we use K to denote the ## population size [err K S V m] = qncschkparam( varargin{:} ); isempty(err) || error(err); N = length(S); # Number of service centers ## This implementation is based on G. Bolch, S. Greiner, H. de Meer ## and K. Trivedi, Queueing Networks and Markov Chains: Modeling and ## Performance Evaluation with Computer Science Applications, Wiley, ## 1998, pp. 313--317. ## First, we remember the indexes of IS nodes i_delay = find(m<1); m( i_delay ) = K; # IS nodes are handled as if they were M/M/K nodes with number of servers equal to the population size K, such that queueing never occurs. ## Initialization G_n = G_nm1 = zeros(1,K+1); F_n = zeros(N,K+1); F_n(:,1) = 1; k=1:K; G_n(1) = 1; G_n(k+1) = F_n(1,k+1) = F(1,k,V,S,m); ## Main convolution loop for n=2:N G_nm1 = G_n; k=1:K; F_n(n,1+k) = F(n,k,V,S,m); # G_n(1) = 1; G_n = conv( F_n(n,:), G_nm1(:) )(1:K+1); endfor ## Done computation of G(n,k). G = G_n(:)'; # ensure G is a row vector ## Computes performance measures X = V*G(K)/G(K+1); U = X .* S ./ m; ## Adjust utilization of delay centers U(i_delay) = X(i_delay) .* S(i_delay); Q = zeros(1,N); i_multi = find(m>1); for i=i_multi G_N_i = zeros(1,K+1); G_N_i(1) = 1; for k=1:K j=1:k; G_N_i(k+1) = G(k+1)-dot( F_n(i,j+1), G_N_i(k-j+1) ); endfor k=0:K; p_i(k+1) = F_n(i,k+1)./G(K+1).*G_N_i(K-k+1); Q(i) = dot( k, p_i( k+1 ) ); endfor i_single = find(m==1); for i=i_single k=1:K; Q(i) = sum( ( V(i)*S(i) ) .^ k .* G(K+1-k)/G(K+1) ); endfor R = Q ./ X; endfunction %!test %! # Example 8.1 p. 318 Bolch et al. %! K=3; %! S = [ 1/0.8 1/0.6 1/0.4 ]; %! m = [2 3 1]; %! V = [ 1 .667 .2 ]; %! [U R Q X G] = qncsconv( K, S, V, m ); %! assert( G, [1 2.861 4.218 4.465], 5e-3 ); %! assert( X, [0.945 0.630 0.189], 1e-3 ); %! assert( U, [0.590 0.350 0.473], 1e-3 ); %! assert( Q, [1.290 1.050 0.660], 1e-3 ); %! assert( R, [1.366 1.667 3.496], 1e-3 ); %!test %! # Example 8.3 p. 331 Bolch et al. %! # compare results of convolution to those of mva %! S = [ 0.02 0.2 0.4 0.6 ]; %! K = 6; %! V = [ 1 0.4 0.2 0.1 ]; %! [U_mva R_mva Q_mva X_mva G_mva] = qncsmva(K, S, V); %! [U_con R_con Q_con X_con G_con] = qncsconv(K, S, V); %! assert( U_mva, U_con, 1e-5 ); %! assert( R_mva, R_con, 1e-5 ); %! assert( Q_mva, Q_con, 1e-5 ); %! assert( X_mva, X_con, 1e-5 ); %! assert( G_mva, G_con, 1e-5 ); %!test %! # Compare the results of convolution to those of mva %! S = [ 0.02 0.2 0.4 0.6 ]; %! K = 6; %! V = [ 1 0.4 0.2 0.1 ]; %! m = [ 1 -1 2 1 ]; # center 2 is IS %! [U_mva R_mva Q_mva X_mva] = qncsmva(K, S, V, m); %! [U_con R_con Q_con X_con G] = qncsconv(K, S, V, m ); %! assert( U_mva, U_con, 1e-5 ); %! assert( R_mva, R_con, 1e-5 ); %! assert( Q_mva, Q_con, 1e-5 ); %! assert( X_mva, X_con, 1e-5 ); ## result = F(i,j,v,S,m) ## ## Helper fuction to compute F(i,j) as defined in Eq 7.61 p. 289 of ## Bolch, Greiner, de Meer, Trivedi "Queueing Networks and Markov ## Chains: Modeling and Performance Evaluation with Computer Science ## Applications", Wiley, 1998. This function has been vectorized, ## and accepts a vector as parameter j. function result = F(i,j,v,S,m) isscalar(i) || ... error( "i must be a scalar" ); k_i = j; if ( m(i) == 1 ) result = ( v(i)*S(i) ).^k_i; else ii = find(k_i<=m(i)); ## if k_i<=m(i) result(ii) = ( v(i)*S(i) ).^k_i(ii) ./ factorial(k_i(ii)); ii = find(k_i>m(i)); ## if k_i>m(i) result(ii) = ( v(i)*S(i) ).^k_i(ii) ./ ( factorial(m(i))*m(i).^(k_i(ii)-m(i)) ); endif endfunction %!demo %! n = [1 2 0]; %! N = sum(n); # Total population size %! S = [ 1/0.8 1/0.6 1/0.4 ]; %! m = [ 2 3 1 ]; %! V = [ 1 .667 .2 ]; %! [U R Q X G] = qncsconv( N, S, V, m ); %! p = [0 0 0]; # initialize p %! # Compute the probability to have n(k) jobs at service center k %! for k=1:3 %! p(k) = (V(k)*S(k))^n(k) / G(N+1) * ... %! (G(N-n(k)+1) - V(k)*S(k)*G(N-n(k)) ); %! printf("Prob( n(%d) = %d )=%f\n", k, n(k), p(k) ); %! endfor queueing/inst/qncmcb.m0000644000175000017500000001124613633127774014677 0ustar morenomoreno## Copyright (C) 2012, 2016, 2018, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmcb (@var{N}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmcb (@var{N}, @var{S}, @var{V}) ## ## @cindex multiclass network, closed ## @cindex closed multiclass network ## @cindex bounds, composite ## @cindex composite bounds ## ## Compute Composite Bounds (CB) on system throughput and response time for closed multiclass networks. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N}(c) ## number of class @math{c} requests in the system. ## ## @item @var{D}(c, k) ## class @math{c} service demand ## at center @math{k} (@code{@var{S}(c,k) @geq{} 0}). ## ## @item @var{S}(c, k) ## mean service time of class @math{c} ## requests at center @math{k} (@code{@var{S}(c,k) @geq{} 0}). ## ## @item @var{V}(c,k) ## average number of visits of class @math{c} ## requests to center @math{k} (@code{@var{V}(c,k) @geq{} 0}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl}(c) ## @itemx @var{Xu}(c) ## Lower and upper bounds on class @math{c} throughput. ## ## @item @var{Rl}(c) ## @itemx @var{Ru}(c) ## Lower and upper bounds on class @math{c} response time. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Teemu Kerola, @cite{The Composite Bound Method (CBM) for Computing ## Throughput Bounds in Multiple Class Environments}, Performance ## Evaluation Vol. 6, Issue 1, March 1986, DOI ## @uref{http://dx.doi.org/10.1016/0166-5316(86)90002-7, ## 10.1016/0166-5316(86)90002-7}. Also available as ## @uref{http://docs.lib.purdue.edu/cstech/395/, Technical Report ## CSD-TR-475}, Department of Computer Sciences, Purdue University, mar ## 13, 1984 (Revised Aug 27, 1984). ## @end itemize ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [Xl Xu Rl Ru] = qncmcb( varargin ) if ( nargin < 2 || nargin > 3 ) print_usage(); endif [err N S V m Z] = qncmchkparam( varargin{:} ); isempty(err) || error(err); all(m == 1) || ... error("this function only supports single-server FCFS centers"); all(Z == 0) || ... error("this function does not support think time"); [C K] = size(S); D = S .* V; [Xl] = qncmbsb(N, D); Xu = zeros(1,C); D_max = max(D,[],2)'; for r=1:C ## This is equation (13) from T. Kerola, The Composite Bound Method ## (CBM) for Computing Throughput Bounds in Multiple Class ## Environments, Technical Report CSD-TR-475, Purdue University, ## march 13, 1984 (revised august 27, 1984) ## http://docs.lib.purdue.edu/cstech/395/ ## The only modification here is to apply also the upper bound ## 1/D_max(r). s = (1:C != r); # boolean array tmp = (1 .- Xl(s)*D(s,:)) ./ D(r,:); Xu(r) = min([tmp 1/D_max(r)]); endfor Rl = N ./ Xu; Ru = N ./ Xl; endfunction %!demo %! S = [10 7 5 4; ... %! 5 2 4 6]; %! NN=20; %! Xl = Xu = Rl = Ru = Xmva = Rmva = zeros(NN,2); %! for n=1:NN %! N=[n,10]; %! [a b c d] = qncmcb(N,S); %! Xl(n,:) = a; Xu(n,:) = b; Rl(n,:) = c; Ru(n,:) = d; %! [U R Q X] = qncmmva(N,S,ones(size(S))); %! Xmva(n,:) = X(:,1)'; Rmva(n,:) = sum(R,2)'; %! endfor %! subplot(2,2,1); %! plot(1:NN,Xl(:,1), 1:NN,Xu(:,1), 1:NN,Xmva(:,1), ";MVA;", "linewidth", 2); %! ylim([0, 0.2]); %! title("Class 1 throughput"); legend("boxoff"); %! subplot(2,2,2); %! plot(1:NN,Xl(:,2), 1:NN,Xu(:,2), 1:NN,Xmva(:,2), ";MVA;", "linewidth", 2); %! ylim([0, 0.2]); %! title("Class 2 throughput"); legend("boxoff"); %! subplot(2,2,3); %! plot(1:NN,Rl(:,1), 1:NN,Ru(:,1), 1:NN,Rmva(:,1), ";MVA;", "linewidth", 2); %! ylim([0, 700]); %! title("Class 1 response time"); legend("location", "northwest"); legend("boxoff"); %! subplot(2,2,4); %! plot(1:NN,Rl(:,2), 1:NN,Ru(:,2), 1:NN,Rmva(:,2), ";MVA;", "linewidth", 2); %! ylim([0, 700]); %! title("Class 2 response time"); legend("location", "northwest"); legend("boxoff"); queueing/inst/qnmknode.m0000644000175000017500000001243313570046274015242 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{Q} =} qnmknode (@var{"m/m/m-fcfs"}, @var{S}) ## @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"m/m/m-fcfs"}, @var{S}, @var{m}) ## @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"m/m/1-lcfs-pr"}, @var{S}) ## @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/1-ps"}, @var{S}) ## @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/1-ps"}, @var{S}, @var{s2}) ## @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/inf"}, @var{S}) ## @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/inf"}, @var{S}, @var{s2}) ## ## Creates a node; this function can be used together with ## @code{qnsolve}. It is possible to create either single-class nodes ## (where there is only one customer class), or multiple-class nodes ## (where the service time is given per-class). Furthermore, it is ## possible to specify load-dependent service times. String literals ## are case-insensitive, so for example @var{"-/g/inf"}, @var{"-/G/inf"} ## and @var{"-/g/INF"} are all equivalent. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{S} ## Mean service time. ## ## @itemize ## ## @item If @math{S} is a scalar, ## it is assumed to be a load-independent, class-independent service time. ## ## @item If @math{S} is a column vector, then @code{@var{S}(c)} is assumed to ## the the load-independent service time for class @math{c} customers. ## ## @item If @math{S} is a row vector, then @code{@var{S}(n)} is assumed to be ## the class-independent service time at the node, when there are @math{n} ## requests. ## ## @item Finally, if @var{S} is a two-dimensional matrix, then ## @code{@var{S}(c,n)} is assumed to be the class @math{c} service time ## when there are @math{n} requests at the node. ## ## @end itemize ## ## @item @var{m} ## Number of identical servers at the node. Default is @code{@var{m}=1}. ## ## @item @var{s2} ## Squared coefficient of variation for the service time. Default is 1.0. ## ## @end table ## ## The returned struct @var{Q} should be considered opaque to the client. ## ## @c The returned struct @var{Q} has the following fields: ## ## @c @table @var ## ## @c @item Q.node ## @c (String) type of the node; valid values are @code{"m/m/m-fcfs"}, ## @c @code{"-/g/1-lcfs-pr"}, @code{"-/g/1-ps"} (Processor-Sharing) ## @c and @code{"-/g/inf"} (Infinite Server, or delay center). ## ## @c @item Q.S ## @c Average service time. If @code{@var{Q}.S} is a vector, then ## @c @code{@var{Q}.S(i)} is the average service time at that node ## @c if there are @math{i} requests. ## ## @c @item Q.m ## @c Number of identical servers at a @code{"m/m/m-fcfs"}. Default is 1. ## ## @c @item Q.c ## @c Number of customer classes. Default is 1. ## ## @c @end table ## ## @seealso{qnsolve} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function Q = qnmknode( node, S, varargin ) ischar(node) || ... error( "Parameter \"node\" must be a string" ); node = tolower(node); isvector(S) || ismatrix(S) || ... error( "Parameter \"S\" must be a vector" ); m = 1; s2 = ones( size(S) ); if ( strcmp(node, "m/m/m-fcfs") ) ## M/M/k multiserver node if ( nargin > 3 ) print_usage(); endif if ( 3 == nargin ) m = varargin{1}; endif elseif ( strcmp(node, "m/m/1/k-fcfs") ) ## M/M/1/k finite capacity node if ( nargin > 3 ) print_usage(); endif if ( 3 == nargin ) k = varargin{1}; endif elseif ( strcmp(node, "-/g/1-lcfs-pr") ) ## -/G/1-LCFS-PR node ( 2 == nargin || 3 == nargin ) || ... print_usage(); if ( 3 == nargin ) s2 = varargin{1}; endif elseif ( strcmp(node, "-/g/1-ps") ) ## -/G/1-PS (processor sharing) node ( 2 == nargin || 3 == nargin ) || ... print_usage(); if ( 3 == nargin ) s2 = varargin{1}; endif elseif ( strcmp(node, "-/g/inf") ) ## -/G/inf (Infinite Server) node ( 2 == nargin || 3 == nargin ) || ... print_usage(); if ( 3 == nargin ) s2 = varargin{1}; endif else error( "Unknown node type \"%s\". node type must be one of \"m/m/m-fcfs\", \"-/g/1-lcfs-pr\", \"-/g/1-ps\" and \"-/g/inf\"", node ); endif ( isnumeric(m) && m>=1 ) || ... error("m must be >=1"); ( isnumeric(s2) && s2>= 0 ) || ... error("s2 must be >=0"); Q = struct( "node", node, "m", m, "S", S, "s2", s2, "c", rows(S), "comment", "" ); endfunction %!test %! fail( "qnmknode( 'pippo', 1 )", "must be one" ); %! fail( "qnmknode( '-/g/1-ps', 1, 1, 1)", "Invalid call" ); queueing/inst/qncmbsb.m0000644000175000017500000001244613632722337015057 0ustar morenomoreno## Copyright (C) 2012, 2016, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmbsb (@var{N}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmbsb (@var{N}, @var{S}, @var{V}) ## ## @cindex bounds, balanced system ## @cindex balanced system bounds ## @cindex multiclass network, closed ## @cindex closed multiclass network ## ## Compute Balanced System Bounds for closed, multiclass networks ## with @math{K} service centers and @math{C} customer classes. ## Only single-server nodes are supported. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N}(c) ## number of class @math{c} requests in the system (vector of length ## @math{C}). ## ## @item @var{D}(c, k) ## class @math{c} service demand at center @math{k} (@math{C \times K} ## matrix, @code{@var{D}(c,k) @geq{} 0}). ## ## @item @var{S}(c, k) ## mean service time of class @math{c} ## requests at center @math{k} (@math{C \times K} matrix, @code{@var{S}(c,k) @geq{} 0}). ## ## @item @var{V}(c,k) ## average number of visits of class @math{c} ## requests to center @math{k} (@math{C \times K} matrix, @code{@var{V}(c,k) @geq{} 0}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl}(c) ## @itemx @var{Xu}(c) ## Lower and upper class @math{c} throughput bounds (vector of length @math{C}). ## ## @item @var{Rl}(c) ## @itemx @var{Ru}(c) ## Lower and upper class @math{c} response time bounds (vector of length @math{C}). ## ## @end table ## ## @seealso{qncsbsb} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [Xl Xu Rl Ru] = qncmbsb( varargin ) if ( nargin<2 || nargin>5 ) print_usage(); endif [err N S V m Z] = qncmchkparam( varargin{:} ); isempty(err) || error(err); all(m<=1) || ... error("Multiple-server nodes are not supported"); all(Z == 0 ) || ... error("This function only supports Z=0"); if ( sum(N) == 0 ) # handle trivial case of empty network Xl = Xu = Rl = Ru = zeros(size(S)); else D = S .* V; K = columns(S); ## Equations from T. Kerola, The Composite Bound Method (CBM) for ## Computing Throughput Bounds in Multiple Class Environments}, ## Technical Report CSD-TR-475, Department of Computer Sciences, ## Purdue University, mar 13 1984 (Revisted aug 27, 1984), available ## at http://docs.lib.purdue.edu/cstech/395/ Dc = sum(D,2)'; D_max = max(D,[],2)'; D_min = min(D,[],2)'; Xl = N ./ (Dc .+ (sum(N)-1) .* D_max); Xu = min( 1./D_max, N ./ ((K+sum(N)-1) .* D_min)); Rl = N ./ Xu; Ru = N ./ Xl; endif endfunction %!test %! fail("qncmbsb([],[])", "nonempty"); %! fail("qncmbsb([1 0], [1 2 3])", "2 rows"); %! fail("qncmbsb([1 0], [1 2 3; 4 5 -1])", "nonnegative"); %! fail("qncmbsb([1 2], [1 2 3; 4 5 6], [1 2 3])", "2 x 3"); %! fail("qncmbsb([1 2], [1 2 3; 4 5 6], [1 2 3; 4 5 -1])", "nonnegative"); %! fail("qncmbsb([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1])", "3 elements"); %! fail("qncmbsb([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 2])", "not supported"); %! fail("qncmbsb([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 -1],[1 2 3])", "2 elements"); %! fail("qncmbsb([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 -1],[1 -2])", "nonnegative"); %! fail("qncmbsb([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 -1],[1 0])", "only supports"); %!test %! [Xl Xu Rl Ru] = qncmbsb([0 0], [1 2 3; 1 2 3]); %! assert( all(Xl(:) == 0) ); %! assert( all(Xu(:) == 0) ); %! assert( all(Rl(:) == 0) ); %! assert( all(Ru(:) == 0) ); %!demo %! S = [10 7 5 4; ... %! 5 2 4 6]; %! NN=20; %! Xl = Xu = Rl = Ru = Xmva = Rmva = zeros(NN,2); %! for n=1:NN %! N=[n,10]; %! [a b c d] = qncmbsb(N,S); %! Xl(n,:) = a; Xu(n,:) = b; Rl(n,:) = c; Ru(n,:) = d; %! [U R Q X] = qncmmva(N,S,ones(size(S))); %! Xmva(n,:) = X(:,1)'; Rmva(n,:) = sum(R,2)'; %! endfor %! subplot(2,2,1); %! plot(1:NN,Xl(:,1), 1:NN,Xu(:,1), 1:NN,Xmva(:,1),";MVA;", "linewidth", 2); %! ylim([0, 0.2]); %! title("Class 1 throughput"); legend("boxoff"); %! subplot(2,2,2); %! plot(1:NN,Xl(:,2), 1:NN,Xu(:,2), 1:NN,Xmva(:,2),";MVA;", "linewidth", 2); %! ylim([0, 0.2]); %! title("Class 2 throughput"); legend("boxoff"); %! subplot(2,2,3); %! plot(1:NN,Rl(:,1), 1:NN,Ru(:,1), 1:NN,Rmva(:,1),";MVA;", "linewidth", 2); %! ylim([0, 700]); %! title("Class 1 response time"); legend("location", "northwest"); legend("boxoff"); %! subplot(2,2,4); %! plot(1:NN,Rl(:,2), 1:NN,Ru(:,2), 1:NN,Rmva(:,2),";MVA;", "linewidth", 2); %! ylim([0, 700]); %! title("Class 2 response time"); legend("location", "northwest"); legend("boxoff"); queueing/inst/ctmcexps.m0000644000175000017500000001267713570046274015266 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{L} =} ctmcexps (@var{Q}, @var{t}, @var{p} ) ## @deftypefnx {Function File} {@var{L} =} ctmcexps (@var{Q}, @var{p}) ## ## @cindex Markov chain, continuous time ## @cindex continuous time Markov chain ## @cindex expected sojourn time, CTMC ## @cindex CTMC ## ## With three arguments, compute the expected times @code{@var{L}(i)} ## spent in each state @math{i} during the time interval @math{[0,t]}, ## assuming that the initial occupancy vector is @var{p}. With two ## arguments, compute the expected time @code{@var{L}(i)} spent in each ## transient state @math{i} until absorption. ## ## @strong{Note:} In its current implementation, this function ## requires that an absorbing state is reachable from any ## non-absorbing state of @math{Q}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{Q}(i,j) ## @math{N \times N} infinitesimal generator matrix. @code{@var{Q}(i,j)} ## is the transition rate from state @math{i} to state @math{j}, ## @math{1 @leq{} i, j @leq{} N}, @math{i \neq j}. ## The matrix @var{Q} must also satisfy the ## condition @math{\sum_{j=1}^N Q_{i,j} = 0} for every @math{i=1, @dots{}, N}. ## ## @item @var{t} ## If given, compute the expected sojourn times in @math{[0,t]} ## ## @item @var{p}(i) ## Initial occupancy probability vector; @code{@var{p}(i)} is the ## probability the system is in state @math{i} at time 0, @math{i = 1, ## @dots{}, N} ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{L}(i) ## If this function is called with three arguments, @code{@var{L}(i)} is ## the expected time spent in state @math{i} during the interval ## @math{[0,t]}. If this function is called with two arguments ## @code{@var{L}(i)} is the expected time spent in transient state ## @math{i} until absorption; if state @math{i} is absorbing, ## @code{@var{L}(i)} is zero. ## ## @end table ## ## @seealso{dtmcexps} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function L = ctmcexps( Q, varargin ) persistent epsilon = 10*eps; if ( nargin < 2 || nargin > 3 ) print_usage(); endif [N err] = ctmcchkQ(Q); (N>0) || ... error(err); if ( nargin == 2 ) p = varargin{1}; else t = varargin{1}; p = varargin{2}; endif ( isvector(p) && length(p) == size(Q,1) && all(p>=0) && abs(sum(p)-1.0)= 0 ) || ... error( "t must be >= 0" ); ## F(x) are the transient state occupancy probabilities at time x ## F(x) = p*expm(Q*x) (see function ctmc()). F = @(x) (p*expm(Q*x)); L = quadv(F,0,t); else ## FIXME: deprecate this? ( isvector(t) && abs(t(1)) < epsilon ) || ... error( "t must be a vector, and t(1) must be 0.0" ); t = t(:)'; # make t a row vector ff = @(x,t) (x(:)'*Q+p); fj = @(x,t) (Q); L = lsode( {ff, fj}, zeros(size(p)), t ); endif else # absorbing case ## Identify absorbing states. If there are no absorbing states, ## raise an error. N = rows(Q); tr = find( any( abs(Q) > epsilon, 2 ) ); # non-absorbing states if ( length( tr ) == N ) error( "There are no absorbing states" ); endif QN = Q(tr,tr); pN = p(tr); LN = -pN*inv(QN); L = zeros(1,N); L(tr) = LN; endif endfunction %!test %! Q = [-1 1; 1 -1]; %! L = ctmcexps(Q,10,[1 0]); %! L = ctmcexps(Q,linspace(0,10,100),[1 0]); %!test %! Q = ctmcbd( [1 2 3], [3 2 1] ); %! p0 = [1 0 0 0]; %! t = linspace(0,10,10); %! L1 = L2 = zeros(length(t),4); %! # compute L using the differential equation formulation %! ff = @(x,t) (x(:)'*Q+p0); %! fj = @(x,t) (Q); %! L1 = lsode( {ff, fj}, zeros(size(p0)), t ); %! # compute L using ctmcexps (integral formulation) %! for i=1:length(t) %! L2(i,:) = ctmcexps(Q,t(i),p0); %! endfor %! assert( L1, L2, 1e-5); %!demo %! lambda = 0.5; %! N = 4; %! b = lambda*[1:N-1]; %! d = zeros(size(b)); %! Q = ctmcbd(b,d); %! t = linspace(0,10,100); %! p0 = zeros(1,N); p0(1)=1; %! L = zeros(length(t),N); %! for i=1:length(t) %! L(i,:) = ctmcexps(Q,t(i),p0); %! endfor %! plot( t, L(:,1), ";State 1;", "linewidth", 2, ... %! t, L(:,2), ";State 2;", "linewidth", 2, ... %! t, L(:,3), ";State 3;", "linewidth", 2, ... %! t, L(:,4), ";State 4;", "linewidth", 2 ); %! legend("location","northwest"); legend("boxoff"); %! xlabel("Time"); %! ylabel("Expected sojourn time"); %!demo %! lambda = 0.5; %! N = 4; %! b = lambda*[1:N-1]; %! d = zeros(size(b)); %! Q = ctmcbd(b,d); %! p0 = zeros(1,N); p0(1)=1; %! L = ctmcexps(Q,p0); %! disp(L); queueing/inst/qncmmvabs.m0000644000175000017500000002132413570046274015414 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}, @var{iter_max}) ## ## @cindex Mean Value Analysys (MVA), approximate ## @cindex MVA, approximate ## @cindex closed network, multiple classes ## @cindex multiclass network, closed ## ## Approximate Mean Value Analysis (MVA) for closed, multiclass ## queueing networks with @math{K} service centers and @math{C} ## customer classes. ## ## This implementation uses Bard and Schweitzer approximation. It is ## based on the assumption that the queue length at service center ## @math{k} with population set @math{{\bf N}-{\bf 1}_c} is ## approximated with ## ## @tex ## $$Q_k({\bf N}-{\bf 1}_c) \approx {n-1 \over n} Q_k({\bf N})$$ ## @end tex ## @ifnottex ## @example ## @group ## Q_k(N-1c) ~ (n-1)/n Q_k(N) ## @end group ## @end example ## @end ifnottex ## ## where @math{\bf N} is a valid population mix, @math{{\bf N}-{\bf 1}_c} ## is the population mix @math{\bf N} with one class @math{c} customer ## removed, and @math{n = \sum_c N_c} is the total number of requests. ## ## This implementation works for networks with infinite server (IS) ## and single-server nodes only. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N}(c) ## number of class @math{c} requests in the system (@code{@var{N}(c) @geq{} 0}). ## ## @item @var{S}(c,k) ## mean service time for class @math{c} customers at center @math{k} ## (@code{@var{S}(c,k) @geq{} 0}). ## ## @item @var{V}(c,k) ## average number of visits of class @math{c} requests to center ## @math{k} (@code{@var{V}(c,k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. If @code{@var{m}(k) < 1}, ## then the service center @math{k} is assumed to be a delay center ## (IS). If @code{@var{m}(k) == 1}, service center @math{k} is a ## regular queueing center (FCFS, LCFS-PR or PS) with a single server ## node. If omitted, each service center has a single server. Note ## that multiple server nodes are not supported. ## ## @item @var{Z}(c) ## class @math{c} external delay (@code{@var{Z} @geq{} 0}). Default is 0. ## ## @item @var{tol} ## Stopping tolerance (@code{@var{tol}>0}). The algorithm stops if ## the queue length computed on two subsequent iterations are less than ## @var{tol}. Default is @math{10^{-5}}. ## ## @item @var{iter_max} ## Maximum number of iterations (@code{@var{iter_max}>0}. ## The function aborts if convergenge is not reached within the maximum ## number of iterations. Default is 100. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U}(c,k) ## If @math{k} is a FCFS, LCFS-PR or PS node, then @code{@var{U}(c,k)} ## is the utilization of class @math{c} requests on service center ## @math{k}. If @math{k} is an IS node, then @code{@var{U}(c,k)} is the ## class @math{c} @emph{traffic intensity} at device @math{k}, ## defined as @code{@var{U}(c,k) = @var{X}(c)*@var{S}(c,k)} ## ## @item @var{R}(c,k) ## response time of class @math{c} requests at service center @math{k}. ## ## @item @var{Q}(c,k) ## average number of class @math{c} requests at service center @math{k}. ## ## @item @var{X}(c,k) ## class @math{c} throughput at service center @math{k}. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Y. Bard, @cite{Some Extensions to Multiclass Queueing Network Analysis}, ## proc. 4th Int. Symp. on Modelling and Performance Evaluation of ## Computer Systems, Feb 1979, pp. 51--62. ## ## @item ## P. Schweitzer, @cite{Approximate Analysis of Multiclass Closed ## Networks of Queues}, Proc. Int. Conf. on Stochastic Control and ## Optimization, jun 1979, pp. 25--29. ## @end itemize ## ## This implementation is based on Edward D. Lazowska, John Zahorjan, G. ## Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System ## Performance: Computer System Analysis Using Queueing Network Models}, ## Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 7.4.2.2 ("Approximate Solution ## Techniques"). This implementation is slightly different from the one ## described above, as it computes the average response times @math{R} ## instead of the residence times. ## ## @seealso{qncmmva} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncmmvabs( N, S, V, m, Z, tol, iter_max ) if ( nargin < 3 || nargin > 7 ) print_usage(); endif isvector(N) && all( N>=0 ) || ... error( "N must be a vector of positive integers" ); N = N(:)'; # make N a row vector C = length(N); ## Number of classes K = columns(S); ## Number of service centers size(S) == [C,K] || ... error( "S size mismatch" ); size(V) == [C,K] || ... error( "V size mismatch" ); if ( nargin < 4 || isempty(m) ) m = ones(1,K); else isvector(m) || ... error( "m must be a vector"); m = m(:)'; # make m a row vector ( length(m) == K && all( m <= 1 ) ) || ... error( "m must be <= 1 and have %d elements", K ); endif if ( nargin < 5 || isempty(Z) ) Z = zeros(1,C); else isvector(Z) || ... error( "Z must be a vector" ); Z = Z(:)'; # make Z a row vector ( length(Z) == C && all(Z >= 0 ) ) || ... error( "Z must be >= 0 and have %d elements", C ); endif if ( nargin < 6 || isempty(tol) ) tol = 1e-5; endif if ( nargin < 7 || isempty(iter_max) ) iter_max = 100; endif ## Check consistency of parameters all(S(:) >= 0) || ... error( "S contains negative values" ); all(V(:) >= 0) || ... error( "V contains negative values" ); ## Initialize results R = zeros( C, K ); Xc = zeros( 1, C ); # Xc(c) is the class c throughput Q = zeros( C, K ); D = V .* S; ## Initialization of temporaries iter = 0; A = zeros( C, K ); Q = diag(N/K)*ones(C,K); # Q(c,k) = N(c) / K i_single=find(m==1); i_multi=find(m<1); ## Main loop N(N==0)=1; do iter++; Qold = Q; ## A(c,k) = (N(c)-1)/N(c) * Q(c,k) + sum_{j=1, j|=c}^C Qold(j,k) A = diag( (N-1) ./ N )*Q + ( (1 - eye(C)) * Qold ); ## R(c,k) = ## S(c,k) is k is a delay center ## S(c,k) * (1+A(c,k)) if k is a queueing center; R(:,i_multi) = S(:,i_multi); R(:,i_single) = S(:,i_single) .* ( 1 + A(:,i_single)); ## X(c) = N(c) / (sum_k R(c,k) * V(c,k)) Xc = N ./ (Z .+ sum(R.*V,2)'); ## Q(c,k) = X(c) * R(c,k) * V(c,k) Q = (diag(Xc)*R).*V; ## err = norm(Q-Qold); err = norm((Q-Qold)./Qold, "inf"); until (erriter_max); if ( iter > iter_max ) warning( "qncmmvabs(): Convergence not reached after %d iterations", iter_max ); endif X = diag(Xc)*V; # X(c,k) = X(c) * V(c,k) U = diag(Xc)*D; # U(c,k) = X(c) * D(c,k) # U(N==0,:) = R(N==0,:) = Q(N==0,:) = X(N==0,:) = 0; endfunction %!test %! S = [ 1 3 3; 2 4 3]; %! V = [ 1 1 3; 1 1 3]; %! N = [ 1 1 ]; %! m = [1 ; 1 ]; %! Z = [2 2 2]; %! fail( "qncmmvabs(N,S,V,m,Z)", "m must be" ); %! m = [1 ; 1 ; 1]; %! fail( "qncmmvabs(N,S,V,m,Z)", "Z must be" ); %!test %! S = [ 1 3; 2 4]; %! V = [ 1 1; 1 1]; %! N = [ 1 1 ]; %! m = ones(1,2); %! [U R Q X] = qncmmvabs(N,S,V,m); %! assert( Q, [ .192 .808; .248 .752 ], 1e-3 ); %! Xc = ( X(:,1)./V(:,1) )'; %! assert( Xc, [ .154 .104 ], 1e-3 ); %! # Compute the (overall) class-c system response time %! R_c = N ./ Xc; %! assert( R_c, [ 6.508 9.614 ], 5e-3 ); %!demo %! S = [ 1, 1, 1, 1; 2, 1, 3, 1; 4, 2, 3, 3 ]; %! V = ones(3,4); %! N = [10 5 1]; %! m = [1 0 1 1]; %! [U R Q X] = qncmmvabs(N,S,V,m); queueing/inst/ctmcmtta.m0000644000175000017500000000756613570046274015255 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{t} =} ctmcmtta (@var{Q}, @var{p}) ## ## @cindex Markov chain, continuous time ## @cindex continuous time Markov chain ## @cindex CTMC ## @cindex mean time to absorption, CTMC ## ## Compute the Mean-Time to Absorption (MTTA) of the CTMC described by ## the infinitesimal generator matrix @var{Q}, starting from initial ## occupancy probabilities @var{p}. If there are no absorbing states, this ## function fails with an error. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{Q}(i,j) ## @math{N \times N} infinitesimal generator matrix. @code{@var{Q}(i,j)} ## is the transition rate from state @math{i} to state @math{j}, @math{i ## \neq j}. The matrix @var{Q} must satisfy the condition ## @math{\sum_{j=1}^N Q_{i,j} = 0} ## ## @item @var{p}(i) ## probability that the system is in state @math{i} ## at time 0, for each @math{i=1, @dots{}, N} ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{t} ## Mean time to absorption of the process represented by matrix @var{Q}. ## If there are no absorbing states, this function fails. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and ## K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and ## Performance Evaluation with Computer Science Applications}, Wiley, ## 1998. ## @end itemize ## ## @seealso{ctmcexps} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function t = ctmcmtta( Q, p ) persistent epsilon = 10*eps; if ( nargin != 2 ) print_usage(); endif [N err] = ctmcchkQ(Q); (N>0) || ... error(err); ( isvector(p) && length(p) == N && all(p>=0) && abs(sum(p)-1.0). ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## ## @cindex bounds, geometric ## @cindex geometric bounds ## @cindex closed network ## ## Compute Geometric Bounds (GB) on system throughput, system response ## time and server queue lenghts for closed, single-class networks ## with @math{K} service centers and @math{N} requests. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N} ## number of requests in the system (scalar, @code{@var{N} > 0}). ## ## @item @var{D}(k) ## service demand of service center @math{k} (vector of length ## @math{K}, @code{@var{D}(k) @geq{} 0}). ## ## @item @var{S}(k) ## mean service time at center @math{k} (vector of length @math{K}, ## @code{@var{S}(k) @geq{} 0}). ## ## @item @var{V}(k) ## visit ratio to center @math{k} ## (vector of length @math{K}, @code{@var{V}(k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k}. This function only supports ## @math{M/M/1} queues, therefore @var{m} must be ## @code{ones(size(S))}. ## ## @item @var{Z} ## external delay (think time, @code{@var{Z} @geq{} 0}, scalar). Default is 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl} ## @itemx @var{Xu} ## Lower and upper bound on the system throughput. If @code{@var{Z}>0}, ## these bounds are computed using @emph{Geometric Square-root Bounds} ## (GSB). If @code{@var{Z}==0}, these bounds are computed using @emph{Geometric Bounds} (GB) ## ## @item @var{Rl} ## @itemx @var{Ru} ## Lower and upper bound on the system response time. These bounds ## are derived from @var{Xl} and @var{Xu} using Little's Law: ## @code{@var{Rl} = @var{N} / @var{Xu} - @var{Z}}, ## @code{@var{Ru} = @var{N} / @var{Xl} - @var{Z}} ## ## @item @var{Ql}(k) ## @itemx @var{Qu}(k) ## lower and upper bounds of center @math{K} queue length. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Casale, R. R. Muntz, G. Serazzi, @cite{Geometric Bounds: a ## Non-Iterative Analysis Technique for Closed Queueing Networks}, IEEE ## Transactions on Computers, 57(6):780-794, June ## 2008. @uref{http://doi.ieeecomputersociety.org/10.1109/TC.2008.37, ## 10.1109/TC.2008.37} ## @end itemize ## ## In this implementation we set @math{X^+} and @math{X^-} as the upper ## and lower Asymptotic Bounds as computed by the @command{qncsab} ## function, respectively. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [X_lower X_upper R_upper R_lower Q_lower Q_upper] = qncsgb( varargin ) ## This implementation is based on the paper: G.Casale, R.R.Muntz, ## G.Serazzi. Geometric Bounds: a Noniterative Analysis Technique for ## Closed Queueing Networks IEEE Transactions on Computers, ## 57(6):780-794, Jun 2008. ## http://doi.ieeecomputersociety.org/10.1109/TC.2008.37 ## The original paper uses the symbol "L" instead of "D" to denote the ## loadings of service centers. In this function we adopt the same ## notation as the paper. if ( nargin < 2 || ( nargin > 5 && nargin != 7 ) ) print_usage(); endif [err N S V m Z] = qncschkparam( varargin{:} ); isempty(err) || error(err); ## This function requires N>0 N > 0 || ... error( "N must be > 0" ); all(m==1) || ... error("this function only supports single server nodes"); L = S .* V; L_tot = sum(L); L_max = max(L); M = length(L); if ( nargin < 6 ) [X_minus X_plus] = qncsaba(N,L,ones(size(L)),m,Z); else X_minus = varargin{6}; X_plus = varargin{7}; endif ##[X_minus X_plus] = [0 1/L_max]; [Q_lower Q_upper] = __compute_Q( N, L, Z, X_plus, X_minus); [Q_lower_Nm1 Q_upper_Nm1] = __compute_Q( N-1, L, Z, X_plus, X_minus); if ( Z > 0 ) ## Use Geometric Square-root Bounds (GSB) i = find(L=0) ) || error( "N is not valid" ); L_tot = sum(L); L_max = max(L); M = length(L); m_max = sum( L == L_max ); y = Y = zeros(1,M); ## first, handle the case of servers with loading less than the ## maximum that is, L(i) < L_max i=find(L 0" ); %! fail( "qncsgb( -1, [1 2])", "nonnegative" ); %! fail( "qncsgb( 1, [1 2],1,[1 -1])", "single server" ); %!# shared test function %!function test_gb( D, expected, Z=0 ) %! for i=1:rows(expected) %! N = expected(i,1); %! [X_lower X_upper Q_lower Q_upper] = qncsgb(N,D,1,1,Z); %! X_exp_lower = expected(i,2); %! X_exp_upper = expected(i,3); %! assert( [N X_lower X_upper], [N X_exp_lower X_exp_upper], 1e-4 ) %! endfor %!xtest %! # table IV %! D = [ 0.1 0.1 0.09 0.08 ]; %! # N X_lower X_upper %! expected = [ 2 4.3040 4.3174; ... %! 5 6.6859 6.7524; ... %! 10 8.1521 8.2690; ... %! 20 9.0947 9.2431; ... %! 80 9.8233 9.8765 ]; %! test_gb(D, expected); %!xtest %! # table V %! D = [ 0.1 0.1 0.09 0.08 ]; %! Z = 1; %! # N X_lower X_upper %! expected = [ 2 1.4319 1.5195; ... %! 5 3.3432 3.5582; ... %! 10 5.7569 6.1410; ... %! 20 8.0856 8.6467; ... %! 80 9.7147 9.8594]; %! test_gb(D, expected, Z); %!test %! P = [0 0.3 0.7; 1 0 0; 1 0 0]; %! S = [1 0.6 0.2]; %! m = ones(1,3); %! V = qncsvisits(P); %! Z = 2; %! Nmax = 20; %! tol = 1e-5; # compensate for numerical errors %! ## Test case with Z>0 %! for n=1:Nmax %! [X_gb_lower X_gb_upper NC NC Q_gb_lower Q_gb_upper] = qncsgb(n, S.*V, 1, 1, Z); %! [U R Q X] = qnclosed( n, S, V, m, Z ); %! X_mva = X(1)/V(1); %! assert( X_gb_lower <= X_mva+tol ); %! assert( X_gb_upper >= X_mva-tol ); %! assert( Q_gb_lower <= Q+tol ); # compensate for numerical errors %! assert( Q_gb_upper >= Q-tol ); # compensate for numerical errors %! endfor %!test %! P = [0 0.3 0.7; 1 0 0; 1 0 0]; %! S = [1 0.6 0.2]; %! V = qncsvisits(P); %! Nmax = 20; %! tol = 1e-5; # compensate for numerical errors %! %! ## Test case with Z=0 %! for n=1:Nmax %! [X_gb_lower X_gb_upper NC NC Q_gb_lower Q_gb_upper] = qncsgb(n, S.*V); %! [U R Q X] = qnclosed( n, S, V ); %! X_mva = X(1)/V(1); %! assert( X_gb_lower <= X_mva+tol ); %! assert( X_gb_upper >= X_mva-tol ); %! assert( Q_gb_lower <= Q+tol ); %! assert( Q_gb_upper >= Q-tol ); %! endfor queueing/inst/qsmmm.m0000644000175000017500000001357513574752117014574 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pm}] =} qsmmm (@var{lambda}, @var{mu}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pm}] =} qsmmm (@var{lambda}, @var{mu}, @var{m}) ## @deftypefnx {Function File} {@var{pk} =} qsmmm (@var{lambda}, @var{mu}, @var{m}, @var{k}) ## ## @cindex @math{M/M/m} system ## ## Compute utilization, response time, average number of requests in ## service and throughput for a @math{M/M/m} queue, a queueing system ## with @math{m} identical servers connected to a single FCFS ## queue. ## ## @tex ## The steady-state probability @math{\pi_k} that there are @math{k} ## requests in the system, @math{k \geq 0}, can be computed as: ## ## $$ ## \pi_k = \cases{ \displaystyle{\pi_0 { ( m\rho )^k \over k!}} & $0 \leq k \leq m$;\cr\cr ## \displaystyle{\pi_0 { \rho^k m^m \over m!}} & $k>m$.\cr ## } ## $$ ## ## where @math{\rho = \lambda/(m\mu)} is the individual server utilization. ## The steady-state probability @math{\pi_0} that there are no jobs in the ## system is: ## ## $$ ## \pi_0 = \left[ \sum_{k=0}^{m-1} { (m\rho)^k \over k! } + { (m\rho)^m \over m!} {1 \over 1-\rho} \right]^{-1} ## $$ ## ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate (@code{@var{lambda}>0}). ## ## @item @var{mu} ## Service rate (@code{@var{mu}>@var{lambda}}). ## ## @item @var{m} ## Number of servers (@code{@var{m} @geq{} 1}). ## Default is @code{@var{m}=1}. ## ## @item @var{k} ## Number of requests in the system (@code{@var{k} @geq{} 0}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Service center utilization, @math{U = \lambda / (m \mu)}. ## ## @item @var{R} ## Service center mean response time ## ## @item @var{Q} ## Average number of requests in the system ## ## @item @var{X} ## Service center throughput. If the system is ergodic, ## we will always have @code{@var{X} = @var{lambda}} ## ## @item @var{p0} ## Steady-state probability that there are 0 requests in the system ## ## @item @var{pm} ## Steady-state probability that an arriving request has to wait in the ## queue ## ## @item @var{pk} ## Steady-state probability that there are @var{k} requests in the ## system (including the one being served). ## ## @end table ## ## If this function is called with less than four parameters, ## @var{lambda}, @var{mu} and @var{m} can be vectors of the same size. In this ## case, the results will be vectors as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks ## and Markov Chains: Modeling and Performance Evaluation with Computer ## Science Applications}, Wiley, 1998, Section 6.5 ## @end itemize ## ## @seealso{erlangc,qsmm1,qsmminf,qsmmmk} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U_or_pk R Q X p0 pm] = qsmmm( lambda, mu, m, k ) if ( nargin < 2 || nargin > 4 ) print_usage(); endif if ( nargin == 2 ) m = 1; else ( isnumeric(lambda) && isnumeric(mu) && isnumeric(m) ) || ... error( "the parameters must be numeric vectors" ); endif [err lambda mu m] = common_size( lambda, mu, m ); if ( err ) error( "parameters are not of common size" ); endif lambda = lambda(:)'; mu = mu(:)'; m = m(:)'; all( m>0 ) || error( "m must be >0" ); all( lambda>0 ) || error( "lambda must be >0" ); rho = lambda ./ (m .* mu ); all( rho < 1 ) || error( "Processing capacity exceeded" ); for i=1:length(lambda) p0(i) = 1 / ( ... sumexpn( m(i)*rho(i), m(i)-1 ) + ... expn(m(i)*rho(i), m(i))/(1-rho(i)) ... ); endfor if (nargin < 4) X = lambda; U_or_pk = rho; pm = erlangc(lambda ./ mu, m); Q = m .* rho .+ rho ./ (1-rho) .* pm; R = Q ./ X; else (length(lambda) == 1) || error("lambda must be a scalar if this function is called with four arguments"); isvector(k) || error("k must be a vector"); all(k>=0) || error("k must be >= 0"); U_or_pk = 0*k; for idx=1:length(k) if (k(idx) <= m) U_or_pk(idx) = p0 * expn(m * rho, k(idx)); else U_or_pk(idx) = p0 * expn(m * rho, m) * (rho ^ (k(idx)-m)); endif endfor endif endfunction %!demo %! # This is figure 6.4 on p. 220 Bolch et al. %! rho = 0.9; %! ntics = 21; %! lambda = 0.9; %! m = linspace(1,ntics,ntics); %! mu = lambda./(rho .* m); %! [U R Q X] = qsmmm(lambda, mu, m); %! qlen = X.*(R-1./mu); %! plot(m,Q,"o",qlen,"*"); %! axis([0,ntics,0,25]); %! legend("Jobs in the system","Queue Length","location","northwest"); %! legend("boxoff"); %! xlabel("Number of servers (m)"); %! title("M/M/m system, \\lambda = 0.9, \\mu = 0.9"); %!demo %! ## Given a M/M/m queue, compute the steady-state probability pk of %! ## having k jobs in the systen. %! lambda = 0.5; %! mu = 0.15; %! m = 5; %! k = 0:10; %! pk = qsmmm(lambda, mu, m, k); %! plot(k, pk, "-o", "linewidth", 2); %! xlabel("N. of jobs (k)"); %! ylabel("P_k"); %! title(sprintf("M/M/%d system, \\lambda = %g, \\mu = %g", m, lambda, mu)); queueing/inst/qsammm.m0000644000175000017500000000651513570046274014725 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qsammm (@var{lambda}, @var{mu}) ## ## @cindex asymmetric @math{M/M/m} system ## ## Compute @emph{approximate} utilization, response time, average ## number of requests in service and throughput for an asymmetric ## @math{M/M/m} queue. In this type of system there are @math{m} different ## servers connected to a single queue. Each server has its own ## (possibly different) service rate. If there is more than one server ## available, requests are routed to a randomly-chosen one. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate (@code{@var{lambda}>0}) ## ## @item @var{mu} ## @code{@var{mu}(i)} is the service rate of server ## @math{i}, @math{1 @leq{} i @leq{} m}. ## The system must be ergodic (@code{@var{lambda} < sum(@var{mu})}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Approximate service center utilization, ## @math{U = \lambda / ( \sum_{i=1}^m \mu_i )}. ## ## @item @var{R} ## Approximate service center response time ## ## @item @var{Q} ## Approximate number of requests in the system ## ## @item @var{X} ## Approximate system throughput. If the system is ergodic, ## @code{@var{X} = @var{lambda}} ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks ## and Markov Chains: Modeling and Performance Evaluation with Computer ## Science Applications}, Wiley, 1998 ## @end itemize ## ## @seealso{qsmmm} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X p0 pm] = qsammm( lambda, mu ) if ( nargin != 2 ) print_usage(); endif isscalar(lambda) || ... error( "lambda must be scalar" ); isvector(mu) || ... error( "mu must be a vector" ); m = length(mu); # number of servers all( lambda < sum(mu) ) || ... error( "Processing capacity exceeded" ); X = lambda; U = rho = lambda / sum(mu); Q = p0 = 0; #{ k=[0:m-1]; p0 = 1 / ( ... sum( (m*rho).^k ./ factorial(k)) + ... (m*rho)^m / (factorial(m)*(1-rho)) ... ); pm = (m*rho)^m/(factorial(m)*(1-rho))*p0; #} p0 = 1 / ( ... sumexpn( m*rho, m-1 ) + ... expn(m*rho, m) / (1-rho) ... ); pm = expn(m*rho, m)*p0/(1-rho); Q = m*rho+rho / (1-rho) * pm; R = Q / X; endfunction %!test %! [U R Q X] = qsammm( 73,[10,15,20,20,25] ); %! assert( U, 0.81, 1e-2 ); %! assert( Q, 6.5278, 1e-4 ); queueing/inst/private/0000755000175000017500000000000013635377242014723 5ustar morenomorenoqueueing/inst/private/scc.m0000644000175000017500000000402013635373160015640 0ustar morenomoreno## Copyright (C) 2018, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{s} =} scc (@var{G}) ## ## Compute the Strongly Connected Components (SCC) of a graph with ## adjacency matrix @var{G}. Any positive value in @var{G} denotes ## an edge; zero or negative values denote the absence of an edge. ## ## @end deftypefn function s = scc( G ) ## It might be possible to use a different algorithm for SCC (e.g., ## http://pmtksupport.googlecode.com/svn/trunk/gaimc1.0-graphAlgo/scomponents.m assert(issquare(G)); N = rows(G); GF = (G>0); GB = (G'>0); s = zeros(N,1); c=1; for n=1:N if (s(n) == 0) fw = __dfs(GF,n); bw = __dfs(GB,n); r = (fw & bw); s(r) = c++; endif endfor endfunction ## This is essentially the same code from qncmvisits.m function v = __dfs(G, s) assert( issquare(G) ); N = rows(G); v = stack = zeros(1,N); ## v(i) == 1 iff node i has been visited q = 1; # first empty slot in queue stack(q++) = s; v(s) = 1; while( q>1 ) n = stack(--q); ## explore neighbors of n: all f in G(n,:) such that v(f) == 0 ## The following instruction is equivalent to: ## for f=find(G(n,:)) ## if ( v(f) == 0 ) for f = find ( G(n,:) & (v==0) ) stack(q++) = f; v(f) = 1; endfor endwhile endfunction queueing/inst/private/expn.m0000644000175000017500000000223613572237526016056 0ustar morenomoreno## Copyright (C) 2012, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{r} =} expn (@var{a}, @var{n}) ## ## Compute @code{r = a^n / n!}, with @math{a>0} and @math{n @geq{} 0}. ## ## @end deftypefn function r = expn( a, n ) (isscalar(n) && (n>=0)) || error("n must be nonnegative"); (isscalar(a) && (a>0)) || error("a must be positive"); r = prod( a./(1:n) ); # for n=0, prod([]) returns 1 which is correct endfunction queueing/inst/private/qnoschkparam.m0000644000175000017500000000437513570046274017575 0ustar morenomoreno## Copyright (C) 2012, 2013, 2014 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{err} @var{lambda} @var{S} @var{V} @var{m}] = } qnoschkparam( lambda, S, V, m ) ## ## Validate input parameters for open, single class networks. ## @var{err} is the empty string on success, or a suitable error message ## string on failure. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [err lambda S V m] = qnoschkparam( varargin ) err = ""; assert( nargin >= 2 ); lambda = varargin{1}; S = varargin{2}; [V m] = deal(0); if ! ( isscalar(lambda) && lambda > 0 ) err = "lambda must be a positive scalar"; return; endif if ! ( isnumeric(S) && isvector(S) && length(S)>0 ) err = "S must be a nonempty vector"; return; endif if ( any(S<0) ) err = "S must contain nonnegative values"; return; endif S = S(:)'; if ( nargin < 3 ) V = ones(size(S)); else V = varargin{3}; if ! ( isnumeric(V) && isvector(V) ) err = "V must be a vector"; return; endif if ( any(V<0) ) err = "V must contain nonnegative values"; return; endif V = V(:)'; endif if ( nargin < 4 ) m = ones(size(S)); else m = varargin{4}; if ! ( isnumeric(m) && isvector(m) ) err = "m must be a vector"; return; endif m = m(:)'; endif [er S V m] = common_size(S, V, m); if (er != 0 ) err = "S, V and m are of incompatible size"; return; endif endfunction queueing/inst/private/qncmchkparam.m0000644000175000017500000000634213570046274017547 0ustar morenomoreno## Copyright (C) 2012, 2013 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{err} @var{Nout} @var{Sout} @var{Vout} @var{mout} @var{Zout}] = } qncmchkparam( N, S ) ## @deftypefnx {Function File} {[@var{err} @var{Nout} @var{Sout} @var{Vout} @var{mout} @var{Zout}] = } qncmchkparam( N, S, V ) ## @deftypefnx {Function File} {[@var{err} @var{Nout} @var{Sout} @var{Vout} @var{mout} @var{Zout}] = } qncmchkparam( N, S, V, m ) ## @deftypefnx {Function File} {[@var{err} @var{Nout} @var{Sout} @var{Vout} @var{mout} @var{Zout}] = } qncmchkparam( N, S, V, m, Z ) ## ## Validate input parameters for closed, multiclass network. ## @var{err} is the empty string on success, or a suitable error message ## string on failure. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [err Nout Sout Vout mout Zout] = qncmchkparam( N, S, V, m, Z ) err = ""; [Nout Sout Vout mout Zout] = deal(0); if ( nargin < 2 || nargin > 5 ) err = "Wrong number of parameters (min 2, max 5)"; return; endif if ! ( isnumeric(N) && isvector(N) && length(N)>0 ) err = "N must be a nonempty vector"; return; endif if ! ( all(N>=0) && all( fix(N) == N ) ) err = "N must contain nonnegative integers"; return; endif Nout = N(:)'; C = length(Nout); ## Number of classes if ! ( isnumeric(S) && ismatrix(S) && ndims(S) == 2 && rows(S) == C ) err = sprintf("S must be a 2-dimensional matrix with %d rows",C); return; endif if ( any(S(:)<0) ) err = "S must contain nonnegative values"; return; endif Sout = S; K = columns(Sout); if ( nargin < 3 ) Vout = ones(size(Sout)); else if ! ( isnumeric(V) && ismatrix(V) && ndims(V) == 2 && rows(V) == C && columns(V) == K ) err = sprintf("V must be a %d x %d matrix", C, K ); return; endif if ( any(V(:)<0) ) err = "V must contain nonnegative values"; return; endif Vout = V; endif if ( nargin < 4 ) mout = ones(1,K); else if ! ( isnumeric(m) && isvector(m) && length(m) == K ) err = sprintf("m must be a vector with %d elements", K ); return; endif mout = m(:)'; endif if ( nargin < 5 ) Zout = zeros(1,C); else if ! ( isnumeric(Z) && isvector(Z) && length(Z) == C ) err = sprintf("Z must be a vector with %d elements", C); return; endif if ( any(Z<0) ) err = "Z must contain nonnegative values"; return; endif Zout = Z(:)'; endif endfunction queueing/inst/private/sumexpn.m0000644000175000017500000000505013570046274016574 0ustar morenomoreno## Copyright (C) 2012, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{r} =} sumexpn (@var{a}, @var{n}) ## ## Compute the sum: ## ## @iftex ## @tex ## $$ S(a,n) = \sum_{k=0}^n {a^k \over k!} $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## n ## __ ## \ a^k ## S(a,n) = > ----- ## /__ k! ## k=0 ## @end example ## @end ifnottex ## ## with @math{a>0} and @math{n @geq{} 0}. ## ## @end deftypefn function r = sumexpn( a, n ) n>=0 || ... error("n must be nonnegative"); a>0 || ... error("a must be positive"); #{ A direct calculation of the summation yields the expression: r = sum(cumprod([1 a./(1:n)])); However, we can apply an approach similar to Horner's rule and rewrite a^0 a^1 a^2 a^n --- + --- + --- + ... + --- 0! 1! 2! n! as a / a / a / / a \ \\\ 1 + - | 1 + - | 1 + - | 1 + ... | 1 + - | ... ||| 1 \ 2 \ 3 \ \ n / /// from which we can use the following iterative code that is numerically more stable: #} r = 1; for k=n:-1:1 r = (1+(a/k)*r); endfor endfunction %!test %! a = 0.8; %! n = 0; %! assert( sumexpn(a,n), sum(a.^(0:n) ./ factorial(0:n)), 1e-6 ); %!test %! a = 1.2; %! n = 6; %! assert( sumexpn(a,n), sum(a.^(0:n) ./ factorial(0:n)), 1e-6 ); %!test %! a = 18; %! assert( sumexpn(a,0), 1, 1e-6 ); %!test %! a = 18; %! assert( sumexpn(a,1), 1 + a, 1e-6 ); %!test %! a = 1.75; %! assert( sumexpn(a, 2), 1 + a + (a^2)/2, 1e-6 ); %!demo %! function r = sumexpn_direct(a, n) %! r = sum(cumprod([1 a./(1:n)])); %! endfunction %! %! a = 0.1:0.05:30; %! n = 20; %! d = zeros(size(a)); %! for idx=1:length(a) %! d(idx) = sumexpn_direct(a(idx),n) - sumexpn(a(idx),n); %! endfor %! plot(a,d, ";sumexpn_direct - sumexpn;"); queueing/inst/private/qncschkparam.m0000644000175000017500000000501513570046274017551 0ustar morenomoreno## Copyright (C) 2012, 2013, 2014 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{err} @var{Nout} @var{Sout} @var{Vout} @var{mout} @var{Zout}] = } qncschkparam( N, S, V, m, Z ) ## ## Validate input parameters for closed, single class networks. ## @var{err} is the empty string on success, or a suitable error message ## string on failure. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [err Nout Sout Vout mout Zout] = qncschkparam( N, S, V, m, Z ) err = ""; [Nout Sout Vout mout Zout] = deal(0); if ( nargin < 2 || nargin > 5) err = "Wrong number of parameters (min 2, max 5)"; return; endif if ! ( isscalar(N) && N>=0 && N == fix(N) ) err = "N must be a nonnegative integer"; return; endif Nout = N; if ! ( isnumeric(S) && isvector(S) && length(S)>0 ) err = "S must be a nonempty vector"; return; endif if ( any(S<0) ) err = "S must contain nonnegative values"; return; endif Sout = S(:)'; if ( nargin < 3 ) Vout = ones(size(Sout)); else if ! ( isnumeric(V) && isvector(V) ) err = "V must be a vector"; return; endif if ( any(V<0) ) err = "V must contain nonnegative values"; return; endif Vout = V(:)'; endif if ( nargin < 4 ) mout = ones(size(Sout)); else if ! ( isnumeric(V) && isvector(m) ) err = "m must be a vector"; return; endif mout = m(:)'; endif [er Sout Vout mout] = common_size(Sout, Vout, mout); if (er != 0 ) err = "S, V and m are of incompatible size"; return; endif if ( nargin < 5 ) Zout = 0; else if ! ( isscalar(Z) && Z >= 0 ) err = "Z must be a nonnegative scalar"; return; endif Zout = Z; endif endfunction queueing/inst/private/qnomchkparam.m0000644000175000017500000000474113570046274017564 0ustar morenomoreno## Copyright (C) 2012, 2013, 2014 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{err} @var{lambda} @var{S} @var{V} @var{m} @var{Z}] = } qnomchkparam( lambda, S, ... ) ## ## Validate input parameters for open, multiclass network. ## @var{err} is the empty string on success, or a suitable error message ## string on failure. ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [err lambda S V m] = qnomchkparam( varargin ) err = ""; assert( nargin >= 2 ); lambda = varargin{1}; S = varargin{2}; [V m] = deal(0); if ! ( isnumeric(lambda) && isvector(lambda) && length(lambda)>0 ) err = "lambda must be a nonempty vector"; return; endif if ( any(lambda<0) ) err = "lambda must contain nonnegative values"; return; endif lambda = lambda(:)'; C = length(lambda); ## Number of classes if ! ( isnumeric(S) && ismatrix(S) && ndims(S) == 2 && rows(S) == C ) err = sprintf("S must be a 2-dimensional matrix with %d rows",C); return; endif if ( any(S(:)<0) ) err = "S must contain nonnegative values"; return; endif K = columns(S); if ( nargin < 3 ) V = ones(size(S)); else V = varargin{3}; if ! ( isnumeric(V) && ismatrix(V) & ndims(V) == 2 & rows(V) == C & columns(V) == K ) err = sprintf("V must be a %d x %d matrix", C, K ); return; endif if ( any(V(:)<0) ) err = "V must contain nonnegative values"; return; endif endif if ( nargin < 4 ) m = ones(1,K); else m = varargin{4}; if ! ( isnumeric(m) && isvector(m) && length(m) == K ) err = sprintf("m must be a vector with %d elements", K ); return; endif m = m(:)'; endif endfunction queueing/inst/qncmmvaap.m0000644000175000017500000000414513570046274015412 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvaap (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvaap (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}) ## @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}, @var{iter_max}) ## ## @cindex Mean Value Analysys (MVA), approximate ## @cindex MVA, approximate ## @cindex closed network, multiple classes ## @cindex multiclass network, closed ## ## This function is deprecated. Plase use @code{qncmmvabs} instead. ## ## @seealso{qncmmvabs} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U R Q X] = qncmmvaap( N, S, V, m, Z, tol, iter_max ) persistent warned = false; if (!warned) warned = true; warning("qn:deprecated-function", "qncmmvaap is deprecated. Please use qncmmvabs instead"); endif [U R Q X] = qncmmvabs( varargin{:} ); endfunction queueing/inst/qnosvisits.m0000644000175000017500000000563713570046274015660 0ustar morenomoreno## Copyright (C) 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{V} =} qnosvisits (@var{P}, @var{lambda}) ## ## Compute the average number of visits to the service centers of a single ## class open Queueing Network with @math{K} service centers. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## is the probability that a request which completed service at center ## @math{i} is routed to center @math{j} (@math{K \times K} matrix). ## ## @item @var{lambda}(k) ## external arrival rate to center @math{k}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{V}(k) ## average number of visits to server @math{k}. ## ## @end table ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function V = qnosvisits( P, lambda ) if ( nargin != 2 ) print_usage(); endif issquare(P) || ... error("P must be a square matrix"); K = rows(P); all(P(:)>=0) && all( sum(P,2)<=1+1e-5 ) || ... error( "invalid transition probability matrix P" ); ( isvector(lambda) && length(lambda) == K ) || ... error( "lambda must be a vector with %d elements", K ); all( lambda>= 0 ) || ... error( "lambda contains negative values" ); lambda = lambda(:)'; V = zeros(size(P)); A = eye(K)-P; b = lambda / sum(lambda); V = b/A; ## Make sure that no negative values appear (sometimes, numerical ## errors produce tiny negative values instead of zeros) V = max(0,V); endfunction %!test %! fail( "qnosvisits([0 .5; .5 0],[0 -1])", "contains negative" ); %! fail( "qnosvisits([1 1 1; 1 1 1], [1 1])", "square" ); %!test %! %! ## Open, single class network %! %! P = [0 0.2 0.5; 1 0 0; 1 0 0]; %! lambda = [ 0.1 0.3 0.2 ]; %! V = qnosvisits(P,lambda); %! assert( V*P+lambda/sum(lambda),V,1e-5 ); %!demo %! p = 0.3; %! lambda = 1.2 %! P = [0 0.3 0.5; ... %! 1 0 0 ; ... %! 1 0 0 ]; %! V = qnosvisits(P,[1.2 0 0]) %!demo %! P = [ 0 0.4 0.6 0; ... %! 0.2 0 0.2 0.6; ... %! 0 0 0 1; ... %! 0 0 0 0 ]; %! lambda = [0.1 0 0 0.3]; %! V = qnosvisits(P,lambda); %! S = [2 1 2 1.8]; %! m = [3 1 1 2]; %! [U R Q X] = qnos( sum(lambda), S, V, m ) queueing/inst/dtmcfpt.m0000644000175000017500000001021713570046274015065 0ustar morenomoreno## Copyright (C) 2011, 2012, 2016, 2018 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {@var{M} =} dtmcfpt (@var{P}) ## ## @cindex first passage times ## @cindex mean recurrence times ## @cindex discrete time Markov chain ## @cindex Markov chain, discrete time ## @cindex DTMC ## ## Compute mean first passage times and mean recurrence times ## for an irreducible discrete-time Markov chain over the state space ## @math{@{1, @dots{}, N@}}. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{P}(i,j) ## transition probability from state @math{i} to state @math{j}. ## @var{P} must be an irreducible stochastic matrix, which means that ## the sum of each row must be 1 (@math{\sum_{j=1}^N P_{i j} = 1}), ## and the rank of @var{P} must be @math{N}. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{M}(i,j) ## For all @math{1 @leq{} i, j @leq{} N}, @math{i \neq j}, @code{@var{M}(i,j)} is ## the average number of transitions before state @var{j} is entered ## for the first time, starting from state @var{i}. ## @code{@var{M}(i,i)} is the @emph{mean recurrence time} of state ## @math{i}, and represents the average time needed to return to state ## @var{i}. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item Grinstead, Charles M.; Snell, J. Laurie (July ## 1997). @cite{Introduction to Probability}, Ch. 11: Markov ## Chains. American Mathematical Society. ISBN 978-0821807491. ## @end itemize ## ## @seealso{ctmcfpt} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function result = dtmcfpt( P ) if ( nargin != 1 ) print_usage(); endif [N err] = dtmcchkP(P); ( N>0 ) || ... error(err); if ( any(diag(P) == 1) ) error("Cannot compute first passage times for absorbing chains"); endif w = dtmc(P); # steady state probability vector W = repmat(w,N,1); ## Z = (I - P + W)^-1 where W is the matrix where each row is the ## steady-state probability vector for P Z = inv(eye(N)-P+W); ## m_ij = (z_jj - z_ij) / w_j result = (repmat(diag(Z)',N,1) - Z) ./ repmat(w,N,1) + diag(1./w); endfunction %!test %! P = [1 1 1; 1 1 1]; %! fail( "dtmcfpt(P)" ); %!test %! P = dtmcbd([1 1 1], [0 0 0] ); %! fail( "dtmcfpt(P)", "absorbing" ); %!test %! P = [ 0.0 0.9 0.1; ... %! 0.1 0.0 0.9; ... %! 0.9 0.1 0.0 ]; %! p = dtmc(P); %! M = dtmcfpt(P); %! assert( diag(M)', 1./p, 1e-8 ); ## Example on p. 461 of [GrSn97] %!test %! P = [ 0 1 0 0 0; ... %! .25 .0 .75 0 0; ... %! 0 .5 0 .5 0; ... %! 0 0 .75 0 .25; ... %! 0 0 0 1 0 ]; %! M = dtmcfpt(P); %! assert( M, [16 1 2.6667 6.3333 21.3333; ... %! 15 4 1.6667 5.3333 20.3333; ... %! 18.6667 3.6667 2.6667 3.6667 18.6667; ... %! 20.3333 5.3333 1.6667 4 15; ... %! 21.3333 6.3333 2.6667 1 16 ], 1e-4 ); %!test %! sz = 10; %! P = reshape( 1:sz^2, sz, sz ); %! normP = repmat(sum(P,2),1,columns(P)); %! P = P./normP; %! M = dtmcfpt(P); %! for i=1:rows(P) %! for j=1:columns(P) %! assert( M(i,j), 1 + dot(P(i,:), M(:,j)) - P(i,j)*M(j,j), 1e-8); %! endfor %! endfor ## "Rat maze" problem (p. 453 of [GrSn97]); %!test %! P = zeros(9,9); %! P(1,[2 4]) = .5; %! P(2,[1 5 3]) = 1/3; %! P(3,[2 6]) = .5; %! P(4,[1 5 7]) = 1/3; %! P(5,[2 4 6 8]) = 1/4; %! P(6,[3 5 9]) = 1/3; %! P(7,[4 8]) = .5; %! P(8,[7 5 9]) = 1/3; %! P(9,[6 8]) = .5; %! M = dtmcfpt(P); %! assert( M(1:9 != 5,5)', [6 5 6 5 5 6 5 6], 100*eps ); queueing/inst/qncmaba.m0000644000175000017500000001423213632722374015030 0ustar morenomoreno## Copyright (C) 2012, 2016, 2018, 2020 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{D}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{S}, @var{V}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{S}, @var{V}, @var{m}) ## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) ## ## @cindex bounds, asymptotic ## @cindex asymptotic bounds ## @cindex closed network ## @cindex multiclass network, closed ## @cindex closed multiclass network ## ## Compute Asymptotic Bounds for closed, multiclass networks ## with @math{K} service centers and @math{C} customer classes. ## Single-server and infinite-server nodes are supported. ## Multiple-server nodes and general load-dependent servers are not ## supported. ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{N}(c) ## number of class @math{c} requests in the system ## (vector of length @math{C}, @code{@var{N}(c) @geq{} 0}). ## ## @item @var{D}(c, k) ## class @math{c} service demand ## at center @math{k} (@math{C \times K} matrix, @code{@var{D}(c,k) @geq{} 0}). ## ## @item @var{S}(c, k) ## mean service time of class @math{c} ## requests at center @math{k} (@math{C \times K} matrix, @code{@var{S}(c,k) @geq{} 0}). ## ## @item @var{V}(c,k) ## average number of visits of class @math{c} ## requests to center @math{k} (@math{C \times K} matrix, @code{@var{V}(c,k) @geq{} 0}). ## ## @item @var{m}(k) ## number of servers at center @math{k} ## (if @var{m} is a scalar, all centers have that number of servers). If ## @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); ## if @code{@var{m}(k) = 1}, center @math{k} is a M/M/1-FCFS server. ## This function does not support multiple-server nodes. Default ## is 1. ## ## @item @var{Z}(c) ## class @math{c} external delay ## (vector of length @math{C}, @code{@var{Z}(c) @geq{} 0}). Default is 0. ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{Xl}(c) ## @itemx @var{Xu}(c) ## Lower and upper bounds for class @math{c} throughput. ## ## @item @var{Rl}(c) ## @itemx @var{Ru}(c) ## Lower and upper bounds for class @math{c} response time. ## ## @end table ## ## @strong{REFERENCES} ## ## @itemize ## @item ## Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth ## C. Sevcik, @cite{Quantitative System Performance: Computer System ## Analysis Using Queueing Network Models}, Prentice Hall, ## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In ## particular, see section 5.2 ("Asymptotic Bounds"). ## @end itemize ## ## @seealso{qncsaba} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [Xl Xu Rl Ru] = qncmaba( varargin ) if ( nargin<2 || nargin>5 ) print_usage(); endif [err N S V m Z] = qncmchkparam( varargin{:} ); isempty(err) || error(err); all(m<=1) || ... error("Multiple-server nodes are not supported"); if ( sum(N) == 0 ) # handle trivial case of empty network Xl = Xu = Rl = Ru = zeros(size(S)); else D = S .* V; Dc_single = sum(D(:,(m==1)),2)'; # class c demand on single-server nodes Dc_delay = sum(D(:,(m<1)),2)'; # class c demand on delay centers Dc = sum(D,2)'; # class c total demand Dcmax = max(D,[],2)'; # maximum class c demand at any server Xl = N ./ ( dot(N,Dc_single) .+ Dc_delay .+ Z); Xu = min( 1./Dcmax, N ./ (Dc .+ Z) ); Rl = N ./ Xu .- Z; Ru = N ./ Xl .- Z; endif endfunction %!test %! fail("qncmaba([],[])", "nonempty"); %! fail("qncmaba([1 0], [1 2 3])", "2 rows"); %! fail("qncmaba([1 0], [1 2 3; 4 5 -1])", "nonnegative"); %! fail("qncmaba([1 2], [1 2 3; 4 5 6], [1 2 3])", "2 x 3"); %! fail("qncmaba([1 2], [1 2 3; 4 5 6], [1 2 3; 4 5 -1])", "nonnegative"); %! fail("qncmaba([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1])", "3 elements"); %! fail("qncmaba([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 2])", "not supported"); %! fail("qncmaba([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 -1],[1 2 3])", "2 elements"); %! fail("qncmaba([1 2], [1 2 3; 1 2 3], [1 2 3; 1 2 3], [1 1 -1],[1 -2])", "nonnegative"); %!test %! [Xl Xu Rl Ru] = qncmaba([0 0], [1 2 3; 1 2 3]); %! assert( all(Xl(:) == 0) ); %! assert( all(Xu(:) == 0) ); %! assert( all(Rl(:) == 0) ); %! assert( all(Ru(:) == 0) ); %!demo %! S = [10 7 5 4; ... %! 5 2 4 6]; %! NN=20; %! Xl = Xu = Rl = Ru = Xmva = Rmva = zeros(NN,2); %! for n=1:NN %! N=[n,10]; %! [a b c d] = qncmaba(N,S); %! Xl(n,:) = a; Xu(n,:) = b; Rl(n,:) = c; Ru(n,:) = d; %! [U R Q X] = qncmmva(N,S,ones(size(S))); %! Xmva(n,:) = X(:,1)'; Rmva(n,:) = sum(R,2)'; %! endfor %! subplot(2,2,1); %! plot(1:NN,Xl(:,1), 1:NN,Xu(:,1), 1:NN,Xmva(:,1), ";MVA;", "linewidth", 2); %! ylim([0, 0.2]); %! title("Class 1 throughput"); legend("boxoff"); %! subplot(2,2,2); %! plot(1:NN,Xl(:,2), 1:NN,Xu(:,2), 1:NN,Xmva(:,2), ";MVA;", "linewidth", 2); %! ylim([0, 0.2]); %! title("Class 2 throughput"); legend("boxoff"); %! subplot(2,2,3); %! plot(1:NN,Rl(:,1), 1:NN,Ru(:,1), 1:NN,Rmva(:,1), ";MVA;", "linewidth", 2); %! ylim([0, 700]); %! title("Class 1 response time"); legend("location", "northwest"); legend("boxoff"); %! subplot(2,2,4); %! plot(1:NN,Rl(:,2), 1:NN,Ru(:,2), 1:NN,Rmva(:,2), ";MVA;", "linewidth", 2); %! ylim([0, 700]); %! title("Class 2 response time"); legend("location", "northwest"); legend("boxoff"); queueing/inst/qsmminf.m0000644000175000017500000001143613601746530015077 0ustar morenomoreno## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2018, 2019 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see . ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmminf (@var{lambda}, @var{mu}) ## @deftypefnx {Function File} {@var{pk} =} qsmminf (@var{lambda}, @var{mu}, @var{k}) ## ## Compute utilization, response time, average number of requests and throughput for an infinite-server queue. ## ## The @math{M/M/\infty} system has an infinite number of identical ## servers. Such a system is always stable (i.e., the mean queue ## length is always finite) for any arrival and service rates. ## ## @cindex @math{M/M/}inf system ## ## @tex ## The steady-state probability @math{\pi_k} that there are @math{k} ## requests in the system, @math{k @geq{} 0}, can be computed as: ## ## $$ ## \pi_k = {1 \over k!} \left( \lambda \over \mu \right)^k e^{-\lambda / \mu} ## $$ ## @end tex ## ## @strong{INPUTS} ## ## @table @code ## ## @item @var{lambda} ## Arrival rate (@code{@var{lambda}>0}). ## ## @item @var{mu} ## Service rate (@code{@var{mu}>0}). ## ## @item @var{k} ## Number of requests in the system (@code{@var{k} @geq{} 0}). ## ## @end table ## ## @strong{OUTPUTS} ## ## @table @code ## ## @item @var{U} ## Traffic intensity (defined as @math{\lambda/\mu}). Note that this is ## different from the utilization, which in the case of @math{M/M/\infty} ## centers is always zero. ## ## @cindex traffic intensity ## ## @item @var{R} ## Service center response time. ## ## @item @var{Q} ## Average number of requests in the system (which is equal to the ## traffic intensity @math{\lambda/\mu}). ## ## @item @var{X} ## Throughput (which is always equal to @code{@var{X} = @var{lambda}}). ## ## @item @var{p0} ## Steady-state probability that there are no requests in the system ## ## @item @var{pk} ## Steady-state probability that there are @var{k} requests in the ## system (including the one being served). ## ## @end table ## ## If this function is called with less than three arguments, ## @var{lambda} and @var{mu} can be vectors of the same size. In this ## case, the results will be vectors as well. ## ## @strong{REFERENCES} ## ## @itemize ## @item ## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks ## and Markov Chains: Modeling and Performance Evaluation with Computer ## Science Applications}, Wiley, 1998, Section 6.4 ## @end itemize ## ## @seealso{qsmm1,qsmmm,qsmmmk} ## ## @end deftypefn ## Author: Moreno Marzolla ## Web: http://www.moreno.marzolla.name/ function [U_or_pk R Q X p0] = qsmminf( lambda, mu, k ) if ( nargin < 2 || nargin > 3 ) print_usage(); endif ( isvector(lambda) && isvector(mu) ) || ... error( "lambda and mu must be vectors" ); [ err lambda mu ] = common_size( lambda, mu ); if ( err ) error( "Parameters are of incompatible size" ); endif lambda = lambda(:)'; mu = mu(:)'; ( all( lambda>0 ) && all( mu>0 ) ) || ... error( "lambda and mu must be >0" ); if (nargin < 3) U_or_pk = Q = lambda ./ mu; # Traffic intensity. p0 = exp(-lambda./mu); # probability that there are 0 requests in the system R = 1 ./ mu; X = lambda; else (length(lambda) == 1) || error("lambda must be a scalar if this function is called with three arguments"); isvector(k) || error("k must be a vector"); all(k>=0 )|| error("k must be >= 0"); ## expn does not support array arguments, hence we must use arrayfun() U_or_pk = arrayfun(@(x) expn(lambda/mu, x) * exp(-lambda/mu), k); endif endfunction %!test %! fail( "qsmminf( [1 2], [1 2 3] )", "incompatible size"); %! fail( "qsmminf( [-1 -1], [1 1] )", ">0" ); %!demo %! ## Given a M/M/inf and M/M/m queue, compute the steady-state probability pk %! ## of having k requests in the systen. %! lambda = 5; %! mu = 1.1; %! m = 5; %! k = 0:20; %! pk_inf = qsmminf(lambda, mu, k); %! pk_m = qsmmm(lambda, mu, 5, k); %! plot(k, pk_inf, "-o;M/M/\\infty;", "linewidth", 2, ... %! k, pk_m, "-x;M/M/5;", "linewidth", 2); %! xlabel("N. of requests (k)"); %! ylabel("P_k"); %! title(sprintf("M/M/\\infty and M/M/%d systems, \\lambda = %g, \\mu = %g", m, lambda, mu)); queueing/CITATION0000644000175000017500000000234113570046274013425 0ustar morenomorenoTo cite the Octave queueing package use: Moreno Marzolla, The qnetworks Toolbox: A Software Package for Queueing Networks Analysis. Khalid Al-Begain, Dieter Fiems and William J. Knottenbelt, Editors, Proceedings 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications (ASMTA 2010) Cardiff, UK, June 14--16, 2010, volume 6148 of Lecture Notes in Computer Science, Springer, pp. 102--116, ISBN 978-3-642-13567-5, DOI 10.1007/978-3-642-13568-2_8 A BibTeX entry for LaTeX users is: @inproceedings{queueing, author = {Moreno Marzolla}, title = {The qnetworks Toolbox: A Software Package for Queueing Networks Analysis}, booktitle = {Analytical and Stochastic Modeling Techniques and Applications, 17th International Conference, ASMTA 2010, Cardiff, UK, June 14--16, 2010. Proceedings}, editor = {Khalid Al-Begain and Dieter Fiems and William J. Knottenbelt}, year = {2010}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, volume = {6148}, pages = {102--116}, ee = {http://dx.doi.org/10.1007/978-3-642-13568-2_8}, doi = {10.1007/978-3-642-13568-2_8}, isbn = {978-3-642-13567-5} } queueing/doc/0000755000175000017500000000000013635377242013041 5ustar morenomorenoqueueing/doc/conf.texi0000644000175000017500000000005713635377222014661 0ustar morenomoreno@set VERSION 1.2.7 @set VERSIONDATE 2020-03-21 queueing/doc/qn_closed_single.fig0000644000175000017500000000300513570046274017032 0ustar morenomoreno#FIG 3.2 Produced by xfig version 3.2.5b Landscape Center Metric A4 100.00 Single -2 1200 2 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 6930 5580 270 270 6930 5580 6930 5310 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 6930 4500 270 270 6930 4500 6930 4230 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 5317 5043 270 270 5317 5043 5317 4773 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 4 0 0 1.00 60.00 120.00 5580 5040 5850 5040 6030 4500 6300 4500 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 3 0 0 1.00 60.00 120.00 5850 5040 6030 5580 6300 5580 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 5 0 0 1.00 60.00 120.00 7470 5580 7470 3960 4770 3960 4770 5040 5040 5040 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 0 0 1.00 60.00 120.00 7200 5580 7470 5580 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 4 6300 5310 6660 5310 6660 5850 6300 5850 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 4 6300 4230 6660 4230 6660 4770 6300 4770 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 0 0 1.00 60.00 120.00 7200 4500 7470 4500 4 0 0 50 -1 0 12 0.0000 4 135 255 5895 5760 0.7\001 4 0 0 50 -1 0 12 0.0000 4 135 255 5895 4455 0.3\001 4 0 0 50 -1 0 12 0.0000 4 135 210 5175 5490 PS\001 4 0 0 50 -1 0 12 0.0000 4 135 450 6705 4950 FCFS\001 4 0 0 50 -1 0 12 0.0000 4 135 450 6705 6030 FCFS\001 4 0 0 50 -1 0 12 0.0000 4 135 375 5130 4725 CPU\001 4 0 0 50 -1 0 12 0.0000 4 135 480 6705 4185 Disk1\001 4 0 0 50 -1 0 12 0.0000 4 135 480 6705 5265 Disk2\001 4 0 0 50 -1 0 12 0.0000 4 135 105 5265 5085 1\001 4 0 0 50 -1 0 12 0.0000 4 135 105 6885 4545 2\001 4 0 0 50 -1 0 12 0.0000 4 135 105 6885 5625 3\001 queueing/doc/singlestation.texi0000644000175000017500000006177213635377223016633 0ustar morenomoreno@c This file has been automatically generated from singlestation.txi @c by proc.m. Do not edit this file, all changes will be lost @c -*- texinfo -*- @c Copyright (C) 2008, 2009, 2010, 2011, 2012, 2014, 2016, 2018 Moreno Marzolla @c @c This file is part of the queueing package. @c @c The queueing package is free software; you can redistribute it @c and/or modify it under the terms of the GNU General Public License @c as published by the Free Software Foundation; either version 3 of @c the License, or (at your option) any later version. @c @c The queueing package is distributed in the hope that it will be @c useful, but WITHOUT ANY WARRANTY; without even the implied warranty @c of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the @c GNU General Public License for more details. @c @c You should have received a copy of the GNU General Public License @c along with the queueing package; see the file COPYING. If not, see @c . @node Single Station Queueing Systems @chapter Single Station Queueing Systems Single Station Queueing Systems contain a single station, and can usually be analyzed easily. The @code{queueing} package contains functions for handling the following types of queues: @ifnottex @menu * The M/M/1 System:: Single-server queueing station. * The M/M/m System:: Multiple-server queueing station. * The Erlang-B Formula:: * The Erlang-C Formula:: * The Engset Formula:: * The M/M/inf System:: Infinite-server (delay center) station. * The M/M/1/K System:: Single-server, finite-capacity queueing station. * The M/M/m/K System:: Multiple-server, finite-capacity queueing station. * The Asymmetric M/M/m System:: Asymmetric multiple-server queueing station. * The M/G/1 System:: Single-server with general service time distribution. * The M/Hm/1 System:: Single-server with hyperexponential service time distribution. @end menu @end ifnottex @iftex @itemize @item @math{M/M/1} single-server queueing station; @item @math{M/M/m} multiple-server queueing station; @item Asymmetric @math{M/M/m}; @item @math{M/M/\infty} infinite-server station (delay center); @item @math{M/M/1/K} single-server, finite-capacity queueing station; @item @math{M/M/m/K} multiple-server, finite-capacity queueing station; @item @math{M/G/1} single-server with general service time distribution; @item @math{M/H_m/1} single-server with hyperexponential service time distribution. @end itemize @end iftex @c @c M/M/1 @c @node The M/M/1 System @section The @math{M/M/1} System The @math{M/M/1} system contains a single server connected to an unbounded FCFS queue. Requests arrive according to a Poisson process with rate @math{\lambda}; the service time is exponentially distributed with average service rate @math{\mu}. The system is stable if @math{\lambda < \mu}. @anchor{doc-qsmm1} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmm1 (@var{lambda}, @var{mu}) @deftypefnx {Function File} {@var{pk} =} qsmm1 (@var{lambda}, @var{mu}, @var{k}) @cindex @math{M/M/1} system Compute utilization, response time, average number of requests and throughput for a @math{M/M/1} queue. @tex The steady-state probability @math{\pi_k} that there are @math{k} jobs in the system, @math{k \geq 0}, can be computed as: $$ \pi_k = (1-\rho)\rho^k $$ where @math{\rho = \lambda/\mu} is the server utilization. @end tex @strong{INPUTS} @table @code @item @var{lambda} Arrival rate (@code{@var{lambda} @geq{} 0}). @item @var{mu} Service rate (@code{@var{mu} > @var{lambda}}). @item @var{k} Number of requests in the system (@code{@var{k} @geq{} 0}). @end table @strong{OUTPUTS} @table @code @item @var{U} Server utilization @item @var{R} Server response time @item @var{Q} Average number of requests in the system @item @var{X} Server throughput. If the system is ergodic (@code{@var{mu} > @var{lambda}}), we always have @code{@var{X} = @var{lambda}} @item @var{p0} Steady-state probability that there are no requests in the system. @item @var{pk} Steady-state probability that there are @var{k} requests in the system. (including the one being served). @end table If this function is called with less than three input parameters, @var{lambda} and @var{mu} can be vectors of the same size. In this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, Section 6.3 @end itemize @seealso{qsmmm, qsmminf, qsmmmk} @end deftypefn @c @c M/M/m @c @node The M/M/m System @section The @math{M/M/m} System The @math{M/M/m} system is similar to the @math{M/M/1} system, except that there are @math{m \geq 1} identical servers connected to a shared FCFS queue. Thus, at most @math{m} requests can be served at the same time. The @math{M/M/m} system can be seen as a single server with load-dependent service rate @math{\mu(n)}, which is a function of the number @math{n} of requests in the system: @iftex @tex $$\mu(n) = \mu \times \min(m,n)$$ @end tex @end iftex @ifnottex @example mu(n) = min(m,n)*mu @end example @end ifnottex @noindent where @math{\mu} is the service rate of each individual server. @anchor{doc-qsmmm} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pm}] =} qsmmm (@var{lambda}, @var{mu}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pm}] =} qsmmm (@var{lambda}, @var{mu}, @var{m}) @deftypefnx {Function File} {@var{pk} =} qsmmm (@var{lambda}, @var{mu}, @var{m}, @var{k}) @cindex @math{M/M/m} system Compute utilization, response time, average number of requests in service and throughput for a @math{M/M/m} queue, a queueing system with @math{m} identical servers connected to a single FCFS queue. @tex The steady-state probability @math{\pi_k} that there are @math{k} requests in the system, @math{k \geq 0}, can be computed as: $$ \pi_k = \cases{ \displaystyle{\pi_0 { ( m\rho )^k \over k!}} & $0 \leq k \leq m$;\cr\cr \displaystyle{\pi_0 { \rho^k m^m \over m!}} & $k>m$.\cr } $$ where @math{\rho = \lambda/(m\mu)} is the individual server utilization. The steady-state probability @math{\pi_0} that there are no jobs in the system is: $$ \pi_0 = \left[ \sum_{k=0}^{m-1} { (m\rho)^k \over k! } + { (m\rho)^m \over m!} {1 \over 1-\rho} \right]^{-1} $$ @end tex @strong{INPUTS} @table @code @item @var{lambda} Arrival rate (@code{@var{lambda}>0}). @item @var{mu} Service rate (@code{@var{mu}>@var{lambda}}). @item @var{m} Number of servers (@code{@var{m} @geq{} 1}). Default is @code{@var{m}=1}. @item @var{k} Number of requests in the system (@code{@var{k} @geq{} 0}). @end table @strong{OUTPUTS} @table @code @item @var{U} Service center utilization, @math{U = \lambda / (m \mu)}. @item @var{R} Service center mean response time @item @var{Q} Average number of requests in the system @item @var{X} Service center throughput. If the system is ergodic, we will always have @code{@var{X} = @var{lambda}} @item @var{p0} Steady-state probability that there are 0 requests in the system @item @var{pm} Steady-state probability that an arriving request has to wait in the queue @item @var{pk} Steady-state probability that there are @var{k} requests in the system (including the one being served). @end table If this function is called with less than four parameters, @var{lambda}, @var{mu} and @var{m} can be vectors of the same size. In this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, Section 6.5 @end itemize @seealso{erlangc,qsmm1,qsmminf,qsmmmk} @end deftypefn @c @c Erlang-B @c @node The Erlang-B Formula @section The Erlang-B Formula @anchor{doc-erlangb} @deftypefn {Function File} {@var{B} =} erlangb (@var{A}, @var{m}) @cindex Erlang-B formula Compute the steady-state blocking probability in the Erlang loss model. The Erlang-B formula @math{E_B(A, m)} gives the probability that an open system with @math{m} identical servers, arrival rate @math{\lambda}, individual service rate @math{\mu} and offered load @math{A = \lambda / \mu} has all servers busy. This corresponds to the rejection probability of an @math{M/M/m/0} system with @math{m} servers and no queue. @tex @math{E_B(A, m)} is defined as: $$ E_B(A, m) = \displaystyle{{A^m \over m!} \left( \sum_{k=0}^m {A^k \over k!} \right) ^{-1}} $$ @end tex @strong{INPUTS} @table @code @item @var{A} Offered load, defined as @math{A = \lambda / \mu} where @math{\lambda} is the mean arrival rate and @math{\mu} the mean service rate of each individual server (real, @math{A > 0}). @item @var{m} Number of identical servers (integer, @math{m @geq{} 1}). Default @math{m = 1} @end table @strong{OUTPUTS} @table @code @item @var{B} The value @math{E_B(A, m)} @end table @var{A} or @var{m} can be vectors, and in this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Zeng, @cite{Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 @end itemize @seealso{erlangc,engset,qsmmm} @end deftypefn @c @c Erlang-c @c @node The Erlang-C Formula @section The Erlang-C Formula @anchor{doc-erlangc} @deftypefn {Function File} {@var{C} =} erlangc (@var{A}, @var{m}) @cindex Erlang-C formula Compute the steady-state probability of delay in the Erlang delay model. The Erlang-C formula @math{E_C(A, m)} gives the probability that an open queueing system with @math{m} identical servers, infinite wating space, arrival rate @math{\lambda}, individual service rate @math{\mu} and offered load @math{A = \lambda / \mu} has all the servers busy. This is the waiting probability in an @math{M/M/m/\infty} system with @math{m} servers and an infinite queue. @tex @math{E_C(A, m)} is defined as: $$ E_C(A, m) = \displaystyle{ {A^m \over m!} {1 \over 1-\rho} \left( \sum_{k=0}^{m-1} {A^k \over k!} + {A^m \over m!} {1 \over 1 - \rho} \right) ^{-1}} $$ where @math{\rho = A / m = \lambda / (m \mu)}. @end tex @strong{INPUTS} @table @code @item @var{A} Offered load. @math{A = \lambda / \mu} where @math{\lambda} is the mean arrival rate and @math{\mu} the mean service rate of each individual server (real, @math{0 < A < m}). @item @var{m} Number of identical servers (integer, @math{m @geq{} 1}). Default @math{m = 1} @end table @strong{OUTPUTS} @table @code @item @var{B} The value @math{E_C(A, m)} @end table @var{A} or @var{m} can be vectors, and in this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Zeng, @cite{Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 @end itemize @seealso{erlangb,engset,qsmmm} @end deftypefn @c @c Engset @c @node The Engset Formula @section The Engset Formula @anchor{doc-engset} @deftypefn {Function File} {@var{B} =} engset (@var{A}, @var{m}, @var{n}) @cindex Engset loss formula Evaluate the Engset loss formula. The Engset formula computes the blocking probability @math{P_b(A,m,n)} for a system with a finite population of @math{n} users, @math{m} identical servers, no queue, individual service rate @math{\mu}, individual arrival rate @math{\lambda} (i.e., the time until a user tries to request service is exponentially distributed with mean @math{1/\lambda}), and offered load @math{A=\lambda/\mu}. @tex @math{P_b(A, m, n)} is defined for @math{n > m} as: $$ P_b(A, m, n) = {{\displaystyle{A^m {n \choose m}}} \over {\displaystyle{\sum_{k=0}^m A^k {n \choose k}}}} $$ and is 0 if @math{n @leq{} m}. @end tex @strong{INPUTS} @table @code @item @var{A} Offered load, defined as @math{A = \lambda / \mu} where @math{\lambda} is the mean arrival rate and @math{\mu} the mean service rate of each individual server (real, @math{A > 0}). @item @var{m} Number of identical servers (integer, @math{m @geq{} 1}). Default @math{m = 1} @item @var{n} Number of requests (integer, @math{n @geq{} 1}). Default @math{n = 1} @end table @strong{OUTPUTS} @table @code @item @var{B} The value @math{P_b(A, m, n)} @end table @var{A}, @var{m} or @math{n} can be vectors, and in this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Zeng, @cite{Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 @end itemize @seealso{erlangb, erlangc} @end deftypefn @c @c M/M/inf @c @node The M/M/inf System @section The @math{M/M/}inf System The @math{M/M/\infty} system is a special case of @math{M/M/m} system with infinitely many identical servers (i.e., @math{m = \infty}). Each new request is always assigned to a new server, so that queueing never occurs. The @math{M/M/\infty} system is always stable. @anchor{doc-qsmminf} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmminf (@var{lambda}, @var{mu}) @deftypefnx {Function File} {@var{pk} =} qsmminf (@var{lambda}, @var{mu}, @var{k}) Compute utilization, response time, average number of requests and throughput for an infinite-server queue. The @math{M/M/\infty} system has an infinite number of identical servers. Such a system is always stable (i.e., the mean queue length is always finite) for any arrival and service rates. @cindex @math{M/M/}inf system @tex The steady-state probability @math{\pi_k} that there are @math{k} requests in the system, @math{k @geq{} 0}, can be computed as: $$ \pi_k = {1 \over k!} \left( \lambda \over \mu \right)^k e^{-\lambda / \mu} $$ @end tex @strong{INPUTS} @table @code @item @var{lambda} Arrival rate (@code{@var{lambda}>0}). @item @var{mu} Service rate (@code{@var{mu}>0}). @item @var{k} Number of requests in the system (@code{@var{k} @geq{} 0}). @end table @strong{OUTPUTS} @table @code @item @var{U} Traffic intensity (defined as @math{\lambda/\mu}). Note that this is different from the utilization, which in the case of @math{M/M/\infty} centers is always zero. @cindex traffic intensity @item @var{R} Service center response time. @item @var{Q} Average number of requests in the system (which is equal to the traffic intensity @math{\lambda/\mu}). @item @var{X} Throughput (which is always equal to @code{@var{X} = @var{lambda}}). @item @var{p0} Steady-state probability that there are no requests in the system @item @var{pk} Steady-state probability that there are @var{k} requests in the system (including the one being served). @end table If this function is called with less than three arguments, @var{lambda} and @var{mu} can be vectors of the same size. In this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, Section 6.4 @end itemize @seealso{qsmm1,qsmmm,qsmmmk} @end deftypefn @c @c M/M/1/k @c @node The M/M/1/K System @section The @math{M/M/1/K} System In a @math{M/M/1/K} finite capacity system there is a single server, and there can be at most @math{K} jobs at any time (including the job currently in service), @math{K > 1}. If a new request tries to join the system when there are already @math{K} other requests, the request is lost. The queue has @math{K-1} slots. The @math{M/M/1/K} system is always stable, regardless of the arrival and service rates. @anchor{doc-qsmm1k} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pK}] =} qsmm1k (@var{lambda}, @var{mu}, @var{K}) @deftypefnx {Function File} {@var{pn} =} qsmm1k (@var{lambda}, @var{mu}, @var{K}, @var{n}) @cindex @math{M/M/1/K} system Compute utilization, response time, average number of requests and throughput for a @math{M/M/1/K} finite capacity system. In a @math{M/M/1/K} queue there is a single server and a queue with finite capacity: the maximum number of requests in the system (including the request being served) is @math{K}, and the maximum queue length is therefore @math{K-1}. @tex The steady-state probability @math{\pi_n} that there are @math{n} jobs in the system, @math{0 @leq{} n @leq{} K}, is: $$ \pi_n = {(1-a)a^n \over 1-a^{K+1}} $$ where @math{a = \lambda/\mu}. @end tex @strong{INPUTS} @table @code @item @var{lambda} Arrival rate (@code{@var{lambda}>0}). @item @var{mu} Service rate (@code{@var{mu}>0}). @item @var{K} Maximum number of requests allowed in the system (@code{@var{K} @geq{} 1}). @item @var{n} Number of requests in the (@code{0 @leq{} @var{n} @leq{} K}). @end table @strong{OUTPUTS} @table @code @item @var{U} Service center utilization, which is defined as @code{@var{U} = 1-@var{p0}} @item @var{R} Service center response time @item @var{Q} Average number of requests in the system @item @var{X} Service center throughput @item @var{p0} Steady-state probability that there are no requests in the system @item @var{pK} Steady-state probability that there are @math{K} requests in the system (i.e., that the system is full) @item @var{pn} Steady-state probability that there are @math{n} requests in the system (including the one being served). @end table If this function is called with less than four arguments, @var{lambda}, @var{mu} and @var{K} can be vectors of the same size. In this case, the results will be vectors as well. @seealso{qsmm1,qsmminf,qsmmm} @end deftypefn @c @c M/M/m/k @c @node The M/M/m/K System @section The @math{M/M/m/K} System The @math{M/M/m/K} finite capacity system is similar to the @math{M/M/1/k} system except that the number of servers is @math{m}, where @math{1 \leq m \leq K}. The queue has @math{K-m} slots. The @math{M/M/m/K} system is always stable. @anchor{doc-qsmmmk} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}, @var{pK}] =} qsmmmk (@var{lambda}, @var{mu}, @var{m}, @var{K}) @deftypefnx {Function File} {@var{pn} =} qsmmmk (@var{lambda}, @var{mu}, @var{m}, @var{K}, @var{n}) @cindex @math{M/M/m/K} system Compute utilization, response time, average number of requests and throughput for a @math{M/M/m/K} finite capacity system. In a @math{M/M/m/K} system there are @math{m \geq 1} identical service centers sharing a fixed-capacity queue. At any time, at most @math{K @geq{} m} requests can be in the system, including those being served. The maximum queue length is @math{K-m}. This function generates and solves the underlying CTMC. @tex The steady-state probability @math{\pi_n} that there are @math{n} jobs in the system, @math{0 @leq{} n @leq{} K}, is: $$ \pi_n = \cases{ \displaystyle{{\rho^n \over n!} \pi_0} & if $0 \leq n \leq m$;\cr\cr \displaystyle{{\rho^m \over m!} \left( \rho \over m \right)^{n-m} \pi_0} & if $m < n \leq K$\cr} $$ where @math{\rho = \lambda/\mu} is the offered load. The probability @math{\pi_0} that the system is empty can be computed by considering that all probabilities must sum to one: @math{\sum_{k=0}^K \pi_k = 1}, that gives: $$ \pi_0 = \left[ \sum_{k=0}^m {\rho^k \over k!} + {\rho^m \over m!} \sum_{k=m+1}^K \left( {\rho \over m}\right)^{k-m} \right]^{-1} $$ @end tex @strong{INPUTS} @table @code @item @var{lambda} Arrival rate (@code{@var{lambda}>0}) @item @var{mu} Service rate (@code{@var{mu}>0}) @item @var{m} Number of servers (@code{@var{m} @geq{} 1}) @item @var{K} Maximum number of requests allowed in the system, including those being served (@code{@var{K} @geq{} @var{m}}) @item @var{n} Number of requests in the (@code{0 @leq{} @var{n} @leq{} K}). @end table @strong{OUTPUTS} @table @code @item @var{U} Service center utilization @item @var{R} Service center response time @item @var{Q} Average number of requests in the system @item @var{X} Service center throughput @item @var{p0} Steady-state probability that there are no requests in the system. @item @var{pK} Steady-state probability that there are @var{K} requests in the system (i.e., probability that the system is full). @item @var{pn} Steady-state probability that there are @var{n} requests in the system (including those being served). @end table If this function is called with less than five arguments, @var{lambda}, @var{mu}, @var{m} and @var{K} can be either scalars, or vectors of the same size. In this case, the results will be vectors as well. @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, Section 6.6 @end itemize @seealso{qsmm1,qsmminf,qsmmm} @end deftypefn @c @c Asymmetric M/M/m @c @node The Asymmetric M/M/m System @section The Asymmetric @math{M/M/m} System The Asymmetric @math{M/M/m} system contains @math{m} servers connected to a single queue. Differently from the @math{M/M/m} system, in the asymmetric @math{M/M/m} each server may have a different service time. @anchor{doc-qsammm} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qsammm (@var{lambda}, @var{mu}) @cindex asymmetric @math{M/M/m} system Compute @emph{approximate} utilization, response time, average number of requests in service and throughput for an asymmetric @math{M/M/m} queue. In this type of system there are @math{m} different servers connected to a single queue. Each server has its own (possibly different) service rate. If there is more than one server available, requests are routed to a randomly-chosen one. @strong{INPUTS} @table @code @item @var{lambda} Arrival rate (@code{@var{lambda}>0}) @item @var{mu} @code{@var{mu}(i)} is the service rate of server @math{i}, @math{1 @leq{} i @leq{} m}. The system must be ergodic (@code{@var{lambda} < sum(@var{mu})}). @end table @strong{OUTPUTS} @table @code @item @var{U} Approximate service center utilization, @math{U = \lambda / ( \sum_{i=1}^m \mu_i )}. @item @var{R} Approximate service center response time @item @var{Q} Approximate number of requests in the system @item @var{X} Approximate system throughput. If the system is ergodic, @code{@var{X} = @var{lambda}} @end table @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998 @end itemize @seealso{qsmmm} @end deftypefn @c @c @c @node The M/G/1 System @section The @math{M/G/1} System @anchor{doc-qsmg1} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmg1 (@var{lambda}, @var{xavg}, @var{x2nd}) @cindex @math{M/G/1} system Compute utilization, response time, average number of requests and throughput for a @math{M/G/1} system. The service time distribution is described by its mean @var{xavg}, and by its second moment @var{x2nd}. The computations are based on results from L. Kleinrock, @cite{Queuing Systems}, Wiley, Vol 2, and Pollaczek-Khinchine formula. @strong{INPUTS} @table @code @item @var{lambda} Arrival rate @item @var{xavg} Average service time @item @var{x2nd} Second moment of service time distribution @end table @strong{OUTPUTS} @table @code @item @var{U} Service center utilization @item @var{R} Service center response time @item @var{Q} Average number of requests in the system @item @var{X} Service center throughput @item @var{p0} Probability that there is not any request at system @end table @var{lambda}, @var{xavg}, @var{t2nd} can be vectors of the same size. In this case, the results will be vectors as well. @seealso{qsmh1} @end deftypefn @c @c @c @node The M/Hm/1 System @section The @math{M/H_m/1} System @anchor{doc-qsmh1} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{p0}] =} qsmh1 (@var{lambda}, @var{mu}, @var{alpha}) @cindex @math{M/H_m/1} system Compute utilization, response time, average number of requests and throughput for a @math{M/H_m/1} system. In this system, the customer service times have hyper-exponential distribution: @tex $$ B(x) = \sum_{j=1}^m \alpha_j(1-e^{-\mu_j x}),\quad x>0 $$ @end tex @ifnottex @example @group ___ m \ B(x) = > alpha(j) * (1-exp(-mu(j)*x)) x>0 /__ j=1 @end group @end example @end ifnottex where @math{\alpha_j} is the probability that the request is served at phase @math{j}, in which case the average service rate is @math{\mu_j}. After completing service at phase @math{j}, for some @math{j}, the request exits the system. @strong{INPUTS} @table @code @item @var{lambda} Arrival rate @item @var{mu} @code{@var{mu}(j)} is the phase @math{j} service rate. The total number of phases @math{m} is @code{length(@var{mu})}. @item @var{alpha} @code{@var{alpha}(j)} is the probability that a request is served at phase @math{j}. @var{alpha} must have the same size as @var{mu}. @end table @strong{OUTPUTS} @table @code @item @var{U} Service center utilization @item @var{R} Service center response time @item @var{Q} Average number of requests in the system @item @var{X} Service center throughput @end table @end deftypefn queueing/doc/demo_web.m0000644000175000017500000000552713570046274015005 0ustar morenomoreno## Copyright (C) 2012, 2016 Moreno Marzolla ## ## This file is part of the queueing toolbox. ## ## The queueing toolbox is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## The queueing toolbox is distributed in the hope that it will be ## useful, but WITHOUT ANY WARRANTY; without even the implied warranty ## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with the queueing toolbox. If not, see 1; # not a function file # figure (1, "visible", "off"); # do not show plot window page_screen_output(0); # avoid output pagination N = 100; # total population size beta = linspace(0.1,0.9,18); # population mix for class 1 D = [12 14 23 20 80 31; ... 2 20 14 90 30 33 ]; V = ones(size(D)); X1 = X1 = XX = zeros(size(beta)); R1 = R2 = RR = zeros(size(beta)); for i=1:length(beta) pop = [fix(beta(i)*N) N-fix(beta(i)*N)]; [U R Q X] = qncmmva( pop, D, V ); X1(i) = X(1,1) / V(1,1); X2(i) = X(2,1) / V(2,1); XX(i) = X1(i) + X2(i); R1(i) = dot(R(1,:), V(1,:)); R2(i) = dot(R(2,:), V(2,:)); RR(i) = N / XX(i); endfor ## Plot throughput and response times ## set(gcf,"paperorientation","landscape"); ## papersize=[4 3] * 2; margin=[0 0]; ## set(gcf,"papersize",papersize); ## set(gcf,"paperposition", [margin papersize-margin*2]); subplot(2,1,1); plot(beta, X1, "--;Class 1;", "linewidth", 2, ... beta, X2, ":;Class 2;", "linewidth", 2, ... beta, XX, "-;System;", "linewidth", 2 ); ylabel("Throughput"); legend("location", "south", "orientation", "horizontal"); legend("boxoff"); title("Throughput and response time vs population mix"); subplot(2,1,2); plot(beta, R1, "--;Class 1;", "linewidth", 2, ... beta, R2, ":;Class 2;", "linewidth", 2, ... beta, RR, "-;System;", "linewidth", 2 ); ax = axis(); ax(3) = 0; axis(ax); legend("location", "south", "orientation", "horizontal"); legend("boxoff"); xlabel("Class 1 population mix"); ylabel("Response Time"); print("web.eps", "-deps2", "-mono", "-tight"); clf; ## Plot system power ## set(gcf,"paperorientation","landscape"); ## papersize=[4 2.5] * 1.2; margin=[0 0]; ## set(gcf,"papersize",papersize); ## set(gcf,"paperposition", [margin papersize-margin*2]); plot(beta, X1./R1, "--;Class 1;", "linewidth", 2, ... beta, X2./R2, ":;Class 2;", "linewidth", 2, ... beta, XX./RR, "-;System;", "linewidth", 2); legend("location","south", "orientation", "horizontal"); legend("boxoff"); xlabel("Class 1 population mix"); ylabel("Power"); title("Power as a function of the population mix"); print("power.eps", "-deps2", "-mono", "-tight"); queueing/doc/qn_closed_multi_apl.png0000644000175000017500000001606213635377233017571 0ustar morenomoreno‰PNG  IHDR’ݸîh 2iCCPdefault_rgb.iccH‰••gP“YÇïó<é…@B‡PC‘*%€”Z(Ò«¨@èPElˆ¸+Šˆ4EE\•"kE ‹‚tƒ,ʺqQAYpß÷?¼ÿ™{ÏoþsæÞsÏùp ˆƒeÁË{bRºÀÛÉŽÌß(ŒŸ–ÂñôtßÕ»­Ä{ºßÏù®‘iü常¼rù)‚t ìeÖÌJOYá£ËLÿÂgWX°\à2ßXáèyìKο,ú’ãëÍ]~ )úÿ†ÿsïŠT8‚ôبÈl¦OrTzV˜ ’™¶Ò —Ëô$GÅ&D~Sðÿ•ü¥Gf§¯DnrÊ&AltL:ó5204_gñÆëK!FÿÏgE_½äzØs û¾zá•tî@úÑWOm¹¯”|:îð3™ÿz¨• €è@(U  t0–À8à|AØø $ȹ`(E`8ª@-hM œà<¸®ƒÛà.L‚—@Þ‚°¢A2¤é@F²† 7È ‚B¡h( Ê€r¡PT UAuPô tºÝ„¡‡Ð84ý }„˜ÓaXÖ‡Ù0v…}áõp4œ çÀùð^¸®‡OÂðø6< á—ð"Â@”]„p$‰BÈV¤)Gê‘V¤éCî!Bdù€Â h(&Je‰rFù¡ø¨TÔVT1ª uÕêEÝC£D¨Ïh2Z­ƒ¶@óÐèhtº]ŽnD·£¯¡‡Ñ“èw †aaÌ0Θ Lf3¦sÓ†¹ŒÄL`æ°X¬ Vk…õÀ†aÓ±ØJìIì%ìvûGÄ)áŒpŽ¸`\.WŽkÆ]Ä á¦p xq¼:ÞïÀo—àðÝø;øIüA‚À"X| q„„ B+áaŒð†H$ª͉^ÄXâvbññqœøD%i“¸¤Ri/é8é2é!é ™LÖ Û’ƒÉéä½ä&òUòSò{1š˜žO,Bl›XµX‡ØØ+ ž¢NáP6Pr(å”3”;”Yq¼¸†8WJ)Hq¤"¥öHµJ 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PURPOSE. See the GNU General Public License % for more details. % % You should have received a copy of the GNU General Public License % along with the queueing package; see the file COPYING. If not, see % . % This info file is taken from the GNU Octave info file \input texinfo @setfilename queueing.info @c The following macro is used for the on-line help system, but we don't @c want lots of `See also: foo, bar, and baz' strings cluttering the @c printed manual (that information should be in the supporting text for @c each group of functions and variables). @macro seealso {args} @iftex @vskip 2pt @end iftex @ifnottex @sp 1 @end ifnottex @noindent @strong{See also:} \args\. @end macro @c @macro examplefile{file} @c @example @c @group @c @verbatiminclude @value{top_srcdir}/examples/\file\ @c @end group @c @end example @c @end macro @c @macro GETHELP{file} @c @end macro @c @macro GETDEMO{file,n} @c @end macro @c FIXME: The following macros are workaround to fix erratic behavior of texinfo @iftex @macro lambdack @lambda_{c,k} @end macro @end iftex @ifnottex @macro lambda lambda @end macro @macro lambdack lambda_@{c, k@} @end macro @end ifnottex @ifinfo @format START-INFO-DIR-ENTRY * queueing: (octave). Queueing Networks and Markov chains analysis package. END-INFO-DIR-ENTRY @end format @end ifinfo @c Settings for printing on 8-1/2 by 11 inch paper: @c ----------------------------------------------- @setchapternewpage odd @c Settings for small book format: @c ------------------------------ @ignore @smallbook @setchapternewpage odd @finalout @iftex @cropmarks @end iftex @end ignore @defindex op @c Things like the Octave version number are defined in conf.texi. @c This file doesn't include a chapter, so it must not be included @c if you want to run the Emacs function texinfo-multiple-files-update. @include conf.texi @settitle queueing @documentdescription User manual for queueing, a GNU Octave package for queueing networks and Markov chains analysis. This package supports single-station queueing systems, queueing networks and Markov chains. The queueing package implements, among others, the Mean Value Analysis (MVA) and convolution algorithms for steady-state analysis of product-form queueing networks. Transient and steady-state analysis of Markov chains is also implemented. @end documentdescription @ifnottex Copyright @copyright{} 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2020 Moreno Marzolla. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. @ignore Permission is granted to process this file through TeX and print the results, provided the printed document carries copying permission notice identical to this one except for the removal of this paragraph (this paragraph not being relevant to the printed manual). @end ignore Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions. @end ifnottex @titlepage @title The Octave Queueing Package @subtitle User's Guide, Edition 1 for release @value{VERSION} @subtitle @value{VERSIONDATE} @author Moreno Marzolla @page @vskip 0pt plus 1filll Copyright @copyright{} 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2020 Moreno Marzolla (@email{moreno.marzolla@@unibo.it}). This is the first edition of the Queueing package documentation, and is consistent with version @value{VERSION} of the package. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the same conditions as for modified versions. Portions of this document have been adapted from the @code{octave} manual, Copyright @copyright{} John W. Eaton. @end titlepage @contents @ifnottex @node Top @top This manual documents how to install and run the Queueing package. It corresponds to version @value{VERSION} of the package. @end ifnottex @c ------------------------------------------------------------------------ @menu * Summary:: * Installation and Getting Started:: Installation of the queueing package. * Markov Chains:: Functions for Markov chains analysis. * Single Station Queueing Systems:: Functions for single-station queueing systems. * Queueing Networks:: Functions for queueing networks analysis. * References:: References. * Copying:: The GNU General Public License. * Concept Index:: An item for each concept. * Function Index:: An item for each function. @end menu @c ------------------------------------------------------------------------ @include summary.texi @include installation.texi @include markovchains.texi @include singlestation.texi @include queueingnetworks.texi @include references.texi @c @c Appendix starts here @c @include gpl.texi @c @c INDEX @c @node Concept Index @unnumbered Concept Index @printindex cp @node Function Index 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âésN…×íþüÔƧX𨞆ÐU²,ª3*TODäö\{0ÏîÇå¿2ë©#ÉÇýñ†BB*TÅŠéåYâZczj™‚„kÈÇ@CDä4jÎÛ\{ê9ÕæW£‘qv´5~=ŠK§³#ÛsCbò±x@UÊDä÷?EŽþËsJ ²fHÉ‘ùØ3°R6.úº^JÛÓÂáïìJóý¯v.@»gòëONm>èµ?(KDN,r]áð·î.Ì»X¥çÀη's¬³KÆ@»DäôÂ]m¶x†Ùwûí¢Å‡€„÷øÛSGþù/fYµ‘Ÿ²›Pãàqß³?¿¿ñ£;(EDÎDjh…Eß`AD€DdX‘`AD€…Ú}‹|EÌ’jùw³·:w¿·:w³·:w¿·îw³·:w¿·:w³·:w¿·:w³·:w¿·:w³·:w¿·:w¹·RŒ"ÀB™QäšoÜûóùLãá’+cšGb-`AD€DdX‘`AD€]uo³ÚIDATDdX‘`AD€DdX‘`AD€DdX‘`AD€DdX‘`AD€DdX‘`AD€DdX‘`AD€DdX‘`AD€…?óÿ¼ßïRç•0Š "2,¼?ŸOés€ŠE€DdX‘`AD€DdX‘`AD€¹jï÷»ô) çOéèY8àNþ~¿ï8_šgž•‘±Š¹«PøýínþÆåþRr¯蘈œX8ϸ›¿¿ê)%wóB€qˆÈ)Íãàát‚žrpÀ|>IÙ3ˆäëz%™• P!9™SCȶ 9kù€ ‰È°àëzå]u,ˆ¿ý¯?9µ1@gDä2¯Õ0nûÀ‘ÕSHV¿Ý]žùæÆý1Ñ¢1»ùø\[íÔ3í<@DN¬¡pyêTz]7‰È£85S´ `d"rc>ÿ\øÛmð=õõ>ÉÀ DäÄJ¿F×X2À!¹mïC¼Ûµ)Ù÷ÔÆ]‘(2aJÆ1ó.¶²ï©ú#"·g ¬g—¤Ø;÷7茈œÞ£Qò׺ȑ¶Ù7<éÂD€‰ÈiÔ¼Úƒ…,N‘qv´õ‰õ(Â<7à9™ËsÂëQŸÏ. @node Markov Chains @chapter Markov Chains @menu * Discrete-Time Markov Chains:: * Continuous-Time Markov Chains:: @end menu @node Discrete-Time Markov Chains @section Discrete-Time Markov Chains Let @math{X_0, X_1, @dots{}, X_n, @dots{} } be a sequence of random variables defined over the discrete state space @math{1, 2, @dots{}}. The sequence @math{X_0, X_1, @dots{}, X_n, @dots{}} is a @emph{stochastic process} with discrete time @math{0, 1, 2, @dots{}}. A @emph{Markov chain} is a stochastic process @math{@{X_n, n=0, 1, @dots{}@}} which satisfies the following Markov property: @iftex @tex $$\eqalign{P\left(X_{n+1} = x_{n+1}\ |\ X_n = x_n, X_{n-1} = x_{n-1}, \ldots, X_0 = x_0 \right) \cr & = P\left(X_{n+1} = x_{n+1}\ |\ X_n = x_n\right)}$$ @end tex @end iftex @ifnottex @math{P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, @dots{}, X_0 = x_0) = P(X_{n+1} = x_{n+1} | X_n = x_n)} @end ifnottex @noindent which basically means that the probability that the system is in a particular state at time @math{n+1} only depends on the state the system was at time @math{n}. The evolution of a Markov chain with finite state space @math{@{1, @dots{}, N@}} can be fully described by a stochastic matrix @math{{\bf P}(n) = [ P_{i,j}(n) ]} where @math{P_{i, j}(n) = P( X_{n+1} = j\ |\ X_n = i )}. If the Markov chain is homogeneous (that is, the transition probability matrix @math{{\bf P}(n)} is time-independent), we can write @math{{\bf P} = [P_{i, j}]}, where @math{P_{i, j} = P( X_{n+1} = j\ |\ X_n = i )} for all @math{n=0, 1, @dots{}}. @cindex stochastic matrix The transition probability matrix @math{\bf P} must be a @emph{stochastic matrix}, meaning that it must satisfy the following two properties: @enumerate @item @math{P_{i, j} @geq{} 0} for all @math{1 @leq{} i, j @leq{} N}; @item @math{\sum_{j=1}^N P_{i,j} = 1} for all @math{i} @end enumerate Property 1 requires that all probabilities are nonnegative; property 2 requires that the outgoing transition probabilities from any state @math{i} sum to one. @c @anchor{doc-dtmcchkP} @deftypefn {Function File} {[@var{r} @var{err}] =} dtmcchkP (@var{P}) @cindex Markov chain, discrete time @cindex DTMC @cindex discrete time Markov chain Check whether @var{P} is a valid transition probability matrix. If @var{P} is valid, @var{r} is the size (number of rows or columns) of @var{P}. If @var{P} is not a transition probability matrix, @var{r} is set to zero, and @var{err} to an appropriate error string. @end deftypefn A DTMC is @emph{irreducible} if every state can be reached with non-zero probability starting from every other state. @anchor{doc-dtmcisir} @deftypefn {Function File} {[@var{r} @var{s}] =} dtmcisir (@var{P}) @cindex Markov chain, discrete time @cindex discrete time Markov chain @cindex DTMC @cindex irreducible Markov chain Check if @var{P} is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with transition matrix @var{P}. @strong{INPUTS} @table @code @item @var{P}(i,j) transition probability from state @math{i} to state @math{j}. @var{P} must be an @math{N \times N} stochastic matrix. @end table @strong{OUTPUTS} @table @code @item @var{r} 1 if @var{P} is irreducible, 0 otherwise (scalar) @item @var{s}(i) strongly connected component (SCC) that state @math{i} belongs to (vector of length @math{N}). SCCs are numbered @math{1, 2, @dots{}}. The number of SCCs is @code{max(s)}. If the graph is strongly connected, then there is a single SCC and the predicate @code{all(s == 1)} evaluates to true @end table @end deftypefn @menu * State occupancy probabilities (DTMC):: * Birth-death process (DTMC):: * Expected number of visits (DTMC):: * Time-averaged expected sojourn times (DTMC):: * Mean time to absorption (DTMC):: * First passage times (DTMC):: @end menu @c @c @c @node State occupancy probabilities (DTMC) @subsection State occupancy probabilities Given a discrete-time Markov chain with state space @math{@{1, @dots{}, N@}}, we denote with @math{{\bf \pi}(n) = \left[\pi_1(n), @dots{} \pi_N(n) \right]} the @emph{state occupancy probability vector} at step @math{n = 0, 1, @dots{}}. @math{\pi_i(n)} is the probability that the system is in state @math{i} after @math{n} transitions. Given the transition probability matrix @math{\bf P} and the initial state occupancy probability vector @math{{\bf \pi}(0) = \left[\pi_1(0), @dots{}, \pi_N(0)\right]}, @math{{\bf \pi}(n)} can be computed as: @iftex @tex $${\bf \pi}(n) = {\bf \pi}(0) {\bf P}^n$$ @end tex @end iftex @ifnottex @math{\pi(n) = \pi(0) P^n} @end ifnottex Under certain conditions, there exists a @emph{stationary state occupancy probability} @math{{\bf \pi} = \lim_{n \rightarrow +\infty} {\bf \pi}(n)}, which is independent from @math{{\bf \pi}(0)}. The vector @math{\bf \pi} is the solution of the following linear system: @iftex @tex $$ \left\{ \eqalign{ {\bf \pi P} & = {\bf \pi} \cr {\bf \pi 1}^T & = 1 } \right. $$ @end tex @end iftex @ifnottex @example @group / | \pi P = \pi | \pi 1^T = 1 \ @end group @end example @end ifnottex @noindent where @math{\bf 1} is the row vector of ones, and @math{( \cdot )^T} the transpose operator. @c @anchor{doc-dtmc} @deftypefn {Function File} {@var{p} =} dtmc (@var{P}) @deftypefnx {Function File} {@var{p} =} dtmc (@var{P}, @var{n}, @var{p0}) @cindex Markov chain, discrete time @cindex discrete time Markov chain @cindex DTMC @cindex Markov chain, stationary probabilities @cindex Markov chain, transient probabilities Compute stationary or transient state occupancy probabilities for a discrete-time Markov chain. With a single argument, compute the stationary state occupancy probabilities @code{@var{p}(1), @dots{}, @var{p}(N)} for a discrete-time Markov chain with finite state space @math{@{1, @dots{}, N@}} and with @math{N \times N} transition matrix @var{P}. With three arguments, compute the transient state occupancy probabilities @code{@var{p}(1), @dots{}, @var{p}(N)} that the system is in state @math{i} after @var{n} steps, given initial occupancy probabilities @var{p0}(1), @dots{}, @var{p0}(N). @strong{INPUTS} @table @code @item @var{P}(i,j) transition probabilities from state @math{i} to state @math{j}. @var{P} must be an @math{N \times N} irreducible stochastic matrix, meaning that the sum of each row must be 1 (@math{\sum_{j=1}^N P_{i, j} = 1}), and the rank of @var{P} must be @math{N}. @item @var{n} Number of transitions after which state occupancy probabilities are computed (scalar, @math{n @geq{} 0}) @item @var{p0}(i) probability that at step 0 the system is in state @math{i} (vector of length @math{N}). @end table @strong{OUTPUTS} @table @code @item @var{p}(i) If this function is called with a single argument, @code{@var{p}(i)} is the steady-state probability that the system is in state @math{i}. If this function is called with three arguments, @code{@var{p}(i)} is the probability that the system is in state @math{i} after @var{n} transitions, given the probabilities @code{@var{p0}(i)} that the initial state is @math{i}. @end table @seealso{ctmc} @end deftypefn @noindent @strong{EXAMPLE} The following example is from @ref{GrSn97}. Let us consider a maze with nine rooms, as shown in the following figure @example @group +-----+-----+-----+ | | | | | 1 2 3 | | | | | +- -+- -+- -+ | | | | | 4 5 6 | | | | | +- -+- -+- -+ | | | | | 7 8 9 | | | | | +-----+-----+-----+ @end group @end example A mouse is placed in one of the rooms and can wander around. At each step, the mouse moves from the current room to a neighboring one with equal probability. For example, if it is in room 1, it can move to room 2 and 4 with probability @math{1/2}, respectively; if the mouse is in room 8, it can move to either 7, 5 or 9 with probability @math{1/3}. The transition probabilities @math{P_{i, j}} from room @math{i} to room @math{j} can be summarized in the following matrix: @iftex @tex $$ {\bf P} = \pmatrix{ 0 & 1/2 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 \cr 1/3 & 0 & 1/3 & 0 & 1/3 & 0 & 0 & 0 & 0 \cr 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 \cr 1/3 & 0 & 0 & 0 & 1/3 & 0 & 1/3 & 0 & 0 \cr 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0 \cr 0 & 0 & 1/3 & 0 & 1/3 & 0 & 0 & 0 & 1/3 \cr 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 \cr 0 & 0 & 0 & 0 & 1/3 & 0 & 1/3 & 0 & 1/3 \cr 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 1/2 & 0 } $$ @end tex @end iftex @ifnottex @example @group / 0 1/2 0 1/2 0 0 0 0 0 \ | 1/3 0 1/3 0 1/3 0 0 0 0 | | 0 1/2 0 0 0 1/2 0 0 0 | | 1/3 0 0 0 1/3 0 1/3 0 0 | P = | 0 1/4 0 1/4 0 1/4 0 1/4 0 | | 0 0 1/3 0 1/3 0 0 0 1/3 | | 0 0 0 1/2 0 0 0 1/2 0 | | 0 0 0 0 1/3 0 1/3 0 1/3 | \ 0 0 0 0 0 1/2 0 1/2 0 / @end group @end example @end ifnottex The stationary state occupancy probabilities can then be computed with the following code: @example @group @verbatim P = zeros(9,9); P(1,[2 4] ) = 1/2; P(2,[1 5 3] ) = 1/3; P(3,[2 6] ) = 1/2; P(4,[1 5 7] ) = 1/3; P(5,[2 4 6 8]) = 1/4; P(6,[3 5 9] ) = 1/3; P(7,[4 8] ) = 1/2; P(8,[7 5 9] ) = 1/3; P(9,[6 8] ) = 1/2; p = dtmc(P); disp(p) @end verbatim @end group @result{} 0.083333 0.125000 0.083333 0.125000 0.166667 0.125000 0.083333 0.125000 0.083333 @end example @c @node Birth-death process (DTMC) @subsection Birth-death process @anchor{doc-dtmcbd} @deftypefn {Function File} {@var{P} =} dtmcbd (@var{b}, @var{d}) @cindex Markov chain, discrete time @cindex DTMC @cindex discrete time Markov chain @cindex birth-death process, DTMC Returns the transition probability matrix @math{P} for a discrete birth-death process over state space @math{@{1, @dots{}, N@}}. For each @math{i=1, @dots{}, (N-1)}, @code{@var{b}(i)} is the transition probability from state @math{i} to @math{(i+1)}, and @code{@var{d}(i)} is the transition probability from state @math{(i+1)} to @math{i}. Matrix @math{\bf P} is defined as: @tex $$ \pmatrix{ (1-\lambda_1) & \lambda_1 & & & & \cr \mu_1 & (1 - \mu_1 - \lambda_2) & \lambda_2 & & \cr & \mu_2 & (1 - \mu_2 - \lambda_3) & \lambda_3 & & \cr \cr & & \ddots & \ddots & \ddots & & \cr \cr & & & \mu_{N-2} & (1 - \mu_{N-2}-\lambda_{N-1}) & \lambda_{N-1} \cr & & & & \mu_{N-1} & (1-\mu_{N-1}) } $$ @end tex @ifnottex @example @group / \ | 1-b(1) b(1) | | d(1) (1-d(1)-b(2)) b(2) | | d(2) (1-d(2)-b(3)) b(3) | | | | ... ... ... | | | | d(N-2) (1-d(N-2)-b(N-1)) b(N-1) | | d(N-1) 1-d(N-1) | \ / @end group @end example @end ifnottex @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and death probabilities, respectively. @seealso{ctmcbd} @end deftypefn @c @node Expected number of visits (DTMC) @subsection Expected Number of Visits Given a @math{N} state discrete-time Markov chain with transition matrix @math{\bf P} and an integer @math{n @geq{} 0}, we let @math{L_i(n)} be the the expected number of visits to state @math{i} during the first @math{n} transitions. The vector @math{{\bf L}(n) = \left[ L_1(n), @dots{}, L_N(n) \right]} is defined as @iftex @tex $$ {\bf L}(n) = \sum_{i=0}^n {\bf \pi}(i) = \sum_{i=0}^n {\bf \pi}(0) {\bf P}^i $$ @end tex @end iftex @ifnottex @example @group n n ___ ___ \ \ i L(n) = > pi(i) = > pi(0) P /___ /___ i=0 i=0 @end group @end example @end ifnottex @noindent where @math{{\bf \pi}(i) = {\bf \pi}(0){\bf P}^i} is the state occupancy probability after @math{i} transitions, and @math{{\bf \pi}(0) = \left[\pi_1(0), @dots{}, \pi_N(0) \right]} are the initial state occupancy probabilities. If @math{\bf P} is absorbing, i.e., the stochastic process eventually enters a state with no outgoing transitions, then we can compute the expected number of visits until absorption @math{\bf L}. To do so, we first rearrange the states by rewriting @math{\bf P} as @iftex @tex $$ {\bf P} = \pmatrix{ {\bf Q} & {\bf R} \cr {\bf 0} & {\bf I} }$$ @end tex @end iftex @ifnottex @example @group / Q | R \ P = |---+---| \ 0 | I / @end group @end example @end ifnottex @noindent where the first @math{t} states are transient and the last @math{r} states are absorbing (@math{t+r = N}). The matrix @math{{\bf N} = ({\bf I} - {\bf Q})^{-1}} is called the @emph{fundamental matrix}; @math{N_{i,j}} is the expected number of times the process is in the @math{j}-th transient state assuming it started in the @math{i}-th transient state. If we reshape @math{\bf N} to the size of @math{\bf P} (filling missing entries with zeros), we have that, for absorbing chains, @math{{\bf L} = {\bf \pi}(0){\bf N}}. @anchor{doc-dtmcexps} @deftypefn {Function File} {@var{L} =} dtmcexps (@var{P}, @var{n}, @var{p0}) @deftypefnx {Function File} {@var{L} =} dtmcexps (@var{P}, @var{p0}) @cindex expected sojourn times, DTMC @cindex DTMC @cindex discrete time Markov chain @cindex Markov chain, discrete time Compute the expected number of visits to each state during the first @var{n} transitions, or until abrosption. @strong{INPUTS} @table @code @item @var{P}(i,j) @math{N \times N} transition matrix. @code{@var{P}(i,j)} is the transition probability from state @math{i} to state @math{j}. @item @var{n} Number of steps during which the expected number of visits are computed (@math{@var{n} @geq{} 0}). If @code{@var{n}=0}, returns @var{p0}. If @code{@var{n} > 0}, returns the expected number of visits after exactly @var{n} transitions. @item @var{p0}(i) Initial state occupancy probabilities; @code{@var{p0}(i)} is the probability that the system is in state @math{i} at step 0. @end table @strong{OUTPUTS} @table @code @item @var{L}(i) When called with two arguments, @code{@var{L}(i)} is the expected number of visits to state @math{i} before absorption. When called with three arguments, @code{@var{L}(i)} is the expected number of visits to state @math{i} during the first @var{n} transitions. @end table @strong{REFERENCES} @itemize @item Grinstead, Charles M.; Snell, J. Laurie (July 1997). @cite{Introduction to Probability}, Ch. 11: Markov Chains. American Mathematical Society. ISBN 978-0821807491. @end itemize @seealso{ctmcexps} @end deftypefn @c @node Time-averaged expected sojourn times (DTMC) @subsection Time-averaged expected sojourn times @anchor{doc-dtmctaexps} @deftypefn {Function File} {@var{M} =} dtmctaexps (@var{P}, @var{n}, @var{p0}) @deftypefnx {Function File} {@var{M} =} dtmctaexps (@var{P}, @var{p0}) @cindex time-alveraged sojourn time, DTMC @cindex discrete time Markov chain @cindex Markov chain, discrete time @cindex DTMC Compute the @emph{time-averaged sojourn times} @code{@var{M}(i)}, defined as the fraction of time spent in state @math{i} during the first @math{n} transitions (or until absorption), assuming that the state occupancy probabilities at time 0 are @var{p0}. @strong{INPUTS} @table @code @item @var{P}(i,j) @math{N \times N} transition probability matrix. @item @var{n} Number of transitions during which the time-averaged expected sojourn times are computed (scalar, @math{@var{n} @geq{} 0}). if @math{@var{n} = 0}, returns @var{p0}. @item @var{p0}(i) Initial state occupancy probabilities (vector of length @math{N}). @end table @strong{OUTPUTS} @table @code @item @var{M}(i) If this function is called with three arguments, @code{@var{M}(i)} is the expected fraction of steps @math{@{0, @dots{}, n@}} spent in state @math{i}, assuming that the state occupancy probabilities at time zero are @var{p0}. If this function is called with two arguments, @code{@var{M}(i)} is the expected fraction of steps spent in state @math{i} until absorption. @var{M} is a vector of length @math{N}. @end table @seealso{dtmcexps} @end deftypefn @c @node Mean time to absorption (DTMC) @subsection Mean Time to Absorption The @emph{mean time to absorption} is defined as the average number of transitions that are required to enter an absorbing state, starting from a transient state or given initial state occupancy probabilities @math{{\bf \pi}(0)}. Let @math{t_i} be the expected number of transitions before being absorbed in any absorbing state, starting from state @math{i}. The vector @math{{\bf t} = [t_1, @dots{}, t_N]} can be computed from the fundamental matrix @math{\bf N} (@pxref{Expected number of visits (DTMC)}) as @iftex @tex $$ {\bf t} = {\bf N c} $$ @end tex @end iftex @ifnottex @math{t = N c} @end ifnottex @noindent where @math{\bf c} is a column vector of 1's. Let @math{{\bf B} = [ B_{i, j} ]} be a matrix where @math{B_{i, j}} is the probability of being absorbed in state @math{j}, starting from transient state @math{i}. Again, using matrices @math{\bf N} and @math{\bf R} (@pxref{Expected number of visits (DTMC)}) we can write @iftex @tex $$ {\bf B} = {\bf N R} $$ @end tex @end iftex @ifnottex @math{B = N R} @end ifnottex @anchor{doc-dtmcmtta} @deftypefn {Function File} {[@var{t} @var{N} @var{B}] =} dtmcmtta (@var{P}) @deftypefnx {Function File} {[@var{t} @var{N} @var{B}] =} dtmcmtta (@var{P}, @var{p0}) @cindex mean time to absorption, DTMC @cindex absorption probabilities, DTMC @cindex fundamental matrix @cindex DTMC @cindex discrete time Markov chain @cindex Markov chain, discrete time Compute the expected number of steps before absorption for a DTMC with state space @math{@{1, @dots{}, N@}} and transition probability matrix @var{P}. @strong{INPUTS} @table @code @item @var{P}(i,j) @math{N \times N} transition probability matrix. @code{@var{P}(i,j)} is the transition probability from state @math{i} to state @math{j}. @item @var{p0}(i) Initial state occupancy probabilities (vector of length @math{N}). @end table @strong{OUTPUTS} @table @code @item @var{t} @itemx @var{t}(i) When called with a single argument, @var{t} is a vector of length @math{N} such that @code{@var{t}(i)} is the expected number of steps before being absorbed in any absorbing state, starting from state @math{i}; if @math{i} is absorbing, @code{@var{t}(i) = 0}. When called with two arguments, @var{t} is a scalar, and represents the expected number of steps before absorption, starting from the initial state occupancy probability @var{p0}. @item @var{N}(i) @itemx @var{N}(i,j) When called with a single argument, @var{N} is the @math{N \times N} fundamental matrix for @var{P}. @code{@var{N}(i,j)} is the expected number of visits to transient state @var{j} before absorption, if the system started in transient state @var{i}. The initial state is counted if @math{i = j}. When called with two arguments, @var{N} is a vector of length @math{N} such that @code{@var{N}(j)} is the expected number of visits to transient state @var{j} before absorption, given initial state occupancy probability @var{P0}. @item @var{B}(i) @itemx @var{B}(i,j) When called with a single argument, @var{B} is a @math{N \times N} matrix where @code{@var{B}(i,j)} is the probability of being absorbed in state @math{j}, starting from transient state @math{i}; if @math{j} is not absorbing, @code{@var{B}(i,j) = 0}; if @math{i} is absorbing, @code{@var{B}(i,i) = 1} and @code{@var{B}(i,j) = 0} for all @math{i \neq j}. When called with two arguments, @var{B} is a vector of length @math{N} where @code{@var{B}(j)} is the probability of being absorbed in state @var{j}, given initial state occupancy probabilities @var{p0}. @end table @strong{REFERENCES} @itemize @item Grinstead, Charles M.; Snell, J. Laurie (July 1997). @cite{Introduction to Probability}, Ch. 11: Markov Chains. American Mathematical Society. ISBN 978-0821807491. @end itemize @seealso{ctmcmtta} @end deftypefn @c @node First passage times (DTMC) @subsection First Passage Times The First Passage Time @math{M_{i, j}} is the average number of transitions needed to enter state @math{j} for the first time, starting from state @math{i}. Matrix @math{\bf M} satisfies the property @iftex @tex $$ M_{i, j} = 1 + \sum_{k \neq j} P_{i, k} M_{k, j}$$ @end tex @end iftex @ifnottex @example @group ___ \ M_ij = 1 + > P_ij * M_kj /___ k!