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| date | Wed, 14 Mar 2012 09:00:49 +0000 |
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<html lang="en"> <head> <title>queueing</title> <meta http-equiv="Content-Type" content="text/html"> <meta name="description" content="User manual for the queueing toolbox, 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 toolbox implements, among others, the Mean Value Analysis (MVA) and convolution algorithms for product-form queueing networks. Transient and steady-state analysis of Markov chains is also implemented."> <meta name="generator" content="makeinfo 4.13"> <link title="Top" rel="top" href="#Top"> <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> <meta http-equiv="Content-Style-Type" content="text/css"> <style type="text/css"><!-- pre.display { font-family:inherit } pre.format { font-family:inherit } pre.smalldisplay { font-family:inherit; font-size:smaller } pre.smallformat { font-family:inherit; font-size:smaller } pre.smallexample { font-size:smaller } pre.smalllisp { font-size:smaller } span.sc { font-variant:small-caps } span.roman { font-family:serif; font-weight:normal; } span.sansserif { font-family:sans-serif; font-weight:normal; } --></style> </head> <body> Copyright © 2008, 2009, 2010, 2011, 2012 Moreno Marzolla. <p>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. <p>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. <p>Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions. <div class="contents"> <h2>Table of Contents</h2> <ul> <li><a name="toc_Top" href="#Top">queueing</a> <li><a name="toc_Summary" href="#Summary">1 Summary</a> <li><a name="toc_Installation" href="#Installation">2 Installing the queueing toolbox</a> <ul> <li><a href="#Installation-through-Octave-package-management-system">2.1 Installation through Octave package management system</a> <li><a href="#Manual-installation">2.2 Manual installation</a> <li><a href="#Content-of-the-source-distribution">2.3 Content of the source distribution</a> <li><a href="#Using-the-queueing-toolbox">2.4 Using the queueing toolbox</a> </li></ul> <li><a name="toc_Getting-Started" href="#Getting-Started">3 Introduction and Getting Started</a> <ul> <li><a href="#Analysis-of-Closed-Networks">3.1 Analysis of Closed Networks</a> <li><a href="#Analysis-of-Open-Networks">3.2 Analysis of Open Networks</a> </li></ul> <li><a name="toc_Markov-Chains" href="#Markov-Chains">4 Markov Chains</a> <ul> <li><a href="#Discrete_002dTime-Markov-Chains">4.1 Discrete-Time Markov Chains</a> <ul> <li><a href="#Discrete_002dTime-Markov-Chains">4.1.1 State occupancy probabilities</a> <li><a href="#Discrete_002dTime-Markov-Chains">4.1.2 Birth-Death process</a> <li><a href="#Discrete_002dTime-Markov-Chains">4.1.3 First passage times</a> <li><a href="#Discrete_002dTime-Markov-Chains">4.1.4 Mean Time to Absorption</a> </li></ul> <li><a href="#Continuous_002dTime-Markov-Chains">4.2 Continuous-Time Markov Chains</a> <ul> <li><a href="#State-occupancy-probabilities">4.2.1 State occupancy probabilities</a> <li><a href="#Birth_002dDeath-process">4.2.2 Birth-Death process</a> <li><a href="#Expected-Sojourn-Time">4.2.3 Expected Sojourn Time</a> <li><a href="#Time_002dAveraged-Expected-Sojourn-Time">4.2.4 Time-Averaged Expected Sojourn Time</a> <li><a href="#Mean-Time-to-Absorption">4.2.5 Mean Time to Absorption</a> <li><a href="#First-Passage-Times">4.2.6 First Passage Times</a> </li></ul> </li></ul> <li><a name="toc_Single-Station-Queueing-Systems" href="#Single-Station-Queueing-Systems">5 Single Station Queueing Systems</a> <ul> <li><a href="#The-M_002fM_002f1-System">5.1 The M/M/1 System</a> <li><a href="#The-M_002fM_002fm-System">5.2 The M/M/m System</a> <li><a href="#The-M_002fM_002finf-System">5.3 The M/M/inf System</a> <li><a href="#The-M_002fM_002f1_002fK-System">5.4 The M/M/1/K System</a> <li><a href="#The-M_002fM_002fm_002fK-System">5.5 The M/M/m/K System</a> <li><a href="#The-Asymmetric-M_002fM_002fm-System">5.6 The Asymmetric M/M/m System</a> <li><a href="#The-M_002fG_002f1-System">5.7 The M/G/1 System</a> <li><a href="#The-M_002fHm_002f1-System">5.8 The M/H_m/1 System</a> </li></ul> <li><a name="toc_Queueing-Networks" href="#Queueing-Networks">6 Queueing Networks</a> <ul> <li><a href="#Introduction-to-QNs">6.1 Introduction to QNs</a> <ul> <li><a href="#Introduction-to-QNs">6.1.1 Single class models</a> <li><a href="#Introduction-to-QNs">6.1.2 Multiple class models</a> </li></ul> <li><a href="#Generic-Algorithms">6.2 Generic Algorithms</a> <li><a href="#Algorithms-for-Product_002dForm-QNs">6.3 Algorithms for Product-Form QNs</a> <ul> <li><a href="#Algorithms-for-Product_002dForm-QNs">6.3.1 Jackson Networks</a> <li><a href="#Algorithms-for-Product_002dForm-QNs">6.3.2 The Convolution Algorithm</a> <li><a href="#Algorithms-for-Product_002dForm-QNs">6.3.3 Open networks</a> <li><a href="#Algorithms-for-Product_002dForm-QNs">6.3.4 Closed Networks</a> <li><a href="#Algorithms-for-Product_002dForm-QNs">6.3.5 Mixed Networks</a> </li></ul> <li><a href="#Algorithms-for-non-Product_002dform-QNs">6.4 Algorithms for non Product-Form QNs</a> <li><a href="#Bounds-on-performance">6.5 Bounds on performance</a> <li><a href="#Utility-functions">6.6 Utility functions</a> <ul> <li><a href="#Utility-functions">6.6.1 Open or closed networks</a> <li><a href="#Utility-functions">6.6.2 Computation of the visit counts</a> <li><a href="#Utility-functions">6.6.3 Other utility functions</a> </li></ul> </li></ul> <li><a name="toc_Contributing-Guidelines" href="#Contributing-Guidelines">Appendix A Contributing Guidelines</a> <li><a name="toc_Acknowledgements" href="#Acknowledgements">Appendix B Acknowledgements</a> <li><a name="toc_Copying" href="#Copying">Appendix C GNU GENERAL PUBLIC LICENSE</a> <li><a name="toc_Concept-Index" href="#Concept-Index">Concept Index</a> <li><a name="toc_Function-Index" href="#Function-Index">Function Index</a> <li><a name="toc_Author-Index" href="#Author-Index">Author Index</a> </li></ul> </div> <div class="node"> <a name="Top"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Summary">Summary</a>, Up: <a rel="up" accesskey="u" href="#dir">(dir)</a> </div> <h2 class="unnumbered">queueing</h2> <p>This manual documents how to install and run the Queueing Toolbox. It corresponds to version 1.X.0 of the package. <!-- --> <ul class="menu"> <li><a accesskey="1" href="#Summary">Summary</a> <li><a accesskey="2" href="#Installation">Installation</a>: Installation of the queueing toolbox. <li><a accesskey="3" href="#Getting-Started">Getting Started</a>: Getting started with the queueing toolbox. <li><a accesskey="4" href="#Markov-Chains">Markov Chains</a>: Functions for Markov Chains. <li><a accesskey="5" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a>: Functions for single-station queueing systems. <li><a accesskey="6" href="#Queueing-Networks">Queueing Networks</a>: Functions for queueing networks. <li><a accesskey="7" href="#Contributing-Guidelines">Contributing Guidelines</a>: How to contribute. <li><a accesskey="8" href="#Acknowledgements">Acknowledgements</a>: People who contributed to the queueing toolbox. <li><a accesskey="9" href="#Copying">Copying</a>: The GNU General Public License. <li><a href="#Concept-Index">Concept Index</a>: An item for each concept. <li><a href="#Function-Index">Function Index</a>: An item for each function. <li><a href="#Author-Index">Author Index</a>: An item for each author. </ul> <!