=j @end group @end example @end ifnottex To compute @math{{\bf M} = [ M_{i, j}]} a different formulation is used. Let @math{\bf W} be the @math{N \times N} matrix having each row equal to the stationary state occupancy probability vector @math{\bf \pi} for @math{\bf P}; let @math{\bf I} be the @math{N \times N} identity matrix (i.e., the matrix of all ones). Define @math{\bf Z} as follows: @iftex @tex $$ {\bf Z} = \left( {\bf I} - {\bf P} + {\bf W} \right)^{-1} $$ @end tex @end iftex @ifnottex @example @group -1 Z = (I - P + W) @end group @end example @end ifnottex @noindent Then, we have that @iftex @tex $$ M_{i, j} = {Z_{j, j} - Z_{i, j} \over \pi_j} $$ @end tex @end iftex @ifnottex @example @group Z_jj - Z_ij M_ij = ----------- \pi_j @end group @end example @end ifnottex According to the definition above, @math{M_{i,i} = 0}. We arbitrarily set @math{M_{i,i}} to the @emph{mean recurrence time} @math{r_i} for state @math{i}, that is the average number of transitions needed to return to state @math{i} starting from it. @math{r_i} is: @iftex @tex $$ r_i = {1 \over \pi_i} $$ @end tex @end iftex @ifnottex @example @group 1 r_i = ----- \pi_i @end group @end example @end ifnottex @anchor{doc-dtmcfpt} @deftypefn {Function File} {@var{M} =} dtmcfpt (@var{P}) @cindex first passage times @cindex mean recurrence times @cindex discrete time Markov chain @cindex Markov chain, discrete time @cindex DTMC Compute mean first passage times and mean recurrence times for an irreducible discrete-time Markov chain over the state space @math{@{1, @dots{}, N@}}. @strong{INPUTS} @table @code @item @var{P}(i,j) transition probability from state @math{i} to state @math{j}. @var{P} must be an irreducible stochastic matrix, which means that the sum of each row must be 1 (@math{\sum_{j=1}^N P_{i j} = 1}), and the rank of @var{P} must be @math{N}. @end table @strong{OUTPUTS} @table @code @item @var{M}(i,j) For all @math{1 @leq{} i, j @leq{} N}, @math{i \neq j}, @code{@var{M}(i,j)} is the average number of transitions before state @var{j} is entered for the first time, starting from state @var{i}. @code{@var{M}(i,i)} is the @emph{mean recurrence time} of state @math{i}, and represents the average time needed to return to state @var{i}. @end table @strong{REFERENCES} @itemize @item Grinstead, Charles M.; Snell, J. Laurie (July 1997). @cite{Introduction to Probability}, Ch. 11: Markov Chains. American Mathematical Society. ISBN 978-0821807491. @end itemize @seealso{ctmcfpt} @end deftypefn @c @c @c @node Continuous-Time Markov Chains @section Continuous-Time Markov Chains A stochastic process @math{@{X(t), t @geq{} 0@}} is a continuous-time Markov chain if, for all integers @math{n}, and for any sequence @math{t_0, t_1 , @dots{}, t_n, t_{n+1}} such that @math{t_0 < t_1 < @dots{} < t_n < t_{n+1}}, we have @iftex @tex $$\eqalign{P(X(t_{n+1}) = x_{n+1}\ |\ X(t_n) = x_n, X(t_{n-1}) = x_{n-1}, \ldots, X(t_0) = x_0) \cr &\hskip -8 em = P(X(t_{n+1}) = x_{n+1}\ |\ X(t_n) = x_n)}$$ @end tex @end iftex @ifnottex @math{P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, ..., X_0 = x_0) = P(X_{n+1} = x_{n+1} | X_n = x_n)} @end ifnottex A continuous-time Markov chain is defined according to an @emph{infinitesimal generator matrix} @math{{\bf Q} = [Q_{i,j}]}, where for each @math{i \neq j}, @math{Q_{i, j}} is the transition rate from state @math{i} to state @math{j}. The matrix @math{\bf Q} must satisfy the property that, for all @math{i}, @math{\sum_{j=1}^N Q_{i, j} = 0}. @anchor{doc-ctmcchkQ} @deftypefn {Function File} {[@var{result} @var{err}] =} ctmcchkQ (@var{Q}) @cindex Markov chain, continuous time If @var{Q} is a valid infinitesimal generator matrix, return the size (number of rows or columns) of @var{Q}. If @var{Q} is not an infinitesimal generator matrix, set @var{result} to zero, and @var{err} to an appropriate error string. @end deftypefn Similarly to the DTMC case, a CTMC is @emph{irreducible} if every state is eventually reachable from every other state in finite time. @anchor{doc-ctmcisir} @deftypefn {Function File} {[@var{r} @var{s}] =} ctmcisir (@var{P}) @cindex Markov chain, continuous time @cindex continuous time Markov chain @cindex CTMC @cindex irreducible Markov chain Check if @var{Q} is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with infinitesimal generator matrix @var{Q}. @strong{INPUTS} @table @code @item @var{Q}(i,j) Infinitesimal generator matrix. @var{Q} is a @math{N \times N} square matrix where @code{@var{Q}(i,j)} is the transition rate from state @math{i} to state @math{j}, for @math{1 @leq{} i, j @leq{} N}, @math{i \neq j}. @end table @strong{OUTPUTS} @table @code @item @var{r} 1 if @var{Q} is irreducible, 0 otherwise. @item @var{s}(i) strongly connected component (SCC) that state @math{i} belongs to. SCCs are numbered @math{1, 2, @dots{}}. If the graph is strongly connected, then there is a single SCC and the predicate @code{all(s == 1)} evaluates to true. @end table @end deftypefn @menu * State occupancy probabilities (CTMC):: * Birth-death process (CTMC):: * Expected sojourn times (CTMC):: * Time-averaged expected sojourn times (CTMC):: * Mean time to absorption (CTMC):: * First passage times (CTMC):: @end menu @node State occupancy probabilities (CTMC) @subsection State occupancy probabilities Similarly to the discrete case, we denote with @math{{\bf \pi}(t) = \left[\pi_1(t), @dots{}, \pi_N(t) \right]} the @emph{state occupancy probability vector} at time @math{t}. @math{\pi_i(t)} is the probability that the system is in state @math{i} at time @math{t @geq{} 0}. Given the infinitesimal generator matrix @math{\bf Q} and initial state occupancy probabilities @math{{\bf \pi}(0) = \left[\pi_1(0), @dots{}, \pi_N(0)\right]}, the occupancy probabilities @math{{\bf \pi}(t)} at time @math{t} can be computed as: @iftex @tex $${\bf \pi}(t) = {\bf \pi}(0) \exp( {\bf Q} t )$$ @end tex @end iftex @ifnottex @example @group \pi(t) = \pi(0) exp(Qt) @end group @end example @end ifnottex @noindent where @math{\exp( {\bf Q} t )} is the matrix exponential of @math{{\bf Q} t}. Under certain conditions, there exists a @emph{stationary state occupancy probability} @math{{\bf \pi} = \lim_{t \rightarrow +\infty} {\bf \pi}(t)} that is independent from @math{{\bf \pi}(0)}. @math{\bf \pi} is the solution of the following linear system: @iftex @tex $$ \left\{ \eqalign{ {\bf \pi Q} & = {\bf 0} \cr {\bf \pi 1}^T & = 1 } \right. $$ @end tex @end iftex @ifnottex @example @group / | \pi Q = 0 | \pi 1^T = 1 \ @end group @end example @end ifnottex @anchor{doc-ctmc} @deftypefn {Function File} {@var{p} =} ctmc (@var{Q}) @deftypefnx {Function File} {@var{p} =} ctmc (@var{Q}, @var{t}, @var{p0}) @cindex Markov chain, continuous time @cindex continuous time Markov chain @cindex Markov chain, state occupancy probabilities @cindex stationary probabilities @cindex CTMC Compute stationary or transient state occupancy probabilities for a continuous-time Markov chain. With a single argument, compute the stationary state occupancy probabilities @math{@var{p}(1), @dots{}, @var{p}(N)} for a continuous-time Markov chain with finite state space @math{@{1, @dots{}, N@}} and @math{N \times N} infinitesimal generator matrix @var{Q}. With three arguments, compute the state occupancy probabilities @math{@var{p}(1), @dots{}, @var{p}(N)} that the system is in state @math{i} at time @var{t}, given initial state occupancy probabilities @math{@var{p0}(1), @dots{}, @var{p0}(N)} at time 0. @strong{INPUTS} @table @code @item @var{Q}(i,j) Infinitesimal generator matrix. @var{Q} is a @math{N \times N} square matrix where @code{@var{Q}(i,j)} is the transition rate from state @math{i} to state @math{j}, for @math{1 @leq{} i \neq j @leq{} N}. @var{Q} must satisfy the property that @math{\sum_{j=1}^N Q_{i, j} = 0} @item @var{t} Time at which to compute the transient probability (@math{t @geq{} 0}). If omitted, the function computes the steady state occupancy probability vector. @item @var{p0}(i) probability that the system is in state @math{i} at time 0. @end table @strong{OUTPUTS} @table @code @item @var{p}(i) If this function is invoked with a single argument, @code{@var{p}(i)} is the steady-state probability that the system is in state @math{i}, @math{i = 1, @dots{}, N}. If this function is invoked with three arguments, @code{@var{p}(i)} is the probability that the system is in state @math{i} at time @var{t}, given the initial occupancy probabilities @var{p0}(1), @dots{}, @var{p0}(N). @end table @seealso{dtmc} @end deftypefn @noindent @strong{EXAMPLE} Consider a two-state CTMC where all transition rates between states are equal to 1. The stationary state occupancy probabilities can be computed as follows: @example @group @verbatim Q = [ -1 1; ... 1 -1 ]; q = ctmc(Q) @end verbatim @result{} q = 0.50000 0.50000 @end group @end example @c @c @c @node Birth-death process (CTMC) @subsection Birth-Death Process @anchor{doc-ctmcbd} @deftypefn {Function File} {@var{Q} =} ctmcbd (@var{b}, @var{d}) @cindex Markov chain, continuous time @cindex continuous time Markov chain @cindex CTMC @cindex birth-death process, CTMC Returns the infinitesimal generator matrix @math{Q} for a continuous birth-death process over the finite state space @math{@{1, @dots{}, N@}}. For each @math{i=1, @dots{}, (N-1)}, @code{@var{b}(i)} is the transition rate from state @math{i} to state @math{(i+1)}, and @code{@var{d}(i)} is the transition rate from state @math{(i+1)} to state @math{i}. Matrix @math{\bf Q} is therefore defined as: @tex $$ \pmatrix{ -\lambda_1 & \lambda_1 & & & & \cr \mu_1 & -(\mu_1 + \lambda_2) & \lambda_2 & & \cr & \mu_2 & -(\mu_2 + \lambda_3) & \lambda_3 & & \cr \cr & & \ddots & \ddots & \ddots & & \cr \cr & & & \mu_{N-2} & -(\mu_{N-2}+\lambda_{N-1}) & \lambda_{N-1} \cr & & & & \mu_{N-1} & -\mu_{N-1} } $$ @end tex @ifnottex @example @group / \ | -b(1) b(1) | | d(1) -(d(1)+b(2)) b(2) | | d(2) -(d(2)+b(3)) b(3) | | | | ... ... ... | | | | d(N-2) -(d(N-2)+b(N-1)) b(N-1) | | d(N-1) -d(N-1) | \ / @end group @end example @end ifnottex @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and death rates, respectively. @seealso{dtmcbd} @end deftypefn @c @c @c @node Expected sojourn times (CTMC) @subsection Expected Sojourn Times Given a @math{N} state continuous-time Markov Chain with infinitesimal generator matrix @math{\bf Q}, we define the vector @math{{\bf L}(t) = \left[L_1(t), @dots{}, L_N(t)\right]} such that @math{L_i(t)} is the expected sojourn time in state @math{i} during the interval @math{[0,t)}, assuming that the initial occupancy probabilities at time 0 were @math{{\bf \pi}(0)}. @math{{\bf L}(t)} can be expressed as the solution of the following differential equation: @iftex @tex $$ { d{\bf L}(t) \over dt} = {\bf L}(t){\bf Q} + {\bf \pi}(0), \qquad {\bf L}(0) = {\bf 0} $$ @end tex @end iftex @ifnottex @example @group dL --(t) = L(t) Q + pi(0), L(0) = 0 dt @end group @end example @end ifnottex Alternatively, @math{{\bf L}(t)} can also be expressed in integral form as: @iftex @tex $$ {\bf L}(t) = \int_0^t {\bf \pi}(u) du$$ @end tex @end iftex @ifnottex @example @group / t L(t) = | pi(u) du / 0 @end group @end example @end ifnottex @noindent where @math{{\bf \pi}(t) = {\bf \pi}(0) \exp({\bf Q}t)} is the state occupancy probability at time @math{t}; @math{\exp({\bf Q}t)} is the matrix exponential of @math{{\bf Q}t}. If there are absorbing states, we can define the vector of @emph{expected sojourn times until absorption} @math{{\bf L}(\infty)}, where for each transient state @math{i}, @math{L_i(\infty)} is the expected total time spent in state @math{i} until absorption, assuming that the system started with given state occupancy probabilities @math{{\bf \pi}(0)}. Let @math{\tau} be the set of transient (i.e., non absorbing) states; let @math{{\bf Q}_\tau} be the restriction of @math{\bf Q} to the transient sub-states only. Similarly, let @math{{\bf \pi}_\tau(0)} be the restriction of the initial state occupancy probability vector @math{{\bf \pi}(0)} to transient states @math{\tau}. The expected time to absorption @math{{\bf L}_\tau(\infty)} is defined as the solution of the following equation: @iftex @tex $$ {\bf L}_\tau(\infty){\bf Q}_\tau = -{\bf \pi}_\tau(0) $$ @end tex @end iftex @ifnottex @example @group L_T( inf ) Q_T = -pi_T(0) @end group @end example @end ifnottex @anchor{doc-ctmcexps} @deftypefn {Function File} {@var{L} =} ctmcexps (@var{Q}, @var{t}, @var{p} ) @deftypefnx {Function File} {@var{L} =} ctmcexps (@var{Q}, @var{p}) @cindex Markov chain, continuous time @cindex continuous time Markov chain @cindex expected sojourn time, CTMC @cindex CTMC With three arguments, compute the expected times @code{@var{L}(i)} spent in each state @math{i} during the time interval @math{[0,t]}, assuming that the initial occupancy vector is @var{p}. With two arguments, compute the expected time @code{@var{L}(i)} spent in each transient state @math{i} until absorption. @strong{Note:} In its current implementation, this function requires that an absorbing state is reachable from any non-absorbing state of @math{Q}. @strong{INPUTS} @table @code @item @var{Q}(i,j) @math{N \times N} infinitesimal generator matrix. @code{@var{Q}(i,j)} is the transition rate from state @math{i} to state @math{j}, @math{1 @leq{} i, j @leq{} N}, @math{i \neq j}. The matrix @var{Q} must also satisfy the condition @math{\sum_{j=1}^N Q_{i,j} = 0} for every @math{i=1, @dots{}, N}. @item @var{t} If given, compute the expected sojourn times in @math{[0,t]} @item @var{p}(i) Initial occupancy probability vector; @code{@var{p}(i)} is the probability the system is in state @math{i} at time 0, @math{i = 1, @dots{}, N} @end table @strong{OUTPUTS} @table @code @item @var{L}(i) If this function is called with three arguments, @code{@var{L}(i)} is the expected time spent in state @math{i} during the interval @math{[0,t]}. If this function is called with two arguments @code{@var{L}(i)} is the expected time spent in transient state @math{i} until absorption; if state @math{i} is absorbing, @code{@var{L}(i)} is zero. @end table @seealso{dtmcexps} @end deftypefn @noindent @strong{EXAMPLE} Let us consider a 4-states pure birth continuous process where the transition rate from state @math{i} to state @math{(i+1)} is @math{\lambda_i = i \lambda} (@math{i=1, 2, 3}), with @math{\lambda = 0.5}. The following code computes the expected sojourn time for each state @math{i}, given initial occupancy probabilities @math{{\bf \pi}_0=[1, 0, 0, 0]}. @example @group @verbatim lambda = 0.5; N = 4; b = lambda*[1:N-1]; d = zeros(size(b)); Q = ctmcbd(b,d); t = linspace(0,10,100); p0 = zeros(1,N); p0(1)=1; L = zeros(length(t),N); for i=1:length(t) L(i,:) = ctmcexps(Q,t(i),p0); endfor plot( t, L(:,1), ";State 1;", "linewidth", 2, ... t, L(:,2), ";State 2;", "linewidth", 2, ... t, L(:,3), ";State 3;", "linewidth", 2, ... t, L(:,4), ";State 4;", "linewidth", 2 ); legend("location","northwest"); legend("boxoff"); xlabel("Time"); ylabel("Expected sojourn time"); @end verbatim @end group @end example @c @c @c @node Time-averaged expected sojourn times (CTMC) @subsection Time-Averaged Expected Sojourn Times @anchor{doc-ctmctaexps} @deftypefn {Function File} {@var{M} =} ctmctaexps (@var{Q}, @var{t}, @var{p0}) @deftypefnx {Function File} {@var{M} =} ctmctaexps (@var{Q}, @var{p0}) @cindex Markov chain, continuous time @cindex time-alveraged sojourn time, CTMC @cindex continuous time Markov chain @cindex CTMC Compute the @emph{time-averaged sojourn time} @code{@var{M}(i)}, defined as the fraction of the time interval @math{[0,t]} (or until absorption) spent in state @math{i}, assuming that the state occupancy probabilities at time 0 are @var{p}. @strong{INPUTS} @table @code @item @var{Q}(i,j) Infinitesimal generator matrix. @code{@var{Q}(i,j)} is the transition rate from state @math{i} to state @math{j}, @math{1 @leq{} i,j @leq{} N}, @math{i \neq j}. The matrix @var{Q} must also satisfy the condition @math{\sum_{j=1}^N Q_{i,j} = 0} @item @var{t} Time. If omitted, the results are computed until absorption. @item @var{p0}(i) initial state occupancy probabilities. @code{@var{p0}(i)} is the probability that the system is in state @math{i} at time 0, @math{i = 1, @dots{}, N} @end table @strong{OUTPUTS} @table @code @item @var{M}(i) When called with three arguments, @code{@var{M}(i)} is the expected fraction of the interval @math{[0,t]} spent in state @math{i} assuming that the state occupancy probability at time zero is @var{p}. When called with two arguments, @code{@var{M}(i)} is the expected fraction of time until absorption spent in state @math{i}; in this case the mean time to absorption is @code{sum(@var{M})}. @end table @seealso{ctmcexps} @end deftypefn @noindent @strong{EXAMPLE} @example @group @verbatim lambda = 0.5; N = 4; birth = lambda*linspace(1,N-1,N-1); death = zeros(1,N-1); Q = diag(birth,1)+diag(death,-1); Q -= diag(sum(Q,2)); t = linspace(1e-5,30,100); p = zeros(1,N); p(1)=1; M = zeros(length(t),N); for i=1:length(t) M(i,:) = ctmctaexps(Q,t(i),p); endfor clf; plot(t, M(:,1), ";State 1;", "linewidth", 2, ... t, M(:,2), ";State 2;", "linewidth", 2, ... t, M(:,3), ";State 3;", "linewidth", 2, ... t, M(:,4), ";State 4 (absorbing);", "linewidth", 2 ); legend("location","east"); legend("boxoff"); xlabel("Time"); ylabel("Time-averaged Expected sojourn time"); @end verbatim @end group @end example @c @c @c @node Mean time to absorption (CTMC) @subsection Mean Time to Absorption @anchor{doc-ctmcmtta} @deftypefn {Function File} {@var{t} =} ctmcmtta (@var{Q}, @var{p}) @cindex Markov chain, continuous time @cindex continuous time Markov chain @cindex CTMC @cindex mean time to absorption, CTMC Compute the Mean-Time to Absorption (MTTA) of the CTMC described by the infinitesimal generator matrix @var{Q}, starting from initial occupancy probabilities @var{p}. If there are no absorbing states, this function fails with an error. @strong{INPUTS} @table @code @item @var{Q}(i,j) @math{N \times N} infinitesimal generator matrix. @code{@var{Q}(i,j)} is the transition rate from state @math{i} to state @math{j}, @math{i \neq j}. The matrix @var{Q} must satisfy the condition @math{\sum_{j=1}^N Q_{i,j} = 0} @item @var{p}(i) probability that the system is in state @math{i} at time 0, for each @math{i=1, @dots{}, N} @end table @strong{OUTPUTS} @table @code @item @var{t} Mean time to absorption of the process represented by matrix @var{Q}. If there are no absorbing states, this function fails. @end table @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998. @end itemize @seealso{ctmcexps} @end deftypefn @noindent @strong{EXAMPLE} Let us consider a simple model of redundant disk array. We assume that the array is made of 5 independent disks and can tolerate up to 2 disk failures without losing data. If three or more disks break, the array is dead and unrecoverable. We want to estimate the Mean-Time-To-Failure (MTTF) of the disk array. We model this system as a 4 states continuous Markov chain with state space @math{@{ 2, 3, 4, 5 @}}. In state @math{i} there are exactly @math{i} active (i.e., non failed) disks; state @math{2} is absorbing. Let @math{\mu} be the failure rate of a single disk. The system starts in state @math{5} (all disks are operational). We use a pure death process, where the death rate from state @math{i} to state @math{(i-1)} is @math{\mu i}, for @math{i = 3, 4, 5}). The MTTF of the disk array is the MTTA of the Markov Chain, and can be computed as follows: @example @group @verbatim mu = 0.01; death = [ 3 4 5 ] * mu; birth = 0*death; Q = ctmcbd(birth,death); t = ctmcmtta(Q,[0 0 0 1]) @end verbatim @result{} t = 78.333 @end group @end example @c @c @c @node First passage times (CTMC) @subsection First Passage Times @anchor{doc-ctmcfpt} @deftypefn {Function File} {@var{M} =} ctmcfpt (@var{Q}) @deftypefnx {Function File} {@var{m} =} ctmcfpt (@var{Q}, @var{i}, @var{j}) @cindex first passage times, CTMC @cindex CTMC @cindex continuous time Markov chain @cindex Markov chain, continuous time Compute mean first passage times for an irreducible continuous-time Markov chain. @strong{INPUTS} @table @code @item @var{Q}(i,j) Infinitesimal generator matrix. @var{Q} is a @math{N \times N} square matrix where @code{@var{Q}(i,j)} is the transition rate from state @math{i} to state @math{j}, for @math{1 @leq{} i, j @leq{} N}, @math{i \neq j}. Transition rates must be nonnegative, and @math{\sum_{j=1}^N Q_{i,j} = 0} @item @var{i} Initial state. @item @var{j} Destination state. @end table @strong{OUTPUTS} @table @code @item @var{M}(i,j) average time before state @var{j} is visited for the first time, starting from state @var{i}. We let @code{@var{M}(i,i) = 0}. @item m @var{m} is the average time before state @var{j} is visited for the first time, starting from state @var{i}. @end table @seealso{ctmcmtta} @end deftypefn queueing/doc/references.texi0000644000175000017500000001541413635377222016060 0ustar morenomoreno@c This file has been automatically generated from references.txi @c by proc.m. Do not edit this file, all changes will be lost @c -*- texinfo -*- @c Copyright (C) 2012, 2016 Moreno Marzolla @c @c This file is part of the queueing package. @c @c The queueing package is free software; you can redistribute it @c and/or modify it under the terms of the GNU General Public License @c as published by the Free Software Foundation; either version 3 of @c the License, or (at your option) any later version. @c @c The queueing package is distributed in the hope that it will be @c useful, but WITHOUT ANY WARRANTY; without even the implied warranty @c of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the @c GNU General Public License for more details. @c @c You should have received a copy of the GNU General Public License @c along with the queueing package; see the file COPYING. If not, see @c . @node References @chapter References @table @asis @item [Aky88] Ian F. Akyildiz, @cite{Mean Value Analysis for Blocking Queueing Networks}, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418--428. DOI @uref{http://dx.doi.org/10.1109/32.4663, 10.1109/32.4663} @item [Bar79] Y. Bard, @cite{Some Extensions to Multiclass Queueing Network Analysis}, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, feb. 1979, pp. 51--62. @item [BCMP75] F. Baskett, K. Mani Chandy, R. R. Muntz, and F. G. Palacios. 1975. @cite{Open, Closed, and Mixed Networks of Queues with Different Classes of Customers}. J. ACM 22, 2 (April 1975), 248—260, DOI @uref{http://doi.acm.org/10.1145/321879.321887, 10.1145/321879.321887} @item [BGMT98] G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998. @item [Buz73] J. P. Buzen, @cite{Computational Algorithms for Closed Queueing Networks with Exponential Servers}, Communications of the ACM, volume 16, number 9, september 1973, pp. 527--531. DOI @uref{http://doi.acm.org/10.1145/362342.362345, 10.1145/362342.362345} @item [C08] G. Casale, @cite{A note on stable flow-equivalent aggregation in closed networks}. Queueing Syst. Theory Appl., 60:193–-202, December 2008, DOI @uref{http://dx.doi.org/10.1007/s11134-008-9093-6, 10.1007/s11134-008-9093-6} @item [CMS08] G. Casale, R. R. Muntz, G. Serazzi, @cite{Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks}, IEEE Transactions on Computers, 57(6):780-794, June 2008. DOI @uref{http://doi.ieeecomputersociety.org/10.1109/TC.2008.37, 10.1109/TC.2008.37} @item @anchor{GrSn97}[GrSn97] C. M. Grinstead, J. L. Snell, (July 1997). @cite{Introduction to Probability}. American Mathematical Society. ISBN 978-0821807491; this excellent textbook is @uref{http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf, available in PDF format} and can be used under the terms of the @uref{http://www.gnu.org/copyleft/fdl.html, GNU Free Documentation License (FDL)} @item [Jac04] J. R. Jackson, @cite{Jobshop-Like Queueing Systems}, Vol. 50, No. 12, Ten Most Influential Titles of "Management Science's" First Fifty Years (Dec., 2004), pp. 1796-1802, @uref{http://www.jstor.org/stable/30046149, available online} @item [Jai91] R. Jain, @cite{The Art of Computer Systems Performance Analysis}, Wiley, 1991, p. 577. @item [HsLa87] C. H. Hsieh and S. Lam, @cite{Two classes of performance bounds for closed queueing networks}, PEVA, vol. 7, n. 1, pp. 3--30, 1987 @item [Ker84] T. Kerola, @cite{The Composite Bound Method (CBM) for Computing Throughput Bounds in Multiple Class Environments}, Performance Evaluation, Vol. 6 Isue 1, March 1986, DOI @uref{http://dx.doi.org/10.1016/0166-5316(86)90002-7, 10.1016/0166-5316(86)90002-7}; also available as @uref{http://docs.lib.purdue.edu/cstech/395/, Technical Report CSD-TR-475}, Department of Computer Sciences, Purdue University, mar 13, 1984 (Revised aug 27, 1984). @item [LZGS84] E. D. Lazowska, J. Zahorjan, G. Scott Graham, and K. C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @uref{http://www.cs.washington.edu/homes/lazowska/qsp/, available online}. @item [ReKo76] M. Reiser, H. Kobayashi, @cite{On The Convolution Algorithm for Separable Queueing Networks}, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29--31, 1976). SIGMETRICS '76. ACM, New York, NY, pp. 109--117. DOI @uref{http://doi.acm.org/10.1145/800200.806187, 10.1145/800200.806187} @item [ReLa80] M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313--322. DOI @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} @item [Sch79] P. Schweitzer, @cite{Approximate Analysis of Multiclass Closed Networks of Queues}, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25—29 @item [Sch80] H. D. Schwetman, @cite{Testing Network-of-Queues Software}, @uref{http://docs.lib.purdue.edu/cstech/259/, Technical Report CSD-TR 330}, Department of computer Sciences, Purdue University, 1980 @item [Sch81] H. D. Schwetman, @cite{Some Computational Aspects of Queueing Network Models}, @uref{http://docs.lib.purdue.edu/cstech/285/, Technical Report CSD-TR-354}, Department of Computer Sciences, Purdue University, feb, 1981 (revised). @item @anchor{Sch82}[Sch82] H. D. Schwetman, @cite{Implementing the Mean Value Algorithm for the Solution of Queueing Network Models}, @uref{http://docs.lib.purdue.edu/cstech/286/, Technical Report CSD-TR-355}, Department of Computer Sciences, Purdue University, feb 15, 1982. @item [Sch84] T. Kerola, H. D. Schwetman, @cite{Performance Bounds for Multiclass Models}, @uref{http://docs.lib.purdue.edu/cstech/399/, Technical Report CSD-TR-479}, Department of Computer Sciences, Purdue University, 1984. @item [Tij03] H. C. Tijms, @cite{A first course in stochastic models}, John Wiley and Sons, 2003, ISBN 0471498807, ISBN 9780471498803, DOI @uref{http://dx.doi.org/10.1002/047001363X, 10.1002/047001363X} @item [ZaWo81] J. Zahorjan and E. Wong, @cite{The solution of separable queueing network models using mean value analysis}. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI @uref{http://doi.acm.org/10.1145/1010629.805477, 10.1145/1010629.805477} @item [Zeng03] G. Zeng, @cite{Two common properties of the erlang-B function, erlang-C function, and Engset blocking function}, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 DOI @uref{http://dx.doi.org/10.1016/S0895-7177(03)90040-9, 10.1016/S0895-7177(03)90040-9} @end table queueing/doc/qn_closed_multi.pdf0000644000175000017500000001316213635377235016722 0ustar morenomoreno%PDF-1.7 %Çì¢ 5 0 obj <> stream xœ•UÉ®S1 Ýç+²¤š8ŽãdË ¶@‹X ¨ù]$x ~'½N.¥T JmsâáØ9q¾ûèCû¬¿ÇÅ=|)þãK 1Eö?]ß÷/Ÿ9F’’/€ûe *TŸB®ì3Dò”Dlut‘©›¯Štkó¶µÅ?ºTiæ_E „-¾-Îò2ŸüÏ 8ºOnï©@,>eä¬eB¹vž)G@M_ BÉ)ë6ŽµòŒ’šÃ°ÀZš‡X—–AV„YÔ!z `Y3ŒÎÛ_ ÷³~/ 1$Þ†!Uà8*³~YÞÙQ#6,Vâ#‚­×³4Œ$@e”†„Ô=Ö¶žÅ ciÎê8UW•ÑèÑ2¯ÀíìTn‡ÏDfË>2#Yf¤‰ìÝ÷~UP¢~e þÇ{5c‚ºJ¯ik"T!$u,…!HR$&à<ÈyÔ»5ÑNQC>¹×ü·y%¹B­-~.'¢Î!VÙ„%dï^ü‘d¹Æè^¬‹ž§Þ¯’¤Ð[j3ðé(”B-ÊS¯l¨¥59g¨¥l˜)P*β/([BÚrH*-šŽggË™lÌš´¦œSX®Ñ¥\Ö‚J QÒK*À¹kÁ.Ð5T8ƒèœHzcr^%= ÌãbOmhàÞx½,R“«yoædN2ÒF†LaÌàU›"m.ŒßS.WX âWu!Œ=ÍÔ…¤1Ц™µä.&ña“UMÿÚ„ “}Åd>dµôkÕÉÙzfÏÖ@¶B1NË5Þ£¶ýúºþø¨ýÂÓÿõç¸øG}›ÊH§üáƒ;½Ò:ÀTäTD_4M‘Š?,îÞãç¯î¾¸ªONÐì(iz7îÞ“Ïw_cÛÝéäãv'¹oN}wßvZ¦(sËÔ8§5ŒÚ Ñ 7”*Cj*ÚQ€TtªüWDºq‡ ˆÒNI'$˨ §ïu*ÇÖçPÿ1Š³ÆKéôX2ë(ÞaÐç4vÃÇ·ïîî|7×~sÐ7ÉÓƒžû ÷ Jå×endstream endobj 6 0 obj 796 endobj 4 0 obj <> /Contents 5 0 R >> endobj 3 0 obj << /Type /Pages /Kids [ 4 0 R ] /Count 1 >> endobj 1 0 obj <> endobj 7 0 obj <>endobj 10 0 obj <> endobj 11 0 obj <> endobj 8 0 obj <> endobj 9 0 obj <> endobj 12 0 obj <>stream xœT{PSW¾—$÷žºljÉFI‘$ÓÚ¾(âZ]§ ¦Zyˆ(- ! r  'šHDBˆFå!`|´ÅE©j-UW¡>»µÓiÇeí̹ÌegöêìÌîßûß9ç7ç;ßù¾ß÷Ã1n†ãxhªB)/‹JV+sTÏ÷oÑ8½,„Žä@j>m~‡b0”C¹ÝËHnêy™_FÚÅÇ×lOo3-9ý­wÞyw“º¤²T‘_ ‘®\³Jº¯RúŸŠT&/S䫤d:y±ºD)WiÊ}Ú2é‹g¥Éò|mqNéÿžýíÿÃÇ0Œ¿IV–“¤H)J+ŽYù')†-ÇÞÀR±8,Û†­Ââ±ÕX† °?`8ö2+ÆÅtØ\²8¤6d†cà.â¸Aî3ÞVÞ4ï²ó磩 ýÏüœyŒNš–§#̺*c94AƒÝк§3Ù ×ÁµÅ ©%È£!˜l$b–¡½ˆ÷ôì½ËâÊÃq²•k¡(¦x³‡S?/~ ­?#€Þ¸4®Û?"î/ò¨»¶þ|¢ áÏæháœCKÐüíÄÕ‹_5ô#ü>QͤLlù–Á™H˜ž b¶ÞYgÛ’HkœÑ¸ÑP2=%ìs;Ç~˜ªRœϤ¾ 2‹ö­ù s·j§q¹FW¿„O‹j}F/ý~OØ—WѺ饂3èúFØ€"y5„¥ÎÒh†fXÛ\ÓªqjZË XŸœ¾)¥?÷Ñ^Édá@yg,e©wo.žÓˆ«˜ ‡»¥¹Áî6ØÞÔc‚3æÀÑoÄ̵Éé/ÊNì”0Ø”ºÓà‡Ñh_àì¥ã…1=b>T{)y’÷‡]|€Þ¼Ç¡¯ õ”Ñb€@Sß” tŽ[z ~í`vWQ±{â5]å~·××bmµ:$M.«:@ ß7òy@•&N$™¨ÄJS¶\[N)`!Ë)>˜u%8Ú=~E,8¸«õpùhÄIØÝ>0Ì–˜èB¨´Ô茥TI‚õÉÓ’Òv98Œ^|úô¢îôÅþtU=7g~©ðvËd‹ ‘ÖÕõ¦ õ è"×7SÖÛm#6uZÂYxÁ}Ú¹ð ¼Ǩá’ÞÜàªcQ¬}J” GE\Í\þvRC–S¶f˜ú —žœ ›™[ýp©à[¶j=ÄÖÖz—õ@ˆl°3Ñ©<é´µ7·ÛÜPäwUeIÙkªÝÒw~…°D£Q«|ÚcÇ}¾Þ^OÅz~ÔzifTá ûî1¢þ±Tð.2¿V¨#Ë´ 5F}D-Ô·è@p¢#/ÏY‘³ô…ª‚âʘÿLAܤ›ûÇsÈ]Õ­:X ¶f|“0øb qœ)àùA|—­½ÃÑ;ºêY$ÃðhÝ©þ6ûì~ê~SòÚ¥œ œcCWÏŒUäˆûŠ:JG>ÝÈ~ý=/~ÿ!DZU{ˆr[ŽÝØf¡xI™-»ÅÞ(rÖÚÍÐL&³Q\a@*‘e/îÊù’‰B»Â{èÄ šjy˜lŠ#Ûlí6'ýn*S²p€´¦QTBx!Ýüb/Ž9hÑÔë!>„ú6ëy@ÿBÖ53¯à±RxüÝ.C‘d!ƒlŠ×S¬/îÞ¡{nãè§ÇZ"…Õ­y;kê3­"¦Ž@ë‘ìÇŸOÝ3¢§Ü}=5]››'.RŠô²nKøȯýÓ<º˜¼fý'ÑkVJ˜ÍÌNž‘~•dQ©Aú§Ëaè»YùM¶‘Qz ü ö4õ˜®i†Sàf°:e{¬ºÜáÏ«ëz6ðJWpZ;}göèèyɹÑ#“ð8^uVݧóU´gw±Í}ãÜ‘¡¿¼z?~⽌,}Q¾X©6”êvxÂGnŸé¿Áõ`öö|CA鉒-m)cEµó©z—}ÚA /zð»a{hèlKèï1ìß_ÃB} endstream endobj 13 0 obj <>stream 2020-03-21T12:46:37+01:00 2020-03-21T12:46:37+01:00 fig2dev Version 3.2.6a qn_closed_multi.fig endstream endobj 2 0 obj <>endobj xref 0 14 0000000000 65535 f 0000001110 00000 n 0000005129 00000 n 0000001051 00000 n 0000000900 00000 n 0000000015 00000 n 0000000881 00000 n 0000001175 00000 n 0000001276 00000 n 0000001627 00000 n 0000001216 00000 n 0000001246 00000 n 0000001897 00000 n 0000003693 00000 n trailer << /Size 14 /Root 1 0 R /Info 2 0 R /ID [<29BF9823BC70065D233987C3492E02B4><29BF9823BC70065D233987C3492E02B4>] >> startxref 5313 %%EOF queueing/doc/gpl.texi0000644000175000017500000010452213635377223014521 0ustar morenomoreno@c This file has been automatically generated from gpl.txi @c by proc.m. Do not edit this file, all changes will be lost @node Copying @appendix GNU GENERAL PUBLIC LICENSE @cindex warranty @cindex copyright @center Version 3, 29 June 2007 @display Copyright @copyright{} 2007 Free Software Foundation, Inc. @url{http://fsf.org/} Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. @end display @heading Preamble The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program---to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. 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For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it. @item Patents. A ``contributor'' is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's ``contributor version''. A contributor's ``essential patent claims'' are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. 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If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. ``Knowingly relying'' means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid. If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it. A patent license is ``discriminatory'' if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007. Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law. @item No Surrender of Others' Freedom. If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program. @item Use with the GNU Affero General Public License. Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such. @item Revised Versions of this License. The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License ``or any later version'' applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation. If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program. Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version. @item Disclaimer of Warranty. THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM ``AS IS'' WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. @item Limitation of Liability. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. @item Interpretation of Sections 15 and 16. If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee. @end enumerate @heading END OF TERMS AND CONDITIONS @heading How to Apply These Terms to Your New Programs If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms. To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the ``copyright'' line and a pointer to where the full notice is found. @smallexample @var{one line to give the program's name and a brief idea of what it does.} Copyright (C) @var{year} @var{name of author} This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see @url{http://www.gnu.org/licenses/}. @end smallexample Also add information on how to contact you by electronic and paper mail. If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: @smallexample @var{program} Copyright (C) @var{year} @var{name of author} This program comes with ABSOLUTELY NO WARRANTY; for details type @samp{show w}. This is free software, and you are welcome to redistribute it under certain conditions; type @samp{show c} for details. @end smallexample The hypothetical commands @samp{show w} and @samp{show c} should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an ``about box''. You should also get your employer (if you work as a programmer) or school, if any, to sign a ``copyright disclaimer'' for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see @url{http://www.gnu.org/licenses/}. The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. 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Do not edit this file, all changes will be lost @c -*- texinfo -*- @c Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2016 Moreno Marzolla @c @c This file is part of the queueing package. @c @c The queueing package is free software; you can redistribute it @c and/or modify it under the terms of the GNU General Public License @c as published by the Free Software Foundation; either version 3 of @c the License, or (at your option) any later version. @c @c The queueing package is distributed in the hope that it will be @c useful, but WITHOUT ANY WARRANTY; without even the implied warranty @c of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the @c GNU General Public License for more details. @c @c You should have received a copy of the GNU General Public License @c along with the queueing package; see the file COPYING. If not, see @c . @node Summary @chapter Summary @menu * About the Queueing Package:: What is the Queueing package. * Contributing Guidelines:: How to contribute. * Acknowledgments:: @end menu @node About the Queueing Package @section About the Queueing Package This document describes the @code{queueing} package for GNU Octave (@code{queueing} in short). The @code{queueing} package, previously known as @code{qnetworks} toolbox, is a collection of functions for analyzing queueing networks and Markov chains written for GNU Octave. Specifically, @code{queueing} contains functions for analyzing Jackson networks, open, closed or mixed product-form BCMP networks, and computing performance bounds. The following algorithms are available @itemize @item Convolution for closed, single-class product-form networks with load-dependent service centers; @item Exact and approximate Mean Value Analysis (MVA) for single and multiple class product-form closed networks; @item MVA for mixed, multiple class product-form networks with load-independent service centers; @item Approximate MVA for closed, single-class networks with blocking (MVABLO algorithm by F. Akyildiz); @item Asymptotic Bounds, Balanced System Bounds and Geometric Bounds; @end itemize @noindent @code{queueing} provides functions for analyzing the following types of single-station queueing systems: @itemize @item @math{M/M/1} @item @math{M/M/m} @item @math{M/M/\infty} @item @math{M/M/1/k} single-server, finite capacity system @item @math{M/M/m/k} multiple-server, finite capacity system @item Asymmetric @math{M/M/m} @item @math{M/G/1} (general service time distribution) @item @math{M/H_m/1} (Hyperexponential service time distribution) @end itemize Functions for Markov chain analysis are also provided (discrete- and continuous-time chains are supported): @itemize @item Birth-death processes; @item Transient and stationary state occupancy probabilities; @item Mean time to absorption; @item Expected sojourn times and time-averaged sojourn times; @item Mean first passage times; @end itemize The @code{queueing} package is distributed under the terms of the GNU General Public License (GPL), version 3 or later (@pxref{Copying}). You are encouraged to share this software with others, and improve this package by contributing additional functions and reporting bugs. @xref{Contributing Guidelines}. If you use the @code{queueing} package in a technical paper, please cite it as: @quotation Moreno Marzolla, @emph{The qnetworks Toolbox: A Software Package for Queueing Networks Analysis}. Khalid Al-Begain, Dieter Fiems and William J. Knottenbelt, Editors, Proceedings 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications (ASMTA 2010) Cardiff, UK, June 14--16, 2010, volume 6148 of Lecture Notes in Computer Science, Springer, pp. 102--116, ISBN 978-3-642-13567-5 @end quotation If you use BibTeX, this is the citation block: @verbatim @inproceedings{queueing, author = {Moreno Marzolla}, title = {The qnetworks Toolbox: A Software Package for Queueing Networks Analysis}, booktitle = {Analytical and Stochastic Modeling Techniques and Applications, 17th International Conference, ASMTA 2010, Cardiff, UK, June 14-16, 2010. Proceedings}, editor = {Khalid Al-Begain and Dieter Fiems and William J. Knottenbelt}, year = {2010}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, volume = {6148}, pages = {102--116}, ee = {http://dx.doi.org/10.1007/978-3-642-13568-2_8}, isbn = {978-3-642-13567-5} } @end verbatim An early draft of the paper above is available as Technical Report @uref{http://www.informatica.unibo.it/it/ricerca/technical-report/2010/UBLCS-2010-04, UBLCS-2010-04}, February 2010, Department of Computer Science, University of Bologna, Italy. @node Contributing Guidelines @section Contributing Guidelines Contributions and bug reports are @emph{always} welcome. If you want to contribute to the @code{queueing} package, here are some guidelines: @itemize @item If you are contributing a new function, please embed proper documentation within the function itself. The documentation must be in @code{texinfo} format, so that it can be extracted and included into the printable manual. See the existing functions for the documentation style. @item Make sure that each new function validates its input parameters. For example, a function accepting vectors should check whether the dimensions match. @item Provide bibliographic references for each new algorithm you contribute. Document any significant difference from the reference. Update the @file{doc/references.txi} file if appropriate. @item Include test and demo blocks. Test blocks are particularly important, since most algorithms are tricky to implement correctly. If appropriate, test blocks should also verify that the function fails on incorrect inputs. @end itemize Send your contribution to Moreno Marzolla (@email{moreno.marzolla@@unibo.it}). If you are a user of this package and find it useful, let me know by dropping me a line. Thanks. @node Acknowledgments @section Acknowledgments The following people (listed alphabetically) contributed to the @code{queueing} package, either by providing feedback, reporting bugs or contributing code: Philip Carinhas, Phil Colbourn, Diego Didona, Yves Durand, Marco Guazzone, Dmitry Kolesnikov, Michele Mazzucco, Marco Paolieri. queueing/doc/qn_open_single.pdf0000644000175000017500000001536113635377235016544 0ustar morenomoreno%PDF-1.7 %Çì¢ 5 0 obj <> stream xœ}VËŽ7 ¼ë+t‹`hQOê?rÉÁÞ 0‰ñ¶Û‡ü~JÝ¢ZÛ3,°=*‘E–D²û»uÄÖµ¿þ¼,æÕC±Ÿ~šHÅGŸìfÝ·¿.5‘]$NÙ. ìÈË…+q°>Dr’-gmë‹á¸9 ‹˜RóP‚¾ÔpP¤ ¼U‚TâêÐ#ô%ì{ º¯)«ûAÂÅ|6C;‰Ó$¬VGiRV9R(iÄÕõ®lXôÄC_kŒ]s 2¤±O.#†®wqÃBóV†ƒŽM]É…•- ô!B"9 ®Ú\™JÎ6 ãQ IÔÕÅ$VóD_VkõÖµò_Œ"ÓÊ×ý«g8$å×åÅh|E4¿î°Éú¾&WÁ¿ýña<•U¿¬º#Ž\´O†9zrHN”RC<Bt=µÉ†‘yhÈgóׯö›)¨*‰È$Æ5ÑÅTüZsëœ]ÂeÌlÞ(òh>\ÅXîå1r½+;:¡”gÙ¡V*3SHö“jí„Ý&ú@.MªÕ8ùv;2kTÆåNÔ=³Ûk†”m–ÏRAU¢Ûð‡Fc84»×ïÿl–E´mDaî¬[o¾üüÊm3·ãØÿ±þå/꨸T7þ²‘­ÖŒ2pQ©dÝXe¾ÍOEžæ,Þ½~÷¸É]cÕð =áôSÄ!œfÎ÷ëÞÛ3nòƒùôÒ*Øendstream endobj 6 0 obj 915 endobj 4 0 obj <> /Contents 5 0 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Do not edit this file, all changes will be lost @c -*- texinfo -*- @c Copyright (C) 2008, 2009, 2010, 2011, 2012, 2014, 2016, 2018 Moreno Marzolla @c @c This file is part of the queueing package. @c @c The queueing package is free software; you can redistribute it @c and/or modify it under the terms of the GNU General Public License @c as published by the Free Software Foundation; either version 3 of @c the License, or (at your option) any later version. @c @c The queueing package is distributed in the hope that it will be @c useful, but WITHOUT ANY WARRANTY; without even the implied warranty @c of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the @c GNU General Public License for more details. @c @c You should have received a copy of the GNU General Public License @c along with the queueing package; see the file COPYING. If not, see @c . @ifset INSTALLONLY @include conf.texi This file documents the installation procedure of the Octave @code{queueing} package. @code{queueing} is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License, version 3 or later, as published by the Free Software Foundation. @quotation Note This file (@file{INSTALL}) is automatically generated from @file{doc/installation.txi} in the @code{queueing} source tree. Do not modify this document directly, as changes will be lost. Modify @file{doc/installation.txi} instead. @end quotation @end ifset @node Installation and Getting Started @chapter Installation and Getting Started @menu * Installation through Octave package management system:: * Manual installation:: * Development sources:: * Naming Conventions:: * Quick start Guide:: @end menu @c @c @c @node Installation through Octave package management system @section Installation through Octave package management system The most recent version of @code{queueing} is @value{VERSION} and can be downloaded from Octave-Forge @url{https://octave.sourceforge.io/queueing/} Additional information can be found at @url{http://www.moreno.marzolla.name/software/queueing/} To install @code{queueing}, follow these steps: @enumerate @item If you have a recent version of GNU Octave and a network connection, you can install @code{queueing} from Octave command prompt using this command: @example octave:1> @kbd{pkg install -forge queueing} @end example The command above will download and install the latest version of the @code{queueing} package from Octave Forge, and install it on your machine. If you do not have root access, you can perform a local install with: @example octave:1> @kbd{pkg install -local -forge queueing} @end example This will install @code{queueing} in your home directory, and the package will be available to the current user only. @item Alternatively, you can first download the @code{queueing} tarball from Octave-Forge; to install the package in the system-wide location issue this command at the Octave prompt: @example octave:1> @kbd{pkg install @emph{queueing-@value{VERSION}.tar.gz}} @end example @noindent (you may need to start Octave as root in order to allow the installation to copy the files to the target locations). After this, all functions will be available each time Octave starts, without the need to tweak the search path. If you do not have root access, you can do a local install using: @example octave:1> @kbd{pkg install -local queueing-@value{VERSION}.tar.gz} @end example @item Use the @kbd{pkg list} command at the Octave prompt to check that the @code{queueing} package has been succesfully installed; you should see something like: @example octave:1>@kbd{pkg list queueing} Package Name | Version | Installation directory --------------+---------+----------------------- queueing | @value{VERSION} | /home/moreno/octave/queueing-@value{VERSION} @end example @item Starting from version 1.1.1, @code{queueing} is no longer automatically loaded on Octave start. To make the functions available for use, you need to issue the command @example octave:1>@kbd{pkg load queueing} @end example @noindent at the Octave prompt. To automatically load @code{queueing} each time Octave starts, you can add the command above to the startup script (usually, @file{~/.octaverc} on Unix systems). @item To completely remove @code{queueing} from your system, use the @kbd{pkg uninstall} command: @example octave:1> @kbd{pkg uninstall queueing} @end example @end enumerate @c @c @c @node Manual installation @section Manual installation If you want to manually install @code{queueing} in a custom location, you can download the tarball and unpack it somewhere: @example @kbd{tar xvfz queueing-@value{VERSION}.tar.gz} @kbd{cd queueing-@value{VERSION}/queueing/} @end example Copy all @code{.m} files from the @file{inst/} directory to some target location. Then, start Octave with the @option{-p} option to add the target location to the search path, so that Octave will find all @code{queueing} functions automatically: @example @kbd{octave -p @emph{/path/to/queueing}} @end example For example, if all @code{queueing} m-files are in @file{/usr/local/queueing}, you can start Octave as follows: @example @kbd{octave -p @emph{/usr/local/queueing}} @end example If you want, you can add the following line to @file{~/.octaverc}: @example @kbd{addpath("@emph{/path/to/queueing}");} @end example @noindent so that the path @file{/path/to/queueing} is automatically added to the search path each time Octave is started, and you no longer need to specify the @option{-p} option on the command line. @c @c The following will not appear in the INSTALL text file @c @ifclear INSTALLONLY @node Development sources @section Development sources The source code of the @code{queueing} package can be found in the Mercurial repository at the URL: @url{https://sourceforge.net/p/octave/queueing/ci/default/tree/} The source distribution contains additional development files that are not present in the installation tarball. This section briefly describes the content of the source tree. This is only relevant for developers who want to modify the code or the documentation. The source distribution contains the following directories: @table @file @item doc/ Documentation sources. Most of the documentation is extracted from the comment blocks of function files from the @file{inst/} directory. @item inst/ This directory contains the @verb{|m|}-files which implement the various algorithms provided by @code{queueing}. As a notational convention, the names of functions for Queueing Networks begin with the @samp{qn} prefix; the name of functions for Continuous-Time Markov Chains (CTMCs) begin with the @samp{ctmc} prefix, and the names of functions for Discrete-Time Markov Chains (DTMCs) begin with the @samp{dtmc} prefix. @item test/ This directory contains the test scripts used to run all function tests. @item devel/ This directory contains functions that are either not working properly, or need additional testing before they are moved to the @file{inst/} directory. @end table The @code{queueing} package ships with a Makefile which can be used to produce the documentation (in PDF and HTML format), and automatically execute all function tests. The following targets are defined: @table @code @item all Running @samp{make} (or @samp{make all}) on the top-level directory builds the programs used to extract the documentation from the comments embedded in the @verb{|m|}-files, and then produce the documentation in PDF and HTML format (@file{doc/queueing.pdf} and @file{doc/queueing.html}, respectively). @item check Running @samp{make check} will execute all tests contained in the @verb{|m|}-files. If you modify the code of any function in the @file{inst/} directory, you should run the tests to ensure that no errors have been introduced. You are also encouraged to contribute new tests, especially for functions that are not adequately validated. @item clean @itemx distclean @itemx dist The @samp{make clean}, @samp{make distclean} and @samp{make dist} commands are used to clean up the source directory and prepare the distribution archive in compressed tar format. @end table @node Naming Conventions @section Naming Conventions Most of the functions in the @code{queueing} package obey a common naming convention. Function names are made of several parts; the first part is a prefix which indicates the class of problems the function addresses: @table @asis @item @strong{ctmc-} Functions for continuous-time Markov chains @item @strong{dtmc-} Functions for discrete-time Markov chains @item @strong{qs-} Functions for analyzing single-station queueing systems (individual service centers) @item @strong{qn-} Functions for analyzing queueing networks @end table Functions dealing with Markov chains start with either the @code{ctmc} or @code{dtmc} prefix; the prefix is optionally followed by an additional string which hints at what the function does: @table @asis @item @strong{-bd} Birth-Death process @item @strong{-mtta} Mean Time to Absorption @item @strong{-fpt} First Passage Times @item @strong{-exps} Expected Sojourn Times @item @strong{-taexps} Time-Averaged Expected Sojourn Times @end table For example, function @code{ctmcbd} returns the infinitesimal generator matrix for a continuous birth-death process, while @code{dtmcbd} returns the transition probability matrix for a discrete birth-death process. Note that there exist functions @code{ctmc} and @code{dtmc} (without any suffix) that compute steady-state and transient state occupancy probabilities for CTMCs and DTMCs, respectively. @xref{Markov Chains}. Functions whose name starts with @code{qs-} deal with single station queueing systems. The suffix describes the type of system, e.g., @code{qsmm1} for @math{M/M/1}, @code{qnmmm} for @math{M/M/m} and so on. @xref{Single Station Queueing Systems}. Finally, functions whose name starts with @code{qn-} deal with queueing networks. The character that follows indicates whether the function handles open (@code{'o'}) or closed (@code{'c'}) networks, and whether there is a single customer class (@code{'s'}) or multiple classes (@code{'m'}). The string @code{mix} indicates that the function supports mixed networks with both open and closed customer classes. @table @asis @item @strong{-os-} Open, single-class network: open network with a single class of customers @item @strong{-om-} Open, multiclass network: open network with multiple job classes @item @strong{-cs-} Closed, single-class network @item @strong{-cm-} Closed, multiclass network @item @strong{-mix-} Mixed network with open and closed classes of customers @end table The last part of the function name indicates the algorithm implemented by the function. @xref{Queueing Networks}. @table @asis @item @strong{-aba} Asymptotic Bounds Analysis @item @strong{-bsb} Balanced System Bounds @item @strong{-gb} Geometric Bounds @item @strong{-pb} PB Bounds @item @strong{-cb} Composite Bounds (CB) @item @strong{-mva} Mean Value Analysis (MVA) algorithm @item @strong{-cmva} Conditional MVA @item @strong{-mvald} MVA with general load-dependent servers @item @strong{-mvaap} Approximate MVA @item @strong{-mvablo} MVABLO approximation for blocking queueing networks @item @strong{-conv} Convolution algorithm @item @strong{-convld} Convolution algorithm with general load-dependent servers @end table @cindex deprecated functions Some deprecated functions may be present in the @code{queueing} package; generally, these are functions that have been renamed, and the old name is kept for a while for backward compatibility. Deprecated functions are not documented and will be removed in future releases. Calling a deprecated functions displays a warning message that appears only once per session. The warning message can be turned off with the command: @example @group octave:1> @kbd{warning ("off", "qn:deprecated-function");} @end group @end example However, you are strongly recommended to update your code to the new API. To help catching usages of deprecated functions, you can transform warnings into errors so that your application stops immediately: @example @group octave:1> @kbd{warning ("error", "qn:deprecated-function");} @end group @end example @node Quick start Guide @section Quick start Guide You can use all functions by simply invoking their name with the appropriate parameters; an error is shown in case of missing/wrong parameters. Extensive documentation is provided for each function, and can be displayed with the @command{help} command. For example: @example octave:2> @kbd{help qncsmvablo} @end example @noindent shows the documentation for the @command{qncsmvablo} function. Additional information can be found in the @code{queueing} manual, that is available in PDF format in @file{doc/queueing.pdf} and in HTML format in @file{doc/queueing.html}. Many functions have demo blocks showing usage examples. To execute the demos for the @command{qnclosed} function, use the @command{demo} command: @example octave:4> @kbd{demo qnclosed} @end example We now illustrate a few examples of how the @code{queueing} package can be used. More examples are provided in the manual. @noindent @strong{Example 1} Compute the stationary state occupancy probabilities of a continuous-time Markov chain with infinitesimal generator matrix @iftex @tex $$ {\bf Q} = \pmatrix{ -0.8 & 0.6 & 9,2 \cr 0.3 & -0.7 & 0.4 \cr 0.2 & 0.2 & -0.4 } $$ @end tex @end iftex @ifnottex @example @group / -0.8 0.6 0.2 \ Q = | 0.3 -0.7 0.4 | \ 0.2 0.2 -0.4 / @end group @end example @end ifnottex @example @group Q = [ -0.8 0.6 0.2; \ 0.3 -0.7 0.4; \ 0.2 0.2 -0.4 ]; q = ctmc(Q) @result{} q = 0.23256 0.32558 0.44186 @end group @end example @noindent @strong{Example 2} Compute the transient state occupancy probability after @math{n=3} transitions of a three state discrete-time birth-death process, with birth probabilities @math{\lambda_{01} = 0.3} and @math{\lambda_{12} = 0.5} and death probabilities @math{\mu_{10} = 0.5} and @math{\mu_{21} = 0.7}, assuming that the system is initially in state zero (i.e., the initial state occupancy probabilities are @math{[1, 0, 0]}). @example @group n = 3; p0 = [1 0 0]; P = dtmcbd( [0.3 0.5], [0.5 0.7] ); p = dtmc(P,n,p0) @result{} p = 0.55300 0.29700 0.15000 @end group @end example @noindent @strong{Example 3} Compute server utilization, response time, mean number of requests and throughput of a closed queueing network with @math{N=4} requests and three @math{M/M/1}--FCFS queues with mean service times @math{S = [1.0, 0.8, 1.4]} and average number of visits @math{V = [1.0, 0.8, 0.8]} @example @group S = [1.0 0.8 1.4]; V = [1.0 0.8 0.8]; N = 4; [U R Q X] = qncsmva(N, S, V) @result{} U = 0.70064 0.44841 0.78471 R = 2.1030 1.2642 3.2433 Q = 1.47346 0.70862 1.81792 X = 0.70064 0.56051 0.56051 @end group @end example @noindent @strong{Example 4} Compute server utilization, response time, mean number of requests and throughput of an open queueing network with three @math{M/M/1}--FCFS queues with mean service times @math{S = [1.0, 0.8, 1.4]} and average number of visits @math{V = [1.0, 0.8, 0.8]}. 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queueing/doc/INSTALL0000644000175000017500000001001113635377236014066 0ustar morenomorenoThis file documents the installation procedure of the Octave 'queueing' package. 'queueing' is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License, version 3 or later, as published by the Free Software Foundation. Note: This file ('INSTALL') is automatically generated from 'doc/installation.txi' in the 'queueing' source tree. Do not modify this document directly, as changes will be lost. Modify 'doc/installation.txi' instead. 1 Installation and Getting Started ********************************** 1.1 Installation through Octave package management system ========================================================= The most recent version of 'queueing' is 1.2.7 and can be downloaded from Octave-Forge Additional information can be found at To install 'queueing', follow these steps: 1. If you have a recent version of GNU Octave and a network connection, you can install 'queueing' from Octave command prompt using this command: octave:1> pkg install -forge queueing The command above will download and install the latest version of the 'queueing' package from Octave Forge, and install it on your machine. If you do not have root access, you can perform a local install with: octave:1> pkg install -local -forge queueing This will install 'queueing' in your home directory, and the package will be available to the current user only. 2. Alternatively, you can first download the 'queueing' tarball from Octave-Forge; to install the package in the system-wide location issue this command at the Octave prompt: octave:1> pkg install _queueing-1.2.7.tar.gz_ (you may need to start Octave as root in order to allow the installation to copy the files to the target locations). After this, all functions will be available each time Octave starts, without the need to tweak the search path. If you do not have root access, you can do a local install using: octave:1> pkg install -local queueing-1.2.7.tar.gz 3. Use the 'pkg list' command at the Octave prompt to check that the 'queueing' package has been succesfully installed; you should see something like: octave:1>pkg list queueing Package Name | Version | Installation directory --------------+---------+----------------------- queueing | 1.2.7 | /home/moreno/octave/queueing-1.2.7 4. Starting from version 1.1.1, 'queueing' is no longer automatically loaded on Octave start. To make the functions available for use, you need to issue the command octave:1>pkg load queueing at the Octave prompt. To automatically load 'queueing' each time Octave starts, you can add the command above to the startup script (usually, '~/.octaverc' on Unix systems). 5. To completely remove 'queueing' from your system, use the 'pkg uninstall' command: octave:1> pkg uninstall queueing 1.2 Manual installation ======================= If you want to manually install 'queueing' in a custom location, you can download the tarball and unpack it somewhere: tar xvfz queueing-1.2.7.tar.gz cd queueing-1.2.7/queueing/ Copy all '.m' files from the 'inst/' directory to some target location. Then, start Octave with the '-p' option to add the target location to the search path, so that Octave will find all 'queueing' functions automatically: octave -p _/path/to/queueing_ For example, if all 'queueing' m-files are in '/usr/local/queueing', you can start Octave as follows: octave -p _/usr/local/queueing_ If you want, you can add the following line to '~/.octaverc': addpath("_/path/to/queueing_"); so that the path '/path/to/queueing' is automatically added to the search path each time Octave is started, and you no longer need to specify the '-p' option on the command line. queueing/doc/README0000644000175000017500000000211213570046274013711 0ustar morenomoreno=============================================================================== The Octave queueing toolbox =============================================================================== Copyright (C) 2008, 2009, 2010, 2011, 2012, 2013, 2018 Moreno Marzolla The queueing toolbox ("queueing", in short) is a collection of GNU Octave scripts for numerical evaluation of queueing network models. Open, closed and mixed networks are supported, with single or multiple classes of customers. The queueing toolbox also provides functions for steady-state and transient analysis of Markov chains, as well as for single station queueing systems. The Web page of the queueing toolbox is http://www.moreno.marzolla.name/software/queueing/ The latest version can be downloaded from Octave forge https://octave.sourceforge.io/ This package requires GNU Octave; version 4.0.0 or later should work. The Octave queueing toolbox is distributed under the terms of the GNU General Public License, version 3 or later. 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queueing

Copyright © 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2020 Moreno Marzolla.

Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies.

Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one.

Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions.

Table of Contents

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This manual documents how to install and run the Queueing package. It corresponds to version 1.2.7 of the package.

• Summary:  
• Installation and Getting Started:  Installation of the queueing package.
• Markov Chains:  Functions for Markov chains analysis.
• Single Station Queueing Systems:  Functions for single-station queueing systems.
• Queueing Networks:  Functions for queueing networks analysis.
• References:  References.
• Copying:  The GNU General Public License.
• Concept Index:  An item for each concept.
• Function Index:  An item for each function.

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1 Summary

• About the Queueing Package:  What is the Queueing package.
• Contributing Guidelines:  How to contribute.
• Acknowledgments:  

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1.1 About the Queueing Package

This document describes the queueing package for GNU Octave (queueing in short). The queueing package, previously known as qnetworks toolbox, is a collection of functions for analyzing queueing networks and Markov chains written for GNU Octave. Specifically, queueing contains functions for analyzing Jackson networks, open, closed or mixed product-form BCMP networks, and computing performance bounds. The following algorithms are available

  • Convolution for closed, single-class product-form networks with load-dependent service centers;
  • Exact and approximate Mean Value Analysis (MVA) for single and multiple class product-form closed networks;
  • MVA for mixed, multiple class product-form networks with load-independent service centers;
  • Approximate MVA for closed, single-class networks with blocking (MVABLO algorithm by F. Akyildiz);
  • Asymptotic Bounds, Balanced System Bounds and Geometric Bounds;

queueing provides functions for analyzing the following types of single-station queueing systems:

  • M/M/1
  • M/M/m
  • M/M/\infty
  • M/M/1/k single-server, finite capacity system
  • M/M/m/k multiple-server, finite capacity system
  • Asymmetric M/M/m
  • M/G/1 (general service time distribution)
  • M/H_m/1 (Hyperexponential service time distribution)

Functions for Markov chain analysis are also provided (discrete- and continuous-time chains are supported):

  • Birth-death processes;
  • Transient and stationary state occupancy probabilities;
  • Mean time to absorption;
  • Expected sojourn times and time-averaged sojourn times;
  • Mean first passage times;

The queueing package is distributed under the terms of the GNU General Public License (GPL), version 3 or later (see Copying). You are encouraged to share this software with others, and improve this package by contributing additional functions and reporting bugs. See Contributing Guidelines.

If you use the queueing package in a technical paper, please cite it as:

Moreno Marzolla, The qnetworks Toolbox: A Software Package for Queueing Networks Analysis. Khalid Al-Begain, Dieter Fiems and William J. Knottenbelt, Editors, Proceedings 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications (ASMTA 2010) Cardiff, UK, June 14–16, 2010, volume 6148 of Lecture Notes in Computer Science, Springer, pp. 102–116, ISBN 978-3-642-13567-5

If you use BibTeX, this is the citation block: 

@inproceedings{queueing,
  author    = {Moreno Marzolla},
  title     = {The qnetworks Toolbox: A Software Package for Queueing 
               Networks Analysis},
  booktitle = {Analytical and Stochastic Modeling Techniques and 
               Applications, 17th International Conference, 
               ASMTA 2010, Cardiff, UK, June 14-16, 2010. Proceedings},
  editor    = {Khalid Al-Begain and Dieter Fiems and William J. Knottenbelt},
  year      = {2010},
  publisher = {Springer},
  series    = {Lecture Notes in Computer Science},
  volume    = {6148},
  pages     = {102--116},
  ee        = {http://dx.doi.org/10.1007/978-3-642-13568-2_8},
  isbn      = {978-3-642-13567-5}
}

An early draft of the paper above is available as Technical Report UBLCS-2010-04, February 2010, Department of Computer Science, University of Bologna, Italy.

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1.2 Contributing Guidelines

Contributions and bug reports are always welcome. If you want to contribute to the queueing package, here are some guidelines:

  • If you are contributing a new function, please embed proper documentation within the function itself. The documentation must be in texinfo format, so that it can be extracted and included into the printable manual. See the existing functions for the documentation style.
  • Make sure that each new function validates its input parameters. For example, a function accepting vectors should check whether the dimensions match.
  • Provide bibliographic references for each new algorithm you contribute. Document any significant difference from the reference. Update the doc/references.txi file if appropriate.
  • Include test and demo blocks. Test blocks are particularly important, since most algorithms are tricky to implement correctly. If appropriate, test blocks should also verify that the function fails on incorrect inputs.

Send your contribution to Moreno Marzolla (moreno.marzolla@unibo.it). If you are a user of this package and find it useful, let me know by dropping me a line. Thanks.

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1.3 Acknowledgments

The following people (listed alphabetically) contributed to the queueing package, either by providing feedback, reporting bugs or contributing code: Philip Carinhas, Phil Colbourn, Diego Didona, Yves Durand, Marco Guazzone, Dmitry Kolesnikov, Michele Mazzucco, Marco Paolieri.

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2 Installation and Getting Started

• Installation through Octave package management system:  
• Manual installation:  
• Development sources:  
• Naming Conventions:  
• Quick start Guide:  

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2.1 Installation through Octave package management system

The most recent version of queueing is 1.2.7 and can be downloaded from Octave-Forge

https://octave.sourceforge.io/queueing/

Additional information can be found at

http://www.moreno.marzolla.name/software/queueing/

To install queueing, follow these steps:

  1. If you have a recent version of GNU Octave and a network connection, you can install queueing from Octave command prompt using this command: 
    octave:1> pkg install -forge queueing

    The command above will download and install the latest version of the queueing package from Octave Forge, and install it on your machine.

    If you do not have root access, you can perform a local install with: 

    octave:1> pkg install -local -forge queueing

    This will install queueing in your home directory, and the package will be available to the current user only.

  2. Alternatively, you can first download the queueing tarball from Octave-Forge; to install the package in the system-wide location issue this command at the Octave prompt: 
    octave:1> pkg install queueing-1.2.7.tar.gz

    (you may need to start Octave as root in order to allow the installation to copy the files to the target locations). After this, all functions will be available each time Octave starts, without the need to tweak the search path.

    If you do not have root access, you can do a local install using: 

    octave:1> pkg install -local queueing-1.2.7.tar.gz
  3. Use the pkg list command at the Octave prompt to check that the queueing package has been succesfully installed; you should see something like: 
    octave:1>pkg list queueing
Package Name  | Version | Installation directory
--------------+---------+-----------------------
    queueing  |   1.2.7 | /home/moreno/octave/queueing-1.2.7
  4. Starting from version 1.1.1, queueing is no longer automatically loaded on Octave start. To make the functions available for use, you need to issue the command 
    octave:1>pkg load queueing

    at the Octave prompt. To automatically load queueing each time Octave starts, you can add the command above to the startup script (usually, ~/.octaverc on Unix systems).

  5. To completely remove queueing from your system, use the pkg uninstall command: 
    octave:1> pkg uninstall queueing

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2.2 Manual installation

If you want to manually install queueing in a custom location, you can download the tarball and unpack it somewhere: 

tar xvfz queueing-1.2.7.tar.gz
cd queueing-1.2.7/queueing/

Copy all .m files from the inst/ directory to some target location. Then, start Octave with the -p option to add the target location to the search path, so that Octave will find all queueing functions automatically: 

octave -p /path/to/queueing

For example, if all queueing m-files are in /usr/local/queueing, you can start Octave as follows: 

octave -p /usr/local/queueing

If you want, you can add the following line to ~/.octaverc: 

addpath("/path/to/queueing");

so that the path /path/to/queueing is automatically added to the search path each time Octave is started, and you no longer need to specify the -p option on the command line.

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2.3 Development sources

The source code of the queueing package can be found in the Mercurial repository at the URL:

https://sourceforge.net/p/octave/queueing/ci/default/tree/

The source distribution contains additional development files that are not present in the installation tarball. This section briefly describes the content of the source tree. This is only relevant for developers who want to modify the code or the documentation.

The source distribution contains the following directories:

doc/

Documentation sources. Most of the documentation is extracted from the comment blocks of function files from the inst/ directory.

inst/

This directory contains the m-files which implement the various algorithms provided by queueing. As a notational convention, the names of functions for Queueing Networks begin with the ‘qn’ prefix; the name of functions for Continuous-Time Markov Chains (CTMCs) begin with the ‘ctmc’ prefix, and the names of functions for Discrete-Time Markov Chains (DTMCs) begin with the ‘dtmc’ prefix.

test/

This directory contains the test scripts used to run all function tests.

devel/

This directory contains functions that are either not working properly, or need additional testing before they are moved to the inst/ directory.

The queueing package ships with a Makefile which can be used to produce the documentation (in PDF and HTML format), and automatically execute all function tests. The following targets are defined:

all

Running ‘make’ (or ‘make all’) on the top-level directory builds the programs used to extract the documentation from the comments embedded in the m-files, and then produce the documentation in PDF and HTML format (doc/queueing.pdf and doc/queueing.html, respectively).

check

Running ‘make check’ will execute all tests contained in the m-files. If you modify the code of any function in the inst/ directory, you should run the tests to ensure that no errors have been introduced. You are also encouraged to contribute new tests, especially for functions that are not adequately validated.

clean
distclean
dist

The ‘make clean’, ‘make distclean’ and ‘make dist’ commands are used to clean up the source directory and prepare the distribution archive in compressed tar format.

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2.4 Naming Conventions

Most of the functions in the queueing package obey a common naming convention. Function names are made of several parts; the first part is a prefix which indicates the class of problems the function addresses:

ctmc-

Functions for continuous-time Markov chains

dtmc-

Functions for discrete-time Markov chains

qs-

Functions for analyzing single-station queueing systems (individual service centers)

qn-

Functions for analyzing queueing networks

Functions dealing with Markov chains start with either the ctmc or dtmc prefix; the prefix is optionally followed by an additional string which hints at what the function does:

-bd

Birth-Death process

-mtta

Mean Time to Absorption

-fpt

First Passage Times

-exps

Expected Sojourn Times

-taexps

Time-Averaged Expected Sojourn Times

For example, function ctmcbd returns the infinitesimal generator matrix for a continuous birth-death process, while dtmcbd returns the transition probability matrix for a discrete birth-death process. Note that there exist functions ctmc and dtmc (without any suffix) that compute steady-state and transient state occupancy probabilities for CTMCs and DTMCs, respectively. See Markov Chains.

Functions whose name starts with qs- deal with single station queueing systems. The suffix describes the type of system, e.g., qsmm1 for M/M/1, qnmmm for M/M/m and so on. See Single Station Queueing Systems.

Finally, functions whose name starts with qn- deal with queueing networks. The character that follows indicates whether the function handles open ('o') or closed ('c') networks, and whether there is a single customer class ('s') or multiple classes ('m'). The string mix indicates that the function supports mixed networks with both open and closed customer classes.

-os-

Open, single-class network: open network with a single class of customers

-om-

Open, multiclass network: open network with multiple job classes

-cs-

Closed, single-class network

-cm-

Closed, multiclass network

-mix-

Mixed network with open and closed classes of customers

The last part of the function name indicates the algorithm implemented by the function. See Queueing Networks.

-aba

Asymptotic Bounds Analysis

-bsb

Balanced System Bounds

-gb

Geometric Bounds

-pb

PB Bounds

-cb

Composite Bounds (CB)

-mva

Mean Value Analysis (MVA) algorithm

-cmva

Conditional MVA

-mvald

MVA with general load-dependent servers

-mvaap

Approximate MVA

-mvablo

MVABLO approximation for blocking queueing networks

-conv

Convolution algorithm

-convld

Convolution algorithm with general load-dependent servers

Some deprecated functions may be present in the queueing package; generally, these are functions that have been renamed, and the old name is kept for a while for backward compatibility. Deprecated functions are not documented and will be removed in future releases. Calling a deprecated functions displays a warning message that appears only once per session. The warning message can be turned off with the command: 

octave:1> warning ("off", "qn:deprecated-function");

However, you are strongly recommended to update your code to the new API. To help catching usages of deprecated functions, you can transform warnings into errors so that your application stops immediately: 

octave:1> warning ("error", "qn:deprecated-function");

Previous: , Up: Installation and Getting Started   [Contents][Index]

2.5 Quick start Guide

You can use all functions by simply invoking their name with the appropriate parameters; an error is shown in case of missing/wrong parameters. Extensive documentation is provided for each function, and can be displayed with the help command. For example: 

octave:2> help qncsmvablo

shows the documentation for the qncsmvablo function. Additional information can be found in the queueing manual, that is available in PDF format in doc/queueing.pdf and in HTML format in doc/queueing.html.

Many functions have demo blocks showing usage examples. To execute the demos for the qnclosed function, use the demo command: 

octave:4> demo qnclosed

We now illustrate a few examples of how the queueing package can be used. More examples are provided in the manual.

Example 1 Compute the stationary state occupancy probabilities of a continuous-time Markov chain with infinitesimal generator matrix 

    / -0.8   0.6   0.2 \
Q = |  0.3  -0.7   0.4 |
    \  0.2   0.2  -0.4 /

Q = [ -0.8  0.6  0.2; \
       0.3 -0.7  0.4; \
       0.2  0.2 -0.4 ];
q = ctmc(Q)
    ⇒ q = 0.23256   0.32558   0.44186

Example 2 Compute the transient state occupancy probability after n=3 transitions of a three state discrete-time birth-death process, with birth probabilities \lambda_{01} = 0.3 and \lambda_{12} = 0.5 and death probabilities \mu_{10} = 0.5 and \mu_{21} = 0.7, assuming that the system is initially in state zero (i.e., the initial state occupancy probabilities are [1, 0, 0]). 

n = 3;
p0 = [1 0 0];
P = dtmcbd( [0.3 0.5], [0.5 0.7] );
p = dtmc(P,n,p0)
    ⇒ p = 0.55300   0.29700   0.15000

Example 3 Compute server utilization, response time, mean number of requests and throughput of a closed queueing network with N=4 requests and three M/M/1–FCFS queues with mean service times S = [1.0, 0.8, 1.4] and average number of visits V = [1.0, 0.8, 0.8] 

S = [1.0 0.8 1.4];
V = [1.0 0.8 0.8];
N = 4;
[U R Q X] = qncsmva(N, S, V)
    ⇒ 
     U = 0.70064   0.44841   0.78471
     R = 2.1030    1.2642    3.2433
     Q = 1.47346   0.70862   1.81792
     X = 0.70064   0.56051   0.56051

Example 4 Compute server utilization, response time, mean number of requests and throughput of an open queueing network with three M/M/1–FCFS queues with mean service times S = [1.0, 0.8, 1.4] and average number of visits V = [1.0, 0.8, 0.8]. The overall arrival rate is \lambda = 0.8 requests/second. 

S = [1.0 0.8 1.4];
V = [1.0 0.8 0.8];
lambda = 0.8;
[U R Q X] = qnos(lambda, S, V)
    ⇒ 
     U = 0.80000   0.51200   0.89600
     R = 5.0000    1.6393   13.4615
     Q = 4.0000    1.0492    8.6154
     X = 0.80000   0.64000   0.64000

Next: , Previous: , Up: Top   [Contents][Index]

3 Markov Chains

• Discrete-Time Markov Chains:  
• Continuous-Time Markov Chains:  

Next: , Up: Markov Chains   [Contents][Index]

3.1 Discrete-Time Markov Chains

Let X_0, X_1, …, X_n, … be a sequence of random variables defined over the discrete state space 1, 2, …. The sequence X_0, X_1, …, X_n, … is a stochastic process with discrete time 0, 1, 2, …. A Markov chain is a stochastic process {X_n, n=0, 1, …} which satisfies the following Markov property:

P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, …, X_0 = x_0) = P(X_{n+1} = x_{n+1} | X_n = x_n)

which basically means that the probability that the system is in a particular state at time n+1 only depends on the state the system was at time n.

The evolution of a Markov chain with finite state space {1, …, N} can be fully described by a stochastic matrix {\bf P}(n) = [ P_{i,j}(n) ] where P_{i, j}(n) = P( X_{n+1} = j\ |\ X_n = i ). If the Markov chain is homogeneous (that is, the transition probability matrix {\bf P}(n) is time-independent), we can write {\bf P} = [P_{i, j}], where P_{i, j} = P( X_{n+1} = j\ |\ X_n = i ) for all n=0, 1, ….

The transition probability matrix \bf P must be a stochastic matrix, meaning that it must satisfy the following two properties:

  1. P_{i, j} ≥ 0 for all 1 ≤ i, j ≤ N;
  2. \sum_{j=1}^N P_{i,j} = 1 for all i

Property 1 requires that all probabilities are nonnegative; property 2 requires that the outgoing transition probabilities from any state i sum to one.

Function File: [r err] = dtmcchkP (P)

Check whether P is a valid transition probability matrix.

If P is valid, r is the size (number of rows or columns) of P. If P is not a transition probability matrix, r is set to zero, and err to an appropriate error string.

A DTMC is irreducible if every state can be reached with non-zero probability starting from every other state.

Function File: [r s] = dtmcisir (P)

Check if P is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with transition matrix P.

INPUTS

P(i,j)

transition probability from state i to state j. P must be an N \times N stochastic matrix.

OUTPUTS

r

1 if P is irreducible, 0 otherwise (scalar)

s(i)

strongly connected component (SCC) that state i belongs to (vector of length N). SCCs are numbered 1, 2, …. The number of SCCs is max(s). If the graph is strongly connected, then there is a single SCC and the predicate all(s == 1) evaluates to true

• State occupancy probabilities (DTMC):  
• Birth-death process (DTMC):  
• Expected number of visits (DTMC):  
• Time-averaged expected sojourn times (DTMC):  
• Mean time to absorption (DTMC):  
• First passage times (DTMC):  

Next: , Up: Discrete-Time Markov Chains   [Contents][Index]

3.1.1 State occupancy probabilities

Given a discrete-time Markov chain with state space {1, …, N}, we denote with {\bf \pi}(n) = \left[\pi_1(n), … \pi_N(n) \right] the state occupancy probability vector at step n = 0, 1, …. \pi_i(n) is the probability that the system is in state i after n transitions.

Given the transition probability matrix \bf P and the initial state occupancy probability vector {\bf \pi}(0) = \left[\pi_1(0), …, \pi_N(0)\right], {\bf \pi}(n) can be computed as:

\pi(n) = \pi(0) P^n

Under certain conditions, there exists a stationary state occupancy probability {\bf \pi} = \lim_{n \rightarrow +\infty} {\bf \pi}(n), which is independent from {\bf \pi}(0). The vector \bf \pi is the solution of the following linear system: 

/
| \pi P   = \pi
| \pi 1^T = 1
\

where \bf 1 is the row vector of ones, and ( \cdot )^T the transpose operator.

Function File: p = dtmc (P)
Function File: p = dtmc (P, n, p0)

Compute stationary or transient state occupancy probabilities for a discrete-time Markov chain.

With a single argument, compute the stationary state occupancy probabilities p(1), …, p(N) for a discrete-time Markov chain with finite state space {1, …, N} and with N \times N transition matrix P. With three arguments, compute the transient state occupancy probabilities p(1), …, p(N) that the system is in state i after n steps, given initial occupancy probabilities p0(1), …, p0(N).

INPUTS

P(i,j)

transition probabilities from state i to state j. P must be an N \times N irreducible stochastic matrix, meaning that the sum of each row must be 1 (\sum_{j=1}^N P_{i, j} = 1), and the rank of P must be N.

n

Number of transitions after which state occupancy probabilities are computed (scalar, n ≥ 0)

p0(i)

probability that at step 0 the system is in state i (vector of length N).

OUTPUTS

p(i)

If this function is called with a single argument, p(i) is the steady-state probability that the system is in state i. If this function is called with three arguments, p(i) is the probability that the system is in state i after n transitions, given the probabilities p0(i) that the initial state is i.


See also: ctmc.

EXAMPLE

The following example is from GrSn97. Let us consider a maze with nine rooms, as shown in the following figure 

+-----+-----+-----+
|     |     |     |
|  1     2     3  |
|     |     |     |
+-   -+-   -+-   -+
|     |     |     |
|  4     5     6  |
|     |     |     |
+-   -+-   -+-   -+
|     |     |     |
|  7     8     9  |
|     |     |     |
+-----+-----+-----+

A mouse is placed in one of the rooms and can wander around. At each step, the mouse moves from the current room to a neighboring one with equal probability. For example, if it is in room 1, it can move to room 2 and 4 with probability 1/2, respectively; if the mouse is in room 8, it can move to either 7, 5 or 9 with probability 1/3.

The transition probabilities P_{i, j} from room i to room j can be summarized in the following matrix: 

        / 0     1/2   0     1/2   0     0     0     0     0   \
        | 1/3   0     1/3   0     1/3   0     0     0     0   |
        | 0     1/2   0     0     0     1/2   0     0     0   |
        | 1/3   0     0     0     1/3   0     1/3   0     0   |
    P = | 0     1/4   0     1/4   0     1/4   0     1/4   0   |
        | 0     0     1/3   0     1/3   0     0     0     1/3 |
        | 0     0     0     1/2   0     0     0     1/2   0   |
        | 0     0     0     0     1/3   0     1/3   0     1/3 |
        \ 0     0     0     0     0     1/2   0     1/2   0   /

The stationary state occupancy probabilities can then be computed with the following code: 

 P = zeros(9,9);
 P(1,[2 4]    ) = 1/2;
 P(2,[1 5 3]  ) = 1/3;
 P(3,[2 6]    ) = 1/2;
 P(4,[1 5 7]  ) = 1/3;
 P(5,[2 4 6 8]) = 1/4;
 P(6,[3 5 9]  ) = 1/3;
 P(7,[4 8]    ) = 1/2;
 P(8,[7 5 9]  ) = 1/3;
 P(9,[6 8]    ) = 1/2;
 p = dtmc(P);
 disp(p)

    ⇒ 0.083333   0.125000   0.083333   0.125000   
       0.166667   0.125000   0.083333   0.125000   
       0.083333

Next: , Previous: , Up: Discrete-Time Markov Chains   [Contents][Index]

3.1.2 Birth-death process

Function File: P = dtmcbd (b, d)

Returns the transition probability matrix P for a discrete birth-death process over state space {1, …, N}. For each i=1, …, (N-1), b(i) is the transition probability from state i to (i+1), and d(i) is the transition probability from state (i+1) to i.

Matrix \bf P is defined as: 

/                                                             \
| 1-b(1)     b(1)                                             |
|  d(1)  (1-d(1)-b(2))     b(2)                               |
|            d(2)      (1-d(2)-b(3))     b(3)                 |
|                                                             |
|                 ...           ...          ...              |
|                                                             |
|                         d(N-2)   (1-d(N-2)-b(N-1))  b(N-1)  |
|                                        d(N-1)      1-d(N-1) |
\                                                             /

where \lambda_i and \mu_i are the birth and death probabilities, respectively.


See also: ctmcbd.

Next: , Previous: , Up: Discrete-Time Markov Chains   [Contents][Index]

3.1.3 Expected Number of Visits

Given a N state discrete-time Markov chain with transition matrix \bf P and an integer n ≥ 0, we let L_i(n) be the the expected number of visits to state i during the first n transitions. The vector {\bf L}(n) = \left[ L_1(n), …, L_N(n) \right] is defined as 

         n            n
        ___          ___
       \            \           i
L(n) =  >   pi(i) =  >   pi(0) P
       /___         /___
        i=0          i=0

where {\bf \pi}(i) = {\bf \pi}(0){\bf P}^i is the state occupancy probability after i transitions, and {\bf \pi}(0) = \left[\pi_1(0), …, \pi_N(0) \right] are the initial state occupancy probabilities.

If \bf P is absorbing, i.e., the stochastic process eventually enters a state with no outgoing transitions, then we can compute the expected number of visits until absorption \bf L. To do so, we first rearrange the states by rewriting \bf P as 

    / Q | R \
P = |---+---|
    \ 0 | I /

where the first t states are transient and the last r states are absorbing (t+r = N). The matrix {\bf N} = ({\bf I} - {\bf Q})^{-1} is called the fundamental matrix; N_{i,j} is the expected number of times the process is in the j-th transient state assuming it started in the i-th transient state. If we reshape \bf N to the size of \bf P (filling missing entries with zeros), we have that, for absorbing chains, {\bf L} = {\bf \pi}(0){\bf N}.

Function File: L = dtmcexps (P, n, p0)
Function File: L = dtmcexps (P, p0)

Compute the expected number of visits to each state during the first n transitions, or until abrosption.

INPUTS

P(i,j)

N \times N transition matrix. P(i,j) is the transition probability from state i to state j.

n

Number of steps during which the expected number of visits are computed (n ≥ 0). If n=0, returns p0. If n > 0, returns the expected number of visits after exactly n transitions.

p0(i)

Initial state occupancy probabilities; p0(i) is the probability that the system is in state i at step 0.

OUTPUTS

L(i)

When called with two arguments, L(i) is the expected number of visits to state i before absorption. When called with three arguments, L(i) is the expected number of visits to state i during the first n transitions.

REFERENCES

  • Grinstead, Charles M.; Snell, J. Laurie (July 1997). Introduction to Probability, Ch. 11: Markov Chains. American Mathematical Society. ISBN 978-0821807491.

See also: ctmcexps.

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3.1.4 Time-averaged expected sojourn times

Function File: M = dtmctaexps (P, n, p0)
Function File: M = dtmctaexps (P, p0)

Compute the time-averaged sojourn times M(i), defined as the fraction of time spent in state i during the first n transitions (or until absorption), assuming that the state occupancy probabilities at time 0 are p0.

INPUTS

P(i,j)

N \times N transition probability matrix.

n

Number of transitions during which the time-averaged expected sojourn times are computed (scalar, n ≥ 0). if n = 0, returns p0.

p0(i)

Initial state occupancy probabilities (vector of length N).

OUTPUTS

M(i)

If this function is called with three arguments, M(i) is the expected fraction of steps {0, …, n} spent in state i, assuming that the state occupancy probabilities at time zero are p0. If this function is called with two arguments, M(i) is the expected fraction of steps spent in state i until absorption. M is a vector of length N.


See also: dtmcexps.

Next: , Previous: , Up: Discrete-Time Markov Chains   [Contents][Index]

3.1.5 Mean Time to Absorption

The mean time to absorption is defined as the average number of transitions that are required to enter an absorbing state, starting from a transient state or given initial state occupancy probabilities {\bf \pi}(0).

Let t_i be the expected number of transitions before being absorbed in any absorbing state, starting from state i. The vector {\bf t} = [t_1, …, t_N] can be computed from the fundamental matrix \bf N (see Expected number of visits (DTMC)) as

t = N c

where \bf c is a column vector of 1’s.

Let {\bf B} = [ B_{i, j} ] be a matrix where B_{i, j} is the probability of being absorbed in state j, starting from transient state i. Again, using matrices \bf N and \bf R (see Expected number of visits (DTMC)) we can write

B = N R

Function File: [t N B] = dtmcmtta (P)
Function File: [t N B] = dtmcmtta (P, p0)

Compute the expected number of steps before absorption for a DTMC with state space {1, …, N} and transition probability matrix P.

INPUTS

P(i,j)

N \times N transition probability matrix. P(i,j) is the transition probability from state i to state j.

p0(i)

Initial state occupancy probabilities (vector of length N).

OUTPUTS

t
t(i)

When called with a single argument, t is a vector of length N such that t(i) is the expected number of steps before being absorbed in any absorbing state, starting from state i; if i is absorbing, t(i) = 0. When called with two arguments, t is a scalar, and represents the expected number of steps before absorption, starting from the initial state occupancy probability p0.

N(i)
N(i,j)

When called with a single argument, N is the N \times N fundamental matrix for P. N(i,j) is the expected number of visits to transient state j before absorption, if the system started in transient state i. The initial state is counted if i = j. When called with two arguments, N is a vector of length N such that N(j) is the expected number of visits to transient state j before absorption, given initial state occupancy probability P0.

B(i)
B(i,j)

When called with a single argument, B is a N \times N matrix where B(i,j) is the probability of being absorbed in state j, starting from transient state i; if j is not absorbing, B(i,j) = 0; if i is absorbing, B(i,i) = 1 and B(i,j) = 0 for all i \neq j. When called with two arguments, B is a vector of length N where B(j) is the probability of being absorbed in state j, given initial state occupancy probabilities p0.

REFERENCES

  • Grinstead, Charles M.; Snell, J. Laurie (July 1997). Introduction to Probability, Ch. 11: Markov Chains. American Mathematical Society. ISBN 978-0821807491.

See also: ctmcmtta.

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3.1.6 First Passage Times

The First Passage Time M_{i, j} is the average number of transitions needed to enter state j for the first time, starting from state i. Matrix \bf M satisfies the property 

           ___
          \
M_ij = 1 + >   P_ij * M_kj
          /___
          k!=j

To compute {\bf M} = [ M_{i, j}] a different formulation is used. Let \bf W be the N \times N matrix having each row equal to the stationary state occupancy probability vector \bf \pi for \bf P; let \bf I be the N \times N identity matrix (i.e., the matrix of all ones). Define \bf Z as follows: 

               -1
Z = (I - P + W)

Then, we have that 

       Z_jj - Z_ij
M_ij = -----------
          \pi_j

According to the definition above, M_{i,i} = 0. We arbitrarily set M_{i,i} to the mean recurrence time r_i for state i, that is the average number of transitions needed to return to state i starting from it. r_i is: 

        1
r_i = -----
      \pi_i
Function File: M = dtmcfpt (P)

Compute mean first passage times and mean recurrence times for an irreducible discrete-time Markov chain over the state space {1, …, N}.

INPUTS

P(i,j)

transition probability from state i to state j. P must be an irreducible stochastic matrix, which means that the sum of each row must be 1 (\sum_{j=1}^N P_{i j} = 1), and the rank of P must be N.

OUTPUTS

M(i,j)

For all 1 ≤ i, j ≤ N, i \neq j, M(i,j) is the average number of transitions before state j is entered for the first time, starting from state i. M(i,i) is the mean recurrence time of state i, and represents the average time needed to return to state i.

REFERENCES

  • Grinstead, Charles M.; Snell, J. Laurie (July 1997). Introduction to Probability, Ch. 11: Markov Chains. American Mathematical Society. ISBN 978-0821807491.

See also: ctmcfpt.

Previous: , Up: Markov Chains   [Contents][Index]

3.2 Continuous-Time Markov Chains

A stochastic process {X(t), t ≥ 0} is a continuous-time Markov chain if, for all integers n, and for any sequence t_0, t_1 , …, t_n, t_{n+1} such that t_0 < t_1 < … < t_n < t_{n+1}, we have

P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, ..., X_0 = x_0) = P(X_{n+1} = x_{n+1} | X_n = x_n)

A continuous-time Markov chain is defined according to an infinitesimal generator matrix {\bf Q} = [Q_{i,j}], where for each i \neq j, Q_{i, j} is the transition rate from state i to state j. The matrix \bf Q must satisfy the property that, for all i, \sum_{j=1}^N Q_{i, j} = 0.

Function File: [result err] = ctmcchkQ (Q)

If Q is a valid infinitesimal generator matrix, return the size (number of rows or columns) of Q. If Q is not an infinitesimal generator matrix, set result to zero, and err to an appropriate error string.

Similarly to the DTMC case, a CTMC is irreducible if every state is eventually reachable from every other state in finite time.

Function File: [r s] = ctmcisir (P)

Check if Q is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with infinitesimal generator matrix Q.

INPUTS

Q(i,j)

Infinitesimal generator matrix. Q is a N \times N square matrix where Q(i,j) is the transition rate from state i to state j, for 1 ≤ i, j ≤ N, i \neq j.

OUTPUTS

r

1 if Q is irreducible, 0 otherwise.

s(i)

strongly connected component (SCC) that state i belongs to. SCCs are numbered 1, 2, …. If the graph is strongly connected, then there is a single SCC and the predicate all(s == 1) evaluates to true.