-- --> <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Summary"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Installation">Installation</a>, Previous: <a rel="previous" accesskey="p" href="#Top">Top</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="chapter">1 Summary</h2> <p>This document describes the <code>queueing</code> toolbox for GNU Octave (<code>queueing</code> in short). The <code>queueing</code> toolbox, previously known as <code>qnetworks</code>, is a collection of functions written in GNU Octave for analyzing queueing networks and Markov chains. Specifically, <code>queueing</code> contains functions for analyzing Jackson networks, open, closed or mixed product-form BCMP networks, and computation of performance bounds. The following algorithms have been implemented <ul> <li>Convolution for closed, single-class product-form networks with load-dependent service centers; <li>Exact and approximate Mean Value Analysis (MVA) for single and multiple class product-form closed networks; <li>MVA for mixed, multiple class product-form networks with load-independent service centers; <li>Approximate MVA for closed, single-class networks with blocking (MVABLO algorithm by F. Akyildiz); <li>Asymptotic Bounds, Balanced System Bounds and Geometric Bounds; </ul> <p class="noindent"><code>queueing</code> provides functions for analyzing the following kind of single-station queueing systems: <ul> <li>M/M/1 <li>M/M/m <li>M/M/\infty <li>M/M/1/k single-server, finite capacity system <li>M/M/m/k multiple-server, finite capacity system <li>Asymmetric M/M/m <li>M/G/1 (general service time distribution) <li>M/H_m/1 (Hyperexponential service time distribution) </ul> <p>Functions for Markov chain analysis are also provided, for discrete-time chains (DTMC) or continuous-time chains (CTMC): <ul> <li>Birth-death process; <li>Transient and steady-state occupancy probabilities; <li>Mean times to absorption; <li>Expected sojourn times and time-averaged sojourn times (CTMC only); <li>Mean first passage times; </ul> <p>The <code>queueing</code> toolbox is distributed under the terms of the GNU General Public License (GPL), version 3 or later (see <a href="#Copying">Copying</a>). You are encouraged to share this software with others, and make this package more useful by contributing additional functions and reporting problems. See <a href="#Contributing-Guidelines">Contributing Guidelines</a>. <p>If you use the <code>queueing</code> toolbox in a technical paper, please cite it as: <blockquote> Moreno Marzolla, <em>The qnetworks Toolbox: A Software Package for Queueing Networks Analysis</em>. 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 </blockquote> <p>If you use BibTeX, this is the citation block: <pre class="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} } </pre> <p>An early draft of the paper above is available as Technical Report <a href="http://www.informatica.unibo.it/ricerca/ublcs/2010/UBLCS-2010-04">UBLCS-2010-04</a>, February 2010, Department of Computer Science, University of Bologna, Italy. <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Installation"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Getting-Started">Getting Started</a>, Previous: <a rel="previous" accesskey="p" href="#Summary">Summary</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="chapter">2 Installing the queueing toolbox</h2> <ul class="menu"> <li><a accesskey="1" href="#Installation-through-Octave-package-management-system">Installation through Octave package management system</a> <li><a accesskey="2" href="#Manual-installation">Manual installation</a> <li><a accesskey="3" href="#Content-of-the-source-distribution">Content of the source distribution</a> <li><a accesskey="4" href="#Using-the-queueing-toolbox">Using the queueing toolbox</a> </ul> <div class="node"> <a name="Installation-through-Octave-package-management-system"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Manual-installation">Manual installation</a>, Up: <a rel="up" accesskey="u" href="#Installation">Installation</a> </div> <h3 class="section">2.1 Installation through Octave package management system</h3> <p>The most recent version of <code>queueing</code> is 1.X.0 and can be downloaded from Octave-Forge <p><a href="http://octave.sourceforge.net/queueing/">http://octave.sourceforge.net/queueing/</a> <p>The package Web page is <p><a href="http://www.moreno.marzolla.name/software/queueing/">http://www.moreno.marzolla.name/software/queueing/</a> <p>If you have a recent version of GNU Octave and a network connection, you can install <code>queueing</code> directly from the prompt using this command: <pre class="example"> octave:1> <kbd>pkg install -forge queueing</kbd> </pre> <p>The command above will automaticall download and install the latest version of the queueing toolbox from Octave Forge, and install it on your machine. You can verify that the package is indeed installed: <pre class="example"> octave:1><kbd>pkg list queueing</kbd> Package Name | Version | Installation directory --------------+---------+----------------------- queueing *| 1.X.0 | /home/moreno/octave/queueing-1.X.0 </pre> <p>Alternatively, you can first download <code>queueing</code> from Octave-Forge; then, to install the package in the system-wide location issue this command at the Octave prompt: <pre class="example"> octave:1> <kbd>pkg install </kbd><em>queueing-1.X.0.tar.gz</em> </pre> <p class="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 readily available each time Octave starts, without the need to tweak the search path. <p>If you do not have root access, you can do a local install using: <pre class="example"> octave:1> <kbd>pkg install -local queueing-1.X.0.tar.gz</kbd> </pre> <p>This will install <code>queueing</code> within your home directory, and the package will be available to your user only. <strong>Note:</strong> Octave version 3.2.3 as shipped with Ubuntu 10.04 seems to ignore <code>-local</code> and always tries to install the package on the system directory. <p>To remove <code>queueing</code> you can use <pre class="example"> octave:1> <kbd>pkg uninstall queueing</kbd> </pre> <div class="node"> <a name="Manual-installation"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Content-of-the-source-distribution">Content of the source distribution</a>, Previous: <a rel="previous" accesskey="p" href="#Installation-through-Octave-package-management-system">Installation through Octave package management system</a>, Up: <a rel="up" accesskey="u" href="#Installation">Installation</a> </div> <h3 class="section">2.2 Manual installation</h3> <p>If you want to manually install <code>queueing</code> in a custom location, you can download the tarball and unpack it somewhere: <pre class="example"> <kbd>tar xvfz queueing-1.X.0.tar.gz</kbd> <kbd>cd queueing-1.X.0/queueing/</kbd> </pre> <p>Copy all <code>.m</code> files from the <samp><span class="file">inst/</span></samp> directory to some target location. Then, start Octave with the <samp><span class="option">-p</span></samp> option to add the target location to the search path, so that Octave will find all <code>queueing</code> functions automatically: <pre class="example"> <kbd>octave -p </kbd><em>/path/to/queueing</em> </pre> <p>For example, if all <code>queueing</code> m-files are in <samp><span class="file">/usr/local/queueing</span></samp>, you can start Octave as follows: <pre class="example"> <kbd>octave -p </kbd><em>/usr/local/queueing</em> </pre> <p>If you want, you can add the following line to <samp><span class="file">~/.octaverc</span></samp>: <pre class="example"> <kbd>addpath("</kbd><em>/path/to/queueing</em><kbd>");</kbd> </pre> <p class="noindent">so that the path <samp><span class="file">/usr/local/queueing</span></samp> is automatically added to the search path each time Octave is started, and you no longer need to specify the <samp><span class="option">-p</span></samp> option on the command line. <div class="node"> <a name="Content-of-the-source-distribution"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Using-the-queueing-toolbox">Using the queueing toolbox</a>, Previous: <a rel="previous" accesskey="p" href="#Manual-installation">Manual installation</a>, Up: <a rel="up" accesskey="u" href="#Installation">Installation</a> </div> <h3 class="section">2.3 Content of the source distribution</h3> <p>The source code of the latest version of the <code>queueing</code> package can be found in the Subversion repository at the URL: <p><a href="http://octave.svn.sourceforge.net/viewvc/octave/trunk/octave-forge/main/queueing/">http://octave.svn.sourceforge.net/viewvc/octave/trunk/octave-forge/main/queueing/</a> <p>The source distribution contains the following directories (some of which are not included in the installation tarball): <dl> <dt><samp><span class="file">doc/</span></samp><dd>Documentation source. Most of the documentation is extracted from the comment blocks of individual function files from the <samp><span class="file">inst/</span></samp> directory. <br><dt><samp><span class="file">inst/</span></samp><dd>This directory contains the <tt>m</tt>-files which implement the various Queueing Network algorithms provided by <code>queueing</code>. As a notational convention, the names of source files containing functions for Queueing Networks start with the ‘<samp><span class="samp">qn</span></samp>’ prefix; the name of source files containing functions for Continuous-Time Markov Chains (CTMSs) start with the ‘<samp><span class="samp">ctmc</span></samp>’ prefix, and the names of files containing functions for Discrete-Time Markov Chains (DTMCs) start with the ‘<samp><span class="samp">dtmc</span></samp>’ prefix. <br><dt><samp><span class="file">test/</span></samp><dd>This directory contains the test functions used to invoke all tests on all function files. <br><dt><samp><span class="file">scripts/</span></samp><dd>This directory contains some utility scripts mostly from GNU Octave, which extract the documentation from the specially-formatted comments in the <tt>m</tt>-files. <br><dt><samp><span class="file">examples/</span></samp><dd>This directory contains examples which are automatically extracted from the ‘<samp><span class="samp">demo</span></samp>’ blocks of the function files. <br><dt><samp><span class="file">devel/</span></samp><dd>This directory contains function files which are either not working properly, or need additional testing before they are moved to the <samp><span class="file">inst/</span></samp> directory. </dl> <p>The <code>queueing</code> package ships with a Makefile which can be used to produce the documentation (in PDF and HTML format), and automatically execute all function tests. Specifically, the following targets are defined: <dl> <dt><code>all</code><dd>Running ‘<samp><span class="samp">make</span></samp>’ (or ‘<samp><span class="samp">make all</span></samp>’) on the top-level directory builds the programs used to extract the documentation from the comments embedded in the <tt>m</tt>-files, and then produce the documentation in PDF and HTML format (<samp><span class="file">doc/queueing.pdf</span></samp> and <samp><span class="file">doc/queueing.html</span></samp>, respectively). <br><dt><code>check</code><dd>Running ‘<samp><span class="samp">make check</span></samp>’ will execute all tests contained in the <tt>m</tt>-files. If you modify the code of any function in the <samp><span class="file">inst/</span></samp> 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 which are not adequately validated. <br><dt><code>clean</code><dt><code>distclean</code><dt><code>dist</code><dd>The ‘<samp><span class="samp">make clean</span></samp>’, ‘<samp><span class="samp">make distclean</span></samp>’ and ‘<samp><span class="samp">make dist</span></samp>’ commands are used to clean up the source directory and prepare the distribution archive in compressed tar format. </dl> <div class="node"> <a name="Using-the-queueing-toolbox"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#Content-of-the-source-distribution">Content of the source distribution</a>, Up: <a rel="up" accesskey="u" href="#Installation">Installation</a> </div> <h3 class="section">2.4 Using the queueing toolbox</h3> <p>You can use all functions by simply invoking their name with the appropriate parameters; the <code>queueing</code> package should display an error message in case of missing/wrong parameters. You can display the help text for any function using the <samp><span class="command">help</span></samp> command. For example: <pre class="example"> octave:2> <kbd>help qnmvablo</kbd> </pre> <p>prints the documentation for the <samp><span class="command">qnmvablo</span></samp> function. Additional information can be found in the <code>queueing</code> manual, which is available in PDF format in <samp><span class="file">doc/queueing.pdf</span></samp> and in HTML format in <samp><span class="file">doc/queueing.html</span></samp>. <p>Within GNU Octave, you can also run the test and demo blocks associated to the functions, using the <samp><span class="command">test</span></samp> and <samp><span class="command">demo</span></samp> commands respectively. To run all the tests of, say, the <samp><span class="command">qnmvablo</span></samp> function: <pre class="example"> octave:3> <kbd>test qnmvablo</kbd> -| PASSES 4 out of 4 tests </pre> <p>To execute the demos of the <samp><span class="command">qnclosed</span></samp> function, use the following: <pre class="example"> octave:4> <kbd>demo qnclosed</kbd> </pre> <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Getting-Started"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Markov-Chains">Markov Chains</a>, Previous: <a rel="previous" accesskey="p" href="#Installation">Installation</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="chapter">3 Introduction and Getting Started</h2> <ul class="menu"> <li><a accesskey="1" href="#Analysis-of-Closed-Networks">Analysis of Closed Networks</a> <li><a accesskey="2" href="#Analysis-of-Open-Networks">Analysis of Open Networks</a> </ul> <p>In this chapter we give some usage examples of the <code>queueing</code> package. The reader is assumed to be familiar with Queueing Networks (although some basic terminology and notation will be given here). Additional usage examples are embedded in most of the function files; to display and execute the demos associated with function <em>fname</em> you can type <samp><span class="command">demo </span><em>fname</em></samp> at the Octave prompt. For example <pre class="example"> <kbd>demo qnclosed</kbd> </pre> <p class="noindent">executes all demos (if any) for the <samp><span class="command">qnclosed</span></samp> function. <div class="node"> <a name="Analysis-of-Closed-Networks"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Analysis-of-Open-Networks">Analysis of Open Networks</a>, Up: <a rel="up" accesskey="u" href="#Getting-Started">Getting Started</a> </div> <h3 class="section">3.1 Analysis of Closed Networks</h3> <p>Let us consider a simple closed network with K=3 service centers. Each center is of type M/M/1–FCFS. We denote with S_i the average service time at center i, i=1, 2, 3. Let S_1 = 1.0, S_2 = 2.0 and S_3 = 0.8. The routing of jobs within the network is described with a <em>routing probability matrix</em> P. Specifically, a request completing service at center i is enqueued at center j with probability P_ij. Let us assume the following routing probability matrix: <pre class="example"> [ 0 0.3 0.7 ] P = [ 1 0 0 ] [ 1 0 0 ] </pre> <p>For example, according to matric P a job completing service at center 1 is routed to center 2 with probability 0.3, and is routed to center 3 with probability 0.7. <p>The network above can be analyzed with the <samp><span class="command">qnclosed</span></samp> function; if there is just a single class of requests, as in the example above, <samp><span class="command">qnclosed</span></samp> calls <samp><span class="command">qnclosedsinglemva</span></samp> which implements the Mean Value Analysys (MVA) algorithm for single-class, product-form network. <p><samp><span class="command">qnclosed</span></samp> requires the following parameters: <dl> <dt><var>N</var><dd>Number of requests in the network (since we are considering a closed network, the number of requests is fixed) <br><dt><var>S</var><dd>Array of average service times at the centers: <var>S</var><code>(k)</code> is the average service time at center k. <br><dt><var>V</var><dd>Array of visit ratios: <var>V</var><code>(k)</code> is the average number of visits to center k. </dl> <p>As can be seen, we must compute the <em>visit ratios</em> (or visit counts) V_k for each center k. The visit counts satisfy the following equations: <pre class="example"> V_j = sum_i V_i P_ij </pre> <p>We can compute V_k from the routing probability matrix P_ij using the <samp><span class="command">qnvisits</span></samp> function: <pre class="example"> <kbd>P = [0 0.3 0.7; 1 0 0; 1 0 0];</kbd> <kbd>V = qnvisits(P)</kbd> ⇒ V = 1.00000 