• State occupancy probabilities (CTMC):  
• Birth-death process (CTMC):  
• Expected sojourn times (CTMC):  
• Time-averaged expected sojourn times (CTMC):  
• Mean time to absorption (CTMC):  
• First passage times (CTMC):  

Next: , Up: Continuous-Time Markov Chains   [Contents][Index]

3.2.1 State occupancy probabilities

Similarly to the discrete case, we denote with {\bf \pi}(t) = \left[\pi_1(t), …, \pi_N(t) \right] the state occupancy probability vector at time t. \pi_i(t) is the probability that the system is in state i at time t ≥ 0.

Given the infinitesimal generator matrix \bf Q and initial state occupancy probabilities {\bf \pi}(0) = \left[\pi_1(0), …, \pi_N(0)\right], the occupancy probabilities {\bf \pi}(t) at time t can be computed as: 

\pi(t) = \pi(0) exp(Qt)

where \exp( {\bf Q} t ) is the matrix exponential of {\bf Q} t. Under certain conditions, there exists a stationary state occupancy probability {\bf \pi} = \lim_{t \rightarrow +\infty} {\bf \pi}(t) that is independent from {\bf \pi}(0). \bf \pi is the solution of the following linear system: 

/
| \pi Q   = 0
| \pi 1^T = 1
\
Function File: p = ctmc (Q)
Function File: p = ctmc (Q, t, p0)

Compute stationary or transient state occupancy probabilities for a continuous-time Markov chain.

With a single argument, compute the stationary state occupancy probabilities p(1), …, p(N) for a continuous-time Markov chain with finite state space {1, …, N} and N \times N infinitesimal generator matrix Q. With three arguments, compute the state occupancy probabilities p(1), …, p(N) that the system is in state i at time t, given initial state occupancy probabilities p0(1), …, p0(N) at time 0.

INPUTS

Q(i,j)

Infinitesimal generator matrix. Q is a N \times N square matrix where Q(i,j) is the transition rate from state i to state j, for 1 ≤ i \neq j ≤ N. Q must satisfy the property that \sum_{j=1}^N Q_{i, j} = 0

t

Time at which to compute the transient probability (t ≥ 0). If omitted, the function computes the steady state occupancy probability vector.

p0(i)

probability that the system is in state i at time 0.

OUTPUTS

p(i)

If this function is invoked with a single argument, p(i) is the steady-state probability that the system is in state i, i = 1, …, N. If this function is invoked with three arguments, p(i) is the probability that the system is in state i at time t, given the initial occupancy probabilities p0(1), …, p0(N).


See also: dtmc.

EXAMPLE

Consider a two-state CTMC where all transition rates between states are equal to 1. The stationary state occupancy probabilities can be computed as follows: 

 Q = [ -1  1; ...
        1 -1  ];
 q = ctmc(Q)

    ⇒ q = 0.50000   0.50000

Next: , Previous: , Up: Continuous-Time Markov Chains   [Contents][Index]

3.2.2 Birth-Death Process

Function File: Q = ctmcbd (b, d)

Returns the infinitesimal generator matrix Q for a continuous birth-death process over the finite state space {1, …, N}. For each i=1, …, (N-1), b(i) is the transition rate from state i to state (i+1), and d(i) is the transition rate from state (i+1) to state i.

Matrix \bf Q is therefore defined as: 

/                                                          \
| -b(1)     b(1)                                           |
|  d(1) -(d(1)+b(2))     b(2)                              |
|           d(2)     -(d(2)+b(3))        b(3)              |
|                                                          |
|                ...           ...          ...            |
|                                                          |
|                       d(N-2)    -(d(N-2)+b(N-1))  b(N-1) |
|                                       d(N-1)     -d(N-1) |
\                                                          /

where \lambda_i and \mu_i are the birth and death rates, respectively.


See also: dtmcbd.

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3.2.3 Expected Sojourn Times

Given a N state continuous-time Markov Chain with infinitesimal generator matrix \bf Q, we define the vector {\bf L}(t) = \left[L_1(t), …, L_N(t)\right] such that L_i(t) is the expected sojourn time in state i during the interval [0,t), assuming that the initial occupancy probabilities at time 0 were {\bf \pi}(0). {\bf L}(t) can be expressed as the solution of the following differential equation: 

 dL
 --(t) = L(t) Q + pi(0),    L(0) = 0
 dt

Alternatively, {\bf L}(t) can also be expressed in integral form as: 

       / t
L(t) = |   pi(u) du
       / 0

where {\bf \pi}(t) = {\bf \pi}(0) \exp({\bf Q}t) is the state occupancy probability at time t; \exp({\bf Q}t) is the matrix exponential of {\bf Q}t.

If there are absorbing states, we can define the vector of expected sojourn times until absorption {\bf L}(\infty), where for each transient state i, L_i(\infty) is the expected total time spent in state i until absorption, assuming that the system started with given state occupancy probabilities {\bf \pi}(0). Let \tau be the set of transient (i.e., non absorbing) states; let {\bf Q}_\tau be the restriction of \bf Q to the transient sub-states only. Similarly, let {\bf \pi}_\tau(0) be the restriction of the initial state occupancy probability vector {\bf \pi}(0) to transient states \tau.

The expected time to absorption {\bf L}_\tau(\infty) is defined as the solution of the following equation: 

L_T( inf ) Q_T = -pi_T(0)
Function File: L = ctmcexps (Q, t, p )
Function File: L = ctmcexps (Q, p)

With three arguments, compute the expected times L(i) spent in each state i during the time interval [0,t], assuming that the initial occupancy vector is p. With two arguments, compute the expected time L(i) spent in each transient state i until absorption.

Note: In its current implementation, this function requires that an absorbing state is reachable from any non-absorbing state of Q.

INPUTS

Q(i,j)

N \times N infinitesimal generator matrix. Q(i,j) is the transition rate from state i to state j, 1 ≤ i, j ≤ N, i \neq j. The matrix Q must also satisfy the condition \sum_{j=1}^N Q_{i,j} = 0 for every i=1, …, N.

t

If given, compute the expected sojourn times in [0,t]

p(i)

Initial occupancy probability vector; p(i) is the probability the system is in state i at time 0, i = 1, …, N

OUTPUTS

L(i)

If this function is called with three arguments, L(i) is the expected time spent in state i during the interval [0,t]. If this function is called with two arguments L(i) is the expected time spent in transient state i until absorption; if state i is absorbing, L(i) is zero.


See also: dtmcexps.

EXAMPLE

Let us consider a 4-states pure birth continuous process where the transition rate from state i to state (i+1) is \lambda_i = i \lambda (i=1, 2, 3), with \lambda = 0.5. The following code computes the expected sojourn time for each state i, given initial occupancy probabilities {\bf \pi}_0=[1, 0, 0, 0]. 

 lambda = 0.5;
 N = 4;
 b = lambda*[1:N-1];
 d = zeros(size(b));
 Q = ctmcbd(b,d);
 t = linspace(0,10,100);
 p0 = zeros(1,N); p0(1)=1;
 L = zeros(length(t),N);
 for i=1:length(t)
   L(i,:) = ctmcexps(Q,t(i),p0);
 endfor
 plot( t, L(:,1), ";State 1;", "linewidth", 2, ...
       t, L(:,2), ";State 2;", "linewidth", 2, ...
       t, L(:,3), ";State 3;", "linewidth", 2, ...
       t, L(:,4), ";State 4;", "linewidth", 2 );
 legend("location","northwest"); legend("boxoff");
 xlabel("Time");
 ylabel("Expected sojourn time");

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3.2.4 Time-Averaged Expected Sojourn Times

Function File: M = ctmctaexps (Q, t, p0)
Function File: M = ctmctaexps (Q, p0)

Compute the time-averaged sojourn time M(i), defined as the fraction of the time interval [0,t] (or until absorption) spent in state i, assuming that the state occupancy probabilities at time 0 are p.

INPUTS

Q(i,j)

Infinitesimal generator matrix. Q(i,j) is the transition rate from state i to state j, 1 ≤ i,j ≤ N, i \neq j. The matrix Q must also satisfy the condition \sum_{j=1}^N Q_{i,j} = 0

t

Time. If omitted, the results are computed until absorption.

p0(i)

initial state occupancy probabilities. p0(i) is the probability that the system is in state i at time 0, i = 1, …, N

OUTPUTS

M(i)

When called with three arguments, M(i) is the expected fraction of the interval [0,t] spent in state i assuming that the state occupancy probability at time zero is p. When called with two arguments, M(i) is the expected fraction of time until absorption spent in state i; in this case the mean time to absorption is sum(M).


See also: ctmcexps.

EXAMPLE 

 lambda = 0.5;
 N = 4;
 birth = lambda*linspace(1,N-1,N-1);
 death = zeros(1,N-1);
 Q = diag(birth,1)+diag(death,-1);
 Q -= diag(sum(Q,2));
 t = linspace(1e-5,30,100);
 p = zeros(1,N); p(1)=1;
 M = zeros(length(t),N);
 for i=1:length(t)
   M(i,:) = ctmctaexps(Q,t(i),p);
 endfor
 clf;
 plot(t, M(:,1), ";State 1;", "linewidth", 2, ...
      t, M(:,2), ";State 2;", "linewidth", 2, ...
      t, M(:,3), ";State 3;", "linewidth", 2, ...
      t, M(:,4), ";State 4 (absorbing);", "linewidth", 2 );
 legend("location","east"); legend("boxoff");
 xlabel("Time");
 ylabel("Time-averaged Expected sojourn time");

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3.2.5 Mean Time to Absorption

Function File: t = ctmcmtta (Q, p)

Compute the Mean-Time to Absorption (MTTA) of the CTMC described by the infinitesimal generator matrix Q, starting from initial occupancy probabilities p. If there are no absorbing states, this function fails with an error.

INPUTS

Q(i,j)

N \times N infinitesimal generator matrix. Q(i,j) is the transition rate from state i to state j, i \neq j. The matrix Q must satisfy the condition \sum_{j=1}^N Q_{i,j} = 0

p(i)

probability that the system is in state i at time 0, for each i=1, …, N

OUTPUTS

t

Mean time to absorption of the process represented by matrix Q. If there are no absorbing states, this function fails.

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998.

See also: ctmcexps.

EXAMPLE

Let us consider a simple model of redundant disk array. We assume that the array is made of 5 independent disks and can tolerate up to 2 disk failures without losing data. If three or more disks break, the array is dead and unrecoverable. We want to estimate the Mean-Time-To-Failure (MTTF) of the disk array.

We model this system as a 4 states continuous Markov chain with state space { 2, 3, 4, 5 }. In state i there are exactly i active (i.e., non failed) disks; state 2 is absorbing. Let \mu be the failure rate of a single disk. The system starts in state 5 (all disks are operational). We use a pure death process, where the death rate from state i to state (i-1) is \mu i, for i = 3, 4, 5).

The MTTF of the disk array is the MTTA of the Markov Chain, and can be computed as follows: 

 mu = 0.01;
 death = [ 3 4 5 ] * mu;
 birth = 0*death;
 Q = ctmcbd(birth,death);
 t = ctmcmtta(Q,[0 0 0 1])

    ⇒ t = 78.333

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3.2.6 First Passage Times

Function File: M = ctmcfpt (Q)
Function File: m = ctmcfpt (Q, i, j)

Compute mean first passage times for an irreducible continuous-time Markov chain.

INPUTS

Q(i,j)

Infinitesimal generator matrix. Q is a N \times N square matrix where Q(i,j) is the transition rate from state i to state j, for 1 ≤ i, j ≤ N, i \neq j. Transition rates must be nonnegative, and \sum_{j=1}^N Q_{i,j} = 0

i

Initial state.

j

Destination state.

OUTPUTS

M(i,j)

average time before state j is visited for the first time, starting from state i. We let M(i,i) = 0.

m

m is the average time before state j is visited for the first time, starting from state i.


See also: ctmcmtta.

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4 Single Station Queueing Systems

Single Station Queueing Systems contain a single station, and can usually be analyzed easily. The queueing package contains functions for handling the following types of queues:

• The M/M/1 System:  Single-server queueing station.
• The M/M/m System:  Multiple-server queueing station.
• The Erlang-B Formula:  
• The Erlang-C Formula:  
• The Engset Formula:  
• The M/M/inf System:  Infinite-server (delay center) station.
• The M/M/1/K System:  Single-server, finite-capacity queueing station.
• The M/M/m/K System:  Multiple-server, finite-capacity queueing station.
• The Asymmetric M/M/m System:  Asymmetric multiple-server queueing station.
• The M/G/1 System:  Single-server with general service time distribution.
• The M/Hm/1 System:  Single-server with hyperexponential service time distribution.

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4.1 The M/M/1 System

The M/M/1 system contains a single server connected to an unbounded FCFS queue. Requests arrive according to a Poisson process with rate \lambda; the service time is exponentially distributed with average service rate \mu. The system is stable if \lambda < \mu.

Function File: [U, R, Q, X, p0] = qsmm1 (lambda, mu)
Function File: pk = qsmm1 (lambda, mu, k)

Compute utilization, response time, average number of requests and throughput for a M/M/1 queue.

INPUTS

lambda

Arrival rate (lambda ≥ 0).

mu

Service rate (mu > lambda).

k

Number of requests in the system (k ≥ 0).

OUTPUTS

U

Server utilization

R

Server response time

Q

Average number of requests in the system

X

Server throughput. If the system is ergodic (mu > lambda), we always have X = lambda

p0

Steady-state probability that there are no requests in the system.

pk

Steady-state probability that there are k requests in the system. (including the one being served).

If this function is called with less than three input parameters, lambda and mu can be vectors of the same size. In this case, the results will be vectors as well.

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.3

See also: qsmmm, qsmminf, qsmmmk.

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4.2 The M/M/m System

The M/M/m system is similar to the M/M/1 system, except that there are m \geq 1 identical servers connected to a shared FCFS queue. Thus, at most m requests can be served at the same time. The M/M/m system can be seen as a single server with load-dependent service rate \mu(n), which is a function of the number n of requests in the system: 

mu(n) = min(m,n)*mu

where \mu is the service rate of each individual server.

Function File: [U, R, Q, X, p0, pm] = qsmmm (lambda, mu)
Function File: [U, R, Q, X, p0, pm] = qsmmm (lambda, mu, m)
Function File: pk = qsmmm (lambda, mu, m, k)

Compute utilization, response time, average number of requests in service and throughput for a M/M/m queue, a queueing system with m identical servers connected to a single FCFS queue.

INPUTS

lambda

Arrival rate (lambda>0).

mu

Service rate (mu>lambda).

m

Number of servers (m ≥ 1). Default is m=1.

k

Number of requests in the system (k ≥ 0).

OUTPUTS

U

Service center utilization, U = \lambda / (m \mu).

R

Service center mean response time

Q

Average number of requests in the system

X

Service center throughput. If the system is ergodic, we will always have X = lambda

p0

Steady-state probability that there are 0 requests in the system

pm

Steady-state probability that an arriving request has to wait in the queue

pk

Steady-state probability that there are k requests in the system (including the one being served).

If this function is called with less than four parameters, lambda, mu and m can be vectors of the same size. In this case, the results will be vectors as well.

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.5

See also: erlangc,qsmm1,qsmminf,qsmmmk.

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4.3 The Erlang-B Formula

Function File: B = erlangb (A, m)

Compute the steady-state blocking probability in the Erlang loss model.

The Erlang-B formula E_B(A, m) gives the probability that an open system with m identical servers, arrival rate \lambda, individual service rate \mu and offered load A = \lambda / \mu has all servers busy. This corresponds to the rejection probability of an M/M/m/0 system with m servers and no queue.

INPUTS

A

Offered load, defined as A = \lambda / \mu where \lambda is the mean arrival rate and \mu the mean service rate of each individual server (real, A > 0).

m

Number of identical servers (integer, m ≥ 1). Default m = 1

OUTPUTS

B

The value E_B(A, m)

A or m can be vectors, and in this case, the results will be vectors as well.

REFERENCES

  • G. Zeng, Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296

See also: erlangc,engset,qsmmm.

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4.4 The Erlang-C Formula

Function File: C = erlangc (A, m)

Compute the steady-state probability of delay in the Erlang delay model.

The Erlang-C formula E_C(A, m) gives the probability that an open queueing system with m identical servers, infinite wating space, arrival rate \lambda, individual service rate \mu and offered load A = \lambda / \mu has all the servers busy. This is the waiting probability in an M/M/m/\infty system with m servers and an infinite queue.

INPUTS

A

Offered load. A = \lambda / \mu where \lambda is the mean arrival rate and \mu the mean service rate of each individual server (real, 0 < A < m).

m

Number of identical servers (integer, m ≥ 1). Default m = 1

OUTPUTS

B

The value E_C(A, m)

A or m can be vectors, and in this case, the results will be vectors as well.

REFERENCES

  • G. Zeng, Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296

See also: erlangb,engset,qsmmm.

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4.5 The Engset Formula

Function File: B = engset (A, m, n)

Evaluate the Engset loss formula.

The Engset formula computes the blocking probability P_b(A,m,n) for a system with a finite population of n users, m identical servers, no queue, individual service rate \mu, individual arrival rate \lambda (i.e., the time until a user tries to request service is exponentially distributed with mean 1/\lambda), and offered load A=\lambda/\mu.

INPUTS

A

Offered load, defined as A = \lambda / \mu where \lambda is the mean arrival rate and \mu the mean service rate of each individual server (real, A > 0).

m

Number of identical servers (integer, m ≥ 1). Default m = 1

n

Number of requests (integer, n ≥ 1). Default n = 1

OUTPUTS

B

The value P_b(A, m, n)

A, m or n can be vectors, and in this case, the results will be vectors as well.

REFERENCES

  • G. Zeng, Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296

See also: erlangb, erlangc.

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4.6 The M/M/inf System

The M/M/\infty system is a special case of M/M/m system with infinitely many identical servers (i.e., m = \infty). Each new request is always assigned to a new server, so that queueing never occurs. The M/M/\infty system is always stable.

Function File: [U, R, Q, X, p0] = qsmminf (lambda, mu)
Function File: pk = qsmminf (lambda, mu, k)

Compute utilization, response time, average number of requests and throughput for an infinite-server queue.

The M/M/\infty system has an infinite number of identical servers. Such a system is always stable (i.e., the mean queue length is always finite) for any arrival and service rates.

INPUTS

lambda

Arrival rate (lambda>0).

mu

Service rate (mu>0).

k

Number of requests in the system (k ≥ 0).

OUTPUTS

U

Traffic intensity (defined as \lambda/\mu). Note that this is different from the utilization, which in the case of M/M/\infty centers is always zero.

R

Service center response time.

Q

Average number of requests in the system (which is equal to the traffic intensity \lambda/\mu).

X

Throughput (which is always equal to X = lambda).

p0

Steady-state probability that there are no requests in the system

pk

Steady-state probability that there are k requests in the system (including the one being served).

If this function is called with less than three arguments, lambda and mu can be vectors of the same size. In this case, the results will be vectors as well.

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.4

See also: qsmm1,qsmmm,qsmmmk.

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4.7 The M/M/1/K System

In a M/M/1/K finite capacity system there is a single server, and there can be at most K jobs at any time (including the job currently in service), K > 1. If a new request tries to join the system when there are already K other requests, the request is lost. The queue has K-1 slots. The M/M/1/K system is always stable, regardless of the arrival and service rates.

Function File: [U, R, Q, X, p0, pK] = qsmm1k (lambda, mu, K)
Function File: pn = qsmm1k (lambda, mu, K, n)

Compute utilization, response time, average number of requests and throughput for a M/M/1/K finite capacity system.

In a M/M/1/K queue there is a single server and a queue with finite capacity: the maximum number of requests in the system (including the request being served) is K, and the maximum queue length is therefore K-1.

INPUTS

lambda

Arrival rate (lambda>0).

mu

Service rate (mu>0).

K

Maximum number of requests allowed in the system (K ≥ 1).

n

Number of requests in the (0 ≤ n ≤ K).

OUTPUTS

U

Service center utilization, which is defined as U = 1-p0

R

Service center response time

Q

Average number of requests in the system

X

Service center throughput

p0

Steady-state probability that there are no requests in the system

pK

Steady-state probability that there are K requests in the system (i.e., that the system is full)

pn

Steady-state probability that there are n requests in the system (including the one being served).

If this function is called with less than four arguments, lambda, mu and K can be vectors of the same size. In this case, the results will be vectors as well.


See also: qsmm1,qsmminf,qsmmm.

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4.8 The M/M/m/K System

The M/M/m/K finite capacity system is similar to the M/M/1/k system except that the number of servers is m, where 1 \leq m \leq K. The queue has K-m slots. The M/M/m/K system is always stable.

Function File: [U, R, Q, X, p0, pK] = qsmmmk (lambda, mu, m, K)
Function File: pn = qsmmmk (lambda, mu, m, K, n)

Compute utilization, response time, average number of requests and throughput for a M/M/m/K finite capacity system. In a M/M/m/K system there are m \geq 1 identical service centers sharing a fixed-capacity queue. At any time, at most K ≥ m requests can be in the system, including those being served. The maximum queue length is K-m. This function generates and solves the underlying CTMC.

INPUTS

lambda

Arrival rate (lambda>0)

mu

Service rate (mu>0)

m

Number of servers (m ≥ 1)

K

Maximum number of requests allowed in the system, including those being served (K ≥ m)

n

Number of requests in the (0 ≤ n ≤ K).

OUTPUTS

U

Service center utilization

R

Service center response time

Q

Average number of requests in the system

X

Service center throughput

p0

Steady-state probability that there are no requests in the system.

pK

Steady-state probability that there are K requests in the system (i.e., probability that the system is full).

pn

Steady-state probability that there are n requests in the system (including those being served).

If this function is called with less than five arguments, lambda, mu, m and K can be either scalars, or vectors of the same size. In this case, the results will be vectors as well.

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.6

See also: qsmm1,qsmminf,qsmmm.

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4.9 The Asymmetric M/M/m System

The Asymmetric M/M/m system contains m servers connected to a single queue. Differently from the M/M/m system, in the asymmetric M/M/m each server may have a different service time.

Function File: [U, R, Q, X] = qsammm (lambda, mu)

Compute approximate utilization, response time, average number of requests in service and throughput for an asymmetric M/M/m queue. In this type of system there are m different servers connected to a single queue. Each server has its own (possibly different) service rate. If there is more than one server available, requests are routed to a randomly-chosen one.

INPUTS

lambda

Arrival rate (lambda>0)

mu

mu(i) is the service rate of server i, 1 ≤ i ≤ m. The system must be ergodic (lambda < sum(mu)).

OUTPUTS

U

Approximate service center utilization, U = \lambda / ( \sum_{i=1}^m \mu_i ).

R

Approximate service center response time

Q

Approximate number of requests in the system

X

Approximate system throughput. If the system is ergodic, X = lambda

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998

See also: qsmmm.

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4.10 The M/G/1 System

Function File: [U, R, Q, X, p0] = qsmg1 (lambda, xavg, x2nd)

Compute utilization, response time, average number of requests and throughput for a M/G/1 system. The service time distribution is described by its mean xavg, and by its second moment x2nd. The computations are based on results from L. Kleinrock, Queuing Systems, Wiley, Vol 2, and Pollaczek-Khinchine formula.

INPUTS

lambda

Arrival rate

xavg

Average service time

x2nd

Second moment of service time distribution

OUTPUTS

U

Service center utilization

R

Service center response time

Q

Average number of requests in the system

X

Service center throughput

p0

Probability that there is not any request at system

lambda, xavg, t2nd can be vectors of the same size. In this case, the results will be vectors as well.


See also: qsmh1.

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4.11 The M/H_m/1 System

Function File: [U, R, Q, X, p0] = qsmh1 (lambda, mu, alpha)

Compute utilization, response time, average number of requests and throughput for a M/H_m/1 system. In this system, the customer service times have hyper-exponential distribution: 

       ___ m
       \
B(x) =  >  alpha(j) * (1-exp(-mu(j)*x))   x>0
       /__ 
           j=1

where \alpha_j is the probability that the request is served at phase j, in which case the average service rate is \mu_j. After completing service at phase j, for some j, the request exits the system.

INPUTS

lambda

Arrival rate

mu

mu(j) is the phase j service rate. The total number of phases m is length(mu).

alpha

alpha(j) is the probability that a request is served at phase j. alpha must have the same size as mu.

OUTPUTS

U

Service center utilization

R

Service center response time

Q

Average number of requests in the system

X

Service center throughput

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5 Queueing Networks

• Introduction to QNs:  A brief introduction to Queueing Networks
• Single Class Models:  Queueing models with a single job class
• Multiple Class Models:  Queueing models with multiple job classes
• Generic Algorithms:  High-level functions for QN analysis
• Bounds Analysis:  Computation of asymptotic performance bounds
• QN Analysis Examples:  Queueing Networks analysis examples

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5.1 Introduction to QNs

Queueing Networks (QN) are a simple modeling notation that can be used to analyze many kinds of systems. In its simplest form, a QN is made of K service centers; center k has a queue connected to m_k (usually identical) servers. Arriving customers (requests) join the queue if there is at least one slot available. Requests are served according to a (de)queueing policy (e.g., FIFO). After service completes, requests leave the server and can join another queue or exit from the system.

Service centers where m_k = \infty are called delay centers or infinite servers. In this kind of centers, there is always one available server, so that queueing never occurs.

Requests join the queue according to a queueing policy, such as:

FCFS

First-Come-First-Served

LCFS-PR

Last-Come-First-Served, Preemptive Resume

PS

Processor Sharing

IS

Infinite Server (m_k = \infty).

Queueing networks can be open or closed. In open networks there is an infinite population of requests; new customers are generated outside the system, and eventually leave the network. In closed networks there is a fixed population of request that never leave the system.

Queueing models can have a single request class (single class models), meaning that all requests behave in the same way (e.g., they spend the same average time on each particular server). In multiple class models there are multiple request classes, each with its own parameters (e.g., with different service times or different routing probabilities). Furthermore, in multiclass models there can be open and closed chains of requests at the same time.

A particular class of QN models, product-form networks, is of particular interest. Product-form networks fulfill the following assumptions:

  • The network can consist of open and closed job classes.
  • The following queueing disciplines are allowed: FCFS, PS, LCFS-PR and IS.
  • Service times for FCFS nodes must be exponentially distributed and class-independent. Service centers at PS, LCFS-PR and IS nodes can have any kind of service time distribution with a rational Laplace transform. Furthermore, for PS, LCFS-PR and IS nodes, different classes of customers can have different service times.
  • The service rate of an FCFS node is only allowed to depend on the number of jobs at this node; in a PS, LCFS-PR and IS node the service rate for a particular job class can also depend on the number of jobs of that class at the node.
  • In open networks two kinds of arrival processes are allowed: i) the arrival process is Poisson, with arrival rate \lambda that can depend on the number of jobs in the network. ii) the arrival process consists of C independent Poisson arrival streams where the C job sources are assigned to the C chains; the arrival rate can be load dependent.

Product-form networks are attractive because steady-state performance measures can be efficiently computed.

Next: , Previous: , Up: Queueing Networks   [Contents][Index]

5.2 Single Class Models

In single class models, all requests are indistinguishable and belong to the same class. This means that every request has the same average service time, and all requests move through the system with the same routing probabilities.

Model Inputs

{lambda}_k

(Open models only) External arrival rate to service center k.

lambda

(Open models only) Overall external arrival rate to the system as a whole: \lambda = \sum_k \lambda_k.

N

(Closed models only) Total number of requests in the system.

S_k

Mean service time at center k. S_k is the average time elapsed from service start to service completion at center k.

P_{i, j}

Routing probability matrix. {\bf P} = [P_{i, j}] is a K \times K matrix where P_{i, j} is the probability that a request completing service at center i is routed to center j. The probability that a request leaves the system after being served at center i is \left(1-\sum_{j=1}^K P_{i, j}\right).

V_k

Mean number of visits to center k (also called visit ratio or relative arrival rate).

Model Outputs

U_k

Utilization of service center k. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). If center k is a single-server or multiserver node, then 0 ≤ U_k ≤ 1. If center k is an infinite server node (delay center), then U_k denotes the traffic intensity and is defined as U_k = X_k S_k; in this case the utilization may be greater than one.

R_k

Average response time of service center k, defined as the mean time between the arrival of a request in the queue and service completion of the same request.

Q_k

Average number of requests in center k; this includes both the requests in the queue and those being served.

X_k

Throughput of service center k. The throughput is the rate of job completions, i.e., the average number of jobs completed over a given time interval.

Given the output parameters above, additional performance measures can be computed:

X

System throughput, X = X_k / V_k for any k for which V_k \neq 0

R

System response time, R = \sum_{k=1}^K R_k V_k

Q

Average number of requests in the system, Q = \sum_{k=1}^K Q_k; for closed systems, this can be written as Q = N-XZ;

For open, single class models, the scalar \lambda denotes the external arrival rate of requests to the system. The average number of visits V_j satisfy the following equation: 

                  K
                 ___
                \
V_j = P_(0, j) + >   V_i P_(i, j)    j=1,...,K
                /___
                 i=1

where P_{0, j} is the probability that an external request goes to center j. If we denote with \lambda_j the external arrival rate to center j, and \lambda = \sum_j \lambda_j the overall external arrival rate, then P_{0, j} = \lambda_j / \lambda.

For closed models, the visit ratios satisfy the following equation: 

/
|         K
|        ___
|       \
| V_j =  >   V_i P_(i, j)     j=1,...,K
|       /___
|        i=1
|
| V_r = 1                     for a selected reference station r
\

Note that the set of traffic equations V_j = \sum_{i=1}^K V_i P_{i, j} alone can only be solved up to a multiplicative constant; to get a unique solution we impose an additional constraint V_r = 1 for some 1 ≤ r ≤ K. This constraint is equivalent to defining station r as the reference station; the default is r=1, see doc-qncsvisits. A job that returns to the reference station is assumed to have completed its activity cycle. The network throughput is set to the throughput of the reference station.

Function File: V = qncsvisits (P)
Function File: V = qncsvisits (P, r)

Compute the mean number of visits to the service centers of a single class, closed network with K service centers.

INPUTS

P(i,j)

probability that a request which completed service at center i is routed to center j (K \times K matrix). For closed networks it must hold that sum(P,2)==1. The routing graph must be strongly connected, meaning that each node must be reachable from every other node.

r

Index of the reference station, r \in {1, …, K}; Default r=1. The traffic equations are solved by imposing the condition V(r) = 1. A request returning to the reference station completes its activity cycle.

OUTPUTS

V(k)

average number of visits to service center k, assuming r as the reference station.

Function File: V = qnosvisits (P, lambda)

Compute the average number of visits to the service centers of a single class open Queueing Network with K service centers.

INPUTS

P(i,j)

is the probability that a request which completed service at center i is routed to center j (K \times K matrix).

lambda(k)

external arrival rate to center k.

OUTPUTS

V(k)

average number of visits to server k.

EXAMPLE

qn_closed_single

Figure 5.1: Closed network with a single class of requests

Figure 5.1 shows a closed queueing network with a single class of requests. The network has three service centers, labeled CPU, Disk1 and Disk2, and is known as a central server model of a computer system. Requests spend some time at the CPU, which is represented by a PS (Processor Sharing) node. After that, requests are routed to Disk1 with probability 0.3, and to Disk2 with probability 0.7. Both Disk1 and Disk2 are FCFS nodes.

If we label the servers as CPU=1, Disk1=2, Disk2=3, we can define the routing matrix as follows: 

    / 0  0.3  0.7 \
P = | 1  0    0   |
    \ 1  0    0   /

The visit ratios V, using station 1 as the reference station, can be computed with: 

 P = [0 0.3 0.7; ...
      1 0   0  ; ...
      1 0   0  ];
 V = qncsvisits(P)

   ⇒ V = 1.00000   0.30000   0.70000

EXAMPLE

qn_open_single

Figure 5.2: Open Queueing Network with a single class of requests

Figure 5.2 shows a open QN with a single class of requests. The network has the same structure as the one in Figure 5.1, with the difference that here we have a stream of jobs arriving from outside the system, at a rate \lambda. After service completion at the CPU, a job can leave the system with probability 0.2, or be transferred to other nodes with the probabilities shown in the figure.

The routing matrix is 

    / 0  0.3  0.5 \
P = | 1  0    0   |
    \ 1  0    0   /

If we let \lambda = 1.2, we can compute the visit ratios V as follows: 

 p = 0.3;
 lambda = 1.2
 P = [0 0.3 0.5; ...
      1 0   0  ; ...
      1 0   0  ];
 V = qnosvisits(P,[1.2 0 0])

   ⇒ V = 5.0000   1.5000   2.5000

Function qnosvisits expects a vector with K elements as a second parameter, for open networks only. The vector contains the arrival rates at each individual node; since in our example external arrivals exist only for node S_1 with rate \lambda = 1.2, the second parameter is [1.2, 0, 0].

5.2.1 Open Networks

Jackson networks satisfy the following conditions:

  • There is only one job class in the network; the total number of jobs in the system is unbounded.
  • There are K service centers in the network. Each service center may have Poisson arrivals from outside the system. A job can leave the system from any node.
  • Arrival rates as well as routing probabilities are independent from the number of nodes in the network.
  • External arrivals and service times at the service centers are exponentially distributed, and in general can be load-dependent.
  • Service discipline at each node is FCFS

We define the joint probability vector \pi(n_1, …, n_K) as the steady-state probability that there are n_k requests at service center k, for all k=1, …, N. Jackson networks have the property that the joint probability is the product of the marginal probabilities \pi_k: 

joint_prob = prod( pi )

where \pi_k(n_k) is the steady-state probability that there are n_k requests at service center k.

Function File: [U, R, Q, X] = qnos (lambda, S, V)
Function File: [U, R, Q, X] = qnos (lambda, S, V, m)

Analyze open, single class BCMP queueing networks with K service centers.

This function works for a subset of BCMP single-class open networks satisfying the following properties:

  • The allowed service disciplines at network nodes are: FCFS, PS, LCFS-PR, IS (infinite server);
  • Service times are exponentially distributed and load-independent;
  • Center k can consist of m(k) ≥ 1 identical servers.
  • Routing is load-independent

INPUTS

lambda

Overall external arrival rate (lambda>0).

S(k)

average service time at center k (S(k)>0).

V(k)

average number of visits to center k (V(k) ≥ 0).

m(k)

number of servers at center i. If m(k) < 1, enter k is a delay center (IS); otherwise it is a regular queueing center with m(k) servers. Default is m(k) = 1 for all k.

OUTPUTS

U(k)

If k is a queueing center, U(k) is the utilization of center k. If k is an IS node, then U(k) is the traffic intensity defined as X(k)*S(k).

R(k)

center k average response time.

Q(k)

average number of requests at center k.

X(k)

center k throughput.

REFERENCES

  • G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998

See also: qnopen,qnclosed,qnosvisits.

From the results computed by this function, it is possible to derive other quantities of interest as follows:

  • System Response Time: The overall system response time can be computed as R_s = dot(V,R);
  • Average number of requests: The average number of requests in the system can be computed as: Q_avg = sum(Q)

EXAMPLE 

 lambda = 3;
 V = [16 7 8];
 S = [0.01 0.02 0.03];
 [U R Q X] = qnos( lambda, S, V );
 R_s = dot(R,V) # System response time
 N = sum(Q) # Average number in system

-| R_s =  1.4062
-| N =  4.2186

5.2.2 Closed Networks

Function File: [U, R, Q, X, G] = qncsmva (N, S, V)
Function File: [U, R, Q, X, G] = qncsmva (N, S, V, m)
Function File: [U, R, Q, X, G] = qncsmva (N, S, V, m, Z)

Analyze closed, single class queueing networks using the exact Mean Value Analysis (MVA) algorithm.

The following queueing disciplines are supported: FCFS, LCFS-PR, PS and IS (Infinite Server). This function supports fixed-rate service centers or multiple server nodes. For general load-dependent service centers, use the function qncsmvald instead.

Additionally, the normalization constant G(n), n=0, …, N is computed; G(n) can be used in conjunction with the BCMP theorem to compute steady-state probabilities.

INPUTS

N

Population size (number of requests in the system, N ≥ 0). If N == 0, this function returns U = R = Q = X = 0

S(k)

mean service time at center k (S(k) ≥ 0).

V(k)

average number of visits to service center k (V(k) ≥ 0).

Z

External delay for customers (Z ≥ 0). Default is 0.

m(k)

number of servers at center k (if m is a scalar, all centers have that number of servers). If m(k) < 1, center k is a delay center (IS); otherwise it is a regular queueing center (FCFS, LCFS-PR or PS) with m(k) servers. Default is m(k) = 1 for all k (each service center has a single server).

OUTPUTS

U(k)

If k is a FCFS, LCFS-PR or PS node (m(k) ≥ 1), then U(k) is the utilization of center k, 0 ≤ U(k) ≤ 1. If k is an IS node (m(k) < 1), then U(k) is the traffic intensity defined as X(k)*S(k). In this case the value of U(k) may be greater than one.

R(k)

center k response time. The Residence Time at center k is R(k) * V(k). The system response time Rsys can be computed either as Rsys = N/Xsys - Z or as Rsys = dot(R,V)

Q(k)

average number of requests at center k. The number of requests in the system can be computed either as sum(Q), or using the formula N-Xsys*Z.

X(k)

center K throughput. The system throughput Xsys can be computed as Xsys = X(1) / V(1)

G(n)

Normalization constants. G(n+1) contains the value of the normalization constant G(n), n=0, …, N as array indexes in Octave start from 1. G(n) can be used in conjunction with the BCMP theorem to compute steady-state probabilities.

NOTES

In presence of load-dependent servers (i.e., if m(k)>1 for some k), the MVA algorithm is known to be numerically unstable. Generally, this issue manifests itself as negative values for the response times or utilizations. This is not a problem of the queueing toolbox, but of the MVA algorithm, and has currently no known solution. This function prints a warning if numerical problems are detected; the warning can be disabled with the command warning("off", "qn:numerical-instability").

REFERENCES

  • M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. 10.1145/322186.322195

This implementation is described in R. Jain , The Art of Computer Systems Performance Analysis, Wiley, 1991, p. 577. Multi-server nodes are treated according to G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 8.2.1, "Single Class Queueing Networks".


See also: qncsmvald,qncscmva.

EXAMPLE 

 S = [ 0.125 0.3 0.2 ];
 V = [ 16 10 5 ];
 N = 20;
 m = ones(1,3);
 Z = 4;
 [U R Q X] = qncsmva(N,S,V,m,Z);
 X_s = X(1)/V(1); # System throughput
 R_s = dot(R,V); # System response time
 printf("\t    Util      Qlen     RespT      Tput\n");
 printf("\t--------  --------  --------  --------\n");
 for k=1:length(S)
   printf("Dev%d\t%8.4f  %8.4f  %8.4f  %8.4f\n", k, U(k), Q(k), R(k), X(k) );
 endfor
 printf("\nSystem\t          %8.4f  %8.4f  %8.4f\n\n", N-X_s*Z, R_s, X_s );
Function File: [U, R, Q, X] = qncsmvald (N, S, V)
Function File: [U, R, Q, X] = qncsmvald (N, S, V, Z)

Mean Value Analysis algorithm for closed, single class queueing networks with K service centers and load-dependent service times. This function supports FCFS, LCFS-PR, PS and IS nodes. For networks with only fixed-rate centers and multiple-server nodes, the function qncsmva is more efficient.

INPUTS

N

Population size (number of requests in the system, N ≥ 0). If N == 0, this function returns U = R = Q = X = 0

S(k,n)

mean service time at center k where there are n requests, 1 ≤ n ≤ N. S(k,n) = 1 / \mu_{k}(n), where \mu_{k}(n) is the service rate of center k when there are n requests.

V(k)

average number of visits to service center k (V(k) ≥ 0).

Z

external delay ("think time", Z ≥ 0); default 0.

OUTPUTS

U(k)

utilization of service center k. The utilization is defined as the probability that service center k is not empty, that is, U_k = 1-\pi_k(0) where \pi_k(0) is the steady-state probability that there are 0 jobs at service center k.

R(k)

response time on service center k.

Q(k)

average number of requests in service center k.

X(k)

throughput of service center k.

NOTES

In presence of load-dependent servers, the MVA algorithm is known to be numerically unstable. Generally this problem manifests itself as negative response times or utilization.

REFERENCES

  • M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. 10.1145/322186.322195

This implementation is described in G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 8.2.4.1, “Networks with Load-Dependent Service: Closed Networks”.


See also: qncsmva.

Function File: [U, R, Q, X] = qncscmva (N, S, Sld, V)
Function File: [U, R, Q, X] = qncscmva (N, S, Sld, V, Z)

Conditional MVA (CMVA) algorithm, a numerically stable variant of MVA. This function supports a network of M ≥ 1 service centers and a single delay center. Servers 1, …, (M-1) are load-independent; server M is load-dependent.

INPUTS

N

Number of requests in the system, N ≥ 0. If N == 0, this function returns U = R = Q = X = 0

S(k)

mean service time on server k = 1, …, (M-1) (S(k) > 0). If there are no fixed-rate servers, then S = []

Sld(n)

inverse service rate at server M (the load-dependent server) when there are n requests, n=1, …, N. Sld(n) = 1 / \mu(n).

V(k)

average number of visits to service center k=1, …, M, where V(k) ≥ 0. V(1:M-1) are the visit rates to the fixed rate servers; V(M) is the visit rate to the load dependent server.

Z

External delay for customers (Z ≥ 0). Default is 0.

OUTPUTS

U(k)

center k utilization (k=1, …, M)

R(k)

response time of center k (k=1, …, M). The system response time Rsys can be computed as Rsys = N/Xsys - Z

Q(k)

average number of requests at center k (k=1, …, M).

X(k)

center k throughput (k=1, …, M).

REFERENCES

  • G. Casale. A note on stable flow-equivalent aggregation in closed networks. Queueing Syst. Theory Appl., 60:193–-202, December 2008, 10.1007/s11134-008-9093-6
Function File: [U, R, Q, X] = qncsmvaap (N, S, V)
Function File: [U, R, Q, X] = qncsmvaap (N, S, V, m)
Function File: [U, R, Q, X] = qncsmvaap (N, S, V, m, Z)
Function File: [U, R, Q, X] = qncsmvaap (N, S, V, m, Z, tol)
Function File: [U, R, Q, X] = qncsmvaap (N, S, V, m, Z, tol, iter_max)

Analyze closed, single class queueing networks using the Approximate Mean Value Analysis (MVA) algorithm. This function is based on approximating the number of customers seen at center k when a new request arrives as Q_k(N) \times (N-1)/N. This function only handles single-server and delay centers; if your network contains general load-dependent service centers, use the function qncsmvald instead.

INPUTS

N

Population size (number of requests in the system, N > 0).

S(k)

mean service time on server k (S(k)>0).

V(k)

average number of visits to service center k (V(k) ≥ 0).

m(k)

number of servers at center k (if m is a scalar, all centers have that number of servers). If m(k) < 1, center k is a delay center (IS); if m(k) == 1, center k is a regular queueing center (FCFS, LCFS-PR or PS) with one server (default). This function does not support multiple server nodes (m(k) > 1).

Z

External delay for customers (Z ≥ 0). Default is 0.

tol

Stopping tolerance. The algorithm stops when the maximum relative difference between the new and old value of the queue lengths Q becomes less than the tolerance. Default is 10^{-5}.

iter_max

Maximum number of iterations (iter_max>0. The function aborts if convergenge is not reached within the maximum number of iterations. Default is 100.

OUTPUTS

U(k)

If k is a FCFS, LCFS-PR or PS node (m(k) == 1), then U(k) is the utilization of center k. If k is an IS node (m(k) < 1), then U(k) is the traffic intensity defined as X(k)*S(k).

R(k)

response time at center k. The system response time Rsys can be computed as Rsys = N/Xsys - Z

Q(k)

average number of requests at center k. The number of requests in the system can be computed either as sum(Q), or using the formula N-Xsys*Z.

X(k)

center k throughput. The system throughput Xsys can be computed as Xsys = X(1) / V(1)

REFERENCES

This implementation is based on Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 6.4.2.2 ("Approximate Solution Techniques").


See also: qncsmva,qncsmvald.

According to the BCMP theorem, the state probability of a closed single class queueing network with K nodes and N requests can be expressed as: 

n = [n1, … nK]; population vector
p = 1/G(N+1) \prod F(k,k);

Here \pi(n_1, …, n_K) is the joint probability of having n_k requests at node k, for all k=1, …, K; we have that \sum_{k=1}^K n_k = N

The convolution algorithms computes the normalization constants {\bf G} = \left[G(0), …, G(N)\right] for single-class, closed networks with N requests. The normalization constants are returned as vector G=[G(1), … G(N+1)] where G(i+1) is the value of G(i) (remember that Octave uses 1-base vectors). The normalization constant can be used to compute all performance measures of interest (utilization, average response time and so on).

queueing implements the convolution algorithm, in the function qncsconv and qncsconvld. The first one supports single-station nodes, multiple-station nodes and IS nodes. The second one supports networks with general load-dependent service centers.

Function File: [U, R, Q, X, G] = qncsconv (N, S, V)
Function File: [U, R, Q, X, G] = qncsconv (N, S, V, m)

Analyze product-form, single class closed networks with K service centers using the convolution algorithm.

Load-independent service centers, multiple servers (M/M/m queues) and IS nodes are supported. For general load-dependent service centers, use qncsconvld instead.

INPUTS

N

Number of requests in the system (N>0).

S(k)

average service time on center k (S(k) ≥ 0).

V(k)

visit count of service center k (V(k) ≥ 0).

m(k)

number of servers at center k. If m(k) < 1, center k is a delay center (IS); if m(k) ≥ 1, center k it is a regular M/M/m queueing center with m(k) identical servers. Default is m(k) = 1 for all k.

OUTPUT

U(k)

center k utilization. For IS nodes, U(k) is the traffic intensity X(k) * S(k).

R(k)

average response time of center k.

Q(k)

average number of customers at center k.

X(k)

throughput of center k.

G(n)

Vector of normalization constants. G(n+1) contains the value of the normalization constant with n requests G(n), n=0, …, N.

NOTE

For a network with K service centers and N requests, this implementation of the convolution algorithm has time and space complexity O(NK).

REFERENCES

  • Jeffrey P. Buzen, Computational Algorithms for Closed Queueing Networks with Exponential Servers, Communications of the ACM, volume 16, number 9, September 1973, pp. 527–531. 10.1145/362342.362345

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, pp. 313–317.


See also: qncsconvld.

EXAMPLE

The normalization constant G can be used to compute the steady-state probabilities for a closed single class product-form Queueing Network with K nodes and N requests. Let n = [n_1, …, n_K] be a valid population vector, \sum_{k=1}^K n_k = N. Then, the steady-state probability p(k) to have n(k) requests at service center k can be computed as: 

 n = [1 2 0];
 N = sum(n); # Total population size
 S = [ 1/0.8 1/0.6 1/0.4 ];
 m = [ 2 3 1 ];
 V = [ 1 .667 .2 ];
 [U R Q X G] = qncsconv( N, S, V, m );
 p = [0 0 0]; # initialize p
 # Compute the probability to have n(k) jobs at service center k
 for k=1:3
   p(k) = (V(k)*S(k))^n(k) / G(N+1) * ...
          (G(N-n(k)+1) - V(k)*S(k)*G(N-n(k)) );
   printf("Prob( n(%d) = %d )=%f\n", k, n(k), p(k) );
 endfor

-| Prob( n(1) = 1 ) = 0.17975
-| Prob( n(2) = 2 ) = 0.48404
-| Prob( n(3) = 0 ) = 0.52779

(recall that G(N+1) represents G(N), since in Octave array indices start at one).

Function File: [U, R, Q, X, G] = qncsconvld (N, S, V)

Convolution algorithm for product-form, single-class closed queueing networks with K general load-dependent service centers.

This function computes steady-state performance measures for single-class, closed networks with load-dependent service centers using the convolution algorithm; the normalization constants are also computed. The normalization constants are returned as vector G=[G(1), …, G(N+1)] where G(i+1) is the value of G(i).

INPUTS

N

Number of requests in the system (N>0).

S(k,n)

mean service time at center k where there are n requests, 1 ≤ n ≤ N. S(k,n) = 1 / \mu_{k,n}, where \mu_{k,n} is the service rate of center k when there are n requests.

V(k)

visit count of service center k (V(k) ≥ 0). The length of V is the number of servers K in the network.

OUTPUT

U(k)

center k utilization.

R(k)

average response time at center k.

Q(k)

average number of requests in center k.

X(k)

center k throughput.

G(n)

Normalization constants (vector). G(n+1) corresponds to G(n), as array indexes in Octave start from 1.

REFERENCES

  • Herb Schwetman, Some Computational Aspects of Queueing Network Models, Technical Report CSD-TR-354, Department of Computer Sciences, Purdue University, February 1981 (revised).
  • M. Reiser, H. Kobayashi, On The Convolution Algorithm for Separable Queueing Networks, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29–31, 1976). SIGMETRICS ’76. ACM, New York, NY, pp. 109–117. 10.1145/800200.806187

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, pp. 313–317. Function qncsconvld is slightly different from the version described in Bolch et al. because it supports general load-dependent centers (while the version in the book does not). The modification is in the definition of function F() in qncsconvld which has been made similar to function f_i defined in Schwetman, Some Computational Aspects of Queueing Network Models.


See also: qncsconv.

5.2.3 Non Product-Form QNs

Function File: [U, R, Q, X] = qncsmvablo (N, S, M, P )

Approximate MVA algorithm for closed queueing networks with blocking.

INPUTS

N

number of requests in the system. N must be strictly greater than zero, and less than the overall network capacity: 0 < N < sum(M).

S(k)

average service time on server k (S(k) > 0).

M(k)

capacity of center k. The capacity is the maximum number of requests in a service center, including the request in service (M(k) ≥ 1).

P(i,j)

probability that a request which completes service at server i will be transferred to server j.

OUTPUTS

U(k)

center k utilization.

R(k)

average response time of service center k.

Q(k)

average number of requests in service center k (including the request in service).

X(k)

center k throughput.

REFERENCES

  • Ian F. Akyildiz, Mean Value Analysis for Blocking Queueing Networks, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418–428. 10.1109/32.4663

See also: qnopen, qnclosed.

Function File: [U, R, Q, X] = qnmarkov (lambda, S, C, P)
Function File: [U, R, Q, X] = qnmarkov (lambda, S, C, P, m)
Function File: [U, R, Q, X] = qnmarkov (N, S, C, P)
Function File: [U, R, Q, X] = qnmarkov (N, S, C, P, m)

Compute utilization, response time, average queue length and throughput for open or closed queueing networks with finite capacity and a single class of requests. Blocking type is Repetitive-Service (RS). This function explicitly generates and solve the underlying Markov chain, and thus might require a large amount of memory.

More specifically, networks which can me analyzed by this function have the following properties:

  • There exists only a single class of customers.
  • The network has K service centers. Center k \in {1, …, K} has m_k > 0 servers, and has a total (finite) capacity of C_k \geq m_k which includes both buffer space and servers. The buffer space at service center k is therefore C_k - m_k.
  • The network can be open, with external arrival rate to center k equal to \lambda_k, or closed with fixed population size N. For closed networks, the population size N must be strictly less than the network capacity: N < \sum_{k=1}^K C_k.
  • Average service times are load-independent.
  • P_{i, j} is the probability that requests completing execution at center i are transferred to center j, i \neq j. For open networks, a request may leave the system from any node i with probability 1-\sum_{j=1}^K P_{i, j}.
  • Blocking type is Repetitive-Service (RS). Service center j is saturated if the number of requests is equal to its capacity C_j. Under the RS blocking discipline, a request completing service at center i which is being transferred to a saturated server j is put back at the end of the queue of i and will receive service again. Center i then processes the next request in queue. External arrivals to a saturated servers are dropped.

INPUTS

lambda(k)
N

If the first argument is a vector lambda, it is considered to be the external arrival rate lambda(k) ≥ 0 to service center k of an open network. If the first argument is a scalar, it is considered as the population size N of a closed network; in this case N must be strictly less than the network capacity: N < sum(C).

S(k)

average service time at service center k

C(k)

capacity of service center k. The capacity includes both the buffer and server space m(k). Thus the buffer space is C(k)-m(k).

P(i,j)

transition probability from service center i to service center j.

m(k)

number of servers at service center k. Note that m(k) ≥ C(k) for each k. If m is omitted, all service centers are assumed to have a single server (m(k) = 1 for all k).

OUTPUTS

U(k)

center k utilization.

R(k)

response time on service center k.

Q(k)

average number of customers in the service center k, including the request in service.

X(k)

throughput of service center k.

NOTES

The space complexity of this implementation is O(\prod_{k=1}^K (C_k + 1)^2). The time complexity is dominated by the time needed to solve a linear system with \prod_{k=1}^K (C_k + 1) unknowns.

Next: , Previous: , Up: Queueing Networks   [Contents][Index]

5.3 Multiple Class Models

In multiple class queueing models, we assume that there exist C different classes of requests. Each request from class c spends on average time S_{c, k} in service at center k. For open models, we denote with {\bf \lambda} = \lambda_{c, k} the arrival rates, where \lambda_{c, k} is the external arrival rate of class c requests at center k. For closed models, we denote with {\bf N} = \left[N_1, …, N_C\right] the population vector, where N_c is the number of class c requests in the system.

The transition probability matrix for multiple class networks is a C \times K \times C \times K matrix {\bf P} = [P_{r, i, s, j}] where P_{r, i, s, j} is the probability that a class r request which completes service at center i will join server j as a class s request.

Model input and outputs can be adjusted by adding additional indexes for the customer classes.

Model Inputs

lambda_{c, k}

(open networks) External arrival rate of class-c requests to service center k

lambda

(open networks) Overall external arrival rate to the whole system: \lambda = \sum_{c=1}^C \sum_{k=1}^K \lambda_{c, k}

N_c

(closed networks) Number of class c requests in the system.

S_{c, k}

Average service time. S_{c, k} is the average service time on service center k for class c requests.

P_{r, i, s, j}

Routing probability matrix. {\bf P} = [P_{r, i, s, j}] is a C \times K \times C \times K matrix such that P_{r, i, s, j} is the probability that a class r request which completes service at server i will move to server j as a class s request.

V_{c, k}

Mean number of visits of class c requests to center k.

Model Outputs

U_{c, k}

Utilization of service center k by class c requests. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). If center k is a single-server or multiserver node, then 0 ≤ U_{c, k} ≤ 1. If center k is an infinite server node (delay center), then U_{c, k} denotes the traffic intensity and is defined as U_{c, k} = X_{c, k} S_{c, k}; in this case the utilization may be greater than one.

R_{c, k}

Average response time experienced by class c requests on service center k. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service.

Q_{c, k}

Average number of class c requests on service center k. This includes both the requests in the queue, and the request being served.

X_{c, k}

Throughput of service center k for class c requests. The throughput is defined as the rate of completion of class c requests.

It is possible to define aggregate performance measures as follows:

U_k

Utilization of service center k: Uk = sum(U,k);

R_c

System response time for class c requests: Rc = sum( V.*R, 1 );

Q_c

Average number of class c requests in the system: Qc = sum( Q, 2 );

X_c

Class c throughput: X(c) = X(c,k) ./ V(c,k); for any k for which V(c,k) != 0

For closed networks, we can define the visit ratios V_{s, j} for class s customers at service center j as follows:

V_sj = sum_r sum_i V_ri P_risj s=1,...,C, j=1,...,K V_s r_s = 1 s=1,...,C

where r_s is the class s reference station. Similarly to single class models, the traffic equation for closed multiclass networks can be solved up to multiplicative constants unless we choose one reference station for each closed class and set its visit ratio to 1.

For open networks the traffic equations are as follows:

V_sj = P_0sj + sum_r sum_i V_ri P_risj s=1,...,C, j=1,...,K

where P_{0, s, j} is the probability that an external arrival goes to service center j as a class-s request. If \lambda_{s, j} is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_{s, j} is the overall external arrival rate, then P_{0, s, j} = \lambda_{s, j} / \lambda.

Function File: [V ch] = qncmvisits (P)
Function File: [V ch] = qncmvisits (P, r)

Compute the average number of visits for the nodes of a closed multiclass network with K service centers and C customer classes.

INPUTS

P(r,i,s,j)

probability that a class r request which completed service at center i is routed to center j as a class s request. Class switching is allowed.

r(c)

index of class c reference station, r(c) \in {1, …, K}, 1 ≤ c ≤ C. The class c visit count to server r(c) (V(c,r(c))) is conventionally set to 1. The reference station serves two purposes: (i) its throughput is assumed to be the system throughput, and (ii) a job returning to the reference station is assumed to have completed one cycle. Default is to consider station 1 as the reference station for all classes.

OUTPUTS

V(c,i)

number of visits of class c requests at center i.

ch(c)

chain number that class c belongs to. Different classes can belong to the same chain. Chains are numbered sequentially starting from 1 (1, 2, …). The total number of chains is max(ch).

Function File: V = qnomvisits (P, lambda)

Compute the visit ratios to the service centers of an open multiclass network with K service centers and C customer classes.

INPUTS

P(r,i,s,j)

probability that a class r request which completed service at center i is routed to center j as a class s request. Class switching is supported.

lambda(r,i)

external arrival rate of class r requests to center i.

OUTPUTS

V(r,i)

visit ratio of class r requests at center i.

5.3.1 Open Networks

Function File: [U, R, Q, X] = qnom (lambda, S, V)
Function File: [U, R, Q, X] = qnom (lambda, S, V, m)
Function File: [U, R, Q, X] = qnom (lambda, S, P)
Function File: [U, R, Q, X] = qnom (lambda, S, P, m)

Exact analysis of open, multiple-class BCMP networks. The network can be made of single-server queueing centers (FCFS, LCFS-PR or PS) or delay centers (IS). This function assumes a network with K service centers and C customer classes.

INPUTS

lambda(c)

If this function is invoked as qnom(lambda, S, V, …), then lambda(c) is the external arrival rate of class c customers (lambda(c) ≥ 0). If this function is invoked as qnom(lambda, S, P, …), then lambda(c,k) is the external arrival rate of class c customers at center k (lambda(c,k) ≥ 0).

S(c,k)

mean service time of class c customers on the service center k (S(c,k)>0). For FCFS nodes, mean service times must be class-independent.