0.30000 0.70000 </pre> <p>We can check that the computed values satisfy the above equation by evaluating the following expression: <pre class="example"> <kbd>V*P</kbd> ⇒ ans = 1.00000 0.30000 0.70000 </pre> <p class="noindent">which is equal to V. Hence, we can analyze the network for a given population size N (for example, N=10) as follows: <pre class="example"> <kbd>N = 10;</kbd> <kbd>S = [1 2 0.8];</kbd> <kbd>P = [0 0.3 0.7; 1 0 0; 1 0 0];</kbd> <kbd>V = qnvisits(P);</kbd> <kbd>[U R Q X] = qnclosed( N, S, V )</kbd> ⇒ 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 </pre> <p>The output of <samp><span class="command">qnclosed</span></samp> includes the vector 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 higher value among the service centers. Tus, center 1 is the <em>bottleneck device</em>. <p>This network can also be analyzed with the <samp><span class="command">qnsolve</span></samp> function. <samp><span class="command">qnsolve</span></samp> can handle open, closed or mixed networks, and allows the network to be described in a very flexible way. First, let <var>Q1</var>, <var>Q2</var> and <var>Q3</var> be the variables describing the service centers. Each variable is instantiated with the <samp><span class="command">qnmknode</span></samp> function. <pre class="example"> <kbd>Q1 = qnmknode( "m/m/m-fcfs", 1 );</kbd> <kbd>Q2 = qnmknode( "m/m/m-fcfs", 2 );</kbd> <kbd>Q3 = qnmknode( "m/m/m-fcfs", 0.8 );</kbd> </pre> <p>The first parameter of <samp><span class="command">qnmknode</span></samp> is a string describing the type of the node. Here we use <code>"m/m/m-fcfs"</code> to denote a M/M/m–FCFS center. 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). <p>Now, the network can be analyzed as follows: <pre class="example"> <kbd>N = 10;</kbd> <kbd>V = [1 0.3 0.7];</kbd> <kbd>[U R Q X] = qnsolve( "closed", N, { Q1, Q2, Q3 }, V )</kbd> ⇒ 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 </pre> <p>Of course, we get exactly the same results. Other functions can be used for closed networks, see <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a>. <div class="node"> <a name="Analysis-of-Open-Networks"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#Analysis-of-Closed-Networks">Analysis of Closed Networks</a>, Up: <a rel="up" accesskey="u" href="#Getting-Started">Getting Started</a> </div> <h3 class="section">3.2 Analysis of Open Networks</h3> <p>Open networks can be analyzed in a similar way. Let us consider an open network with K=3 service centers, and routing probability matrix as follows: <pre class="example"> [ 0 0.3 0.5 ] P = [ 1 0 0 ] [ 1 0 0 ] </pre> <p>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. <p>Similarly to closed networks, we first need to compute the visit counts V_k to center k. Again, we use the <samp><span class="command">qnvisits</span></samp> function as follows: <pre class="example"> <kbd>P = [0 0.3 0.5; 1 0 0; 1 0 0];</kbd> <kbd>lambda = [0.15 0 0];</kbd> <kbd>V = qnvisits(P, lambda)</kbd> ⇒ V = 5.00000 1.50000 2.50000 </pre> <p class="noindent">where <var>lambda</var><code>(k)</code> is the arrival rate at center k, and <var>P</var> is the routing matrix. The visit counts V_k for open networks satisfy the following equation: <pre class="example"> V_j = sum_i V_i P_ij </pre> <p>where P_0j is the probability of an external arrival to center j. This can be computed as: <p>Assuming the same service times as in the previous example, the network can be analyzed with the <samp><span class="command">qnopen</span></samp> function, as follows: <pre class="example"> <kbd>S = [1 2 0.8];</kbd> <kbd>[U R Q X] = qnopen( sum(lambda), S, V )</kbd> ⇒ 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 </pre> <p>The first parameter of the <samp><span class="command">qnopen</span></samp> function is the (scalar) aggregate arrival rate. <p>Again, it is possible to use the <samp><span class="command">qnsolve</span></samp> high-level function: <pre class="example"> <kbd>Q1 = qnmknode( "m/m/m-fcfs", 1 );</kbd> <kbd>Q2 = qnmknode( "m/m/m-fcfs", 2 );</kbd> <kbd>Q3 = qnmknode( "m/m/m-fcfs", 0.8 );</kbd> <kbd>lambda = [0.15 0 0];</kbd> <kbd>[U R Q X] = qnsolve( "open", sum(lambda), { Q1, Q2, Q3 }, V )</kbd> ⇒ 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 </pre> <!-- @node Markov Chains Analysis --> <!-- @section Markov Chains Analysis --> <!-- @subsection Discrete-Time Markov Chains --> <!-- (TODO) --> <!-- @subsection Continuous-Time Markov Chains --> <!-- (TODO) --> <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Markov-Chains"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a>, Previous: <a rel="previous" accesskey="p" href="#Getting-Started">Getting Started</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="chapter">4 Markov Chains</h2> <ul class="menu"> <li><a accesskey="1" href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a> <li><a accesskey="2" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </ul> <div class="node"> <a name="Discrete-Time-Markov-Chains"></a> <a name="Discrete_002dTime-Markov-Chains"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a>, Up: <a rel="up" accesskey="u" href="#Markov-Chains">Markov Chains</a> </div> <h3 class="section">4.1 Discrete-Time Markov Chains</h3> <p>Let X_0, X_1, <small class="dots">...</small>, X_n, <small class="dots">...</small> be a sequence of random variables, each one defined over a discete state space 0, 1, 2, <small class="dots">...</small>. The sequence X_0, X_1, <small class="dots">...</small>, X_n, <small class="dots">...</small> is a <em>stochastic process</em> with discrete time 0, 1, 2, <small class="dots">...</small>. A <em>Markov chain</em> is a stochastic process {X_n, n=0, 1, 2, <small class="dots">...</small>} which satisfies the following Marrkov property: <p>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) <p class="noindent">which 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. <p>The evolution of a Markov chain with finite state space {1, 2, <small class="dots">...</small>, N} can be fully described by a stochastic matrix \bf P(n) = P_i,j(n) such that P_i, j(n) = P( X_n+1 = j\ |\ X_n = j ). If the Markov chain is homogeneous (that is, the transition probability matrix \bf P(n) is time-independent), we can simply write \bf P = P_i, j, where P_i, j = P( X_n+1 = j\ |\ X_n = j ) for all n=0, 1, 2, <small class="dots">...</small>. <p>The transition probability matrix \bf P must satisfy the following two properties: (1) P_i, j ≥ 0 for all i, j, and (2) \sum_j=1^N P_i,j = 1. <p><a name="doc_002ddtmc_005fcheck_005fP"></a> <div class="defun"> — Function File: [<var>result</var> <var>err</var>] = <b>dtmc_check_P</b> (<var>P</var>)<var><a name="index-dtmc_005fcheck_005fP-1"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-discrete-time-2"></a> If <var>P</var> is a valid transition probability matrix, return the size (number of rows or columns) of <var>P</var>. If <var>P</var> is not a transition probability matrix, set <var>result</var> to zero, and <var>err</var> to an appropriate error string. </blockquote></div> <h4 class="subsection">4.1.1 State occupancy probabilities</h4> <p>We denote with \bf \pi(n) = (\pi_1(n), \pi_2(n), <small class="dots">...</small>, \pi_N(n) ) the <em>state occupancy probability vector</em> at step n. \pi_i(n) denotes the probability that the system is in state i at step n. <p>Given the transition probability matrix \bf P and the initial state occupancy probability vector \bf \pi(0) = (\pi_1(0), \pi_2(0), <small class="dots">...