V(c,k)

visit ratio of class c customers to service center k (V(c,k) ≥ 0 ). If you pass this argument, class switching is not allowed

P(r,i,s,j)

probability that a class r job completing service at center i is routed to center j as a class s job. If you pass argument P, class switching is allowed; however, all servers must be fixed-rate or infinite-server nodes (m(k) ≤ 1 for all k).

m(k)

number of servers at center k. If m(k) < 1, enter k is a delay center (IS); otherwise it is a regular queueing center with m(k) servers. Default is m(k) = 1 for all k.

OUTPUTS

U(c,k)

If k is a queueing center, then U(c,k) is the class c utilization of center k. If k is an IS node, then U(c,k) is the class c traffic intensity defined as X(c,k)*S(c,k).

R(c,k)

class c response time at center k. The system response time for class c requests can be computed as dot(R, V, 2).

Q(c,k)

average number of class c requests at center k. The average number of class c requests in the system Qc can be computed as Qc = sum(Q, 2)

X(c,k)

class c throughput at center k.

NOTES

If the function call specifies the visit ratios V, class switching is not allowed. If the function call specifies the routing probability matrix P, then class switching is allowed; however, all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported. Note that the meaning of parameter lambda is different from one case to the other (see below).

REFERENCES

  • Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.1 ("Open Model Solution Techniques").

See also: qnopen,qnos,qnomvisits.

5.3.2 Closed Networks

Function File: pop_mix = qncmpopmix (k, N)

Return the set of population mixes for a closed multiclass queueing network with exactly k customers. Specifically, given a closed multiclass QN with C customer classes, where there are N(c) class c requests, c = 1, …, C a k-mix M is a vector of length C with the following properties:

  • 0 ≤ M_c ≤ N(c) for all c = 1, …, C;
  • \sum_{c=1}^C M_c = k

In other words, a k-mix is an allocation of k requests to C classes such that the number of requests assigned to class c does not exceed the maximum value N(c).

pop_mix is a matrix with C columns, such that each row represents a valid mix.

INPUTS

k

Size of the requested mix (scalar, k ≥ 0).

N(c)

number of class c requests (k ≤ sum(N)).

OUTPUTS

pop_mix(i,c)

number of class c requests in the i-th population mix. The number of mixes is rows(pop_mix).

If you are interested in the number of k-mixes only, you can use the funcion qnmvapop.

REFERENCES

  • Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report 80-355, Department of Computer Sciences, Purdue University, revised February 15, 1982.

The slightly different problem of enumerating all tuples k_1, …, k_N such that \sum_i k_i = k and k_i ≥ 0, for a given k ≥ 0 has been described in S. Santini, Computing the Indices for a Complex Summation, unpublished report, available at http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf


See also: qncmnpop.

EXAMPLE

Let us consider a multiclass network with C=2 customer classes; the maximum number of class 1 requests is 2, and the maximum number of class 2 requests is 3. How is it possible to allocate 3 requests to the two classes so that the maximum number of requests per class is not exceeded? 

 N = [2 3];
 mix = qncmpopmix(3, N)

-| mix = [ [2 1] [1 2] [0 3] ]
Function File: H = qncmnpop (N)

Given a network with C customer classes, this function computes the number of k-mixes H(r,k) that can be constructed by the multiclass MVA algorithm by allocating k customers to the first r classes. See doc-qncmpopmix for the definition of k-mix.

INPUTS

N(c)

number of class-c requests in the system. The total number of requests in the network is sum(N).

OUTPUTS

H(r,k)

is the number of k mixes that can be constructed allocating k customers to the first r classes.

REFERENCES

  • Zahorjan, J. and Wong, E. The solution of separable queueing network models using mean value analysis. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI 10.1145/1010629.805477

See also: qncmmva,qncmpopmix.

Function File: [U, R, Q, X] = qncmmva (N, S )
Function File: [U, R, Q, X] = qncmmva (N, S, V)
Function File: [U, R, Q, X] = qncmmva (N, S, V, m)
Function File: [U, R, Q, X] = qncmmva (N, S, V, m, Z)
Function File: [U, R, Q, X] = qncmmva (N, S, P)
Function File: [U, R, Q, X] = qncmmva (N, S, P, r)
Function File: [U, R, Q, X] = qncmmva (N, S, P, r, m)

Compute steady-state performance measures for closed, multiclass queueing networks using the Mean Value Analysys (MVA) algorithm.

Queueing policies at service centers can be any of the following:

FCFS

(First-Come-First-Served) customers are served in order of arrival; multiple servers are allowed. For this kind of queueing discipline, average service times must be class-independent.

PS

(Processor Sharing) customers are served in parallel by a single server, each customer receiving an equal share of the service rate.

LCFS-PR

(Last-Come-First-Served, Preemptive Resume) customers are served in reverse order of arrival by a single server and the last arrival preempts the customer in service who will later resume service at the point of interruption.

IS

(Infinite Server) customers are delayed independently of other customers at the service center (there is effectively an infinite number of servers).

INPUTS

N(c)

number of class c requests; N(c) ≥ 0. If class c has no requests (N(c) == 0), then for all k, this function returns U(c,k) = R(c,k) = Q(c,k) = X(c,k) = 0

S(c,k)

mean service time for class c requests at center k (S(c,k) ≥ 0). If the service time at center k is class-dependent, then center k is assumed to be of type -/G/1–PS (Processor Sharing). If center k is a FCFS node (m(k)>1), then the service times must be class-independent, i.e., all classes must have the same service time.

V(c,k)

average number of visits of class c requests at center k; V(c,k) ≥ 0, default is 1. If you pass this argument, class switching is not allowed

P(r,i,s,j)

probability that a class r request completing service at center i is routed to center j as a class s request; the reference stations for each class are specified with the paramter r. If you pass argument P, class switching is allowed; however, you can not specify any external delay (i.e., Z must be zero) and all servers must be fixed-rate or infinite-server nodes (m(k) ≤ 1 for all k).

r(c)

reference station for class c. If omitted, station 1 is the reference station for all classes. See qncmvisits.

m(k)

If m(k)<1, then center k is assumed to be a delay center (IS node -/G/\infty). If m(k)==1, then service center k is a regular queueing center (M/M/1–FCFS, -/G/1–LCFS-PR or -/G/1–PS). Finally, if m(k)>1, center k is a M/M/m–FCFS center with m(k) identical servers. Default is m(k)=1 for each k.

Z(c)

class c external delay (think time); Z(c) ≥ 0. Default is 0. This parameter can not be used if you pass a routing matrix as the second parameter of qncmmva.

OUTPUTS

U(c,k)

If k is a FCFS, LCFS-PR or PS node (m(k) ≥ 1), then U(c,k) is the class c utilization at center k, 0 ≤ U(c,k) ≤ 1. If k is an IS node, then U(c,k) is the class c traffic intensity at center k, defined as U(c,k) = X(c,k)*S(c,k). In this case the value of U(c,k) may be greater than one.

R(c,k)

class c response time at center k. The class c residence time at center k is R(c,k) * C(c,k). The total class c system response time is dot(R, V, 2).

Q(c,k)

average number of class c requests at center k. The total number of requests at center k is sum(Q(:,k)). The total number of class c requests in the system is sum(Q(c,:)).

X(c,k)

class c throughput at center k. The class c throughput can be computed as X(c,1) / V(c,1).

NOTES

If the function call specifies the visit ratios V, then class switching is not allowed. If the function call specifies the routing probability matrix P, then class switching is allowed; however, in this case all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported.

In presence of load-dependent servers (e.g., if m(i)>1 for some i), the MVA algorithm is known to be numerically unstable. Generally this problem shows up as negative values for the computed response times or utilizations. This is not a problem with the queueing package, but with the MVA algorithm; as such, there is no known workaround at the moment (aoart from using a different solution technique, if available). This function prints a warning if it detects numerical problems; you can disable the warning with the command warning("off", "qn:numerical-instability").

Given a network with K service centers, C job classes and population vector {\bf N}=\left[N_1, …, N_C\right], the MVA algorithm requires space O(C \prod_i (N_i + 1)). The time complexity is O(CK\prod_i (N_i + 1)). This implementation is slightly more space-efficient (see details in the code). While the space requirement can be mitigated by using some optimizations, the time complexity can not. If you need to analyze large closed networks you should consider the qncmmvaap function, which implements the approximate MVA algorithm. Note however that qncmmvaap will only provide approximate results.

REFERENCES

  • M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. 10.1145/322186.322195

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998 and Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.2.1 ("Exact Solution Techniques").


See also: qnclosed, qncmmvaapprox, qncmvisits.

Function File: [U, R, Q, X] = qncmmvabs (N, S, V)
Function File: [U, R, Q, X] = qncmmvabs (N, S, V, m)
Function File: [U, R, Q, X] = qncmmvabs (N, S, V, m, Z)
Function File: [U, R, Q, X] = qncmmvabs (N, S, V, m, Z, tol)
Function File: [U, R, Q, X] = qncmmvabs (N, S, V, m, Z, tol, iter_max)

Approximate Mean Value Analysis (MVA) for closed, multiclass queueing networks with K service centers and C customer classes.

This implementation uses Bard and Schweitzer approximation. It is based on the assumption that the queue length at service center k with population set {\bf N}-{\bf 1}_c is approximated with 

Q_k(N-1c) ~ (n-1)/n Q_k(N)

where \bf N is a valid population mix, {\bf N}-{\bf 1}_c is the population mix \bf N with one class c customer removed, and n = \sum_c N_c is the total number of requests.

This implementation works for networks with infinite server (IS) and single-server nodes only.

INPUTS

N(c)

number of class c requests in the system (N(c) ≥ 0).

S(c,k)

mean service time for class c customers at center k (S(c,k) ≥ 0).

V(c,k)

average number of visits of class c requests to center k (V(c,k) ≥ 0).

m(k)

number of servers at center k. If m(k) < 1, then the service center k is assumed to be a delay center (IS). If m(k) == 1, service center k is a regular queueing center (FCFS, LCFS-PR or PS) with a single server node. If omitted, each service center has a single server. Note that multiple server nodes are not supported.

Z(c)

class c external delay (Z ≥ 0). Default is 0.

tol

Stopping tolerance (tol>0). The algorithm stops if the queue length computed on two subsequent iterations are less than tol. Default is 10^{-5}.

iter_max

Maximum number of iterations (iter_max>0. The function aborts if convergenge is not reached within the maximum number of iterations. Default is 100.

OUTPUTS

U(c,k)

If k is a FCFS, LCFS-PR or PS node, then U(c,k) is the utilization of class c requests on service center k. If k is an IS node, then U(c,k) is the class c traffic intensity at device k, defined as U(c,k) = X(c)*S(c,k)

R(c,k)

response time of class c requests at service center k.

Q(c,k)

average number of class c requests at service center k.

X(c,k)

class c throughput at service center k.

REFERENCES

  • Y. Bard, Some Extensions to Multiclass Queueing Network Analysis, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, Feb 1979, pp. 51–62.
  • P. Schweitzer, Approximate Analysis of Multiclass Closed Networks of Queues, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25–29.

This implementation is based on Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.2.2 ("Approximate Solution Techniques"). This implementation is slightly different from the one described above, as it computes the average response times R instead of the residence times.


See also: qncmmva.

5.3.3 Mixed Networks

Function File: [U, R, Q, X] = qnmix (lambda, N, S, V, m)

Mean Value Analysis for mixed queueing networks. The network consists of K service centers (single-server or delay centers) and C independent customer chains. Both open and closed chains are possible. lambda is the vector of per-chain arrival rates (open classes); N is the vector of populations for closed chains.

Class switching is not allowed. Each customer class must correspond to an independent chain.

If the network is made of open or closed classes only, then this function calls qnom or qncmmva respectively, and prints a warning message.

INPUTS

lambda(c)
N(c)

For each customer chain c:

  • if c is a closed chain, then N(c)>0 is the number of class c requests and lambda(c) must be zero;
  • If c is an open chain, lambda(c)>0 is the arrival rate of class c requests and N(c) must be zero;

In other words, for each class c the following must hold: 

(lambda(c)>0 && N(c)==0) || (lambda(c)==0 && N(c)>0)
S(c,k)

mean class c service time at center k, S(c,k) ≥ 0. For FCFS nodes, service times must be class-independent.

V(c,k)

average number of visits of class c customers to center k (V(c,k) ≥ 0).

m(k)

number of servers at center k. Only single-server (m(k)==1) or IS (Infinite Server) nodes (m(k)<1) are supported. If omitted, each center is assumed to be of type M/M/1-FCFS. Queueing discipline for single-server nodes can be FCFS, PS or LCFS-PR.

OUTPUTS

U(c,k)

class c utilization at center k.

R(c,k)

class c response time at center k.

Q(c,k)

average number of class c requests at center k.

X(c,k)

class c throughput at center k.

REFERENCES

  • Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.3 ("Mixed Model Solution Techniques"). Note that in this function we compute the mean response time R instead of the mean residence time as in the reference.
  • Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, revised Feb 15, 1982.

See also: qncmmva, qncm.

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5.4 Generic Algorithms

The queueing package provides a high-level function qnsolve for analyzing QN models. qnsolve takes as input a high-level description of the queueing model, and delegates the actual solution of the model to one of the lower-level function. qnsolve supports single or multiclass models, but at the moment only product-form networks can be analyzed. For non product-form networks See Non Product-Form QNs.

qnsolve accepts two input parameters. The first one is the list of nodes, encoded as an Octave cell array. The second parameter is the vector of visit ratios V, which can be either a vector (for single-class models) or a two-dimensional matrix (for multiple-class models).

Individual nodes in the network are structures build using the qnmknode function.

Function File: Q = qnmknode ("m/m/m-fcfs", S)
Function File: Q = qnmknode ("m/m/m-fcfs", S, m)
Function File: Q = qnmknode ("m/m/1-lcfs-pr", S)
Function File: Q = qnmknode ("-/g/1-ps", S)
Function File: Q = qnmknode ("-/g/1-ps", S, s2)
Function File: Q = qnmknode ("-/g/inf", S)
Function File: Q = qnmknode ("-/g/inf", S, s2)

Creates a node; this function can be used together with qnsolve. It is possible to create either single-class nodes (where there is only one customer class), or multiple-class nodes (where the service time is given per-class). Furthermore, it is possible to specify load-dependent service times. String literals are case-insensitive, so for example "-/g/inf", "-/G/inf" and "-/g/INF" are all equivalent.

INPUTS

S

Mean service time.

  • If S is a scalar, it is assumed to be a load-independent, class-independent service time.
  • If S is a column vector, then S(c) is assumed to the the load-independent service time for class c customers.
  • If S is a row vector, then S(n) is assumed to be the class-independent service time at the node, when there are n requests.
  • Finally, if S is a two-dimensional matrix, then S(c,n) is assumed to be the class c service time when there are n requests at the node.
m

Number of identical servers at the node. Default is m=1.

s2

Squared coefficient of variation for the service time. Default is 1.0.

The returned struct Q should be considered opaque to the client.


See also: qnsolve.

After the network has been defined, it is possible to solve it using qnsolve.

Function File: [U, R, Q, X] = qnsolve ("closed", N, QQ, V)
Function File: [U, R, Q, X] = qnsolve ("closed", N, QQ, V, Z)
Function File: [U, R, Q, X] = qnsolve ("open", lambda, QQ, V)
Function File: [U, R, Q, X] = qnsolve ("mixed", lambda, N, QQ, V)

High-level function for analyzing QN models.

  • For closed networks, the following server types are supported: M/M/m–FCFS, -/G/\infty, -/G/1–LCFS-PR, -/G/1–PS and load-dependent variants.
  • For open networks, the following server types are supported: M/M/m–FCFS, -/G/\infty and -/G/1–PS. General load-dependent nodes are not supported. Multiclass open networks do not support multiple server M/M/m nodes, but only single server M/M/1–FCFS.
  • For mixed networks, the following server types are supported: M/M/1–FCFS, -/G/\infty and -/G/1–PS. General load-dependent nodes are not supported.

INPUTS

N
N(c)

Number of requests in the system for closed networks. For single-class networks, N must be a scalar. For multiclass networks, N(c) is the population size of closed class c.

lambda
lambda(c)

External arrival rate (scalar) for open networks. For single-class networks, lambda must be a scalar. For multiclass networks, lambda(c) is the class c overall arrival rate.

QQ{i}

List of queues in the network. This must be a cell array with N elements, such that QQ{i} is a struct produced by the qnmknode function.

Z

External delay ("think time") for closed networks. Default 0.

OUTPUTS

U(k)

If k is a FCFS node, then U(k) is the utilization of service center k. If k is an IS node, then U(k) is the traffic intensity defined as X(k)*S(k).

R(k)

average response time of service center k.

Q(k)

average number of customers in service center k.

X(k)

throughput of service center k.

Note that for multiclass networks, the computed results are per-class utilization, response time, number of customers and throughput: U(c,k), R(c,k), Q(c,k), X(c,k).

String literals are case-insensitive, so "closed", "Closed" and "CLoSEd" are all equivalent.

EXAMPLE

Let us consider a closed, multiclass network with C=2 classes and K=3 service center. Let the population be M=(2, 1) (class 1 has 2 requests, and class 2 has 1 request). The nodes are as follows:

  • Node 1 is a M/M/1–FCFS node, with load-dependent service times. Service times are class-independent, and are defined by the matrix [0.2 0.1 0.1; 0.2 0.1 0.1]. Thus, S(1,2) = 0.2 means that service time for class 1 customers where there are 2 requests in 0.2. Note that service times are class-independent;
  • Node 2 is a -/G/1–PS node, with service times S_{1, 2} = 0.4 for class 1, and S_{2, 2} = 0.6 for class 2 requests;
  • Node 3 is a -/G/\infty node (delay center), with service times S_{1, 3}=1 and S_{2, 3}=2 for class 1 and 2 respectively.

After defining the per-class visit count V such that V(c,k) is the visit count of class c requests to service center k. We can define and solve the model as follows: 




 QQ = { qnmknode( "m/m/m-fcfs", [0.2 0.1 0.1; 0.2 0.1 0.1] ), ...
        qnmknode( "-/g/1-ps", [0.4; 0.6] ), ...
        qnmknode( "-/g/inf", [1; 2] ) };
 V = [ 1 0.6 0.4; ...
       1 0.3 0.7 ];
 N = [ 2 1 ];
 [U R Q X] = qnsolve( "closed", N, QQ, V );
Function File: [U, R, Q, X] = qnclosed (N, S, V, …)

This function computes steady-state performance measures of closed queueing networks using the Mean Value Analysis (MVA) algorithm. The qneneing network is allowed to contain fixed-capacity centers, delay centers or general load-dependent centers. Multiple request classes are supported.

This function dispatches the computation to one of qncsemva, qncsmvald or qncmmva.

  • If N is a scalar, the network is assumed to have a single class of requests; in this case, the exact MVA algorithm is used to analyze the network. If S is a vector, then S(k) is the average service time of center k, and this function calls qncsmva which supports load-independent service centers. If S is a matrix, S(k,i) is the average service time at center k when i=1, …, N jobs are present; in this case, the network is analyzed with the qncmmvald function.
  • If N is a vector, the network is assumed to have multiple classes of requests, and is analyzed using the exact multiclass MVA algorithm as implemented in the qncmmva function.

See also: qncsmva, qncsmvald, qncmmva.

EXAMPLE 

 P = [0 0.3 0.7; 1 0 0; 1 0 0]; # Transition probability matrix
 S = [1 0.6 0.2];               # Average service times
 m = ones(size(S));             # All centers are single-server
 Z = 2;                         # External delay
 N = 15;                        # Maximum population to consider
 V = qncsvisits(P);             # Compute number of visits
 X_bsb_lower = X_bsb_upper = X_ab_lower = X_ab_upper = X_mva = zeros(1,N);
 for n=1:N
   [X_bsb_lower(n) X_bsb_upper(n)] = qncsbsb(n, S, V, m, Z);
   [X_ab_lower(n) X_ab_upper(n)] = qncsaba(n, S, V, m, Z);
   [U R Q X] = qnclosed( n, S, V, m, Z );
   X_mva(n) = X(1)/V(1);
 endfor
 close all;
 plot(1:N, X_ab_lower,"g;Asymptotic Bounds;", ...
      1:N, X_bsb_lower,"k;Balanced System Bounds;", ...
      1:N, X_mva,"b;MVA;", "linewidth", 2, ...
      1:N, X_bsb_upper,"k", 1:N, X_ab_upper,"g" );
 axis([1,N,0,1]); legend("location","southeast"); legend("boxoff");
 xlabel("Number of Requests n"); ylabel("System Throughput X(n)");
Function File: [U, R, Q, X] = qnopen (lambda, S, V, …)

Compute utilization, response time, average number of requests in the system, and throughput for open queueing networks. If lambda is a scalar, the network is considered a single-class QN and is solved using qnopensingle. If lambda is a vector, the network is considered as a multiclass QN and solved using qnopenmulti.


See also: qnos, qnom.

Next: , Previous: , Up: Queueing Networks   [Contents][Index]

5.5 Bounds Analysis

Function File: [Xl, Xu, Rl, Ru] = qnosaba (lambda, D)
Function File: [Xl, Xu, Rl, Ru] = qnosaba (lambda, S, V)
Function File: [Xl, Xu, Rl, Ru] = qnosaba (lambda, S, V, m)

Compute Asymptotic Bounds for open, single-class networks with K service centers.

INPUTS

lambda

Arrival rate of requests (scalar, lambda ≥ 0).

D(k)

service demand at center k. (vector of length K, D(k) ≥ 0).

S(k)

mean service time at center k. (vector of length K, S(k) ≥ 0).

V(k)

mean number of visits to center k. (vector of length K, V(k) ≥ 0).

m(k)

number of servers at center k. This function only supports M/M/1 queues, therefore m must be ones(size(S)).

OUTPUTS

Xl
Xu

Lower and upper bounds on the system throughput. Xl is always set to 0 since there can be no lower bound on the throughput of open networks (scalar).

Rl
Ru

Lower and upper bounds on the system response time. Ru is always set to +inf since there can be no upper bound on the throughput of open networks (scalar).


See also: qnomaba.

Function File: [Xl, Xu, Rl, Ru] = qnomaba (lambda, D)
Function File: [Xl, Xu, Rl, Rl] = qnomaba (lambda, S, V)

Compute Asymptotic Bounds for open, multiclass networks with K service centers and C customer classes.

INPUTS

lambda(c)

class c arrival rate to the system (vector of length C, lambda(c) > 0).

D(c, k)

class c service demand at center k (C \times K matrix, D(c, k) ≥ 0).

S(c, k)

mean service time of class c requests at center k (C \times K matrix, S(c, k) ≥ 0).

V(c, k)

mean number of visits of class c requests at center k (C \times K matrix, V(c, k) ≥ 0).

OUTPUTS

Xl(c)
Xu(c)

lower and upper bounds of class c throughput. Xl(c) is always 0 since there can be no lower bound on the throughput of open networks (vector of length C).

Rl(c)
Ru(c)

lower and upper bounds of class c response time. Ru(c) is always +inf since there can be no upper bound on the response time of open networks (vector of length C).

Function File: [Xl, Xu, Rl, Ru] = qncsaba (N, D)
Function File: [Xl, Xu, Rl, Ru] = qncsaba (N, S, V)
Function File: [Xl, Xu, Rl, Ru] = qncsaba (N, S, V, m)
Function File: [Xl, Xu, Rl, Ru] = qncsaba (N, S, V, m, Z)

Compute Asymptotic Bounds for the system throughput and response time of closed, single-class networks with K service centers.

Single-server and infinite-server nodes are supported. Multiple-server nodes and general load-dependent servers are not supported.

INPUTS

N

number of requests in the system (scalar, N>0).

D(k)

service demand at center k (D(k) ≥ 0).

S(k)

mean service time at center k (S(k) ≥ 0).

V(k)

average number of visits to center k (V(k) ≥ 0).

m(k)

number of servers at center k (if m is a scalar, all centers have that number of servers). If m(k) < 1, center k is a delay center (IS); if m(k) = 1, center k is a M/M/1-FCFS server. This function does not support multiple-server nodes. Default is 1.

Z

External delay (scalar, Z ≥ 0). Default is 0.

OUTPUTS

Xl
Xu

Lower and upper bounds on the system throughput.

Rl
Ru

Lower and upper bounds on the system response time.


See also: qncmaba.

Function File: [Xl, Xu, Rl, Ru] = qncmaba (N, D)
Function File: [Xl, Xu, Rl, Ru] = qncmaba (N, S, V)
Function File: [Xl, Xu, Rl, Ru] = qncmaba (N, S, V, m)
Function File: [Xl, Xu, Rl, Ru] = qncmaba (N, S, V, m, Z)

Compute Asymptotic Bounds for closed, multiclass networks with K service centers and C customer classes. Single-server and infinite-server nodes are supported. Multiple-server nodes and general load-dependent servers are not supported.

INPUTS

N(c)

number of class c requests in the system (vector of length C, N(c) ≥ 0).

D(c, k)

class c service demand at center k (C \times K matrix, D(c,k) ≥ 0).

S(c, k)

mean service time of class c requests at center k (C \times K matrix, S(c,k) ≥ 0).

V(c,k)

average number of visits of class c requests to center k (C \times K matrix, V(c,k) ≥ 0).

m(k)

number of servers at center k (if m is a scalar, all centers have that number of servers). If m(k) < 1, center k is a delay center (IS); if m(k) = 1, center k is a M/M/1-FCFS server. This function does not support multiple-server nodes. Default is 1.

Z(c)

class c external delay (vector of length C, Z(c) ≥ 0). Default is 0.

OUTPUTS

Xl(c)
Xu(c)

Lower and upper bounds for class c throughput.

Rl(c)
Ru(c)

Lower and upper bounds for class c response time.

REFERENCES

  • Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 5.2 ("Asymptotic Bounds").

See also: qncsaba.

Function File: [Xl, Xu, Rl, Ru] = qnosbsb (lambda, D)
Function File: [Xl, Xu, Rl, Ru] = qnosbsb (lambda, S, V)

Compute Balanced System Bounds for single-class, open networks with K service centers.

INPUTS

lambda

overall arrival rate to the system (scalar, lambda ≥ 0).

D(k)

service demand at center k (D(k) ≥ 0).

S(k)

service time at center k (S(k) ≥ 0).

V(k)

mean number of visits at center k (V(k) ≥ 0).

m(k)

number of servers at center k. This function only supports M/M/1 queues, therefore m must be ones(size(S)).

OUTPUTS

Xl
Xu

Lower and upper bounds on the system throughput. Xl is always set to 0, since there can be no lower bound on open networks throughput.

Rl
Ru

Lower and upper bounds on the system response time.


See also: qnosaba.

Function File: [Xl, Xu, Rl, Ru] = qncsbsb (N, D)
Function File: [Xl, Xu, Rl, Ru] = qncsbsb (N, S, V)
Function File: [Xl, Xu, Rl, Ru] = qncsbsb (N, S, V, m)
Function File: [Xl, Xu, Rl, Ru] = qncsbsb (N, S, V, m, Z)

Compute Balanced System Bounds on system throughput and response time for closed, single-class networks with K service centers.

INPUTS

N

number of requests in the system (scalar, N ≥ 0).

D(k)

service demand at center k (D(k) ≥ 0).

S(k)

mean service time at center k (S(k) ≥ 0).

V(k)

average number of visits to center k (V(k) ≥ 0). Default is 1.

m(k)

number of servers at center k. This function supports m(k) = 1 only (single-eserver FCFS nodes); this parameter is only for compatibility with qncsaba. Default is 1.

Z

External delay (Z ≥ 0). Default is 0.

OUTPUTS

Xl
Xu

Lower and upper bound on the system throughput.

Rl
Ru

Lower and upper bound on the system response time.

REFERENCES

  • Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 5.4 ("Balanced Systems Bounds").

See also: qncmbsb.

Function File: [Xl, Xu, Rl, Ru] = qncmbsb (N, D)
Function File: [Xl, Xu, Rl, Ru] = qncmbsb (N, S, V)

Compute Balanced System Bounds for closed, multiclass networks with K service centers and C customer classes. Only single-server nodes are supported.

INPUTS

N(c)

number of class c requests in the system (vector of length C).

D(c, k)

class c service demand at center k (C \times K matrix, D(c,k) ≥ 0).

S(c, k)

mean service time of class c requests at center k (C \times K matrix, S(c,k) ≥ 0).

V(c,k)

average number of visits of class c requests to center k (C \times K matrix, V(c,k) ≥ 0).

OUTPUTS

Xl(c)
Xu(c)

Lower and upper class c throughput bounds (vector of length C).

Rl(c)
Ru(c)

Lower and upper class c response time bounds (vector of length C).


See also: qncsbsb.

Function File: [Xl, Xu, Rl, Ru] = qncmcb (N, D)
Function File: [Xl, Xu, Rl, Ru] = qncmcb (N, S, V)

Compute Composite Bounds (CB) on system throughput and response time for closed multiclass networks.

INPUTS

N(c)

number of class c requests in the system.

D(c, k)

class c service demand at center k (S(c,k) ≥ 0).

S(c, k)

mean service time of class c requests at center k (S(c,k) ≥ 0).

V(c,k)

average number of visits of class c requests to center k (V(c,k) ≥ 0).

OUTPUTS

Xl(c)
Xu(c)

Lower and upper bounds on class c throughput.

Rl(c)
Ru(c)

Lower and upper bounds on class c response time.

REFERENCES

  • Teemu Kerola, The Composite Bound Method (CBM) for Computing Throughput Bounds in Multiple Class Environments, Performance Evaluation Vol. 6, Issue 1, March 1986, DOI 10.1016/0166-5316(86)90002-7. Also available as Technical Report CSD-TR-475, Department of Computer Sciences, Purdue University, mar 13, 1984 (Revised Aug 27, 1984).
Function File: [Xl, Xu, Rl, Ru] = qncspb (N, D )
Function File: [Xl, Xu, Rl, Ru] = qncspb (N, S, V )
Function File: [Xl, Xu, Rl, Ru] = qncspb (N, S, V, m )
Function File: [Xl, Xu, Rl, Ru] = qncspb (N, S, V, m, Z )

Compute PB Bounds (C. H. Hsieh and S. Lam, 1987) for single-class, closed networks with K service centers.

INPUTS

number of requests in the system (scalar, N > 0).

D(k)

service demand of service center k (D(k) ≥ 0).

S(k)

mean service time at center k (S(k) ≥ 0).

V(k)

visit ratio to center k (V(k) ≥ 0).

m(k)

number of servers at center k. This function only supports M/M/1 queues, therefore m must be ones(size(S)).

Z

external delay (think time, Z ≥ 0). Default 0.

OUTPUTS

Xl
Xu

Lower and upper bounds on the system throughput.

Rl
Ru

Lower and upper bounds on the system response time.

REFERENCES

  • C. H. Hsieh and S. Lam, Two classes of performance bounds for closed queueing networks, Performance Evaluation, Vol. 7 Issue 1, pp. 3–30, February 1987, DOI 10.1016/0166-5316(87)90054-X. Also available as Technical Report TR-85-09, Department of Computer Science, University of Texas at Austin, June 1985

This function implements the non-iterative variant described in G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008.


See also: qncsaba, qbcsbsb, qncsgb.

Function File: [Xl, Xu, Rl, Ru, Ql, Qu] = qncsgb (N, D)
Function File: [Xl, Xu, Rl, Ru, Ql, Qu] = qncsgb (N, S, V)
Function File: [Xl, Xu, Rl, Ru, Ql, Qu] = qncsgb (N, S, V, m)
Function File: [Xl, Xu, Rl, Ru, Ql, Qu] = qncsgb (N, S, V, m, Z)

Compute Geometric Bounds (GB) on system throughput, system response time and server queue lenghts for closed, single-class networks with K service centers and N requests.

INPUTS

N

number of requests in the system (scalar, N > 0).

D(k)

service demand of service center k (vector of length K, D(k) ≥ 0).

S(k)

mean service time at center k (vector of length K, S(k) ≥ 0).

V(k)

visit ratio to center k (vector of length K, V(k) ≥ 0).

m(k)

number of servers at center k. This function only supports M/M/1 queues, therefore m must be ones(size(S)).

Z

external delay (think time, Z ≥ 0, scalar). Default is 0.

OUTPUTS

Xl
Xu

Lower and upper bound on the system throughput. If Z>0, these bounds are computed using Geometric Square-root Bounds (GSB). If Z==0, these bounds are computed using Geometric Bounds (GB)

Rl
Ru

Lower and upper bound on the system response time. These bounds are derived from Xl and Xu using Little’s Law: Rl = N / Xu - Z, Ru = N / Xl - Z

Ql(k)
Qu(k)

lower and upper bounds of center K queue length.

REFERENCES

  • G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008. 10.1109/TC.2008.37

In this implementation we set X^+ and X^- as the upper and lower Asymptotic Bounds as computed by the qncsab function, respectively.

Previous: , Up: Queueing Networks   [Contents][Index]

5.6 QN Analysis Examples

In this section we illustrate with a few examples how the queueing package can be used to analyze queueing network models. Further examples can be found in the functions demo blocks, and can be inspected with the demo function Octave command.

5.6.1 Closed, Single Class Network

Let us consider again the network shown in Figure 5.1. We denote with S_k the average service time at center k, k=1, 2, 3. Let the service times be S_1 = 1.0, S_2 = 2.0 and S_3 = 0.8. The routing of jobs within the network is described with a routing probability matrix \bf P: a request completing service at center i is enqueued at center j with probability P_{i, j}. We use the following routing matrix: 

    / 0  0.3  0.7 \
P = | 1  0    0   |
    \ 1  0    0   /

The network above can be analyzed with the qnclosed function see doc-qnclosed. qnclosed requires the following parameters:

N

Number of requests in the network (since we are considering a closed network, the number of requests is fixed)

S

Array of average service times at the centers: S(k) is the average service time at center k.

V

Array of visit ratios: V(k) is the average number of visits to center k.

We can compute V_k from the routing probability matrix P_{i, j} using the qncsvisits function see doc-qncsvisits. Therefore, we can analyze the network for a given population size N (e.g., N=10) as follows: 

N = 10;
S = [1 2 0.8];
P = [0 0.3 0.7; 1 0 0; 1 0 0];
V = qncsvisits(P);
[U R Q X] = qnclosed( N, S, V )
   ⇒ U = 0.99139 0.59483 0.55518
   ⇒ R = 7.4360  4.7531  1.7500
   ⇒ Q = 7.3719  1.4136  1.2144
   ⇒ X = 0.99139 0.29742 0.69397

The output of qnclosed includes the vectors of utilizations U_k at center k, response time R_k, average number of customers Q_k and throughput X_k. In our example, the throughput of center 1 is X_1 = 0.99139, and the average number of requests in center 3 is Q_3 = 1.2144. The utilization of center 1 is U_1 = 0.99139, which is the highest among the service centers. Thus, center 1 is the bottleneck device.

This network can also be analyzed with the qnsolve function see doc-qnsolve. qnsolve can handle open, closed or mixed networks, and allows the network to be described in a very flexible way. First, let Q1, Q2 and Q3 be the variables describing the service centers. Each variable is instantiated with the qnmknode function. 

Q1 = qnmknode( "m/m/m-fcfs", 1 );
Q2 = qnmknode( "m/m/m-fcfs", 2 );
Q3 = qnmknode( "m/m/m-fcfs", 0.8 );

The first parameter of qnmknode is a string describing the type of the node; "m/m/m-fcfs" denotes a M/M/m–FCFS center (this parameter is case-insensitive). The second parameter gives the average service time. An optional third parameter can be used to specify the number m of service centers. If omitted, it is assumed m=1 (single-server node).

Now, the network can be analyzed as follows: 

N = 10;
V = [1 0.3 0.7];
[U R Q X] = qnsolve( "closed", N, { Q1, Q2, Q3 }, V )
   ⇒ U = 0.99139 0.59483 0.55518
   ⇒ R = 7.4360  4.7531  1.7500
   ⇒ Q = 7.3719  1.4136  1.2144
   ⇒ X = 0.99139 0.29742 0.69397

5.6.2 Open, Single Class Network

Let us consider an open network with K=3 service centers and the following routing probabilities: 

    / 0  0.3  0.5 \
P = ! 1  0    0   |
    \ 1  0    0   /

In this network, requests can leave the system from center 1 with probability 1-(0.3+0.5) = 0.2. We suppose that external jobs arrive at center 1 with rate \lambda_1 = 0.15; there are no arrivals at centers 2 and 3.

Similarly to closed networks, we first compute the visit counts V_k to center k, k = 1, 2, 3. We use the qnosvisits function as follows: 

P = [0 0.3 0.5; 1 0 0; 1 0 0];
lambda = [0.15 0 0];
V = qnosvisits(P, lambda)
   ⇒ V = 5.00000 1.50000 2.50000

where lambda(k) is the arrival rate at center k, and \bf P is the routing matrix. Assuming the same service times as in the previous example, the network can be analyzed with the qnopen function see doc-qnopen, as follows: 

S = [1 2 0.8];
[U R Q X] = qnopen( sum(lambda), S, V )
   ⇒ U = 0.75000 0.45000 0.30000
   ⇒ R = 4.0000  3.6364  1.1429
   ⇒ Q = 3.00000 0.81818 0.42857
   ⇒ X = 0.75000 0.22500 0.37500

The first parameter of the qnopen function is the (scalar) aggregate arrival rate.

Again, it is possible to use the qnsolve high-level function: 

Q1 = qnmknode( "m/m/m-fcfs", 1 );
Q2 = qnmknode( "m/m/m-fcfs", 2 );
Q3 = qnmknode( "m/m/m-fcfs", 0.8 );
lambda = [0.15 0 0];
[U R Q X] = qnsolve( "open", sum(lambda), { Q1, Q2, Q3 }, V )
   ⇒ U = 0.75000 0.45000 0.30000
   ⇒ R = 4.0000  3.6364  1.1429
   ⇒ Q = 3.00000 0.81818 0.42857
   ⇒ X = 0.75000 0.22500 0.37500

5.6.3 Closed Multiclass Network/1

The following example is taken from Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, Feb 15, 1982.

Let us consider the following multiclass QN with three servers and two classes

qn_closed_multi_apl

Figure 5.3

Servers 1 and 2 (labeled APL and IMS, respectively) are infinite server nodes; server 3 (labeled SYS) is Processor Sharing (PS). Mean service times are given in the following table:

APLIMSSYS
Class 11-0.025
Class 2-150.500

There is no class switching. If we assume a population of 15 requests for class 1, and 5 requests for class 2, then the model can be analyzed as follows: 

 S = [1 0 .025; 0 15 .5];
 P = zeros(2,3,2,3);
 P(1,1,1,3) = P(1,3,1,1) = 1;
 P(2,2,2,3) = P(2,3,2,2) = 1;
 V = qncmvisits(P,[3 3]); # reference station is station 3
 N = [15 5];
 m = [-1 -1 1];
 [U R Q X] = qncmmva(N,S,V,m)

  ⇒
U =

   14.32312    0.00000    0.35808
    0.00000    4.70699    0.15690

R =

    1.00000    0.00000    0.04726
    0.00000   15.00000    0.93374

Q =

   14.32312    0.00000    0.67688
    0.00000    4.70699    0.29301

X =

   14.32312    0.00000   14.32312
    0.00000    0.31380    0.31380

5.6.4 Closed Multiclass Network/2

The following example is from M. Marzolla, The qnetworks Toolbox: A Software Package for Queueing Networks Analysis, Technical Report UBLCS-2010-04, Department of Computer Science, University of Bologna, Italy, February 2010.

qn_web_model

Figure 5.4: Three-tier enterprise system model

The model shown in Figure 5.4 shows a three-tier enterprise system with K=6 service centers. The first tier contains the Web server (node 1), which is responsible for generating Web pages and transmitting them to clients. The application logic is implemented by nodes 2 and 3, and the storage tier is made of nodes 4–6.The system is subject to two workload classes, both represented as closed populations of N_1 and N_2 requests, respectively. Let D_{c, k} denote the service demand of class c requests at center k. We use the parameter values:

Serv. no.NameClass 1Class 2
1Web Server122
2App. Server 11420
3App. Server 22314
4DB Server 12090
5DB Server 28030
6DB Server 33133

We set the total number of requests to 100, that is N_1 + N_2 = N = 100, and we study how different population mixes (N_1, N_2) affect the system throughput and response time. Let 0 < \beta_1 < 1 denote the fraction of class 1 requests: N_1 = \beta_1 N, N_2 = (1-\beta_1)N. The following Octave code defines the model for \beta_1 = 0.1: 

N = 100;     # total population size
beta1 = 0.1; # fraction of class 1 reqs.
S = [12 14 23 20 80 31; \
      2 20 14 90 30 33 ];
V = ones(size(S));
pop = [fix(beta1*N) N-fix(beta1*N)];
[U R Q X] = qncmmva(pop, S, V);

The qncmmva(pop, S, V) function invocation uses the multiclass MVA algorithm to compute per-class utilizations U_{c, k}, response times R_{c,k}, mean queue lengths Q_{c,k} and throughputs X_{c,k} at each service center k, given a population vector pop, mean service times S and visit ratios V. Since we are given the service demands D_{c, k} = S_{c, k} V_{c,k}, but function qncmmva requires separate service times and visit ratios, we set the service times equal to the demands, and all visit ratios equal to one. Overall class and system throughputs and response times can also be computed: 

X1 = X(1,1) / V(1,1)     # class 1 throughput
        ⇒ X1 =  0.0044219
X2 = X(2,1) / V(2,1)     # class 2 throughput
        ⇒ X2 =  0.010128
XX = X1 + X2             # system throughput
        ⇒ XX =  0.014550
R1 = dot(R(1,:), V(1,:)) # class 1 resp. time
        ⇒ R1 =  2261.5
R2 = dot(R(2,:), V(2,:)) # class 2 resp. time
        ⇒ R2 =  8885.9
RR = N / XX              # system resp. time
        ⇒ RR =  6872.7

dot(X,Y) computes the dot product of two vectors. R(1,:) is the first row of matrix R and V(1,:) is the first row of matrix V, so dot(R(1,:), V(1,:)) computes \sum_k R_{1,k} V_{1,k}.

web

Figure 5.5: Throughput and Response Times as a function of the population mix

We can also compute the system power \Phi = X / R, which defines how efficiently resources are being used: high values of \Phi denote the desirable situation of high throughput and low response time. Figure 5.6 shows \Phi as a function of \beta_1. We observe a “plateau” of the global system power, corresponding to values of \beta_1 which approximately lie between 0.3 and 0.7. The per-class power exhibits an interesting (although not completely surprising) pattern, where the class with higher population exhibits worst efficiency as it produces higher contention on the resources.

power

Figure 5.6: System Power as a function of the population mix

5.6.5 Closed Multiclass Network/3

We now consider an example of multiclass network with class switching. The example is taken from Sch82, and is shown in Figure Figure 5.7.

qn_closed_multi_cs

Figure 5.7: Multiclass Model with Class Switching

The system consists of three devices and two job classes. The CPU node is a PS server, while the two nodes labeled I/O are FCFS. Class 1 mean service time at the CPU is 0.01; class 2 mean service time at the CPU is 0.05. The mean service time at node 2 is 0.1, and is class-independent. Similarly, the mean service time at node 3 is 0.07. Jobs in class 1 leave the CPU and join class 2 with probability 0.1; jobs of class 2 leave the CPU and join class 1 with probability 0.2. There are N=3 jobs, which are initially allocated to class 1. However, note that since class switching is allowed, the total number of jobs in each class does not remain constant; however the total number of jobs does. 

 C = 2; K = 3;
 S = [.01 .07 .10; ...
      .05 .07 .10 ];
 P = zeros(C,K,C,K);
 P(1,1,1,2) = .7; P(1,1,1,3) = .2; P(1,1,2,1) = .1;
 P(2,1,2,2) = .3; P(2,1,2,3) = .5; P(2,1,1,1) = .2;
 P(1,2,1,1) = P(2,2,2,1) = 1;
 P(1,3,1,1) = P(2,3,2,1) = 1;
 N = [3 0];
 [U R Q X] = qncmmva(N, S, P)

  ⇒
U =

   0.12609   0.61784   0.25218
   0.31522   0.13239   0.31522

R =

   0.014653   0.133148   0.163256
   0.073266   0.133148   0.163256

Q =

   0.18476   1.17519   0.41170
   0.46190   0.25183   0.51462

X =

   12.6089    8.8262    2.5218
    6.3044    1.8913    3.1522

Next: , Previous: , Up: Top   [Contents][Index]

6 References

[Aky88]

Ian F. Akyildiz, Mean Value Analysis for Blocking Queueing Networks, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418–428. DOI 10.1109/32.4663

[Bar79]

Y. Bard, Some Extensions to Multiclass Queueing Network Analysis, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, feb. 1979, pp. 51–62.

[BCMP75]

F. Baskett, K. Mani Chandy, R. R. Muntz, and F. G. Palacios. 1975. Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. J. ACM 22, 2 (April 1975), 248—260, DOI 10.1145/321879.321887

[BGMT98]

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998.

[Buz73]

J. P. Buzen, Computational Algorithms for Closed Queueing Networks with Exponential Servers, Communications of the ACM, volume 16, number 9, september 1973, pp. 527–531. DOI 10.1145/362342.362345

[C08]

G. Casale, A note on stable flow-equivalent aggregation in closed networks. Queueing Syst. Theory Appl., 60:193–-202, December 2008, DOI 10.1007/s11134-008-9093-6

[CMS08]

G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008. DOI 10.1109/TC.2008.37

[GrSn97]

C. M. Grinstead, J. L. Snell, (July 1997). Introduction to Probability. American Mathematical Society. ISBN 978-0821807491; this excellent textbook is available in PDF format and can be used under the terms of the GNU Free Documentation License (FDL)

[Jac04]

J. R. Jackson, Jobshop-Like Queueing Systems, Vol. 50, No. 12, Ten Most Influential Titles of "Management Science’s" First Fifty Years (Dec., 2004), pp. 1796-1802, available online

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R. Jain, The Art of Computer Systems Performance Analysis, Wiley, 1991, p. 577.

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C. H. Hsieh and S. Lam, Two classes of performance bounds for closed queueing networks, PEVA, vol. 7, n. 1, pp. 3–30, 1987

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T. Kerola, The Composite Bound Method (CBM) for Computing Throughput Bounds in Multiple Class Environments, Performance Evaluation, Vol. 6 Isue 1, March 1986, DOI 10.1016/0166-5316(86)90002-7; also available as Technical Report CSD-TR-475, Department of Computer Sciences, Purdue University, mar 13, 1984 (Revised aug 27, 1984).

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E. D. Lazowska, J. Zahorjan, G. Scott Graham, and K. C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. available online.

[ReKo76]

M. Reiser, H. Kobayashi, On The Convolution Algorithm for Separable Queueing Networks, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29–31, 1976). SIGMETRICS ’76. ACM, New York, NY, pp. 109–117. DOI 10.1145/800200.806187

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M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. DOI 10.1145/322186.322195

[Sch79]

P. Schweitzer, Approximate Analysis of Multiclass Closed Networks of Queues, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25—29

[Sch80]

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[Sch81]

H. D. Schwetman, Some Computational Aspects of Queueing Network Models, Technical Report CSD-TR-354, Department of Computer Sciences, Purdue University, feb, 1981 (revised).

[Sch82]

H. D. Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, feb 15, 1982.

[Sch84]

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J. Zahorjan and E. Wong, The solution of separable queueing network models using mean value analysis. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI 10.1145/1010629.805477

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G. Zeng, Two common properties of the erlang-B function, erlang-C function, and Engset blocking function, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296 DOI 10.1016/S0895-7177(03)90040-9

Next: , Previous: , Up: Top   [Contents][Index]

Appendix A GNU GENERAL PUBLIC LICENSE

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Next: , Previous: , Up: Top   [Contents][Index]

Concept Index

Jump to:   A   B   C   D   E   F   G   I   L   M   N   O   P   Q   R   S   T   U   W  
Index Entry  Section
A
absorption probabilities, DTMC: Mean time to absorption (DTMC)
approximate MVA: Single Class Models
asymmetric M/M/m system: The Asymmetric M/M/m System
asymptotic bounds: Bounds Analysis
asymptotic bounds: Bounds Analysis
average number of customers: Single Class Models
average number of customers: Multiple Class Models
average number of visits: Single Class Models
average number of visits: Multiple Class Models
B
balanced system bounds: Bounds Analysis
balanced system bounds: Bounds Analysis
BCMP network: Single Class Models
birth-death process, CTMC: Birth-death process (CTMC)
birth-death process, DTMC: Birth-death process (DTMC)
blocking queueing network: Single Class Models
blocking queueing network: Single Class Models
bounds, asymptotic: Bounds Analysis
bounds, asymptotic: Bounds Analysis
bounds, asymptotic: Bounds Analysis
bounds, asymptotic: Bounds Analysis
bounds, balanced system: Bounds Analysis
bounds, balanced system: Bounds Analysis
bounds, balanced system: Bounds Analysis
bounds, composite: Bounds Analysis
bounds, geometric: Bounds Analysis
bounds, PB: Bounds Analysis
C
closed multiclass network: Bounds Analysis
closed multiclass network: Bounds Analysis
closed multiclass network: Bounds Analysis
closed network: Single Class Models
closed network: Bounds Analysis
closed network: Bounds Analysis
closed network, approximate analysis: Single Class Models
closed network, finite capacity: Single Class Models
closed network, finite capacity: Single Class Models
closed network, multiple classes: Multiple Class Models
closed network, multiple classes: Multiple Class Models
closed network, multiple classes: Multiple Class Models
closed network, multiple classes: Multiple Class Models
closed network, multiple classes: Generic Algorithms
closed network, single class: Single Class Models
closed network, single class: Single Class Models
closed network, single class: Single Class Models
closed network, single class: Single Class Models
closed network, single class: Single Class Models
closed network, single class: Single Class Models
closed network, single class: Generic Algorithms
closed network, single class: Bounds Analysis
closed network, single class: Bounds Analysis
closed network, single class: Bounds Analysis
CMVA: Single Class Models
composite bounds: Bounds Analysis
conditional MVA (CMVA): Single Class Models
continuous time Markov chain: Continuous-Time Markov Chains
continuous time Markov chain: State occupancy probabilities (CTMC)
continuous time Markov chain: Birth-death process (CTMC)
continuous time Markov chain: Expected sojourn times (CTMC)
continuous time Markov chain: Time-averaged expected sojourn times (CTMC)
continuous time Markov chain: Mean time to absorption (CTMC)
continuous time Markov chain: First passage times (CTMC)
convolution algorithm: Single Class Models
convolution algorithm: Single Class Models
copyright: Copying
CTMC: Continuous-Time Markov Chains
CTMC: State occupancy probabilities (CTMC)
CTMC: Birth-death process (CTMC)
CTMC: Expected sojourn times (CTMC)
CTMC: Time-averaged expected sojourn times (CTMC)
CTMC: Mean time to absorption (CTMC)
CTMC: First passage times (CTMC)
D
delay center: Introduction to QNs
deprecated functions: Naming Conventions
discrete time Markov chain: Discrete-Time Markov Chains
discrete time Markov chain: Discrete-Time Markov Chains
discrete time Markov chain: State occupancy probabilities (DTMC)
discrete time Markov chain: Birth-death process (DTMC)
discrete time Markov chain: Expected number of visits (DTMC)
discrete time Markov chain: Time-averaged expected sojourn times (DTMC)
discrete time Markov chain: Mean time to absorption (DTMC)
discrete time Markov chain: First passage times (DTMC)
DTMC: Discrete-Time Markov Chains
DTMC: Discrete-Time Markov Chains
DTMC: State occupancy probabilities (DTMC)
DTMC: Birth-death process (DTMC)
DTMC: Expected number of visits (DTMC)
DTMC: Time-averaged expected sojourn times (DTMC)
DTMC: Mean time to absorption (DTMC)
DTMC: First passage times (DTMC)
E
Engset loss formula: The Engset Formula
Erlang-B formula: The Erlang-B Formula
Erlang-C formula: The Erlang-C Formula
expected sojourn time, CTMC: Expected sojourn times (CTMC)
expected sojourn times, DTMC: Expected number of visits (DTMC)
external arrival rate: Single Class Models
external arrival rate: Multiple Class Models
F
FCFS: Introduction to QNs
first passage times: First passage times (DTMC)
first passage times, CTMC: First passage times (CTMC)
First-Come-First-Served: Introduction to QNs
fundamental matrix: Mean time to absorption (DTMC)
G
geometric bounds: Bounds Analysis
I
infinite server: Introduction to QNs
Infinite Server: Introduction to QNs
irreducible Markov chain: Discrete-Time Markov Chains
irreducible Markov chain: Continuous-Time Markov Chains
IS: Introduction to QNs
L
Last-Come-First-Served Preemptive Resume: Introduction to QNs
LCFS-PR: Introduction to QNs
load-dependent service center: Single Class Models
load-dependent service center: Single Class Models
M
M/G/1 system: The M/G/1 System
M/H_m/1 system: The M/Hm/1 System
M/M/1 system: The M/M/1 System
M/M/1/K system: The M/M/1/K System
M/M/inf system: The M/M/inf System
M/M/m system: The M/M/m System
M/M/m/K system: The M/M/m/K System
Markov chain, continuous time: Continuous-Time Markov Chains
Markov chain, continuous time: Continuous-Time Markov Chains
Markov chain, continuous time: State occupancy probabilities (CTMC)
Markov chain, continuous time: Birth-death process (CTMC)
Markov chain, continuous time: Expected sojourn times (CTMC)
Markov chain, continuous time: Time-averaged expected sojourn times (CTMC)
Markov chain, continuous time: Mean time to absorption (CTMC)
Markov chain, continuous time: First passage times (CTMC)
Markov chain, discrete time: Discrete-Time Markov Chains
Markov chain, discrete time: Discrete-Time Markov Chains
Markov chain, discrete time: State occupancy probabilities (DTMC)
Markov chain, discrete time: Birth-death process (DTMC)
Markov chain, discrete time: Expected number of visits (DTMC)
Markov chain, discrete time: Time-averaged expected sojourn times (DTMC)
Markov chain, discrete time: Mean time to absorption (DTMC)
Markov chain, discrete time: First passage times (DTMC)
Markov chain, state occupancy probabilities: State occupancy probabilities (CTMC)
Markov chain, stationary probabilities: State occupancy probabilities (DTMC)
Markov chain, transient probabilities: State occupancy probabilities (DTMC)
mean recurrence times: First passage times (DTMC)
mean time to absorption, CTMC: Mean time to absorption (CTMC)
mean time to absorption, DTMC: Mean time to absorption (DTMC)
Mean Value Analysis, conditional (CMVA): Single Class Models
Mean Value Analysys (MVA): Single Class Models
Mean Value Analysys (MVA): Single Class Models
Mean Value Analysys (MVA): Multiple Class Models
Mean Value Analysys (MVA): Multiple Class Models
Mean Value Analysys (MVA), approximate: Single Class Models
Mean Value Analysys (MVA), approximate: Multiple Class Models
mixed network: Multiple Class Models
multiclass network, closed: Multiple Class Models
multiclass network, closed: Multiple Class Models
multiclass network, closed: Bounds Analysis
multiclass network, closed: Bounds Analysis
multiclass network, closed: Bounds Analysis
multiclass network, open: Multiple Class Models
multiclass network, open: Bounds Analysis
multiple class queueing network: Introduction to QNs
multiple class queueing network: Multiple Class Models
MVA: Single Class Models
MVA, approximate: Single Class Models
MVA, approximate: Multiple Class Models
MVABLO: Single Class Models
N
normalization constant: Single Class Models
normalization constant: Single Class Models
normalization constant: Single Class Models
O
open network: Generic Algorithms
open network: Bounds Analysis
open network: Bounds Analysis
open network: Bounds Analysis
open network, multiple classes: Multiple Class Models
open network, single class: Single Class Models
open network, single class: Single Class Models
P
PB bounds: Bounds Analysis
population mix: Multiple Class Models
population mix: Multiple Class Models
Processor Sharing: Introduction to QNs
product-form queueing network: Introduction to QNs
PS: Introduction to QNs
Q
queueing network with blocking: Single Class Models
queueing network, multiple class: Introduction to QNs
queueing network, multiple class: Multiple Class Models
queueing network, product-form: Introduction to QNs
queueing network, single class: Introduction to QNs
queueing network, single class: Single Class Models
queueing networks: Queueing Networks
R
response time: Single Class Models
response time: Multiple Class Models
routing probability matrix: Single Class Models
routing probability matrix: Multiple Class Models
RS blocking: Single Class Models
S
service time: Single Class Models
service time: Multiple Class Models
single class queueing network: Introduction to QNs
single class queueing network: Single Class Models
stationary probabilities: State occupancy probabilities (CTMC)
stochastic matrix: Discrete-Time Markov Chains
system response time: Single Class Models
system response time: Multiple Class Models
system throughput: Single Class Models
system throughput: Multiple Class Models
T
throughput: Single Class Models
throughput: Multiple Class Models
time-alveraged sojourn time, CTMC: Time-averaged expected sojourn times (CTMC)
time-alveraged sojourn time, DTMC: Time-averaged expected sojourn times (DTMC)
traffic intensity: The M/M/inf System
U
utilization: Single Class Models
utilization: Multiple Class Models
W
warranty: Copying
Jump to:   A   B   C   D   E   F   G   I   L   M   N   O   P   Q   R   S   T   U   W  

Previous: , Up: Top   [Contents][Index]