</small>, \pi_N(0)) at step 0, the state occupancy probability vector \bf \pi(n) at step n can be computed as: <pre class="example"> \pi(n) = \pi(0) P^n </pre> <p>Under certain conditions, there exists a <em>stationary state occupancy probability</em> \bf \pi = \lim_n \rightarrow +\infty \bf \pi(n), which is independent from the initial state occupancy \bf \pi(0). The stationary state occupancy probability vector \bf \pi satisfies \bf \pi = \bf \pi \bf P and \sum_i=1^N \pi_i = 1 <p><a name="doc_002ddtmc"></a> <div class="defun"> — Function File: <var>p</var> = <b>dtmc</b> (<var>P</var>)<var><a name="index-dtmc-3"></a></var><br> — Function File: <var>p</var> = <b>dtmc</b> (<var>P, n, p0</var>)<var><a name="index-dtmc-4"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-discrete-time-5"></a><a name="index-Discrete-time-Markov-chain-6"></a><a name="index-Markov-chain_002c-stationary-probabilities-7"></a><a name="index-Stationary-probabilities-8"></a> With a single argument, compute the steady-state probability vector <var>p</var><code>(1), ..., </code><var>p</var><code>(N)</code> for a Discrete-Time Markov Chain given the N \times N transition probability matrix <var>P</var>. With three arguments, compute the probability vector <var>p</var><code>(1), ..., </code><var>p</var><code>(N)</code> after <var>n</var> steps, given initial probability vector <var>p0</var> at time 0. <p><strong>INPUTS</strong> <dl> <dt><var>P</var><dd><var>P</var><code>(i,j)</code> is the transition probability from state i to state j. <var>P</var> 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 <var>P</var> must be equal to its dimension. <br><dt><var>n</var><dd>Step at which to compute the transient probability <br><dt><var>p0</var><dd><var>p0</var><code>(i)</code> is the probability that at step 0 the system is in state i. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>p</var><dd>If this function is invoked with a single argument, <var>p</var><code>(i)</code> is the steady-state probability that the system is in state i. <var>p</var> satisfies the equations p = p\bf P and \sum_i=1^N p_i = 1. If this function is invoked with three arguments, <var>p</var><code>(i)</code> is the marginal probability that the system is in state i at step <var>n</var>, given the initial probabilities <var>p0</var><code>(i)</code> that the initial state is i. </dl> </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <pre class="example"><pre class="verbatim"> a = 0.2; b = 0.15; P = [ 1-a a; b 1-b]; T = 0:14; pp = zeros(2,length(T)); for i=1:length(T) pp(:,i) = dtmc(P,T(i),[1 0]); endfor ss = dtmc(P); # compute steady state probabilities plot( T, pp(1,:), "b+;p_0(t);", "linewidth", 2, \ T, ss(1)*ones(size(T)), "b;Steady State;", \ T, pp(2,:), "r+;p_1(t);", "linewidth", 2, \ T, ss(2)*ones(size(T)), "r;Steady State;" ); xlabel("Time Step");</pre> </pre> <h4 class="subsection">4.1.2 Birth-Death process</h4> <p><a name="doc_002ddtmc_005fbd"></a> <div class="defun"> — Function File: <var>P</var> = <b>dtmc_bd</b> (<var>birth, death</var>)<var><a name="index-dtmc_005fbd-9"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-discrete-time-10"></a><a name="index-Birth_002ddeath-process-11"></a> Returns the N \times N transition probability matrix P for a birth-death process with given rates. <p><strong>INPUTS</strong> <dl> <dt><var>birth</var><dd>Vector with N-1 elements, where <var>birth</var><code>(i)</code> is the transition probability from state i to state i+1. <br><dt><var>death</var><dd>Vector with N-1 elements, where <var>death</var><code>(i)</code> is the transition probability from state i+1 to state i. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>P</var><dd>Transition probability matrix for the birth-death process. </dl> </blockquote></div> <h4 class="subsection">4.1.3 First passage times</h4> <p>The First Passage Time M_i j is defined as the average number of transitions needed to visit state j for the first time, starting from state i. Matrix \bf M satisfies the property that <pre class="example"> ___ \ M_ij = 1 + > P_ij * M_kj /___ k!=j </pre> <p><a name="doc_002ddtmc_005ffpt"></a> <div class="defun"> — Function File: <var>M</var> = <b>dtmc_fpt</b> (<var>P</var>)<var><a name="index-dtmc_005ffpt-12"></a></var><br> — Function File: <var>m</var> = <b>dtmc_fpt</b> (<var>P, i, j</var>)<var><a name="index-dtmc_005ffpt-13"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-discrete-time-14"></a><a name="index-First-passage-times-15"></a> If called with a single argument, computes the mean first passage times <var>M</var><code>(i,j)</code>, that are the average number of transitions before state <var>j</var> is reached, starting from state <var>i</var>, for all 1 \leq i, j \leq N. If called with three arguments, returns the single value <var>m</var><code> = </code><var>M</var><code>(i,j)</code>. <p><strong>INPUTS</strong> <dl> <dt><var>P</var><dd><var>P</var><code>(i,j)</code> is the transition probability from state i to state j. <var>P</var> 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 <var>P</var> must be equal to its dimension. <br><dt><var>i</var><dd>Initial state. <br><dt><var>j</var><dd>Destination state. If <var>j</var> is a vector, returns the mean first passage time to any state in <var>j</var>. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>M</var><dd>If this function is called with a single argument, <var>M</var><code>(i,j)</code> is the average number of transitions before state <var>j</var> is reached for the first time, starting from state <var>i</var>. <var>M</var><code>(i,i)</code> is the <em>mean recurrence time</em>, and represents the average time needed to return to state <var>i</var>. <br><dt><var>m</var><dd>If this function is called with three arguments, the result <var>m</var> is the average number of transitions before state <var>j</var> is visited for the first time, starting from state <var>i</var>. </dl> </blockquote></div> <h4 class="subsection">4.1.4 Mean Time to Absorption</h4> <p><a name="doc_002ddtmc_005fmtta"></a> <div class="defun"> — Function File: [<var>t</var> <var>B</var>] = <b>dtmc_mtta</b> (<var>P</var>)<var><a name="index-dtmc_005fmtta-16"></a></var><br> — Function File: [<var>t</var> <var>B</var>] = <b>dtmc_mtta</b> (<var>P, p0</var>)<var><a name="index-dtmc_005fmtta-17"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-disctete-time-18"></a><a name="index-Mean-time-to-absorption-19"></a> Compute the expected number of steps before absorption for the DTMC with N \times N transition probability matrix <var>P</var>. <p><strong>INPUTS</strong> <dl> <dt><var>P</var><dd>Transition probability matrix. <br><dt><var>p0</var><dd>Initial state occupancy probabilities. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>t</var><dd>When called with a single argument, <var>t</var> is a vector such that <var>t</var><code>(i)</code> is the expected number of steps before being absorbed, starting from state i. When called with two arguments, <var>t</var> is a scalar and represents the average number of steps before absorption, given initial state occupancy probabilities <var>p0</var>. <br><dt><var>B</var><dd>When called with a single argument, <var>B</var> is a N \times N matrix where <var>B</var><code>(i,j)</code> is the probability of being absorbed in state j, starting from state i; if j is not absorbing, <var>B</var><code>(i,j) = 0</code>; if i is absorbing, then <var>B</var><code>(i,i) = 1</code>. When called with two arguments, <var>B</var> is a vector with N elements where <var>B</var><code>(j)</code> is the probability of being absorbed in state <var>j</var>, given initial state occupancy probabilities <var>p0</var>. </dl> </blockquote></div> <div class="node"> <a name="Continuous-Time-Markov-Chains"></a> <a name="Continuous_002dTime-Markov-Chains"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a>, Up: <a rel="up" accesskey="u" href="#Markov-Chains">Markov Chains</a> </div> <h3 class="section">4.2 Continuous-Time Markov Chains</h3> <p>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 , \ldots , t_n, t_n+1 such that t_0 < t_1 < \ldots < t_n < t_n+1, we have <p>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) <p>A continuous-time Markov chain is defined according to an <em>infinitesimal generator matrix</em> \bf Q = [Q_i,j] such that for each i \neq j, Q_i, j is the transition rate from state i to state j. The elements Q_i, i must be defined in such a way that the infinitesimal generator matrix \bf Q satisfies the property \sum_j=1^N Q_i,j = 0. <p><a name="doc_002dctmc_005fcheck_005fQ"></a> <div class="defun"> — Function File: [<var>result</var> <var>err</var>] = <b>ctmc_check_Q</b> (<var>Q</var>)<var><a name="index-ctmc_005fcheck_005fQ-20"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-21"></a> If <var>Q</var> is a valid infinitesimal generator matrix, return the size (number of rows or columns) of <var>Q</var>. If <var>Q</var> is not an infinitesimal generator matrix, set <var>result</var> to zero, and <var>err</var> to an appropriate error string. </blockquote></div> <ul class="menu"> <li><a accesskey="1" href="#State-occupancy-probabilities">State occupancy probabilities</a> <li><a accesskey="2" href="#Birth_002dDeath-process">Birth-Death process</a> <li><a accesskey="3" href="#Expected-Sojourn-Time">Expected Sojourn Time</a> <li><a accesskey="4" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a> <li><a accesskey="5" href="#Mean-Time-to-Absorption">Mean Time to Absorption</a> <li><a accesskey="6" href="#First-Passage-Times">First Passage Times</a> </ul> <div class="node"> <a name="State-occupancy-probabilities"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Birth_002dDeath-process">Birth-Death process</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> <h4 class="subsection">4.2.1 State occupancy probabilities</h4> <p>Similarly to the discrete case, we denote with \bf \pi(t) = (\pi_1(t), \pi_2(t), <small class="dots">...</small>, \pi_N(t) ) the <em>state occupancy probability vector</em> at time t. \pi_i(t) denotes the probability that the system is in state i at time t ≥ 0. <p>Given the infinitesimal generator matrix \bf Q and the initial state occupancy probability vector \bf \pi(0) = (\pi_1(0), \pi_2(0), <small class="dots">...</small>, \pi_N(0)), the state occupancy probability vector \bf \pi(t) at time t can be computed as: <pre class="example"> \pi(t) = \pi(0) exp(Qt) </pre> <p class="noindent">where \exp( \bf Q t ) is the matrix exponential of \bf Q t. Under certain conditions, there exists a <em>stationary state occupancy probability</em> \bf \pi = \lim_t \rightarrow +\infty \bf \pi(t), which is independent from the initial state occupancy \bf \pi(0). The stationary state occupancy probability vector \bf \pi satisfies \bf \pi \bf Q = \bf 0 and \sum_i=1^N \pi_i = 1. <p><a name="doc_002dctmc"></a> <div class="defun"> — Function File: <var>p</var> = <b>ctmc</b> (<var>Q</var>)<var><a name="index-ctmc-22"></a></var><br> — Function File: <var>p</var> = <b>ctmc</b> (<var>Q, t. p0</var>)<var><a name="index-ctmc-23"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-24"></a><a name="index-Continuous-time-Markov-chain-25"></a><a name="index-Markov-chain_002c-state-occupancy-probabilities-26"></a><a name="index-Stationary-probabilities-27"></a> With a single argument, compute the stationary state occupancy probability vector <var>p</var>(1), <small class="dots">...</small>, <var>p</var>(N) for a Continuous-Time Markov Chain with infinitesimal generator matrix <var>Q</var> of size N \times N. With three arguments, compute the state occupancy probabilities <var>p</var>(1), <small class="dots">...</small>, <var>p</var>(N) at time <var>t</var>, given initial state occupancy probabilities <var>p0</var> at time 0. <p><strong>INPUTS</strong> <dl> <dt><var>Q</var><dd>Infinitesimal generator matrix. <var>Q</var> is a N \times N square matrix where <var>Q</var><code>(i,j)</code> is the transition rate from state i to state j, for 1 ≤ i \neq j ≤ N. Transition rates must be nonnegative, and \sum_j=1^N Q_i j = 0 <br><dt><var>t</var><dd>Time at which to compute the transient probability <br><dt><var>p0</var><dd><var>p0</var><code>(i)</code> is the probability that the system is in state i at time 0 . </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>p</var><dd>If this function is invoked with a single argument, <var>p</var><code>(i)</code> is the steady-state probability that the system is in state i, i = 1, <small class="dots">...</small>, N. The vector <var>p</var> satisfies the equation p\bf Q = 0 and \sum_i=1^N p_i = 1. If this function is invoked with three arguments, <var>p</var><code>(i)</code> is the probability that the system is in state i at time <var>t</var>, given the initial occupancy probabilities <var>p0</var>. </dl> </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <p>Consider a two-state CTMC such that transition rates between states are equal to 1. This can be solved as follows: <pre class="example"><pre class="verbatim"> Q = [ -1 1; \ 1 -1 ]; q = ctmc(Q)</pre> ⇒ q = 0.50000 0.50000 </pre> <div class="node"> <a name="Birth-Death-process"></a> <a name="Birth_002dDeath-process"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Expected-Sojourn-Time">Expected Sojourn Time</a>, Previous: <a rel="previous" accesskey="p" href="#State-occupancy-probabilities">State occupancy probabilities</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> <h4 class="subsection">4.2.2 Birth-Death process</h4> <p><a name="doc_002dctmc_005fbd"></a> <div class="defun"> — Function File: <var>Q</var> = <b>ctmc_bd</b> (<var>birth, death</var>)<var><a name="index-ctmc_005fbd-28"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-29"></a><a name="index-Birth_002ddeath-process-30"></a> Returns the N \times N infinitesimal generator matrix Q for a birth-death process with given rates. <p><strong>INPUTS</strong> <dl> <dt><var>birth</var><dd>Vector with N-1 elements, where <var>birth</var><code>(i)</code> is the transition rate from state i to state i+1. <br><dt><var>death</var><dd>Vector with N-1 elements, where <var>death</var><code>(i)</code> is the transition rate from state i+1 to state i. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Q</var><dd>Infinitesimal generator matrix for the birth-death process. </dl> </blockquote></div> <div class="node"> <a name="Expected-Sojourn-Time"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a>, Previous: <a rel="previous" accesskey="p" href="#Birth_002dDeath-process">Birth-Death process</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> <h4 class="subsection">4.2.3 Expected Sojourn Time</h4> <p>Given a N state continuous-time Markov Chain with infinitesimal generator matrix \bf Q, we define the vector \bf L(t) = (L_1(t), L_2(t), \ldots L_N(t)) such that L_i(t) is the expected sojourn time in state i during the interval [0,t), assuming that the initial occupancy probability at time 0 was \bf \pi(0). \bf L(t) is the solution of the following differential equation: <pre class="example"> dL --(t) = L(t) Q + pi(0), L(0) = 0 dt </pre> <p>Alternatively, \bf L(t) can also be expressed in integral form as: <pre class="example"> / t L(t) = | pi(u) du / u=0 </pre> <p class="noindent">where \bf \pi(t) = \bf \pi(0) \exp(\bf Qt) is the state occupancy probability at time t. <p><a name="doc_002dctmc_005fexps"></a> <div class="defun"> — Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, t, p </var>)<var><a name="index-ctmc_005fexps-31"></a></var><br> — Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fexps-32"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-33"></a><a name="index-Expected-sojourn-time-34"></a> With three arguments, compute the expected times <var>L</var><code>(i)</code> spent in each state i during the time interval [0,t], assuming that the state occupancy probabilities at time 0 are <var>p</var>. With two arguments, compute the expected time <var>L</var><code>(i)</code> spent in each state i until absorption. <p><strong>INPUTS</strong> <dl> <dt><var>Q</var><dd>N \times N infinitesimal generator matrix. <var>Q</var><code>(i,j)</code> is the transition rate from state i to state j, 1 ≤ i \neq j ≤ N. The matrix <var>Q</var> must also satisfy the condition \sum_j=1^N Q_ij = 0. <br><dt><var>t</var><dd>Time <br><dt><var>p</var><dd>Initial occupancy probability vector; <var>p</var><code>(i)</code> is the probability the system is in state i at time 0, i = 1, <small class="dots">...