Function Index

Jump to:   C   D   E   Q  
Index Entry  Section
C
ctmc: State occupancy probabilities (CTMC)
ctmc: State occupancy probabilities (CTMC)
ctmcbd: Birth-death process (CTMC)
ctmcchkQ: Continuous-Time Markov Chains
ctmcexps: Expected sojourn times (CTMC)
ctmcexps: Expected sojourn times (CTMC)
ctmcfpt: First passage times (CTMC)
ctmcfpt: First passage times (CTMC)
ctmcisir: Continuous-Time Markov Chains
ctmcmtta: Mean time to absorption (CTMC)
ctmctaexps: Time-averaged expected sojourn times (CTMC)
ctmctaexps: Time-averaged expected sojourn times (CTMC)
D
dtmc: State occupancy probabilities (DTMC)
dtmc: State occupancy probabilities (DTMC)
dtmcbd: Birth-death process (DTMC)
dtmcchkP: Discrete-Time Markov Chains
dtmcexps: Expected number of visits (DTMC)
dtmcexps: Expected number of visits (DTMC)
dtmcfpt: First passage times (DTMC)
dtmcisir: Discrete-Time Markov Chains
dtmcmtta: Mean time to absorption (DTMC)
dtmcmtta: Mean time to absorption (DTMC)
dtmctaexps: Time-averaged expected sojourn times (DTMC)
dtmctaexps: Time-averaged expected sojourn times (DTMC)
E
engset: The Engset Formula
erlangb: The Erlang-B Formula
erlangc: The Erlang-C Formula
Q
qnclosed: Generic Algorithms
qncmaba: Bounds Analysis
qncmaba: Bounds Analysis
qncmaba: Bounds Analysis
qncmaba: Bounds Analysis
qncmbsb: Bounds Analysis
qncmbsb: Bounds Analysis
qncmcb: Bounds Analysis
qncmcb: Bounds Analysis
qncmmva: Multiple Class Models
qncmmva: Multiple Class Models
qncmmva: Multiple Class Models
qncmmva: Multiple Class Models
qncmmva: Multiple Class Models
qncmmva: Multiple Class Models
qncmmva: Multiple Class Models
qncmmvabs: Multiple Class Models
qncmmvabs: Multiple Class Models
qncmmvabs: Multiple Class Models
qncmmvabs: Multiple Class Models
qncmmvabs: Multiple Class Models
qncmnpop: Multiple Class Models
qncmpopmix: Multiple Class Models
qncmvisits: Multiple Class Models
qncmvisits: Multiple Class Models
qncsaba: Bounds Analysis
qncsaba: Bounds Analysis
qncsaba: Bounds Analysis
qncsaba: Bounds Analysis
qncsbsb: Bounds Analysis
qncsbsb: Bounds Analysis
qncsbsb: Bounds Analysis
qncsbsb: Bounds Analysis
qncscmva: Single Class Models
qncscmva: Single Class Models
qncsconv: Single Class Models
qncsconv: Single Class Models
qncsconvld: Single Class Models
qncsgb: Bounds Analysis
qncsgb: Bounds Analysis
qncsgb: Bounds Analysis
qncsgb: Bounds Analysis
qncsmva: Single Class Models
qncsmva: Single Class Models
qncsmva: Single Class Models
qncsmvaap: Single Class Models
qncsmvaap: Single Class Models
qncsmvaap: Single Class Models
qncsmvaap: Single Class Models
qncsmvaap: Single Class Models
qncsmvablo: Single Class Models
qncsmvald: Single Class Models
qncsmvald: Single Class Models
qncspb: Bounds Analysis
qncspb: Bounds Analysis
qncspb: Bounds Analysis
qncspb: Bounds Analysis
qncsvisits: Single Class Models
qncsvisits: Single Class Models
qnmarkov: Single Class Models
qnmarkov: Single Class Models
qnmarkov: Single Class Models
qnmarkov: Single Class Models
qnmix: Multiple Class Models
qnmknode: Generic Algorithms
qnmknode: Generic Algorithms
qnmknode: Generic Algorithms
qnmknode: Generic Algorithms
qnmknode: Generic Algorithms
qnmknode: Generic Algorithms
qnmknode: Generic Algorithms
qnom: Multiple Class Models
qnom: Multiple Class Models
qnom: Multiple Class Models
qnom: Multiple Class Models
qnomaba: Bounds Analysis
qnomaba: Bounds Analysis
qnomvisits: Multiple Class Models
qnopen: Generic Algorithms
qnos: Single Class Models
qnos: Single Class Models
qnosaba: Bounds Analysis
qnosaba: Bounds Analysis
qnosaba: Bounds Analysis
qnosbsb: Bounds Analysis
qnosbsb: Bounds Analysis
qnosvisits: Single Class Models
qnsolve: Generic Algorithms
qnsolve: Generic Algorithms
qnsolve: Generic Algorithms
qnsolve: Generic Algorithms
qsammm: The Asymmetric M/M/m System
qsmg1: The M/G/1 System
qsmh1: The M/Hm/1 System
qsmm1: The M/M/1 System
qsmm1: The M/M/1 System
qsmm1k: The M/M/1/K System
qsmm1k: The M/M/1/K System
qsmminf: The M/M/inf System
qsmminf: The M/M/inf System
qsmmm: The M/M/m System
qsmmm: The M/M/m System
qsmmm: The M/M/m System
qsmmmk: The M/M/m/K System
qsmmmk: The M/M/m/K System
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with multiple job classes * Generic Algorithms:: High-level functions for QN analysis * Bounds Analysis:: Computation of asymptotic performance bounds * QN Analysis Examples:: Queueing Networks analysis examples @end menu @cindex queueing networks @c @c INTRODUCTION @c @node Introduction to QNs @section Introduction to QNs Queueing Networks (QN) are a simple modeling notation that can be used to analyze many kinds of systems. In its simplest form, a QN is made of @math{K} service centers; center @math{k} has a queue connected to @math{m_k} (usually identical) servers. Arriving customers (requests) join the queue if there is at least one slot available. Requests are served according to a (de)queueing policy (e.g., FIFO). After service completes, requests leave the server and can join another queue or exit from the system. @cindex delay center @cindex infinite server Service centers where @math{m_k = \infty} are called @emph{delay centers} or @emph{infinite servers}. In this kind of centers, there is always one available server, so that queueing never occurs. Requests join the queue according to a @emph{queueing policy}, such as: @cindex First-Come-First-Served @cindex FCFS @cindex Last-Come-First-Served Preemptive Resume @cindex LCFS-PR @cindex Processor Sharing @cindex PS @cindex Infinite Server @cindex IS @table @strong @item FCFS First-Come-First-Served @item LCFS-PR Last-Come-First-Served, Preemptive Resume @item PS Processor Sharing @item IS Infinite Server (@math{m_k = \infty}). @end table Queueing networks can be @emph{open} or @emph{closed}. In open networks there is an infinite population of requests; new customers are generated outside the system, and eventually leave the network. In closed networks there is a fixed population of request that never leave the system. @cindex single class queueing network @cindex multiple class queueing network @cindex queueing network, single class @cindex queueing network, multiple class Queueing models can have a single request class (@emph{single class models}), meaning that all requests behave in the same way (e.g., they spend the same average time on each particular server). In @emph{multiple class models} there are multiple request classes, each with its own parameters (e.g., with different service times or different routing probabilities). Furthermore, in multiclass models there can be open and closed chains of requests at the same time. @cindex product-form queueing network @cindex queueing network, product-form A particular class of QN models, @emph{product-form} networks, is of particular interest. Product-form networks fulfill the following assumptions: @itemize @item The network can consist of open and closed job classes. @item The following queueing disciplines are allowed: FCFS, PS, LCFS-PR and IS. @item Service times for FCFS nodes must be exponentially distributed and class-independent. Service centers at PS, LCFS-PR and IS nodes can have any kind of service time distribution with a rational Laplace transform. Furthermore, for PS, LCFS-PR and IS nodes, different classes of customers can have different service times. @item The service rate of an FCFS node is only allowed to depend on the number of jobs at this node; in a PS, LCFS-PR and IS node the service rate for a particular job class can also depend on the number of jobs of that class at the node. @item In open networks two kinds of arrival processes are allowed: i) the arrival process is Poisson, with arrival rate @math{\lambda} that can depend on the number of jobs in the network. ii) the arrival process consists of @math{C} independent Poisson arrival streams where the @math{C} job sources are assigned to the @math{C} chains; the arrival rate can be load dependent. @end itemize Product-form networks are attractive because steady-state performance measures can be efficiently computed. @c @c Single Class Models @c @node Single Class Models @section Single Class Models @cindex single class queueing network @cindex queueing network, single class In single class models, all requests are indistinguishable and belong to the same class. This means that every request has the same average service time, and all requests move through the system with the same routing probabilities. @noindent @strong{Model Inputs} @cindex external arrival rate @cindex service time @cindex routing probability matrix @cindex average number of visits @table @asis @item @math{{@lambda}_k} (Open models only) External arrival rate to service center @math{k}. @item @math{@lambda} (Open models only) Overall external arrival rate to the system as a whole: @math{\lambda = \sum_k \lambda_k}. @item @math{N} (Closed models only) Total number of requests in the system. @item @math{S_k} Mean service time at center @math{k}. @math{S_k} is the average time elapsed from service start to service completion at center @math{k}. @item @math{P_{i, j}} Routing probability matrix. @math{{\bf P} = [P_{i, j}]} is a @math{K \times K} matrix where @math{P_{i, j}} is the probability that a request completing service at center @math{i} is routed to center @math{j}. The probability that a request leaves the system after being served at center @math{i} is @math{\left(1-\sum_{j=1}^K P_{i, j}\right)}. @item @math{V_k} Mean number of visits to center @math{k} (also called @emph{visit ratio} or @emph{relative arrival rate}). @end table @noindent @strong{Model Outputs} @cindex utilization @cindex response time @cindex average number of customers @cindex throughput @cindex system response time @cindex system throughput @table @math @item U_k Utilization of service center @math{k}. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). If center @math{k} is a single-server or multiserver node, then @math{0 @leq{} U_k @leq{} 1}. If center @math{k} is an infinite server node (delay center), then @math{U_k} denotes the @emph{traffic intensity} and is defined as @math{U_k = X_k S_k}; in this case the utilization may be greater than one. @item R_k Average response time of service center @math{k}, defined as the mean time between the arrival of a request in the queue and service completion of the same request. @item Q_k Average number of requests in center @math{k}; this includes both the requests in the queue and those being served. @item X_k Throughput of service center @math{k}. The throughput is the rate of job completions, i.e., the average number of jobs completed over a given time interval. @end table @noindent Given the output parameters above, additional performance measures can be computed: @table @math @item X System throughput, @math{X = X_k / V_k} for any @math{k} for which @math{V_k \neq 0} @item R System response time, @math{R = \sum_{k=1}^K R_k V_k} @item Q Average number of requests in the system, @math{Q = \sum_{k=1}^K Q_k}; for closed systems, this can be written as @math{Q = N-XZ}; @end table For open, single class models, the scalar @math{\lambda} denotes the external arrival rate of requests to the system. The average number of visits @math{V_j} satisfy the following equation: @iftex @tex $$ V_j = P_{0, j} + \sum_{i=1}^K V_i P_{i, j} \quad j=1, \ldots, K $$ @end tex @end iftex @ifnottex @example @group K ___ \ V_j = P_(0, j) + > V_i P_(i, j) j=1,...,K /___ i=1 @end group @end example @end ifnottex @noindent where @math{P_{0, j}} is the probability that an external request goes to center @math{j}. If we denote with @math{\lambda_j} the external arrival rate to center @math{j}, and @math{\lambda = \sum_j \lambda_j} the overall external arrival rate, then @math{P_{0, j} = \lambda_j / \lambda}. For closed models, the visit ratios satisfy the following equation: @iftex @tex $$\left\{\eqalign{V_j & = \sum_{i=1}^K V_i P_{i, j} \quad j=1, \ldots, K \cr V_r & = 1 \quad \hbox{for a selected reference station $r$}}\right. $$ @end tex @end iftex @ifnottex @example / | K | ___ | \ | V_j = > V_i P_(i, j) j=1,...,K | /___ | i=1 | | V_r = 1 for a selected reference station r \ @end example @end ifnottex Note that the set of traffic equations @math{V_j = \sum_{i=1}^K V_i P_{i, j}} alone can only be solved up to a multiplicative constant; to get a unique solution we impose an additional constraint @math{V_r = 1} for some @math{1 @leq{} r @leq{} K}. This constraint is equivalent to defining station @math{r} as the @emph{reference station}; the default is @math{r=1}, @pxref{doc-qncsvisits}. A job that returns to the reference station is assumed to have completed its activity cycle. The network throughput is set to the throughput of the reference station. @anchor{doc-qncsvisits} @deftypefn {Function File} {@var{V} =} qncsvisits (@var{P}) @deftypefnx {Function File} {@var{V} =} qncsvisits (@var{P}, @var{r}) Compute the mean number of visits to the service centers of a single class, closed network with @math{K} service centers. @strong{INPUTS} @table @code @item @var{P}(i,j) probability that a request which completed service at center @math{i} is routed to center @math{j} (@math{K \times K} matrix). For closed networks it must hold that @code{sum(@var{P},2)==1}. The routing graph must be strongly connected, meaning that each node must be reachable from every other node. @item @var{r} Index of the reference station, @math{r \in @{1, @dots{}, K@}}; Default @code{@var{r}=1}. The traffic equations are solved by imposing the condition @code{@var{V}(r) = 1}. A request returning to the reference station completes its activity cycle. @end table @strong{OUTPUTS} @table @code @item @var{V}(k) average number of visits to service center @math{k}, assuming @math{r} as the reference station. @end table @end deftypefn @anchor{doc-qnosvisits} @deftypefn {Function File} {@var{V} =} qnosvisits (@var{P}, @var{lambda}) Compute the average number of visits to the service centers of a single class open Queueing Network with @math{K} service centers. @strong{INPUTS} @table @code @item @var{P}(i,j) is the probability that a request which completed service at center @math{i} is routed to center @math{j} (@math{K \times K} matrix). @item @var{lambda}(k) external arrival rate to center @math{k}. @end table @strong{OUTPUTS} @table @code @item @var{V}(k) average number of visits to server @math{k}. @end table @end deftypefn @c @c @c @noindent @strong{EXAMPLE} @float Figure,fig:qn_closed_single @center @image{qn_closed_single,3in} @caption{Closed network with a single class of requests} @end float @ref{fig:qn_closed_single} shows a closed queueing network with a single class of requests. The network has three service centers, labeled @emph{CPU}, @emph{Disk1} and @emph{Disk2}, and is known as a @emph{central server} model of a computer system. Requests spend some time at the CPU, which is represented by a PS (Processor Sharing) node. After that, requests are routed to Disk1 with probability @math{0.3}, and to Disk2 with probability @math{0.7}. Both Disk1 and Disk2 are FCFS nodes. If we label the servers as CPU=1, Disk1=2, Disk2=3, we can define the routing matrix as follows: @iftex @tex $$ {\bf P} = \pmatrix{ 0 & 0.3 & 0.7 \cr 1 & 0 & 0 \cr 1 & 0 & 0 } $$ @end tex @end iftex @ifnottex @example / 0 0.3 0.7 \ P = | 1 0 0 | \ 1 0 0 / @end example @end ifnottex The visit ratios @math{V}, using station 1 as the reference station, can be computed with: @example @verbatim P = [0 0.3 0.7; ... 1 0 0 ; ... 1 0 0 ]; V = qncsvisits(P) @end verbatim @result{} V = 1.00000 0.30000 0.70000 @end example @noindent @strong{EXAMPLE} @float Figure,fig:qn_open_single @center @image{qn_open_single,3in} @caption{Open Queueing Network with a single class of requests} @end float @ref{fig:qn_open_single} shows a open QN with a single class of requests. The network has the same structure as the one in @ref{fig:qn_closed_single}, with the difference that here we have a stream of jobs arriving from outside the system, at a rate @math{\lambda}. After service completion at the CPU, a job can leave the system with probability @math{0.2}, or be transferred to other nodes with the probabilities shown in the figure. The routing matrix is @iftex @tex $$ {\bf P} = \pmatrix{ 0 & 0.3 & 0.5 \cr 1 & 0 & 0 \cr 1 & 0 & 0 } $$ @end tex @end iftex @ifnottex @example / 0 0.3 0.5 \ P = | 1 0 0 | \ 1 0 0 / @end example @end ifnottex If we let @math{\lambda = 1.2}, we can compute the visit ratios @math{V} as follows: @example @verbatim p = 0.3; lambda = 1.2 P = [0 0.3 0.5; ... 1 0 0 ; ... 1 0 0 ]; V = qnosvisits(P,[1.2 0 0]) @end verbatim @result{} V = 5.0000 1.5000 2.5000 @end example Function @command{qnosvisits} expects a vector with @math{K} elements as a second parameter, for open networks only. The vector contains the arrival rates at each individual node; since in our example external arrivals exist only for node @math{S_1} with rate @math{\lambda = 1.2}, the second parameter is @code{[1.2, 0, 0]}. @c @c Open Networks @c @subsection Open Networks Jackson networks satisfy the following conditions: @itemize @item There is only one job class in the network; the total number of jobs in the system is unbounded. @item There are @math{K} service centers in the network. Each service center may have Poisson arrivals from outside the system. A job can leave the system from any node. @item Arrival rates as well as routing probabilities are independent from the number of nodes in the network. @item External arrivals and service times at the service centers are exponentially distributed, and in general can be load-dependent. @item Service discipline at each node is FCFS @end itemize We define the @emph{joint probability vector} @math{\pi(n_1, @dots{}, n_K)} as the steady-state probability that there are @math{n_k} requests at service center @math{k}, for all @math{k=1, @dots{}, N}. Jackson networks have the property that the joint probability is the product of the marginal probabilities @math{\pi_k}: @iftex @tex $$ \pi(n_1, \ldots, n_K) = \prod_{k=1}^K \pi_k(n_k) $$ @end tex @end iftex @ifnottex @example @var{joint_prob} = prod( @var{pi} ) @end example @end ifnottex @noindent where @math{\pi_k(n_k)} is the steady-state probability that there are @math{n_k} requests at service center @math{k}. @anchor{doc-qnos} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnos (@var{lambda}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnos (@var{lambda}, @var{S}, @var{V}, @var{m}) @cindex open network, single class @cindex BCMP network Analyze open, single class BCMP queueing networks with @math{K} service centers. This function works for a subset of BCMP single-class open networks satisfying the following properties: @itemize @item The allowed service disciplines at network nodes are: FCFS, PS, LCFS-PR, IS (infinite server); @item Service times are exponentially distributed and load-independent; @item Center @math{k} can consist of @code{@var{m}(k) @geq{} 1} identical servers. @item Routing is load-independent @end itemize @strong{INPUTS} @table @code @item @var{lambda} Overall external arrival rate (@code{@var{lambda}>0}). @item @var{S}(k) average service time at center @math{k} (@code{@var{S}(k)>0}). @item @var{V}(k) average number of visits to center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{i}. If @code{@var{m}(k) < 1}, enter @math{k} is a delay center (IS); otherwise it is a regular queueing center with @code{@var{m}(k)} servers. Default is @code{@var{m}(k) = 1} for all @math{k}. @end table @strong{OUTPUTS} @table @code @item @var{U}(k) If @math{k} is a queueing center, @code{@var{U}(k)} is the utilization of center @math{k}. If @math{k} is an IS node, then @code{@var{U}(k)} is the @emph{traffic intensity} defined as @code{@var{X}(k)*@var{S}(k)}. @item @var{R}(k) center @math{k} average response time. @item @var{Q}(k) average number of requests at center @math{k}. @item @var{X}(k) center @math{k} throughput. @end table @strong{REFERENCES} @itemize @item G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998 @end itemize @seealso{qnopen,qnclosed,qnosvisits} @end deftypefn From the results computed by this function, it is possible to derive other quantities of interest as follows: @itemize @item @strong{System Response Time}: The overall system response time can be computed as @iftex @tex $R_s = \sum_{k=1}^K V_k R_k$ @end tex @end iftex @ifnottex @code{R_s = dot(V,R);} @end ifnottex @item @strong{Average number of requests}: The average number of requests in the system can be computed as: @iftex @tex $$Q_{avg} = \sum_{k=1}^K Q_k$$ @end tex @end iftex @ifnottex @code{Q_avg = sum(Q)} @end ifnottex @end itemize @noindent @strong{EXAMPLE} @example @verbatim lambda = 3; V = [16 7 8]; S = [0.01 0.02 0.03]; [U R Q X] = qnos( lambda, S, V ); R_s = dot(R,V) # System response time N = sum(Q) # Average number in system @end verbatim @print{} R_s = 1.4062 @print{} N = 4.2186 @end example @c @c Closed Networks @c @subsection Closed Networks @anchor{doc-qncsmva} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsmva (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsmva (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsmva (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @cindex Mean Value Analysys (MVA) @cindex closed network, single class @cindex normalization constant Analyze closed, single class queueing networks using the exact Mean Value Analysis (MVA) algorithm. The following queueing disciplines are supported: FCFS, LCFS-PR, PS and IS (Infinite Server). This function supports fixed-rate service centers or multiple server nodes. For general load-dependent service centers, use the function @code{qncsmvald} instead. Additionally, the normalization constant @math{G(n)}, @math{n=0, @dots{}, N} is computed; @math{G(n)} can be used in conjunction with the BCMP theorem to compute steady-state probabilities. @strong{INPUTS} @table @code @item @var{N} Population size (number of requests in the system, @code{@var{N} @geq{} 0}). If @code{@var{N} == 0}, this function returns @code{@var{U} = @var{R} = @var{Q} = @var{X} = 0} @item @var{S}(k) mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). @item @var{V}(k) average number of visits to service center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{Z} External delay for customers (@code{@var{Z} @geq{} 0}). Default is 0. @item @var{m}(k) number of servers at center @math{k} (if @var{m} is a scalar, all centers have that number of servers). If @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); otherwise it is a regular queueing center (FCFS, LCFS-PR or PS) with @code{@var{m}(k)} servers. Default is @code{@var{m}(k) = 1} for all @math{k} (each service center has a single server). @end table @strong{OUTPUTS} @table @code @item @var{U}(k) If @math{k} is a FCFS, LCFS-PR or PS node (@code{@var{m}(k) @geq{} 1}), then @code{@var{U}(k)} is the utilization of center @math{k}, @math{0 @leq{} U(k) @leq{} 1}. If @math{k} is an IS node (@code{@var{m}(k) < 1}), then @code{@var{U}(k)} is the @emph{traffic intensity} defined as @code{@var{X}(k)*@var{S}(k)}. In this case the value of @code{@var{U}(k)} may be greater than one. @item @var{R}(k) center @math{k} response time. The @emph{Residence Time} at center @math{k} is @code{@var{R}(k) * @var{V}(k)}. The system response time @var{Rsys} can be computed either as @code{@var{Rsys} = @var{N}/@var{Xsys} - Z} or as @code{@var{Rsys} = dot(@var{R},@var{V})} @item @var{Q}(k) average number of requests at center @math{k}. The number of requests in the system can be computed either as @code{sum(@var{Q})}, or using the formula @code{@var{N}-@var{Xsys}*@var{Z}}. @item @var{X}(k) center @math{K} throughput. The system throughput @var{Xsys} can be computed as @code{@var{Xsys} = @var{X}(1) / @var{V}(1)} @item @var{G}(n) Normalization constants. @code{@var{G}(n+1)} contains the value of the normalization constant @math{G(n)}, @math{n=0, @dots{}, N} as array indexes in Octave start from 1. @math{G(n)} can be used in conjunction with the BCMP theorem to compute steady-state probabilities. @end table @strong{NOTES} In presence of load-dependent servers (i.e., if @code{@var{m}(k)>1} for some @math{k}), the MVA algorithm is known to be numerically unstable. Generally, this issue manifests itself as negative values for the response times or utilizations. This is not a problem of the @code{queueing} toolbox, but of the MVA algorithm, and has currently no known solution. This function prints a warning if numerical problems are detected; the warning can be disabled with the command @code{warning("off", "qn:numerical-instability")}. @strong{REFERENCES} @itemize @item M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313--322. @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} @end itemize This implementation is described in R. Jain , @cite{The Art of Computer Systems Performance Analysis}, Wiley, 1991, p. 577. Multi-server nodes are treated according to G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, Section 8.2.1, "Single Class Queueing Networks". @seealso{qncsmvald,qncscmva} @end deftypefn @noindent @strong{EXAMPLE} @example @verbatim S = [ 0.125 0.3 0.2 ]; V = [ 16 10 5 ]; N = 20; m = ones(1,3); Z = 4; [U R Q X] = qncsmva(N,S,V,m,Z); X_s = X(1)/V(1); # System throughput R_s = dot(R,V); # System response time printf("\t Util Qlen RespT Tput\n"); printf("\t-------- -------- -------- --------\n"); for k=1:length(S) printf("Dev%d\t%8.4f %8.4f %8.4f %8.4f\n", k, U(k), Q(k), R(k), X(k) ); endfor printf("\nSystem\t %8.4f %8.4f %8.4f\n\n", N-X_s*Z, R_s, X_s ); @end verbatim @end example @c @c MVA for single class, closed networks with load dependent servers @c @anchor{doc-qncsmvald} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvald (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvald (@var{N}, @var{S}, @var{V}, @var{Z}) @cindex Mean Value Analysys (MVA) @cindex MVA @cindex closed network, single class @cindex load-dependent service center Mean Value Analysis algorithm for closed, single class queueing networks with @math{K} service centers and load-dependent service times. This function supports FCFS, LCFS-PR, PS and IS nodes. For networks with only fixed-rate centers and multiple-server nodes, the function @code{qncsmva} is more efficient. @strong{INPUTS} @table @code @item @var{N} Population size (number of requests in the system, @code{@var{N} @geq{} 0}). If @code{@var{N} == 0}, this function returns @code{@var{U} = @var{R} = @var{Q} = @var{X} = 0} @item @var{S}(k,n) mean service time at center @math{k} where there are @math{n} requests, @math{1 @leq{} n @leq{} N}. @code{@var{S}(k,n)} @math{= 1 / \mu_{k}(n)}, where @math{\mu_{k}(n)} is the service rate of center @math{k} when there are @math{n} requests. @item @var{V}(k) average number of visits to service center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{Z} external delay ("think time", @code{@var{Z} @geq{} 0}); default 0. @end table @strong{OUTPUTS} @table @code @item @var{U}(k) utilization of service center @math{k}. The utilization is defined as the probability that service center @math{k} is not empty, that is, @math{U_k = 1-\pi_k(0)} where @math{\pi_k(0)} is the steady-state probability that there are 0 jobs at service center @math{k}. @item @var{R}(k) response time on service center @math{k}. @item @var{Q}(k) average number of requests in service center @math{k}. @item @var{X}(k) throughput of service center @math{k}. @end table @strong{NOTES} In presence of load-dependent servers, the MVA algorithm is known to be numerically unstable. Generally this problem manifests itself as negative response times or utilization. @strong{REFERENCES} @itemize @item M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313--322. @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} @end itemize This implementation is described in G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, Section 8.2.4.1, ``Networks with Load-Dependent Service: Closed Networks''. @seealso{qncsmva} @end deftypefn @c @c CMVA for single class, closed networks with a single load dependent servers @c @anchor{doc-qncscmva} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncscmva (@var{N}, @var{S}, @var{Sld}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncscmva (@var{N}, @var{S}, @var{Sld}, @var{V}, @var{Z}) @cindex conditional MVA (CMVA) @cindex Mean Value Analysis, conditional (CMVA) @cindex closed network, single class @cindex CMVA Conditional MVA (CMVA) algorithm, a numerically stable variant of MVA. This function supports a network of @math{M @geq{} 1} service centers and a single delay center. Servers @math{1, @dots{}, (M-1)} are load-independent; server @math{M} is load-dependent. @strong{INPUTS} @table @code @item @var{N} Number of requests in the system, @code{@var{N} @geq{} 0}. If @code{@var{N} == 0}, this function returns @code{@var{U} = @var{R} = @var{Q} = @var{X} = 0} @item @var{S}(k) mean service time on server @math{k = 1, @dots{}, (M-1)} (@code{@var{S}(k) > 0}). If there are no fixed-rate servers, then @code{S = []} @item @var{Sld}(n) inverse service rate at server @math{M} (the load-dependent server) when there are @math{n} requests, @math{n=1, @dots{}, N}. @code{@var{Sld}(n) = } @math{1 / \mu(n)}. @item @var{V}(k) average number of visits to service center @math{k=1, @dots{}, M}, where @code{@var{V}(k) @geq{} 0}. @code{@var{V}(1:M-1)} are the visit rates to the fixed rate servers; @code{@var{V}(M)} is the visit rate to the load dependent server. @item @var{Z} External delay for customers (@code{@var{Z} @geq{} 0}). Default is 0. @end table @strong{OUTPUTS} @table @code @item @var{U}(k) center @math{k} utilization (@math{k=1, @dots{}, M}) @item @var{R}(k) response time of center @math{k} (@math{k=1, @dots{}, M}). The system response time @var{Rsys} can be computed as @code{@var{Rsys} = @var{N}/@var{Xsys} - Z} @item @var{Q}(k) average number of requests at center @math{k} (@math{k=1, @dots{}, M}). @item @var{X}(k) center @math{k} throughput (@math{k=1, @dots{}, M}). @end table @strong{REFERENCES} @itemize @item G. Casale. @cite{A note on stable flow-equivalent aggregation in closed networks}. Queueing Syst. Theory Appl., 60:193–-202, December 2008, @uref{http://dx.doi.org/10.1007/s11134-008-9093-6, 10.1007/s11134-008-9093-6} @end itemize @end deftypefn @c @c Approximate MVA for single class, closed networks @c @anchor{doc-qncsmvaap} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvaap (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}, @var{iter_max}) @cindex Mean Value Analysys (MVA), approximate @cindex MVA, approximate @cindex approximate MVA @cindex closed network, single class @cindex closed network, approximate analysis Analyze closed, single class queueing networks using the Approximate Mean Value Analysis (MVA) algorithm. This function is based on approximating the number of customers seen at center @math{k} when a new request arrives as @math{Q_k(N) \times (N-1)/N}. This function only handles single-server and delay centers; if your network contains general load-dependent service centers, use the function @code{qncsmvald} instead. @strong{INPUTS} @table @code @item @var{N} Population size (number of requests in the system, @code{@var{N} > 0}). @item @var{S}(k) mean service time on server @math{k} (@code{@var{S}(k)>0}). @item @var{V}(k) average number of visits to service center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k} (if @var{m} is a scalar, all centers have that number of servers). If @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); if @code{@var{m}(k) == 1}, center @math{k} is a regular queueing center (FCFS, LCFS-PR or PS) with one server (default). This function does not support multiple server nodes (@code{@var{m}(k) > 1}). @item @var{Z} External delay for customers (@code{@var{Z} @geq{} 0}). Default is 0. @item @var{tol} Stopping tolerance. The algorithm stops when the maximum relative difference between the new and old value of the queue lengths @var{Q} becomes less than the tolerance. Default is @math{10^{-5}}. @item @var{iter_max} Maximum number of iterations (@code{@var{iter_max}>0}. The function aborts if convergenge is not reached within the maximum number of iterations. Default is 100. @end table @strong{OUTPUTS} @table @code @item @var{U}(k) If @math{k} is a FCFS, LCFS-PR or PS node (@code{@var{m}(k) == 1}), then @code{@var{U}(k)} is the utilization of center @math{k}. If @math{k} is an IS node (@code{@var{m}(k) < 1}), then @code{@var{U}(k)} is the @emph{traffic intensity} defined as @code{@var{X}(k)*@var{S}(k)}. @item @var{R}(k) response time at center @math{k}. The system response time @var{Rsys} can be computed as @code{@var{Rsys} = @var{N}/@var{Xsys} - Z} @item @var{Q}(k) average number of requests at center @math{k}. The number of requests in the system can be computed either as @code{sum(@var{Q})}, or using the formula @code{@var{N}-@var{Xsys}*@var{Z}}. @item @var{X}(k) center @math{k} throughput. The system throughput @var{Xsys} can be computed as @code{@var{Xsys} = @var{X}(1) / @var{V}(1)} @end table @strong{REFERENCES} This implementation is based on Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 6.4.2.2 ("Approximate Solution Techniques"). @seealso{qncsmva,qncsmvald} @end deftypefn @c @c Convolution @c According to the BCMP theorem, the state probability of a closed single class queueing network with @math{K} nodes and @math{N} requests can be expressed as: @iftex @tex $$ \pi(n_1, \ldots, n_K) = {1 \over G(N)} \prod_{k=1}^K F_k(n_k) $$ @end tex @end iftex @ifnottex @example @group n = [n1, @dots{} nK]; @r{population vector} p = 1/G(N+1) \prod F(k,k); @end group @end example @end ifnottex Here @math{\pi(n_1, @dots{}, n_K)} is the joint probability of having @math{n_k} requests at node @math{k}, for all @math{k=1, @dots{}, K}; we have that @math{\sum_{k=1}^K n_k = N} The @emph{convolution algorithms} computes the normalization constants @math{{\bf G} = \left[G(0), @dots{}, G(N)\right]} for single-class, closed networks with @math{N} requests. The normalization constants are returned as vector @code{@var{G}=[@var{G}(1), @dots{} @var{G}(N+1)]} where @code{@var{G}(i+1)} is the value of @math{G(i)} (remember that Octave uses 1-base vectors). The normalization constant can be used to compute all performance measures of interest (utilization, average response time and so on). @command{queueing} implements the convolution algorithm, in the function @command{qncsconv} and @command{qncsconvld}. The first one supports single-station nodes, multiple-station nodes and IS nodes. The second one supports networks with general load-dependent service centers. @anchor{doc-qncsconv} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsconv (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsconv (@var{N}, @var{S}, @var{V}, @var{m}) @cindex closed network, single class @cindex normalization constant @cindex convolution algorithm Analyze product-form, single class closed networks with @math{K} service centers using the convolution algorithm. Load-independent service centers, multiple servers (@math{M/M/m} queues) and IS nodes are supported. For general load-dependent service centers, use @code{qncsconvld} instead. @strong{INPUTS} @table @code @item @var{N} Number of requests in the system (@code{@var{N}>0}). @item @var{S}(k) average service time on center @math{k} (@code{@var{S}(k) @geq{} 0}). @item @var{V}(k) visit count of service center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. If @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); if @code{@var{m}(k) @geq{} 1}, center @math{k} it is a regular @math{M/M/m} queueing center with @code{@var{m}(k)} identical servers. Default is @code{@var{m}(k) = 1} for all @math{k}. @end table @strong{OUTPUT} @table @code @item @var{U}(k) center @math{k} utilization. For IS nodes, @code{@var{U}(k)} is the @emph{traffic intensity} @code{@var{X}(k) * @var{S}(k)}. @item @var{R}(k) average response time of center @math{k}. @item @var{Q}(k) average number of customers at center @math{k}. @item @var{X}(k) throughput of center @math{k}. @item @var{G}(n) Vector of normalization constants. @code{@var{G}(n+1)} contains the value of the normalization constant with @math{n} requests @math{G(n)}, @math{n=0, @dots{}, N}. @end table @strong{NOTE} For a network with @math{K} service centers and @math{N} requests, this implementation of the convolution algorithm has time and space complexity @math{O(NK)}. @strong{REFERENCES} @itemize @item Jeffrey P. Buzen, @cite{Computational Algorithms for Closed Queueing Networks with Exponential Servers}, Communications of the ACM, volume 16, number 9, September 1973, pp. 527--531. @uref{http://doi.acm.org/10.1145/362342.362345, 10.1145/362342.362345} @end itemize This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, pp. 313--317. @seealso{qncsconvld} @end deftypefn @noindent @strong{EXAMPLE} The normalization constant @math{G} can be used to compute the steady-state probabilities for a closed single class product-form Queueing Network with @math{K} nodes and @math{N} requests. Let @code{@var{n} = [@math{n_1, @dots{}, n_K}]} be a valid population vector, @math{\sum_{k=1}^K n_k = N}. Then, the steady-state probability @code{@var{p}(k)} to have @code{@var{n}(k)} requests at service center @math{k} can be computed as: @iftex @tex $$ p_k(n_k) = {(V_k S_k)^{n_k} \over G(N)} \left(G(N-n_k) - V_k S_k G(N-n_k-1)\right), \quad k=1, \ldots, K $$ @end tex @end iftex @example @verbatim n = [1 2 0]; N = sum(n); # Total population size S = [ 1/0.8 1/0.6 1/0.4 ]; m = [ 2 3 1 ]; V = [ 1 .667 .2 ]; [U R Q X G] = qncsconv( N, S, V, m ); p = [0 0 0]; # initialize p # Compute the probability to have n(k) jobs at service center k for k=1:3 p(k) = (V(k)*S(k))^n(k) / G(N+1) * ... (G(N-n(k)+1) - V(k)*S(k)*G(N-n(k)) ); printf("Prob( n(%d) = %d )=%f\n", k, n(k), p(k) ); endfor @end verbatim @print{} Prob( n(1) = 1 ) = 0.17975 @print{} Prob( n(2) = 2 ) = 0.48404 @print{} Prob( n(3) = 0 ) = 0.52779 @end example @noindent (recall that @code{@var{G}(@var{N}+1)} represents @math{G(N)}, since in Octave array indices start at one). @c @anchor{doc-qncsconvld} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}, @var{G}] =} qncsconvld (@var{N}, @var{S}, @var{V}) @cindex closed network @cindex normalization constant @cindex convolution algorithm @cindex load-dependent service center Convolution algorithm for product-form, single-class closed queueing networks with @math{K} general load-dependent service centers. This function computes steady-state performance measures for single-class, closed networks with load-dependent service centers using the convolution algorithm; the normalization constants are also computed. The normalization constants are returned as vector @code{@var{G}=[@var{G}(1), @dots{}, @var{G}(N+1)]} where @code{@var{G}(i+1)} is the value of @math{G(i)}. @strong{INPUTS} @table @code @item @var{N} Number of requests in the system (@code{@var{N}>0}). @item @var{S}(k,n) mean service time at center @math{k} where there are @math{n} requests, @math{1 @leq{} n @leq{} N}. @code{@var{S}(k,n)} @math{= 1 / \mu_{k,n}}, where @math{\mu_{k,n}} is the service rate of center @math{k} when there are @math{n} requests. @item @var{V}(k) visit count of service center @math{k} (@code{@var{V}(k) @geq{} 0}). The length of @var{V} is the number of servers @math{K} in the network. @end table @strong{OUTPUT} @table @code @item @var{U}(k) center @math{k} utilization. @item @var{R}(k) average response time at center @math{k}. @item @var{Q}(k) average number of requests in center @math{k}. @item @var{X}(k) center @math{k} throughput. @item @var{G}(n) Normalization constants (vector). @code{@var{G}(n+1)} corresponds to @math{G(n)}, as array indexes in Octave start from 1. @end table @strong{REFERENCES} @itemize @item Herb Schwetman, @cite{Some Computational Aspects of Queueing Network Models}, Technical Report @uref{http://docs.lib.purdue.edu/cstech/285/, CSD-TR-354}, Department of Computer Sciences, Purdue University, February 1981 (revised). @item M. Reiser, H. Kobayashi, @cite{On The Convolution Algorithm for Separable Queueing Networks}, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29--31, 1976). SIGMETRICS '76. ACM, New York, NY, pp. 109--117. @uref{http://doi.acm.org/10.1145/800200.806187, 10.1145/800200.806187} @end itemize This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998, pp. 313--317. Function @command{qncsconvld} is slightly different from the version described in Bolch et al. because it supports general load-dependent centers (while the version in the book does not). The modification is in the definition of function @code{F()} in @command{qncsconvld} which has been made similar to function @math{f_i} defined in Schwetman, @cite{Some Computational Aspects of Queueing Network Models}. @seealso{qncsconv} @end deftypefn @c @c @c @subsection Non Product-Form QNs @anchor{Non Product-Form QNs} @c @c MVABLO algorithm for approximate analysis of closed, single class @c QN with blocking @c @anchor{doc-qncsmvablo} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncsmvablo (@var{N}, @var{S}, @var{M}, @var{P} ) @cindex queueing network with blocking @cindex blocking queueing network @cindex closed network, finite capacity @cindex MVABLO Approximate MVA algorithm for closed queueing networks with blocking. @strong{INPUTS} @table @code @item @var{N} number of requests in the system. @var{N} must be strictly greater than zero, and less than the overall network capacity: @code{0 < @var{N} < sum(@var{M})}. @item @var{S}(k) average service time on server @math{k} (@code{@var{S}(k) > 0}). @item @var{M}(k) capacity of center @math{k}. The capacity is the maximum number of requests in a service center, including the request in service (@code{@var{M}(k) @geq{} 1}). @item @var{P}(i,j) probability that a request which completes service at server @math{i} will be transferred to server @math{j}. @end table @strong{OUTPUTS} @table @code @item @var{U}(k) center @math{k} utilization. @item @var{R}(k) average response time of service center @math{k}. @item @var{Q}(k) average number of requests in service center @math{k} (including the request in service). @item @var{X}(k) center @math{k} throughput. @end table @strong{REFERENCES} @itemize @item Ian F. Akyildiz, @cite{Mean Value Analysis for Blocking Queueing Networks}, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418--428. @uref{http://dx.doi.org/10.1109/32.4663, 10.1109/32.4663} @end itemize @seealso{qnopen, qnclosed} @end deftypefn @c @anchor{doc-qnmarkov} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{lambda}, @var{S}, @var{C}, @var{P}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{lambda}, @var{S}, @var{C}, @var{P}, @var{m}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{N}, @var{S}, @var{C}, @var{P}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmarkov (@var{N}, @var{S}, @var{C}, @var{P}, @var{m}) @cindex closed network, single class @cindex open network, single class @cindex closed network, finite capacity @cindex blocking queueing network @cindex RS blocking Compute utilization, response time, average queue length and throughput for open or closed queueing networks with finite capacity and a single class of requests. Blocking type is Repetitive-Service (RS). This function explicitly generates and solve the underlying Markov chain, and thus might require a large amount of memory. More specifically, networks which can me analyzed by this function have the following properties: @itemize @bullet @item There exists only a single class of customers. @item The network has @math{K} service centers. Center @math{k \in @{1, @dots{}, K@}} has @math{m_k > 0} servers, and has a total (finite) capacity of @math{C_k \geq m_k} which includes both buffer space and servers. The buffer space at service center @math{k} is therefore @math{C_k - m_k}. @item The network can be open, with external arrival rate to center @math{k} equal to @math{\lambda_k}, or closed with fixed population size @math{N}. For closed networks, the population size @math{N} must be strictly less than the network capacity: @math{N < \sum_{k=1}^K C_k}. @item Average service times are load-independent. @item @math{P_{i, j}} is the probability that requests completing execution at center @math{i} are transferred to center @math{j}, @math{i \neq j}. For open networks, a request may leave the system from any node @math{i} with probability @math{1-\sum_{j=1}^K P_{i, j}}. @item Blocking type is Repetitive-Service (RS). Service center @math{j} is @emph{saturated} if the number of requests is equal to its capacity @math{C_j}. Under the RS blocking discipline, a request completing service at center @math{i} which is being transferred to a saturated server @math{j} is put back at the end of the queue of @math{i} and will receive service again. Center @math{i} then processes the next request in queue. External arrivals to a saturated servers are dropped. @end itemize @strong{INPUTS} @table @code @item @var{lambda}(k) @itemx @var{N} If the first argument is a vector @var{lambda}, it is considered to be the external arrival rate @code{@var{lambda}(k) @geq{} 0} to service center @math{k} of an open network. If the first argument is a scalar, it is considered as the population size @var{N} of a closed network; in this case @var{N} must be strictly less than the network capacity: @code{@var{N} < sum(@var{C})}. @item @var{S}(k) average service time at service center @math{k} @item @var{C}(k) capacity of service center @math{k}. The capacity includes both the buffer and server space @code{@var{m}(k)}. Thus the buffer space is @code{@var{C}(k)-@var{m}(k)}. @item @var{P}(i,j) transition probability from service center @math{i} to service center @math{j}. @item @var{m}(k) number of servers at service center @math{k}. Note that @code{@var{m}(k) @geq{} @var{C}(k)} for each @var{k}. If @var{m} is omitted, all service centers are assumed to have a single server (@code{@var{m}(k) = 1} for all @math{k}). @end table @strong{OUTPUTS} @table @code @item @var{U}(k) center @math{k} utilization. @item @var{R}(k) response time on service center @math{k}. @item @var{Q}(k) average number of customers in the service center @math{k}, @emph{including} the request in service. @item @var{X}(k) throughput of service center @math{k}. @end table @strong{NOTES} The space complexity of this implementation is @math{O(\prod_{k=1}^K (C_k + 1)^2)}. The time complexity is dominated by the time needed to solve a linear system with @math{\prod_{k=1}^K (C_k + 1)} unknowns. @end deftypefn @c @c @c @node Multiple Class Models @section Multiple Class Models @cindex multiple class queueing network @cindex queueing network, multiple class In multiple class queueing models, we assume that there exist @math{C} different classes of requests. Each request from class @math{c} spends on average time @math{S_{c, k}} in service at center @math{k}. For open models, we denote with @math{{\bf \lambda} = \lambda_{c, k}} the arrival rates, where @math{\lambda_{c, k}} is the external arrival rate of class @math{c} requests at center @math{k}. For closed models, we denote with @math{{\bf N} = \left[N_1, @dots{}, N_C\right]} the population vector, where @math{N_c} is the number of class @math{c} requests in the system. The transition probability matrix for multiple class networks is a @math{C \times K \times C \times K} matrix @math{{\bf P} = [P_{r, i, s, j}]} where @math{P_{r, i, s, j}} is the probability that a class @math{r} request which completes service at center @math{i} will join server @math{j} as a class @math{s} request. Model input and outputs can be adjusted by adding additional indexes for the customer classes. @noindent @strong{Model Inputs} @cindex external arrival rate @cindex service time @cindex routing probability matrix @cindex average number of visits @table @asis @item @math{@lambdack} (open networks) External arrival rate of class-@math{c} requests to service center @math{k} @item @math{@lambda} (open networks) Overall external arrival rate to the whole system: @math{\lambda = \sum_{c=1}^C \sum_{k=1}^K \lambda_{c, k}} @item @math{N_c} (closed networks) Number of class @math{c} requests in the system. @item @math{S_{c, k}} Average service time. @math{S_{c, k}} is the average service time on service center @math{k} for class @math{c} requests. @item @math{P_{r, i, s, j}} Routing probability matrix. @math{{\bf P} = [P_{r, i, s, j}]} is a @math{C \times K \times C \times K} matrix such that @math{P_{r, i, s, j}} is the probability that a class @math{r} request which completes service at server @math{i} will move to server @math{j} as a class @math{s} request. @item @math{V_{c, k}} Mean number of visits of class @math{c} requests to center @math{k}. @end table @noindent @strong{Model Outputs} @cindex utilization @cindex response time @cindex average number of customers @cindex throughput @cindex system response time @cindex system throughput @table @asis @item @math{U_{c, k}} Utilization of service center @math{k} by class @math{c} requests. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). If center @math{k} is a single-server or multiserver node, then @math{0 @leq{} U_{c, k} @leq{} 1}. If center @math{k} is an infinite server node (delay center), then @math{U_{c, k}} denotes the @emph{traffic intensity} and is defined as @math{U_{c, k} = X_{c, k} S_{c, k}}; in this case the utilization may be greater than one. @item @math{R_{c, k}} Average response time experienced by class @math{c} requests on service center @math{k}. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service. @item @math{Q_{c, k}} Average number of class @math{c} requests on service center @math{k}. This includes both the requests in the queue, and the request being served. @item @math{X_{c, k}} Throughput of service center @math{k} for class @math{c} requests. The throughput is defined as the rate of completion of class @math{c} requests. @end table @noindent It is possible to define aggregate performance measures as follows: @table @math @item U_k Utilization of service center @math{k}: @iftex @tex $U_k = \sum_{c=1}^C U_{c, k}$ @end tex @end iftex @ifnottex @code{Uk = sum(U,k);} @end ifnottex @item R_c System response time for class @math{c} requests: @iftex @tex $R_c = \sum_{k=1}^K R_{c, k} V_{c, k}$ @end tex @end iftex @ifnottex @code{Rc = sum( V.*R, 1 );} @end ifnottex @item Q_c Average number of class @math{c} requests in the system: @iftex @tex $Q_c = \sum_{k=1}^K Q_{c, k}$ @end tex @end iftex @ifnottex @code{Qc = sum( Q, 2 );} @end ifnottex @item X_c Class @math{c} throughput: @iftex @tex $X_c = X_{c, k} / V_{c, k}$ for any @math{k} for which @math{V_{c,k} \neq 0} @end tex @end iftex @ifnottex @code{X(c) = X(c,k) ./ V(c,k);} for any @math{k} for which @code{V(c,k) != 0} @end ifnottex @end table For closed networks, we can define the visit ratios @math{V_{s, j}} for class @math{s} customers at service center @math{j} as follows: @iftex @tex $$\left\{\eqalign{ V_{s, j} & = \sum_{r=1}^C \sum_{i=1}^K V_{r, i} P_{r, i, s, j}, \quad s=1, \ldots, C, j=1, \ldots, K \cr V_{s, r_s} & = 1, \quad s=1, \ldots, C}\right. $$ @end tex @end iftex @ifnottex @group V_sj = sum_r sum_i V_ri P_risj s=1,...,C, j=1,...,K V_s r_s = 1 s=1,...,C @end group @end ifnottex @noindent where @math{r_s} is the class @math{s} reference station. Similarly to single class models, the traffic equation for closed multiclass networks can be solved up to multiplicative constants unless we choose one reference station for each closed class and set its visit ratio to 1. For open networks the traffic equations are as follows: @iftex @tex $$V_{s, j} = P_{0, s, j} + \sum_{r=1}^C \sum_{i=1}^K V_{r, i} P_{r, i, s, j} \quad s=1, \ldots, C, j=1, \ldots, K$$ @end tex @end iftex @ifnottex @group V_sj = P_0sj + sum_r sum_i V_ri P_risj s=1,...,C, j=1,...,K @end group @end ifnottex @noindent where @math{P_{0, s, j}} is the probability that an external arrival goes to service center @math{j} as a class-@math{s} request. If @math{\lambda_{s, j}} is the external arrival rate of class @math{s} requests to service center @math{j}, and @math{\lambda = \sum_s \sum_j \lambda_{s, j}} is the overall external arrival rate, then @math{P_{0, s, j} = \lambda_{s, j} / \lambda}. @anchor{doc-qncmvisits} @deftypefn {Function File} {[@var{V} @var{ch}] =} qncmvisits (@var{P}) @deftypefnx {Function File} {[@var{V} @var{ch}] =} qncmvisits (@var{P}, @var{r}) Compute the average number of visits for the nodes of a closed multiclass network with @math{K} service centers and @math{C} customer classes. @strong{INPUTS} @table @code @item @var{P}(r,i,s,j) probability that a class @math{r} request which completed service at center @math{i} is routed to center @math{j} as a class @math{s} request. Class switching is allowed. @item @var{r}(c) index of class @math{c} reference station, @math{r(c) \in @{1, @dots{}, K@}}, @math{1 @leq{} c @leq{} C}. The class @math{c} visit count to server @code{@var{r}(c)} (@code{@var{V}(c,r(c))}) is conventionally set to 1. The reference station serves two purposes: (i) its throughput is assumed to be the system throughput, and (ii) a job returning to the reference station is assumed to have completed one cycle. Default is to consider station 1 as the reference station for all classes. @end table @strong{OUTPUTS} @table @code @item @var{V}(c,i) number of visits of class @math{c} requests at center @math{i}. @item @var{ch}(c) chain number that class @math{c} belongs to. Different classes can belong to the same chain. Chains are numbered sequentially starting from 1 (@math{1, 2, @dots{}}). The total number of chains is @code{max(@var{ch})}. @end table @end deftypefn @anchor{doc-qnomvisits} @deftypefn {Function File} {@var{V} =} qnomvisits (@var{P}, @var{lambda}) Compute the visit ratios to the service centers of an open multiclass network with @math{K} service centers and @math{C} customer classes. @strong{INPUTS} @table @code @item @var{P}(r,i,s,j) probability that a class @math{r} request which completed service at center @math{i} is routed to center @math{j} as a class @math{s} request. Class switching is supported. @item @var{lambda}(r,i) external arrival rate of class @math{r} requests to center @math{i}. @end table @strong{OUTPUTS} @table @code @item @var{V}(r,i) visit ratio of class @math{r} requests at center @math{i}. @end table @end deftypefn @c @c Open Networks @c @subsection Open Networks @c @c Open network with multiple classes @c @anchor{doc-qnom} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{P}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnom (@var{lambda}, @var{S}, @var{P}, @var{m}) @cindex open network, multiple classes @cindex multiclass network, open Exact analysis of open, multiple-class BCMP networks. The network can be made of @emph{single-server} queueing centers (FCFS, LCFS-PR or PS) or delay centers (IS). This function assumes a network with @math{K} service centers and @math{C} customer classes. @strong{INPUTS} @table @code @item @var{lambda}(c) If this function is invoked as @code{qnom(lambda, S, V, @dots{})}, then @code{@var{lambda}(c)} is the external arrival rate of class @math{c} customers (@code{@var{lambda}(c) @geq{} 0}). If this function is invoked as @code{qnom(lambda, S, P, @dots{})}, then @code{@var{lambda}(c,k)} is the external arrival rate of class @math{c} customers at center @math{k} (@code{@var{lambda}(c,k) @geq{} 0}). @item @var{S}(c,k) mean service time of class @math{c} customers on the service center @math{k} (@code{@var{S}(c,k)>0}). For FCFS nodes, mean service times must be class-independent. @item @var{V}(c,k) visit ratio of class @math{c} customers to service center @math{k} (@code{@var{V}(c,k) @geq{} 0 }). @strong{If you pass this argument, class switching is not allowed} @item @var{P}(r,i,s,j) probability that a class @math{r} job completing service at center @math{i} is routed to center @math{j} as a class @math{s} job. @strong{If you pass argument @var{P}, class switching is allowed}; however, all servers must be fixed-rate or infinite-server nodes (@code{@var{m}(k) @leq{} 1} for all @math{k}). @item @var{m}(k) number of servers at center @math{k}. If @code{@var{m}(k) < 1}, enter @math{k} is a delay center (IS); otherwise it is a regular queueing center with @code{@var{m}(k)} servers. Default is @code{@var{m}(k) = 1} for all @math{k}. @end table @strong{OUTPUTS} @table @code @item @var{U}(c,k) If @math{k} is a queueing center, then @code{@var{U}(c,k)} is the class @math{c} utilization of center @math{k}. If @math{k} is an IS node, then @code{@var{U}(c,k)} is the class @math{c} @emph{traffic intensity} defined as @code{@var{X}(c,k)*@var{S}(c,k)}. @item @var{R}(c,k) class @math{c} response time at center @math{k}. The system response time for class @math{c} requests can be computed as @code{dot(@var{R}, @var{V}, 2)}. @item @var{Q}(c,k) average number of class @math{c} requests at center @math{k}. The average number of class @math{c} requests in the system @var{Qc} can be computed as @code{Qc = sum(@var{Q}, 2)} @item @var{X}(c,k) class @math{c} throughput at center @math{k}. @end table @strong{NOTES} If the function call specifies the visit ratios @var{V}, class switching is @strong{not} allowed. If the function call specifies the routing probability matrix @var{P}, then class switching @strong{is} allowed; however, all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported. Note that the meaning of parameter @var{lambda} is different from one case to the other (see below). @strong{REFERENCES} @itemize @item Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 7.4.1 ("Open Model Solution Techniques"). @end itemize @seealso{qnopen,qnos,qnomvisits} @end deftypefn @c @c Closed Networks @c @subsection Closed Networks @c @anchor{doc-qncmpopmix} @deftypefn {Function File} {pop_mix =} qncmpopmix (@var{k}, @var{N}) @cindex population mix @cindex closed network, multiple classes Return the set of population mixes for a closed multiclass queueing network with exactly @var{k} customers. Specifically, given a closed multiclass QN with @math{C} customer classes, where there are @code{@var{N}(c)} class @math{c} requests, @math{c = 1, @dots{}, C} a @math{k}-mix @math{M} is a vector of length @math{C} with the following properties: @itemize @item @math{0 @leq{} M_c @leq{} @var{N}(c)} for all @math{c = 1, @dots{}, C}; @item @math{\sum_{c=1}^C M_c = k} @end itemize In other words, a @math{k}-mix is an allocation of @math{k} requests to @math{C} classes such that the number of requests assigned to class @math{c} does not exceed the maximum value @code{@var{N}(c)}. @var{pop_mix} is a matrix with @math{C} columns, such that each row represents a valid mix. @strong{INPUTS} @table @code @item @var{k} Size of the requested mix (scalar, @code{@var{k} @geq{} 0}). @item @var{N}(c) number of class @math{c} requests (@code{@var{k} @leq{} sum(@var{N})}). @end table @strong{OUTPUTS} @table @code @item @var{pop_mix}(i,c) number of class @math{c} requests in the @math{i}-th population mix. The number of mixes is @code{rows(@var{pop_mix})}. @end table If you are interested in the number of @math{k}-mixes only, you can use the funcion @code{qnmvapop}. @strong{REFERENCES} @itemize @item Herb Schwetman, @cite{Implementing the Mean Value Algorithm for the Solution of Queueing Network Models}, Technical Report @uref{http://docs.lib.purdue.edu/cstech/286/, 80-355}, Department of Computer Sciences, Purdue University, revised February 15, 1982. @end itemize The slightly different problem of enumerating all tuples @math{k_1, @dots{}, k_N} such that @math{\sum_i k_i = k} and @math{k_i @geq{} 0}, for a given @math{k @geq{} 0} has been described in S. Santini, @cite{Computing the Indices for a Complex Summation}, unpublished report, available at @url{http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf} @seealso{qncmnpop} @end deftypefn @noindent @strong{EXAMPLE} Let us consider a multiclass network with @math{C=2} customer classes; the maximum number of class 1 requests is 2, and the maximum number of class 2 requests is 3. How is it possible to allocate 3 requests to the two classes so that the maximum number of requests per class is not exceeded? @example @verbatim N = [2 3]; mix = qncmpopmix(3, N) @end verbatim @print{} mix = [ [2 1] [1 2] [0 3] ] @end example @c @anchor{doc-qncmnpop} @deftypefn {Function File} {@var{H} =} qncmnpop (@var{N}) @cindex population mix @cindex closed network, multiple classes Given a network with @math{C} customer classes, this function computes the number of @math{k}-mixes @code{@var{H}(r,k)} that can be constructed by the multiclass MVA algorithm by allocating @math{k} customers to the first @math{r} classes. @xref{doc-qncmpopmix} for the definition of @math{k}-mix. @strong{INPUTS} @table @code @item @var{N}(c) number of class-@math{c} requests in the system. The total number of requests in the network is @code{sum(@var{N})}. @end table @strong{OUTPUTS} @table @code @item @var{H}(r,k) is the number of @math{k} mixes that can be constructed allocating @math{k} customers to the first @math{r} classes. @end table @strong{REFERENCES} @itemize @item Zahorjan, J. and Wong, E. @cite{The solution of separable queueing network models using mean value analysis}. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI @uref{http://doi.acm.org/10.1145/1010629.805477, 10.1145/1010629.805477} @end itemize @seealso{qncmmva,qncmpopmix} @end deftypefn @c @c MVA for multiple class, closed networks @c @anchor{doc-qncmmva} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S} ) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{P}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{P}, @var{r}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmva (@var{N}, @var{S}, @var{P}, @var{r}, @var{m}) @cindex Mean Value Analysys (MVA) @cindex closed network, multiple classes @cindex multiclass network, closed Compute steady-state performance measures for closed, multiclass queueing networks using the Mean Value Analysys (MVA) algorithm. Queueing policies at service centers can be any of the following: @table @strong @item FCFS (First-Come-First-Served) customers are served in order of arrival; multiple servers are allowed. For this kind of queueing discipline, average service times must be class-independent. @item PS (Processor Sharing) customers are served in parallel by a single server, each customer receiving an equal share of the service rate. @item LCFS-PR (Last-Come-First-Served, Preemptive Resume) customers are served in reverse order of arrival by a single server and the last arrival preempts the customer in service who will later resume service at the point of interruption. @item IS (Infinite Server) customers are delayed independently of other customers at the service center (there is effectively an infinite number of servers). @end table @strong{INPUTS} @table @code @item @var{N}(c) number of class @math{c} requests; @code{@var{N}(c) @geq{} 0}. If class @math{c} has no requests (@code{@var{N}(c) == 0}), then for all @var{k}, this function returns @code{@var{U}(c,k) = @var{R}(c,k) = @var{Q}(c,k) = @var{X}(c,k) = 0} @item @var{S}(c,k) mean service time for class @math{c} requests at center @math{k} (@code{@var{S}(c,k) @geq{} 0}). If the service time at center @math{k} is class-dependent, then center @math{k} is assumed to be of type @math{-/G/1}--PS (Processor Sharing). If center @math{k} is a FCFS node (@code{@var{m}(k)>1}), then the service times @strong{must} be class-independent, i.e., all classes @strong{must} have the same service time. @item @var{V}(c,k) average number of visits of class @math{c} requests at center @math{k}; @code{@var{V}(c,k) @geq{} 0}, default is 1. @strong{If you pass this argument, class switching is not allowed} @item @var{P}(r,i,s,j) probability that a class @math{r} request completing service at center @math{i} is routed to center @math{j} as a class @math{s} request; the reference stations for each class are specified with the paramter @var{r}. @strong{If you pass argument @var{P}, class switching is allowed}; however, you can not specify any external delay (i.e., @var{Z} must be zero) and all servers must be fixed-rate or infinite-server nodes (@code{@var{m}(k) @leq{} 1} for all @math{k}). @item @var{r}(c) reference station for class @math{c}. If omitted, station 1 is the reference station for all classes. See @command{qncmvisits}. @item @var{m}(k) If @code{@var{m}(k)<1}, then center @math{k} is assumed to be a delay center (IS node @math{-/G/\infty}). If @code{@var{m}(k)==1}, then service center @math{k} is a regular queueing center (@math{M/M/1}--FCFS, @math{-/G/1}--LCFS-PR or @math{-/G/1}--PS). Finally, if @code{@var{m}(k)>1}, center @math{k} is a @math{M/M/m}--FCFS center with @code{@var{m}(k)} identical servers. Default is @code{@var{m}(k)=1} for each @math{k}. @item @var{Z}(c) class @math{c} external delay (think time); @code{@var{Z}(c) @geq{} 0}. Default is 0. This parameter can not be used if you pass a routing matrix as the second parameter of @code{qncmmva}. @end table @strong{OUTPUTS} @table @code @item @var{U}(c,k) If @math{k} is a FCFS, LCFS-PR or PS node (@code{@var{m}(k) @geq{} 1}), then @code{@var{U}(c,k)} is the class @math{c} utilization at center @math{k}, @math{0 @leq{} U(c,k) @leq{} 1}. If @math{k} is an IS node, then @code{@var{U}(c,k)} is the class @math{c} @emph{traffic intensity} at center @math{k}, defined as @code{@var{U}(c,k) = @var{X}(c,k)*@var{S}(c,k)}. In this case the value of @code{@var{U}(c,k)} may be greater than one. @item @var{R}(c,k) class @math{c} response time at center @math{k}. The class @math{c} @emph{residence time} at center @math{k} is @code{@var{R}(c,k) * @var{C}(c,k)}. The total class @math{c} system response time is @code{dot(@var{R}, @var{V}, 2)}. @item @var{Q}(c,k) average number of class @math{c} requests at center @math{k}. The total number of requests at center @math{k} is @code{sum(@var{Q}(:,k))}. The total number of class @math{c} requests in the system is @code{sum(@var{Q}(c,:))}. @item @var{X}(c,k) class @math{c} throughput at center @math{k}. The class @math{c} throughput can be computed as @code{@var{X}(c,1) / @var{V}(c,1)}. @end table @strong{NOTES} If the function call specifies the visit ratios @var{V}, then class switching is @strong{not} allowed. If the function call specifies the routing probability matrix @var{P}, then class switching @strong{is} allowed; however, in this case all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported. In presence of load-dependent servers (e.g., if @code{@var{m}(i)>1} for some @math{i}), the MVA algorithm is known to be numerically unstable. Generally this problem shows up as negative values for the computed response times or utilizations. This is not a problem with the @code{queueing} package, but with the MVA algorithm; as such, there is no known workaround at the moment (aoart from using a different solution technique, if available). This function prints a warning if it detects numerical problems; you can disable the warning with the command @code{warning("off", "qn:numerical-instability")}. Given a network with @math{K} service centers, @math{C} job classes and population vector @math{{\bf N}=\left[N_1, @dots{}, N_C\right]}, the MVA algorithm requires space @math{O(C \prod_i (N_i + 1))}. The time complexity is @math{O(CK\prod_i (N_i + 1))}. This implementation is slightly more space-efficient (see details in the code). While the space requirement can be mitigated by using some optimizations, the time complexity can not. If you need to analyze large closed networks you should consider the @command{qncmmvaap} function, which implements the approximate MVA algorithm. Note however that @command{qncmmvaap} will only provide approximate results. @strong{REFERENCES} @itemize @item M. Reiser and S. S. Lavenberg, @cite{Mean-Value Analysis of Closed Multichain Queuing Networks}, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313--322. @uref{http://doi.acm.org/10.1145/322186.322195, 10.1145/322186.322195} @end itemize This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications}, Wiley, 1998 and Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 7.4.2.1 ("Exact Solution Techniques"). @seealso{qnclosed, qncmmvaapprox, qncmvisits} @end deftypefn @c @c Approximate MVA, with Bard-Schweitzer approximation @c @anchor{doc-qncmmvabs} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qncmmvabs (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}, @var{tol}, @var{iter_max}) @cindex Mean Value Analysys (MVA), approximate @cindex MVA, approximate @cindex closed network, multiple classes @cindex multiclass network, closed Approximate Mean Value Analysis (MVA) for closed, multiclass queueing networks with @math{K} service centers and @math{C} customer classes. This implementation uses Bard and Schweitzer approximation. It is based on the assumption that the queue length at service center @math{k} with population set @math{{\bf N}-{\bf 1}_c} is approximated with @tex $$Q_k({\bf N}-{\bf 1}_c) \approx {n-1 \over n} Q_k({\bf N})$$ @end tex @ifnottex @example @group Q_k(N-1c) ~ (n-1)/n Q_k(N) @end group @end example @end ifnottex where @math{\bf N} is a valid population mix, @math{{\bf N}-{\bf 1}_c} is the population mix @math{\bf N} with one class @math{c} customer removed, and @math{n = \sum_c N_c} is the total number of requests. This implementation works for networks with infinite server (IS) and single-server nodes only. @strong{INPUTS} @table @code @item @var{N}(c) number of class @math{c} requests in the system (@code{@var{N}(c) @geq{} 0}). @item @var{S}(c,k) mean service time for class @math{c} customers at center @math{k} (@code{@var{S}(c,k) @geq{} 0}). @item @var{V}(c,k) average number of visits of class @math{c} requests to center @math{k} (@code{@var{V}(c,k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. If @code{@var{m}(k) < 1}, then the service center @math{k} is assumed to be a delay center (IS). If @code{@var{m}(k) == 1}, service center @math{k} is a regular queueing center (FCFS, LCFS-PR or PS) with a single server node. If omitted, each service center has a single server. Note that multiple server nodes are not supported. @item @var{Z}(c) class @math{c} external delay (@code{@var{Z} @geq{} 0}). Default is 0. @item @var{tol} Stopping tolerance (@code{@var{tol}>0}). The algorithm stops if the queue length computed on two subsequent iterations are less than @var{tol}. Default is @math{10^{-5}}. @item @var{iter_max} Maximum number of iterations (@code{@var{iter_max}>0}. The function aborts if convergenge is not reached within the maximum number of iterations. Default is 100. @end table @strong{OUTPUTS} @table @code @item @var{U}(c,k) If @math{k} is a FCFS, LCFS-PR or PS node, then @code{@var{U}(c,k)} is the utilization of class @math{c} requests on service center @math{k}. If @math{k} is an IS node, then @code{@var{U}(c,k)} is the class @math{c} @emph{traffic intensity} at device @math{k}, defined as @code{@var{U}(c,k) = @var{X}(c)*@var{S}(c,k)} @item @var{R}(c,k) response time of class @math{c} requests at service center @math{k}. @item @var{Q}(c,k) average number of class @math{c} requests at service center @math{k}. @item @var{X}(c,k) class @math{c} throughput at service center @math{k}. @end table @strong{REFERENCES} @itemize @item Y. Bard, @cite{Some Extensions to Multiclass Queueing Network Analysis}, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, Feb 1979, pp. 51--62. @item P. Schweitzer, @cite{Approximate Analysis of Multiclass Closed Networks of Queues}, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25--29. @end itemize This implementation is based on Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 7.4.2.2 ("Approximate Solution Techniques"). This implementation is slightly different from the one described above, as it computes the average response times @math{R} instead of the residence times. @seealso{qncmmva} @end deftypefn @c @c Mixed networks @c @subsection Mixed Networks @c @c MVA for mixed networks @c @anchor{doc-qnmix} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnmix (@var{lambda}, @var{N}, @var{S}, @var{V}, @var{m}) @cindex Mean Value Analysys (MVA) @cindex mixed network Mean Value Analysis for mixed queueing networks. The network consists of @math{K} service centers (single-server or delay centers) and @math{C} independent customer chains. Both open and closed chains are possible. @var{lambda} is the vector of per-chain arrival rates (open classes); @var{N} is the vector of populations for closed chains. Class switching is @strong{not} allowed. Each customer class @emph{must} correspond to an independent chain. If the network is made of open or closed classes only, then this function calls @code{qnom} or @code{qncmmva} respectively, and prints a warning message. @strong{INPUTS} @table @code @item @var{lambda}(c) @itemx @var{N}(c) For each customer chain @math{c}: @itemize @item if @math{c} is a closed chain, then @code{@var{N}(c)>0} is the number of class @math{c} requests and @code{@var{lambda}(c)} must be zero; @item If @math{c} is an open chain, @code{@var{lambda}(c)>0} is the arrival rate of class @math{c} requests and @code{@var{N}(c)} must be zero; @end itemize @noindent In other words, for each class @math{c} the following must hold: @example (@var{lambda}(c)>0 && @var{N}(c)==0) || (@var{lambda}(c)==0 && @var{N}(c)>0) @end example @item @var{S}(c,k) mean class @math{c} service time at center @math{k}, @code{@var{S}(c,k) @geq{} 0}. For FCFS nodes, service times must be class-independent. @item @var{V}(c,k) average number of visits of class @math{c} customers to center @math{k} (@code{@var{V}(c,k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. Only single-server (@code{@var{m}(k)==1}) or IS (Infinite Server) nodes (@code{@var{m}(k)<1}) are supported. If omitted, each center is assumed to be of type @math{M/M/1}-FCFS. Queueing discipline for single-server nodes can be FCFS, PS or LCFS-PR. @end table @strong{OUTPUTS} @table @code @item @var{U}(c,k) class @math{c} utilization at center @math{k}. @item @var{R}(c,k) class @math{c} response time at center @math{k}. @item @var{Q}(c,k) average number of class @math{c} requests at center @math{k}. @item @var{X}(c,k) class @math{c} throughput at center @math{k}. @end table @strong{REFERENCES} @itemize @item Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 7.4.3 ("Mixed Model Solution Techniques"). Note that in this function we compute the mean response time @math{R} instead of the mean residence time as in the reference. @item Herb Schwetman, @cite{Implementing the Mean Value Algorithm for the Solution of Queueing Network Models}, Technical Report @uref{http://docs.lib.purdue.edu/cstech/286/, CSD-TR-355}, Department of Computer Sciences, Purdue University, revised Feb 15, 1982. @end itemize @seealso{qncmmva, qncm} @end deftypefn @c @c @c @node Generic Algorithms @section Generic Algorithms The @code{queueing} package provides a high-level function @command{qnsolve} for analyzing QN models. @command{qnsolve} takes as input a high-level description of the queueing model, and delegates the actual solution of the model to one of the lower-level function. @command{qnsolve} supports single or multiclass models, but at the moment only product-form networks can be analyzed. For non product-form networks @xref{Non Product-Form QNs}. @command{qnsolve} accepts two input parameters. The first one is the list of nodes, encoded as an Octave @emph{cell array}. The second parameter is the vector of visit ratios @var{V}, which can be either a vector (for single-class models) or a two-dimensional matrix (for multiple-class models). Individual nodes in the network are structures build using the @command{qnmknode} function. @anchor{doc-qnmknode} @deftypefn {Function File} {@var{Q} =} qnmknode (@var{"m/m/m-fcfs"}, @var{S}) @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"m/m/m-fcfs"}, @var{S}, @var{m}) @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"m/m/1-lcfs-pr"}, @var{S}) @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/1-ps"}, @var{S}) @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/1-ps"}, @var{S}, @var{s2}) @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/inf"}, @var{S}) @deftypefnx {Function File} {@var{Q} =} qnmknode (@var{"-/g/inf"}, @var{S}, @var{s2}) Creates a node; this function can be used together with @code{qnsolve}. It is possible to create either single-class nodes (where there is only one customer class), or multiple-class nodes (where the service time is given per-class). Furthermore, it is possible to specify load-dependent service times. String literals are case-insensitive, so for example @var{"-/g/inf"}, @var{"-/G/inf"} and @var{"-/g/INF"} are all equivalent. @strong{INPUTS} @table @code @item @var{S} Mean service time. @itemize @item If @math{S} is a scalar, it is assumed to be a load-independent, class-independent service time. @item If @math{S} is a column vector, then @code{@var{S}(c)} is assumed to the the load-independent service time for class @math{c} customers. @item If @math{S} is a row vector, then @code{@var{S}(n)} is assumed to be the class-independent service time at the node, when there are @math{n} requests. @item Finally, if @var{S} is a two-dimensional matrix, then @code{@var{S}(c,n)} is assumed to be the class @math{c} service time when there are @math{n} requests at the node. @end itemize @item @var{m} Number of identical servers at the node. Default is @code{@var{m}=1}. @item @var{s2} Squared coefficient of variation for the service time. Default is 1.0. @end table The returned struct @var{Q} should be considered opaque to the client. @c The returned struct @var{Q} has the following fields: @c @table @var @c @item Q.node @c (String) type of the node; valid values are @code{"m/m/m-fcfs"}, @c @code{"-/g/1-lcfs-pr"}, @code{"-/g/1-ps"} (Processor-Sharing) @c and @code{"-/g/inf"} (Infinite Server, or delay center). @c @item Q.S @c Average service time. If @code{@var{Q}.S} is a vector, then @c @code{@var{Q}.S(i)} is the average service time at that node @c if there are @math{i} requests. @c @item Q.m @c Number of identical servers at a @code{"m/m/m-fcfs"}. Default is 1. @c @item Q.c @c Number of customer classes. Default is 1. @c @end table @seealso{qnsolve} @end deftypefn After the network has been defined, it is possible to solve it using @command{qnsolve}. @anchor{doc-qnsolve} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"closed"}, @var{N}, @var{QQ}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"closed"}, @var{N}, @var{QQ}, @var{V}, @var{Z}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"open"}, @var{lambda}, @var{QQ}, @var{V}) @deftypefnx {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnsolve (@var{"mixed"}, @var{lambda}, @var{N}, @var{QQ}, @var{V}) High-level function for analyzing QN models. @itemize @item For @strong{closed} networks, the following server types are supported: @math{M/M/m}--FCFS, @math{-/G/\infty}, @math{-/G/1}--LCFS-PR, @math{-/G/1}--PS and load-dependent variants. @item For @strong{open} networks, the following server types are supported: @math{M/M/m}--FCFS, @math{-/G/\infty} and @math{-/G/1}--PS. General load-dependent nodes are @emph{not} supported. Multiclass open networks do not support multiple server @math{M/M/m} nodes, but only single server @math{M/M/1}--FCFS. @item For @strong{mixed} networks, the following server types are supported: @math{M/M/1}--FCFS, @math{-/G/\infty} and @math{-/G/1}--PS. General load-dependent nodes are @emph{not} supported. @end itemize @strong{INPUTS} @table @code @item @var{N} @itemx @var{N}(c) Number of requests in the system for closed networks. For single-class networks, @var{N} must be a scalar. For multiclass networks, @code{@var{N}(c)} is the population size of closed class @math{c}. @item @var{lambda} @itemx @var{lambda}(c) External arrival rate (scalar) for open networks. For single-class networks, @var{lambda} must be a scalar. For multiclass networks, @code{@var{lambda}(c)} is the class @math{c} overall arrival rate. @item @var{QQ}@{i@} List of queues in the network. This must be a cell array with @math{N} elements, such that @code{@var{QQ}@{i@}} is a struct produced by the @code{qnmknode} function. @item @var{Z} External delay ("think time") for closed networks. Default 0. @end table @strong{OUTPUTS} @table @code @item @var{U}(k) If @math{k} is a FCFS node, then @code{@var{U}(k)} is the utilization of service center @math{k}. If @math{k} is an IS node, then @code{@var{U}(k)} is the @emph{traffic intensity} defined as @code{@var{X}(k)*@var{S}(k)}. @item @var{R}(k) average response time of service center @math{k}. @item @var{Q}(k) average number of customers in service center @math{k}. @item @var{X}(k) throughput of service center @math{k}. @end table Note that for multiclass networks, the computed results are per-class utilization, response time, number of customers and throughput: @code{@var{U}(c,k)}, @code{@var{R}(c,k)}, @code{@var{Q}(c,k)}, @code{@var{X}(c,k)}. String literals are case-insensitive, so @var{"closed"}, @var{"Closed"} and @var{"CLoSEd"} are all equivalent. @end deftypefn @noindent @strong{EXAMPLE} Let us consider a closed, multiclass network with @math{C=2} classes and @math{K=3} service center. Let the population be @math{M=(2, 1)} (class 1 has 2 requests, and class 2 has 1 request). The nodes are as follows: @itemize @item Node 1 is a @math{M/M/1}--FCFS node, with load-dependent service times. Service times are class-independent, and are defined by the matrix @code{[0.2 0.1 0.1; 0.2 0.1 0.1]}. Thus, @code{@var{S}(1,2) = 0.2} means that service time for class 1 customers where there are 2 requests in 0.2. Note that service times are class-independent; @item Node 2 is a @math{-/G/1}--PS node, with service times @math{S_{1, 2} = 0.4} for class 1, and @math{S_{2, 2} = 0.6} for class 2 requests; @item Node 3 is a @math{-/G/\infty} node (delay center), with service times @math{S_{1, 3}=1} and @math{S_{2, 3}=2} for class 1 and 2 respectively. @end itemize After defining the per-class visit count @var{V} such that @code{@var{V}(c,k)} is the visit count of class @math{c} requests to service center @math{k}. We can define and solve the model as follows: @example @verbatim QQ = { qnmknode( "m/m/m-fcfs", [0.2 0.1 0.1; 0.2 0.1 0.1] ), ... qnmknode( "-/g/1-ps", [0.4; 0.6] ), ... qnmknode( "-/g/inf", [1; 2] ) }; V = [ 1 0.6 0.4; ... 1 0.3 0.7 ]; N = [ 2 1 ]; [U R Q X] = qnsolve( "closed", N, QQ, V ); @end verbatim @end example @anchor{doc-qnclosed} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnclosed (@var{N}, @var{S}, @var{V}, @dots{}) @cindex closed network, single class @cindex closed network, multiple classes This function computes steady-state performance measures of closed queueing networks using the Mean Value Analysis (MVA) algorithm. The qneneing network is allowed to contain fixed-capacity centers, delay centers or general load-dependent centers. Multiple request classes are supported. This function dispatches the computation to one of @code{qncsemva}, @code{qncsmvald} or @code{qncmmva}. @itemize @item If @var{N} is a scalar, the network is assumed to have a single class of requests; in this case, the exact MVA algorithm is used to analyze the network. If @var{S} is a vector, then @code{@var{S}(k)} is the average service time of center @math{k}, and this function calls @code{qncsmva} which supports load-independent service centers. If @var{S} is a matrix, @code{@var{S}(k,i)} is the average service time at center @math{k} when @math{i=1, @dots{}, N} jobs are present; in this case, the network is analyzed with the @code{qncmmvald} function. @item If @var{N} is a vector, the network is assumed to have multiple classes of requests, and is analyzed using the exact multiclass MVA algorithm as implemented in the @code{qncmmva} function. @end itemize @seealso{qncsmva, qncsmvald, qncmmva} @end deftypefn @noindent @strong{EXAMPLE} @example @verbatim P = [0 0.3 0.7; 1 0 0; 1 0 0]; # Transition probability matrix S = [1 0.6 0.2]; # Average service times m = ones(size(S)); # All centers are single-server Z = 2; # External delay N = 15; # Maximum population to consider V = qncsvisits(P); # Compute number of visits X_bsb_lower = X_bsb_upper = X_ab_lower = X_ab_upper = X_mva = zeros(1,N); for n=1:N [X_bsb_lower(n) X_bsb_upper(n)] = qncsbsb(n, S, V, m, Z); [X_ab_lower(n) X_ab_upper(n)] = qncsaba(n, S, V, m, Z); [U R Q X] = qnclosed( n, S, V, m, Z ); X_mva(n) = X(1)/V(1); endfor close all; plot(1:N, X_ab_lower,"g;Asymptotic Bounds;", ... 1:N, X_bsb_lower,"k;Balanced System Bounds;", ... 1:N, X_mva,"b;MVA;", "linewidth", 2, ... 1:N, X_bsb_upper,"k", 1:N, X_ab_upper,"g" ); axis([1,N,0,1]); legend("location","southeast"); legend("boxoff"); xlabel("Number of Requests n"); ylabel("System Throughput X(n)"); @end verbatim @end example @anchor{doc-qnopen} @deftypefn {Function File} {[@var{U}, @var{R}, @var{Q}, @var{X}] =} qnopen (@var{lambda}, @var{S}, @var{V}, @dots{}) @cindex open network Compute utilization, response time, average number of requests in the system, and throughput for open queueing networks. If @var{lambda} is a scalar, the network is considered a single-class QN and is solved using @code{qnopensingle}. If @var{lambda} is a vector, the network is considered as a multiclass QN and solved using @code{qnopenmulti}. @seealso{qnos, qnom} @end deftypefn @c @c @c @node Bounds Analysis @section Bounds Analysis @c @anchor{doc-qnosaba} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosaba (@var{lambda}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosaba (@var{lambda}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosaba (@var{lambda}, @var{S}, @var{V}, @var{m}) @cindex bounds, asymptotic @cindex open network Compute Asymptotic Bounds for open, single-class networks with @math{K} service centers. @strong{INPUTS} @table @code @item @var{lambda} Arrival rate of requests (scalar, @code{@var{lambda} @geq{} 0}). @item @var{D}(k) service demand at center @math{k}. (vector of length @math{K}, @code{@var{D}(k) @geq{} 0}). @item @var{S}(k) mean service time at center @math{k}. (vector of length @math{K}, @code{@var{S}(k) @geq{} 0}). @item @var{V}(k) mean number of visits to center @math{k}. (vector of length @math{K}, @code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. This function only supports @math{M/M/1} queues, therefore @var{m} must be @code{ones(size(S))}. @end table @strong{OUTPUTS} @table @code @item @var{Xl} @item @var{Xu} Lower and upper bounds on the system throughput. @var{Xl} is always set to @math{0} since there can be no lower bound on the throughput of open networks (scalar). @item @var{Rl} @item @var{Ru} Lower and upper bounds on the system response time. @var{Ru} is always set to @code{+inf} since there can be no upper bound on the throughput of open networks (scalar). @end table @seealso{qnomaba} @end deftypefn @c @anchor{doc-qnomaba} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnomaba (@var{lambda}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Rl}] =} qnomaba (@var{lambda}, @var{S}, @var{V}) @cindex bounds, asymptotic @cindex open network @cindex multiclass network, open Compute Asymptotic Bounds for open, multiclass networks with @math{K} service centers and @math{C} customer classes. @strong{INPUTS} @table @code @item @var{lambda}(c) class @math{c} arrival rate to the system (vector of length @math{C}, @code{@var{lambda}(c) > 0}). @item @var{D}(c, k) class @math{c} service demand at center @math{k} (@math{C \times K} matrix, @code{@var{D}(c, k) @geq{} 0}). @item @var{S}(c, k) mean service time of class @math{c} requests at center @math{k} (@math{C \times K} matrix, @code{@var{S}(c, k) @geq{} 0}). @item @var{V}(c, k) mean number of visits of class @math{c} requests at center @math{k} (@math{C \times K} matrix, @code{@var{V}(c, k) @geq{} 0}). @end table @strong{OUTPUTS} @table @code @item @var{Xl}(c) @item @var{Xu}(c) lower and upper bounds of class @math{c} throughput. @code{@var{Xl}(c)} is always @math{0} since there can be no lower bound on the throughput of open networks (vector of length @math{C}). @item @var{Rl}(c) @item @var{Ru}(c) lower and upper bounds of class @math{c} response time. @code{@var{Ru}(c)} is always @code{+inf} since there can be no upper bound on the response time of open networks (vector of length @math{C}). @end table @end deftypefn @c @anchor{doc-qncsaba} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @cindex bounds, asymptotic @cindex asymptotic bounds @cindex closed network, single class Compute Asymptotic Bounds for the system throughput and response time of closed, single-class networks with @math{K} service centers. Single-server and infinite-server nodes are supported. Multiple-server nodes and general load-dependent servers are not supported. @strong{INPUTS} @table @code @item @var{N} number of requests in the system (scalar, @code{@var{N}>0}). @item @var{D}(k) service demand at center @math{k} (@code{@var{D}(k) @geq{} 0}). @item @var{S}(k) mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). @item @var{V}(k) average number of visits to center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k} (if @var{m} is a scalar, all centers have that number of servers). If @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); if @code{@var{m}(k) = 1}, center @math{k} is a M/M/1-FCFS server. This function does not support multiple-server nodes. Default is 1. @item @var{Z} External delay (scalar, @code{@var{Z} @geq{} 0}). Default is 0. @end table @strong{OUTPUTS} @table @code @item @var{Xl} @itemx @var{Xu} Lower and upper bounds on the system throughput. @item @var{Rl} @itemx @var{Ru} Lower and upper bounds on the system response time. @end table @seealso{qncmaba} @end deftypefn @c @anchor{doc-qncmaba} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmaba (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @cindex bounds, asymptotic @cindex asymptotic bounds @cindex closed network @cindex multiclass network, closed @cindex closed multiclass network Compute Asymptotic Bounds for closed, multiclass networks with @math{K} service centers and @math{C} customer classes. Single-server and infinite-server nodes are supported. Multiple-server nodes and general load-dependent servers are not supported. @strong{INPUTS} @table @code @item @var{N}(c) number of class @math{c} requests in the system (vector of length @math{C}, @code{@var{N}(c) @geq{} 0}). @item @var{D}(c, k) class @math{c} service demand at center @math{k} (@math{C \times K} matrix, @code{@var{D}(c,k) @geq{} 0}). @item @var{S}(c, k) mean service time of class @math{c} requests at center @math{k} (@math{C \times K} matrix, @code{@var{S}(c,k) @geq{} 0}). @item @var{V}(c,k) average number of visits of class @math{c} requests to center @math{k} (@math{C \times K} matrix, @code{@var{V}(c,k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k} (if @var{m} is a scalar, all centers have that number of servers). If @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS); if @code{@var{m}(k) = 1}, center @math{k} is a M/M/1-FCFS server. This function does not support multiple-server nodes. Default is 1. @item @var{Z}(c) class @math{c} external delay (vector of length @math{C}, @code{@var{Z}(c) @geq{} 0}). Default is 0. @end table @strong{OUTPUTS} @table @code @item @var{Xl}(c) @itemx @var{Xu}(c) Lower and upper bounds for class @math{c} throughput. @item @var{Rl}(c) @itemx @var{Ru}(c) Lower and upper bounds for class @math{c} response time. @end table @strong{REFERENCES} @itemize @item Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 5.2 ("Asymptotic Bounds"). @end itemize @seealso{qncsaba} @end deftypefn @c @anchor{doc-qnosbsb} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosbsb (@var{lambda}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qnosbsb (@var{lambda}, @var{S}, @var{V}) @cindex bounds, balanced system @cindex open network Compute Balanced System Bounds for single-class, open networks with @math{K} service centers. @strong{INPUTS} @table @code @item @var{lambda} overall arrival rate to the system (scalar, @code{@var{lambda} @geq{} 0}). @item @var{D}(k) service demand at center @math{k} (@code{@var{D}(k) @geq{} 0}). @item @var{S}(k) service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). @item @var{V}(k) mean number of visits at center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. This function only supports @math{M/M/1} queues, therefore @var{m} must be @code{ones(size(S))}. @end table @strong{OUTPUTS} @table @code @item @var{Xl} @item @var{Xu} Lower and upper bounds on the system throughput. @var{Xl} is always set to @math{0}, since there can be no lower bound on open networks throughput. @item @var{Rl} @itemx @var{Ru} Lower and upper bounds on the system response time. @end table @seealso{qnosaba} @end deftypefn @c @anchor{doc-qncsbsb} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @cindex bounds, balanced system @cindex closed network, single class @cindex balanced system bounds Compute Balanced System Bounds on system throughput and response time for closed, single-class networks with @math{K} service centers. @strong{INPUTS} @table @code @item @var{N} number of requests in the system (scalar, @code{@var{N} @geq{} 0}). @item @var{D}(k) service demand at center @math{k} (@code{@var{D}(k) @geq{} 0}). @item @var{S}(k) mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). @item @var{V}(k) average number of visits to center @math{k} (@code{@var{V}(k) @geq{} 0}). Default is 1. @item @var{m}(k) number of servers at center @math{k}. This function supports @code{@var{m}(k) = 1} only (single-eserver FCFS nodes); this parameter is only for compatibility with @code{qncsaba}. Default is 1. @item @var{Z} External delay (@code{@var{Z} @geq{} 0}). Default is 0. @end table @strong{OUTPUTS} @table @code @item @var{Xl} @itemx @var{Xu} Lower and upper bound on the system throughput. @item @var{Rl} @itemx @var{Ru} Lower and upper bound on the system response time. @end table @strong{REFERENCES} @itemize @item Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, @cite{Quantitative System Performance: Computer System Analysis Using Queueing Network Models}, Prentice Hall, 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In particular, see section 5.4 ("Balanced Systems Bounds"). @end itemize @seealso{qncmbsb} @end deftypefn @c @anchor{doc-qncmbsb} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmbsb (@var{N}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmbsb (@var{N}, @var{S}, @var{V}) @cindex bounds, balanced system @cindex balanced system bounds @cindex multiclass network, closed @cindex closed multiclass network Compute Balanced System Bounds for closed, multiclass networks with @math{K} service centers and @math{C} customer classes. Only single-server nodes are supported. @strong{INPUTS} @table @code @item @var{N}(c) number of class @math{c} requests in the system (vector of length @math{C}). @item @var{D}(c, k) class @math{c} service demand at center @math{k} (@math{C \times K} matrix, @code{@var{D}(c,k) @geq{} 0}). @item @var{S}(c, k) mean service time of class @math{c} requests at center @math{k} (@math{C \times K} matrix, @code{@var{S}(c,k) @geq{} 0}). @item @var{V}(c,k) average number of visits of class @math{c} requests to center @math{k} (@math{C \times K} matrix, @code{@var{V}(c,k) @geq{} 0}). @end table @strong{OUTPUTS} @table @code @item @var{Xl}(c) @itemx @var{Xu}(c) Lower and upper class @math{c} throughput bounds (vector of length @math{C}). @item @var{Rl}(c) @itemx @var{Ru}(c) Lower and upper class @math{c} response time bounds (vector of length @math{C}). @end table @seealso{qncsbsb} @end deftypefn @c @anchor{doc-qncmcb} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmcb (@var{N}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmcb (@var{N}, @var{S}, @var{V}) @cindex multiclass network, closed @cindex closed multiclass network @cindex bounds, composite @cindex composite bounds Compute Composite Bounds (CB) on system throughput and response time for closed multiclass networks. @strong{INPUTS} @table @code @item @var{N}(c) number of class @math{c} requests in the system. @item @var{D}(c, k) class @math{c} service demand at center @math{k} (@code{@var{S}(c,k) @geq{} 0}). @item @var{S}(c, k) mean service time of class @math{c} requests at center @math{k} (@code{@var{S}(c,k) @geq{} 0}). @item @var{V}(c,k) average number of visits of class @math{c} requests to center @math{k} (@code{@var{V}(c,k) @geq{} 0}). @end table @strong{OUTPUTS} @table @code @item @var{Xl}(c) @itemx @var{Xu}(c) Lower and upper bounds on class @math{c} throughput. @item @var{Rl}(c) @itemx @var{Ru}(c) Lower and upper bounds on class @math{c} response time. @end table @strong{REFERENCES} @itemize @item Teemu Kerola, @cite{The Composite Bound Method (CBM) for Computing Throughput Bounds in Multiple Class Environments}, Performance Evaluation Vol. 6, Issue 1, March 1986, DOI @uref{http://dx.doi.org/10.1016/0166-5316(86)90002-7, 10.1016/0166-5316(86)90002-7}. Also available as @uref{http://docs.lib.purdue.edu/cstech/395/, Technical Report CSD-TR-475}, Department of Computer Sciences, Purdue University, mar 13, 1984 (Revised Aug 27, 1984). @end itemize @end deftypefn @c @anchor{doc-qncspb} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{D} ) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{S}, @var{V} ) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{S}, @var{V}, @var{m} ) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncspb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z} ) @cindex bounds, PB @cindex PB bounds @cindex closed network, single class Compute PB Bounds (C. H. Hsieh and S. Lam, 1987) for single-class, closed networks with @math{K} service centers. @strong{INPUTS} @table @code @item @var{} number of requests in the system (scalar, @code{@var{N} > 0}). @item @var{D}(k) service demand of service center @math{k} (@code{@var{D}(k) @geq{} 0}). @item @var{S}(k) mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}). @item @var{V}(k) visit ratio to center @math{k} (@code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. This function only supports @math{M/M/1} queues, therefore @var{m} must be @code{ones(size(S))}. @item @var{Z} external delay (think time, @code{@var{Z} @geq{} 0}). Default 0. @end table @strong{OUTPUTS} @table @code @item @var{Xl} @itemx @var{Xu} Lower and upper bounds on the system throughput. @item @var{Rl} @itemx @var{Ru} Lower and upper bounds on the system response time. @end table @strong{REFERENCES} @itemize @item C. H. Hsieh and S. Lam, @cite{Two classes of performance bounds for closed queueing networks}, Performance Evaluation, Vol. 7 Issue 1, pp. 3--30, February 1987, DOI @uref{http://dx.doi.org/10.1016/0166-5316(87)90054-X, 10.1016/0166-5316(87)90054-X}. Also available as @uref{ftp://ftp.cs.utexas.edu/pub/techreports/tr85-09.pdf, Technical Report TR-85-09}, Department of Computer Science, University of Texas at Austin, June 1985 @end itemize This function implements the non-iterative variant described in G. Casale, R. R. Muntz, G. Serazzi, @cite{Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks}, IEEE Transactions on Computers, 57(6):780-794, June 2008. @seealso{qncsaba, qbcsbsb, qncsgb} @end deftypefn @c @anchor{doc-qncsgb} @deftypefn {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{D}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{S}, @var{V}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{S}, @var{V}, @var{m}) @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}, @var{Ql}, @var{Qu}] =} qncsgb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z}) @cindex bounds, geometric @cindex geometric bounds @cindex closed network Compute Geometric Bounds (GB) on system throughput, system response time and server queue lenghts for closed, single-class networks with @math{K} service centers and @math{N} requests. @strong{INPUTS} @table @code @item @var{N} number of requests in the system (scalar, @code{@var{N} > 0}). @item @var{D}(k) service demand of service center @math{k} (vector of length @math{K}, @code{@var{D}(k) @geq{} 0}). @item @var{S}(k) mean service time at center @math{k} (vector of length @math{K}, @code{@var{S}(k) @geq{} 0}). @item @var{V}(k) visit ratio to center @math{k} (vector of length @math{K}, @code{@var{V}(k) @geq{} 0}). @item @var{m}(k) number of servers at center @math{k}. This function only supports @math{M/M/1} queues, therefore @var{m} must be @code{ones(size(S))}. @item @var{Z} external delay (think time, @code{@var{Z} @geq{} 0}, scalar). Default is 0. @end table @strong{OUTPUTS} @table @code @item @var{Xl} @itemx @var{Xu} Lower and upper bound on the system throughput. If @code{@var{Z}>0}, these bounds are computed using @emph{Geometric Square-root Bounds} (GSB). If @code{@var{Z}==0}, these bounds are computed using @emph{Geometric Bounds} (GB) @item @var{Rl} @itemx @var{Ru} Lower and upper bound on the system response time. These bounds are derived from @var{Xl} and @var{Xu} using Little's Law: @code{@var{Rl} = @var{N} / @var{Xu} - @var{Z}}, @code{@var{Ru} = @var{N} / @var{Xl} - @var{Z}} @item @var{Ql}(k) @itemx @var{Qu}(k) lower and upper bounds of center @math{K} queue length. @end table @strong{REFERENCES} @itemize @item G. Casale, R. R. Muntz, G. Serazzi, @cite{Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks}, IEEE Transactions on Computers, 57(6):780-794, June 2008. @uref{http://doi.ieeecomputersociety.org/10.1109/TC.2008.37, 10.1109/TC.2008.37} @end itemize In this implementation we set @math{X^+} and @math{X^-} as the upper and lower Asymptotic Bounds as computed by the @command{qncsab} function, respectively. @end deftypefn @c @c Examples @c @node QN Analysis Examples @section QN Analysis Examples In this section we illustrate with a few examples how the @code{queueing} package can be used to analyze queueing network models. Further examples can be found in the functions demo blocks, and can be inspected with the @code{demo @emph{function}} Octave command. @subsection Closed, Single Class Network Let us consider again the network shown in @ref{fig:qn_closed_single}. We denote with @math{S_k} the average service time at center @math{k}, @math{k=1, 2, 3}. Let the service times be @math{S_1 = 1.0}, @math{S_2 = 2.0} and @math{S_3 = 0.8}. The routing of jobs within the network is described with a @emph{routing probability matrix} @math{\bf P}: a request completing service at center @math{i} is enqueued at center @math{j} with probability @math{P_{i, j}}. We use the following routing matrix: @iftex @tex $$ {\bf P} = \pmatrix{ 0 & 0.3 & 0.7 \cr 1 & 0 & 0 \cr 1 & 0 & 0 } $$ @end tex @end iftex @ifnottex @example / 0 0.3 0.7 \ P = | 1 0 0 | \ 1 0 0 / @end example @end ifnottex The network above can be analyzed with the @command{qnclosed} function @pxref{doc-qnclosed}. @command{qnclosed} requires the following parameters: @table @var @item N Number of requests in the network (since we are considering a closed network, the number of requests is fixed) @item S Array of average service times at the centers: @code{@var{S}(k)} is the average service time at center @math{k}. @item V Array of visit ratios: @code{@var{V}(k)} is the average number of visits to center @math{k}. @end table We can compute @math{V_k} from the routing probability matrix @math{P_{i, j}} using the @command{qncsvisits} function @pxref{doc-qncsvisits}. Therefore, we can analyze the network for a given population size @math{N} (e.g., @math{N=10}) as follows: @example @group @kbd{N = 10;} @kbd{S = [1 2 0.8];} @kbd{P = [0 0.3 0.7; 1 0 0; 1 0 0];} @kbd{V = qncsvisits(P);} @kbd{[U R Q X] = qnclosed( N, S, V )} @result{} U = 0.99139 0.59483 0.55518 @result{} R = 7.4360 4.7531 1.7500 @result{} Q = 7.3719 1.4136 1.2144 @result{} X = 0.99139 0.29742 0.69397 @end group @end example The output of @command{qnclosed} includes the vectors of utilizations @math{U_k} at center @math{k}, response time @math{R_k}, average number of customers @math{Q_k} and throughput @math{X_k}. In our example, the throughput of center 1 is @math{X_1 = 0.99139}, and the average number of requests in center 3 is @math{Q_3 = 1.2144}. The utilization of center 1 is @math{U_1 = 0.99139}, which is the highest among the service centers. Thus, center 1 is the @emph{bottleneck device}. This network can also be analyzed with the @command{qnsolve} function @pxref{doc-qnsolve}. @command{qnsolve} can handle open, closed or mixed networks, and allows the network to be described in a very flexible way. First, let @var{Q1}, @var{Q2} and @var{Q3} be the variables describing the service centers. Each variable is instantiated with the @command{qnmknode} function. @example @group @kbd{Q1 = qnmknode( "m/m/m-fcfs", 1 );} @kbd{Q2 = qnmknode( "m/m/m-fcfs", 2 );} @kbd{Q3 = qnmknode( "m/m/m-fcfs", 0.8 );} @end group @end example The first parameter of @command{qnmknode} is a string describing the type of the node; @code{"m/m/m-fcfs"} denotes a @math{M/M/m}--FCFS center (this parameter is case-insensitive). The second parameter gives the average service time. An optional third parameter can be used to specify the number @math{m} of service centers. If omitted, it is assumed @math{m=1} (single-server node). Now, the network can be analyzed as follows: @example @group @kbd{N = 10;} @kbd{V = [1 0.3 0.7];} @kbd{[U R Q X] = qnsolve( "closed", N, @{ Q1, Q2, Q3 @}, V )} @result{} U = 0.99139 0.59483 0.55518 @result{} R = 7.4360 4.7531 1.7500 @result{} Q = 7.3719 1.4136 1.2144 @result{} X = 0.99139 0.29742 0.69397 @end group @end example @subsection Open, Single Class Network Let us consider an open network with @math{K=3} service centers and the following routing probabilities: @iftex @tex $$ {\bf P} = \pmatrix{ 0 & 0.3 & 0.5 \cr 1 & 0 & 0 \cr 1 & 0 & 0 } $$ @end tex @end iftex @ifnottex @example / 0 0.3 0.5 \ P = ! 1 0 0 | \ 1 0 0 / @end example @end ifnottex In this network, requests can leave the system from center 1 with probability @math{1-(0.3+0.5) = 0.2}. We suppose that external jobs arrive at center 1 with rate @math{\lambda_1 = 0.15}; there are no arrivals at centers 2 and 3. Similarly to closed networks, we first compute the visit counts @math{V_k} to center @math{k}, @math{k = 1, 2, 3}. We use the @command{qnosvisits} function as follows: @example @group @kbd{P = [0 0.3 0.5; 1 0 0; 1 0 0];} @kbd{lambda = [0.15 0 0];} @kbd{V = qnosvisits(P, lambda)} @result{} V = 5.00000 1.50000 2.50000 @end group @end example @noindent where @code{@var{lambda}(k)} is the arrival rate at center @math{k}, and @math{\bf P} is the routing matrix. Assuming the same service times as in the previous example, the network can be analyzed with the @command{qnopen} function @pxref{doc-qnopen}, as follows: @example @group @kbd{S = [1 2 0.8];} @kbd{[U R Q X] = qnopen( sum(lambda), S, V )} @result{} U = 0.75000 0.45000 0.30000 @result{} R = 4.0000 3.6364 1.1429 @result{} Q = 3.00000 0.81818 0.42857 @result{} X = 0.75000 0.22500 0.37500 @end group @end example The first parameter of the @command{qnopen} function is the (scalar) aggregate arrival rate. Again, it is possible to use the @command{qnsolve} high-level function: @example @group @kbd{Q1 = qnmknode( "m/m/m-fcfs", 1 );} @kbd{Q2 = qnmknode( "m/m/m-fcfs", 2 );} @kbd{Q3 = qnmknode( "m/m/m-fcfs", 0.8 );} @kbd{lambda = [0.15 0 0];} @kbd{[U R Q X] = qnsolve( "open", sum(lambda), @{ Q1, Q2, Q3 @}, V )} @result{} U = 0.75000 0.45000 0.30000 @result{} R = 4.0000 3.6364 1.1429 @result{} Q = 3.00000 0.81818 0.42857 @result{} X = 0.75000 0.22500 0.37500 @end group @end example @subsection Closed Multiclass Network/1 The following example is taken from Herb Schwetman, @cite{Implementing the Mean Value Algorithm for the Solution of Queueing Network Models}, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, Feb 15, 1982. Let us consider the following multiclass QN with three servers and two classes @float Figure,fig:apl @center @image{qn_closed_multi_apl} @end float Servers 1 and 2 (labeled @emph{APL} and @emph{IMS}, respectively) are infinite server nodes; server 3 (labeled @emph{SYS}) is Processor Sharing (PS). Mean service times are given in the following table: @multitable @columnfractions .15 .15 .15 .15 @headitem @tab APL @tab IMS @tab SYS @item Class 1 @tab 1 @tab - @tab 0.025 @item Class 2 @tab - @tab 15 @tab 0.500 @end multitable There is no class switching. If we assume a population of 15 requests for class 1, and 5 requests for class 2, then the model can be analyzed as follows: @example @verbatim S = [1 0 .025; 0 15 .5]; P = zeros(2,3,2,3); P(1,1,1,3) = P(1,3,1,1) = 1; P(2,2,2,3) = P(2,3,2,2) = 1; V = qncmvisits(P,[3 3]); # reference station is station 3 N = [15 5]; m = [-1 -1 1]; [U R Q X] = qncmmva(N,S,V,m) @end verbatim @result{} U = 14.32312 0.00000 0.35808 0.00000 4.70699 0.15690 R = 1.00000 0.00000 0.04726 0.00000 15.00000 0.93374 Q = 14.32312 0.00000 0.67688 0.00000 4.70699 0.29301 X = 14.32312 0.00000 14.32312 0.00000 0.31380 0.31380 @end example @subsection Closed Multiclass Network/2 The following example is from M. Marzolla, @cite{The qnetworks Toolbox: A Software Package for Queueing Networks Analysis}, Technical Report @uref{http://www.informatica.unibo.it/it/ricerca/technical-report/2010/UBLCS-2010-04, UBLCS-2010-04}, Department of Computer Science, University of Bologna, Italy, February 2010. @float Figure,fig:web_model @center @image{qn_web_model,3in} @caption{Three-tier enterprise system model} @end float The model shown in @ref{fig:web_model} shows a three-tier enterprise system with @math{K=6} service centers. The first tier contains the @emph{Web server} (node 1), which is responsible for generating Web pages and transmitting them to clients. The application logic is implemented by nodes 2 and 3, and the storage tier is made of nodes 4--6.The system is subject to two workload classes, both represented as closed populations of @math{N_1} and @math{N_2} requests, respectively. Let @math{D_{c, k}} denote the service demand of class @math{c} requests at center @math{k}. We use the parameter values: @multitable @columnfractions .2 .33 .1 .1 @headitem Serv. no. @tab Name @tab Class 1 @tab Class 2 @item 1 @tab Web Server @tab 12 @tab 2 @item 2 @tab App. Server 1 @tab 14 @tab 20 @item 3 @tab App. Server 2 @tab 23 @tab 14 @item 4 @tab DB Server 1 @tab 20 @tab 90 @item 5 @tab DB Server 2 @tab 80 @tab 30 @item 6 @tab DB Server 3 @tab 31 @tab 33 @end multitable We set the total number of requests to 100, that is @math{N_1 + N_2 = N = 100}, and we study how different population mixes @math{(N_1, N_2)} affect the system throughput and response time. Let @math{0 < \beta_1 < 1} denote the fraction of class 1 requests: @math{N_1 = \beta_1 N}, @math{N_2 = (1-\beta_1)N}. The following Octave code defines the model for @math{\beta_1 = 0.1}: @example @group N = 100; # total population size beta1 = 0.1; # fraction of class 1 reqs. S = [12 14 23 20 80 31; \ 2 20 14 90 30 33 ]; V = ones(size(S)); pop = [fix(beta1*N) N-fix(beta1*N)]; [U R Q X] = qncmmva(pop, S, V); @end group @end example The @command{qncmmva(pop, S, V)} function invocation uses the multiclass MVA algorithm to compute per-class utilizations @math{U_{c, k}}, response times @math{R_{c,k}}, mean queue lengths @math{Q_{c,k}} and throughputs @math{X_{c,k}} at each service center @math{k}, given a population vector @var{pop}, mean service times @var{S} and visit ratios @var{V}. Since we are given the service demands @math{D_{c, k} = S_{c, k} V_{c,k}}, but function @command{qncmmva} requires separate service times and visit ratios, we set the service times equal to the demands, and all visit ratios equal to one. Overall class and system throughputs and response times can also be computed: @example @group X1 = X(1,1) / V(1,1) # class 1 throughput @result{} X1 = 0.0044219 X2 = X(2,1) / V(2,1) # class 2 throughput @result{} X2 = 0.010128 XX = X1 + X2 # system throughput @result{} XX = 0.014550 R1 = dot(R(1,:), V(1,:)) # class 1 resp. time @result{} R1 = 2261.5 R2 = dot(R(2,:), V(2,:)) # class 2 resp. time @result{} R2 = 8885.9 RR = N / XX # system resp. time @result{} RR = 6872.7 @end group @end example @code{dot(X,Y)} computes the dot product of two vectors. @code{R(1,:)} is the first row of matrix @var{R} and @code{V(1,:)} is the first row of matrix @var{V}, so @code{dot(R(1,:), V(1,:))} computes @math{\sum_k R_{1,k} V_{1,k}}. @float Figure,fig:web @center @image{web,5in} @caption{Throughput and Response Times as a function of the population mix} @end float We can also compute the system power @math{\Phi = X / R}, which defines how efficiently resources are being used: high values of @math{\Phi} denote the desirable situation of high throughput and low response time. @ref{fig:power} shows @math{\Phi} as a function of @math{\beta_1}. We observe a ``plateau'' of the global system power, corresponding to values of @math{\beta_1} which approximately lie between @math{0.3} and @math{0.7}. The per-class power exhibits an interesting (although not completely surprising) pattern, where the class with higher population exhibits worst efficiency as it produces higher contention on the resources. @float Figure,fig:power @center @image{power,5in} @caption{System Power as a function of the population mix} @end float @subsection Closed Multiclass Network/3 We now consider an example of multiclass network with class switching. The example is taken from @ref{Sch82}, and is shown in Figure @ref{fig:class_switching}. @float Figure,fig:class_switching @center @image{qn_closed_multi_cs,3in} @caption{Multiclass Model with Class Switching} @end float The system consists of three devices and two job classes. The CPU node is a PS server, while the two nodes labeled I/O are FCFS. Class 1 mean service time at the CPU is @math{0.01}; class 2 mean service time at the CPU is @math{0.05}. The mean service time at node 2 is @math{0.1}, and is class-independent. Similarly, the mean service time at node 3 is @math{0.07}. Jobs in class 1 leave the CPU and join class 2 with probability @math{0.1}; jobs of class 2 leave the CPU and join class 1 with probability @math{0.2}. There are @math{N=3} jobs, which are initially allocated to class 1. However, note that since class switching is allowed, the total number of jobs in each class does not remain constant; however the total number of jobs does. @example @verbatim C = 2; K = 3; S = [.01 .07 .10; ... .05 .07 .10 ]; P = zeros(C,K,C,K); P(1,1,1,2) = .7; P(1,1,1,3) = .2; P(1,1,2,1) = .1; P(2,1,2,2) = .3; P(2,1,2,3) = .5; P(2,1,1,1) = .2; P(1,2,1,1) = P(2,2,2,1) = 1; P(1,3,1,1) = P(2,3,2,1) = 1; N = [3 0]; [U R Q X] = qncmmva(N, S, P) @end verbatim @result{} U = 0.12609 0.61784 0.25218 0.31522 0.13239 0.31522 R = 0.014653 0.133148 0.163256 0.073266 0.133148 0.163256 Q = 0.18476 1.17519 0.41170 0.46190 0.25183 0.51462 X = 12.6089 8.8262 2.5218 6.3044 1.8913 3.1522 @end example queueing/INDEX0000644000175000017500000000134513635374655013076 0ustar morenomorenoqueueing >> Queueing Networks and Markov chains Discrete-time Markov chains dtmcchkP dtmc dtmcbd dtmcexps dtmcisir dtmctaexps dtmcmtta dtmcfpt Continuous-time Markov chains ctmcchkQ ctmc ctmcbd ctmcexps ctmcisir ctmctaexps ctmcmtta ctmcfpt Single Station Queueing Systems qsmm1 qsmmm erlangb erlangc engset qsmminf qsmm1k qsmmmk qsammm qsmg1 qsmh1 Queueing Networks qnclosed qncmmvaap qncmmvabs qncmmva qncmnpop qncmpopmix qncmvisits qncscmva qncsconvld qncsconv qncsmvaap qncsmvablo qncsmvald qncsmva qncsvisits qnmarkov qnmix qnmknode qnom qnomvisits qnopen qnos qnosvisits qnsolve Bounds Analysis qnosaba qnomaba qncsaba qncmaba qnosbsb qncsbsb qncmbsb qncmcb qncspb qncsgb queueing/NEWS0000644000175000017500000001152413635375133012773 0ustar morenomorenoSummary of important user-visible changes for queueing-1.2.7 ------------------------------------------------------------------------------ ** queueing-1.2.7 contains new features ** Added computation of state occupancy probabilities to functions qsmm1(), qsmm1k(), qsmminf(), qsmmm(), qsmmmk() ** Function qncmmvaap() has been deprecated and renamed to qncmmvabs() ** The following deprecated functions have been removed: qnvisits(), qnopensingle(), qnopenmulti(), qnopenbsb(), qnopenab(), qnmvapop(), qnmvablo(), qnjackson(), qnconvolution(), qnconvolutionld(), qncmva(), qnclosedsinglemva(), qnclosedsinglemvald(), qnclosedpb(), qnclosedmultimva(), qnclosedab(), qnclosedbsb(), qnclosedgb(), qnclosedmultimvaapprox(), qnclosedsinglemvaapprox(), qnmg1(), qnmh1(), qnmm1(), qnmminf(), qnmmmk(), qnmmm(), qnmm1k(), qnammm(), ctmc_bd(), ctmc_exps(), ctmc_fpt(), ctmc_mtta(), ctmc_taexps(), dtmc_bd(), dtmc_exps(), dtmc_fpt(), dtmc_mtta(), dtmc_taexps(), population_mix(), dtmc_is_irreducible(), ctmc_check_Q(), dtmc_check_P() Summary of important user-visible changes for queueing-1.2.6 ------------------------------------------------------------------------------ ** queueing-1.2.6 contains new features ** New functions dtmcisir() and ctmcisir() ** General documentation cleanup Summary of important user-visible changes for queueing-1.2.5 ------------------------------------------------------------------------------ ** queueing-1.2.5 is a bug fix release ** Fix for bug in qncmmva ** General documentation cleanup Summary of important user-visible changes for queueing-1.2.4 ------------------------------------------------------------------------------ ** queueing-1.2.4 is a bug fix release ** Fixes for bugs #48958, #48959 and #48960 Summary of important user-visible changes for queueing-1.2.3 ------------------------------------------------------------------------------ ** queueing-1.2.3 contains new features ** Added new functions erlangb (Erlang-B formula), erlangc (Erlang-C formula) and engset Summary of important user-visible changes for queueing-1.2.2 ------------------------------------------------------------------------------ ** queueing-1.2.2 is a bug fix release ** Fixed bug in qncmmva: utilization for M/M/m centers was not correctly scaled to [0,1] when m>1 ** Added warning to qncsmva and qncmmva to detect and report numerical instability problems Summary of important user-visible changes for queueing-1.2.1 ------------------------------------------------------------------------------ ** queueing-1.2.1 contains new features ** Function qnvisits() has been deprecated, and is replaced by qncsvisits(), qnosvisits() (for single-class networks) and qncmvisits(), qnomvisits() (for multiple-class networks). The new functions allow the user to specify the reference station. Summary of important user-visible changes for queueing-1.2.0 ------------------------------------------------------------------------------ ** queueing-1.2.0 includes many bug fixes and new features ** The documentation has been restructured, hopefully improving readability and overall organization ** Added functions to compute various kinds of performance bounds on multiclass networks; balanced system, asymptotic and composite bounds are supported. ** Added function qnom for analyzing open, multiclass product-form networks. ** Adopted a new scheme for naming functions; old names are still available for compatibility with existing scripts, but will be removed in future releases. ** Fixed bugs in qncscmva, qncsconv, qncsconvld, qnopensingle, qnopenmulti and qncmmva Summary of important user-visible changes for queueing-1.1.1 ------------------------------------------------------------------------------ ** queueing-1.1.1 is a bug fix release ** Increased tolerance in tests for dtmc_fpt and dtmc_mtta to avoid spurious failures on some platforms ** Set "Autoload: no" in the DESCRIPTION file. This means that this package is no longer automatically loaded on Octave startup. To use the queueing package you need to issue the command "pkg load queueing" at the Octave prompt. Summary of important user-visible changes for queueing-1.1.0 ------------------------------------------------------------------------------ ** Function ctmc_exps() can now compute the expected sojourn time until absorption for absorbing CTMC ** Functions ctmc_exps() and ctmc_taexps() now accept a scalar as the second argument (time). ** Function ctmc_bd() now returns the infinitesimal generator matrix Q of the birth-death process with given rates, not the steady-state solution. ** The following new functions have been added: dtmc_bd(), dtmc_mtta(), ctmc_check_Q(), dtmc_exps(), dtmc_taexps() ** The following deprecated functions have been removed: ctmc_bd_solve(), ctmc_solve(), dtmc_solve()