</small>, N </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>L</var><dd>If this function is called with three arguments, <var>L</var><code>(i)</code> is the expected time spent in state i during the interval [0,t]. If this function is called with two arguments <var>L</var><code>(i)</code> is either the expected time spent in state i until absorption (if i is a transient state), or zero (if <var>i</var> is an absorbing state). </dl> </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <p>Let us consider a pure-birth, 4-states CTMC such that 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 in state i, given the initial occupancy probability \bf \pi_0=(1,0,0,0). <pre class="example"><pre class="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(0,10,100); p0 = zeros(1,N); p0(1)=1; L = zeros(length(t),N); for i=1:length(t) L(i,:) = ctmc_exps(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"); xlabel("Time"); ylabel("Expected sojourn time");</pre> </pre> <div class="node"> <a name="Time-Averaged-Expected-Sojourn-Time"></a> <a name="Time_002dAveraged-Expected-Sojourn-Time"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Mean-Time-to-Absorption">Mean Time to Absorption</a>, Previous: <a rel="previous" accesskey="p" href="#Expected-Sojourn-Time">Expected Sojourn Time</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> <h4 class="subsection">4.2.4 Time-Averaged Expected Sojourn Time</h4> <p><a name="doc_002dctmc_005ftaexps"></a> <div class="defun"> — Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, t, p</var>)<var><a name="index-ctmc_005ftaexps-35"></a></var><br> — Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, p</var>)<var><a name="index-ctmc_005ftaexps-36"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-37"></a><a name="index-Time_002dalveraged-sojourn-time-38"></a> Compute the <em>time-averaged sojourn time</em> <var>M</var><code>(i)</code>, 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 <var>p</var>. <p><strong>INPUTS</strong> <dl> <dt><var>Q</var><dd>Infinitesimal generator matrix. <var>Q</var><code>(i,j)</code> is the transition rate from state i to state j, 1 ≤ i \neq j ≤ N. The matrix <var>Q</var> must also satisfy the condition \sum_j=1^N Q_ij = 0 <br><dt><var>t</var><dd>Time. If omitted, the results are computed until absorption. <br><dt><var>p</var><dd><var>p</var><code>(i)</code> is the probability that, at time 0, the system was in state i, for all i = 1, <small class="dots">...</small>, N </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>M</var><dd>If this function is called with three parameters, <var>M</var><code>(i)</code> is the expected fraction of the interval 0,t] spent in state i assuming that the state occupancy probability at time zero is <var>p</var>. If this function is called with two parameters, <var>M</var><code>(i)</code> is the expected fraction of time until absorption spent in state i. </dl> </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <pre class="example"><pre class="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,:) = ctmc_taexps(Q,t(i),p); endfor 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"); xlabel("Time"); ylabel("Time-averaged Expected sojourn time");</pre> </pre> <div class="node"> <a name="Mean-Time-to-Absorption"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#First-Passage-Times">First Passage Times</a>, Previous: <a rel="previous" accesskey="p" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> <h4 class="subsection">4.2.5 Mean Time to Absorption</h4> <p>If we consider a Markov Chain with absorbing states, it is possible to define the <em>expected time to absorption</em> as the expected time until the system goes into an absorbing state. More specifically, let us suppose that A is the set of transient (i.e., non-absorbing) states of a CTMC with N states and infinitesimal generator matrix \bf Q. The expected time to absorption \bf L_A(\infty) is defined as the solution of the following equation: <pre class="example"> L_A( inf ) Q_A = -pi_A(0) </pre> <p class="noindent">where \bf Q_A is the restriction of matrix \bf Q to only states in A, and \bf \pi_A(0) is the initial state occupancy probability at time 0, restricted to states in A. <p><a name="doc_002dctmc_005fmtta"></a> <div class="defun"> — Function File: <var>t</var> = <b>ctmc_mtta</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fmtta-39"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-40"></a><a name="index-Mean-time-to-absorption-41"></a> Compute the Mean-Time to Absorption (MTTA) of the CTMC described by the infinitesimal generator matrix <var>Q</var>, starting from initial occupancy probabilities <var>p</var>. If there are no absorbing states, this function fails with an error. <p><strong>INPUTS</strong> <dl> <dt><var>Q</var><dd>N \times N infinitesimal generator matrix. <var>Q</var><code>(i,j)</code> is the transition rate from state i to state j, i \neq j. The matrix <var>Q</var> must satisfy the condition \sum_j=1^N Q_i j = 0 <br><dt><var>p</var><dd><var>p</var><code>(i)</code> is the probability that the system is in state i at time 0, for each i=1, <small class="dots">...</small>, N </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>t</var><dd>Mean time to absorption of the process represented by matrix <var>Q</var>. If there are no absorbing states, this function fails. </dl> </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <p>Let us consider a simple model of a redundant disk array. We assume that the array is made of 5 independent disks, such that the array 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. <p>We model this system as a 4 states Markov chain with state space \ 2, 3, 4, 5 \. State i denotes the fact that exactly i disks are active; 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, with death rate from state i to state i-1 is \mu i, for i = 3, 4, 5). <p>The MTTF of the disk array is the MTTA of the Markov Chain, and can be computed with the following expression: <pre class="example"><pre class="verbatim"> mu = 0.01; death = [ 3 4 5 ] * mu; Q = diag(death,-1); Q -= diag(sum(Q,2)); [t L] = ctmc_mtta(Q,[0 0 0 1])</pre> ⇒ t = 78.333 </pre> <p class="noindent"><strong>REFERENCES</strong> <p>G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998. <p><a name="index-Bolch_002c-G_002e-42"></a><a name="index-Greiner_002c-S_002e-43"></a><a name="index-de-Meer_002c-H_002e-44"></a><a name="index-Trivedi_002c-K_002e-45"></a> <div class="node"> <a name="First-Passage-Times"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#Mean-Time-to-Absorption">Mean Time to Absorption</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> <h4 class="subsection">4.2.6 First Passage Times</h4> <p><a name="doc_002dctmc_005ffpt"></a> <div class="defun"> — Function File: <var>M</var> = <b>ctmc_fpt</b> (<var>Q</var>)<var><a name="index-ctmc_005ffpt-46"></a></var><br> — Function File: <var>m</var> = <b>ctmc_fpt</b> (<var>Q, i, j</var>)<var><a name="index-ctmc_005ffpt-47"></a></var><br> <blockquote> <p><a name="index-Markov-chain_002c-continuous-time-48"></a><a name="index-First-passage-times-49"></a> If called with a single argument, computes the mean first passage times <var>M</var><code>(i,j)</code>, the average times before state <var>j</var> is reached, starting from state <var>i</var>, for all 1 \leq i, j \leq N. If called with three arguments, returns the single value <var>m</var><code> = </code><var>M</var><code>(i,j)</code>. <p><strong>INPUTS</strong> <dl> <dt><var>Q</var><dd>Infinitesimal generator matrix. <var>Q</var> is a N \times N square matrix where <var>Q</var><code>(i,j)</code> is the transition rate from state i to state j, for 1 ≤ i \neq j ≤ N. Transition rates must be nonnegative, and \sum_j=1^N Q_i j = 0 <br><dt><var>i</var><dd>Initial state. <br><dt><var>j</var><dd>Destination state. If <var>j</var> is a vector, returns the mean first passage time to any state in <var>j</var>. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>M</var><dd>If this function is called with a single argument, the result <var>M</var><code>(i,j)</code> is the average time before state <var>j</var> is visited for the first time, starting from state <var>i</var>. <br><dt><var>m</var><dd>If this function is called with three arguments, the result <var>m</var> is the average time before state <var>j</var> is visited for the first time, starting from state <var>i</var>. </dl> </blockquote></div> <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Single-Station-Queueing-Systems"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Queueing-Networks">Queueing Networks</a>, Previous: <a rel="previous" accesskey="p" href="#Markov-Chains">Markov Chains</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="chapter">5 Single Station Queueing Systems</h2> <p>Single Station Queueing Systems contain a single station, and are thus quite easy to analyze. The <code>queueing</code> package contains functions for handling the following types of queues: <ul class="menu"> <li><a accesskey="1" href="#The-M_002fM_002f1-System">The M/M/1 System</a>: Single-server queueing station. <li><a accesskey="2" href="#The-M_002fM_002fm-System">The M/M/m System</a>: Multiple-server queueing station. <li><a accesskey="3" href="#The-M_002fM_002finf-System">The M/M/inf System</a>: Infinite-server (delay center) station. <li><a accesskey="4" href="#The-M_002fM_002f1_002fK-System">The M/M/1/K System</a>: Single-server, finite-capacity queueing station. <li><a accesskey="5" href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a>: Multiple-server, finite-capacity queueing station. <li><a accesskey="6" href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a>: Asymmetric multiple-server queueing station. <li><a accesskey="7" href="#The-M_002fG_002f1-System">The M/G/1 System</a>: Single-server with general service time distribution. <li><a accesskey="8" href="#The-M_002fHm_002f1-System">The M/Hm/1 System</a>: Single-server with hyperexponential service time distribution. </ul> <p>The functions which analyze the queues above can be used as building blocks for analyzing Queueing Networks. For example, Jackson networks can be solved by computing the aggregate arrival rates to each node, and then solving each node in isolation as if it were a single station queueing system. <!-- M/M/1 --> <div class="node"> <a name="The-M%2fM%2f1-System"></a> <a name="The-M_002fM_002f1-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-M_002fM_002fm-System">The M/M/m System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.1 The M/M/1 System</h3> <p>The M/M/1 system is made of a single server connected to an unlimited FCFS queue. The mean arrival rate is Poisson with arrival rate \lambda; the service time is exponentially distributed with average service rate \mu. The system is stable if \lambda < \mu. <p><a name="doc_002dqnmm1"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>] = <b>qnmm1</b> (<var>lambda, mu</var>)<var><a name="index-qnmm1-50"></a></var><br> <blockquote> <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-51"></a> Compute utilization, response time, average number of requests and throughput for a M/M/1 queue. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate (<var>lambda</var><code> > 0</code>). <br><dt><var>mu</var><dd>Service rate (<var>mu</var><code> > </code><var>lambda</var>). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Server utilization <br><dt><var>R</var><dd>Service center response time <br><dt><var>Q</var><dd>Average number of requests in the system <br><dt><var>X</var><dd>Service center throughput. If the system is ergodic, we will always have <var>X</var><code> = </code><var>lambda</var> <br><dt><var>p0</var><dd>Steady-state probability that there are no requests in the system. </dl> <p><var>lambda</var> and <var>mu</var> can be vectors of the same size. In this case, the results will be vectors as well. <pre class="sp"> </pre> <strong>See also:</strong> qnmmm, qnmminf, qnmmmk. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p class="noindent">G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998, Section 6.3. <p><a name="index-Bolch_002c-G_002e-52"></a><a name="index-Greiner_002c-S_002e-53"></a><a name="index-de-Meer_002c-H_002e-54"></a><a name="index-Trivedi_002c-K_002e-55"></a> <!-- M/M/m --> <div class="node"> <a name="The-M%2fM%2fm-System"></a> <a name="The-M_002fM_002fm-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-M_002fM_002finf-System">The M/M/inf System</a>, Previous: <a rel="previous" accesskey="p" href="#The-M_002fM_002f1-System">The M/M/1 System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.2 The M/M/m System</h3> <p>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 single 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 nodes in the center: <pre class="example"> <code>mu(n) = min(m,n)*mu</code> </pre> <p><a name="doc_002dqnmmm"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>, <var>pm</var>] = <b>qnmmm</b> (<var>lambda, mu</var>)<var><a name="index-qnmmm-56"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>, <var>pm</var>] = <b>qnmmm</b> (<var>lambda, mu, m</var>)<var><a name="index-qnmmm-57"></a></var><br> <blockquote> <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-58"></a> Compute utilization, response time, average number of requests in service and throughput for a M/M/m queue, a queueing system with m identical service centers connected to a single queue. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate (<var>lambda</var><code>>0</code>). <br><dt><var>mu</var><dd>Service rate (<var>mu</var><code>></code><var>lambda</var>). <br><dt><var>m</var><dd>Number of servers (<var>m</var><code> ≥ 1</code>). If omitted, it is assumed <var>m</var><code>=1</code>. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Service center utilization, U = \lambda / (m \mu). <br><dt><var>R</var><dd>Service center response time <br><dt><var>Q</var><dd>Average number of requests in the system <br><dt><var>X</var><dd>Service center throughput. If the system is ergodic, we will always have <var>X</var><code> = </code><var>lambda</var> <br><dt><var>p0</var><dd>Steady-state probability that there are 0 requests in the system <br><dt><var>pm</var><dd>Steady-state probability that an arriving request has to wait in the queue </dl> <p><var>lambda</var>, <var>mu</var> and <var>m</var> can be vectors of the same size. In this case, the results will be vectors as well. <pre class="sp"> </pre> <strong>See also:</strong> qnmm1,qnmminf,qnmmmk. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p class="noindent">G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998, Section 6.5. <p><a name="index-Bolch_002c-G_002e-59"></a><a name="index-Greiner_002c-S_002e-60"></a><a name="index-de-Meer_002c-H_002e-61"></a><a name="index-Trivedi_002c-K_002e-62"></a> <!-- M/M/inf --> <div class="node"> <a name="The-M%2fM%2finf-System"></a> <a name="The-M_002fM_002finf-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-M_002fM_002f1_002fK-System">The M/M/1/K System</a>, Previous: <a rel="previous" accesskey="p" href="#The-M_002fM_002fm-System">The M/M/m System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.3 The M/M/inf System</h3> <p>The M/M/\infty system is similar to the M/M/m system, except that there are infinitely many identical servers (that is, m = \infty). Each new request is assigned to a new server, so that queueing never occurs. The M/M/\infty system is always stable. <p><a name="doc_002dqnmminf"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>] = <b>qnmminf</b> (<var>lambda, mu</var>)<var><a name="index-qnmminf-63"></a></var><br> <blockquote> <p>Compute utilization, response time, average number of requests and throughput for a M/M/\infty queue. This is a system with an infinite number of identical servers. Note that a M/M/\infty system is always stable, regardless the values of the arrival and service rates. <p><a name="index-g_t_0040math_007bM_002fM_002f_007dinf-system-64"></a> <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate (<var>lambda</var><code>>0</code>). <br><dt><var>mu</var><dd>Service rate (<var>mu</var><code>>0</code>). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>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. <p><a name="index-traffic-intensity-65"></a> <br><dt><var>R</var><dd>Service center response time. <br><dt><var>Q</var><dd>Average number of requests in the system (which is equal to the traffic intensity \lambda/\mu). <br><dt><var>X</var><dd>Throughput (which is always equal to <var>X</var><code> = </code><var>lambda</var>). <br><dt><var>p0</var><dd>Steady-state probability that there are no requests in the system </dl> <p><var>lambda</var> and <var>mu</var> can be vectors of the same size. In this case, the results will be vectors as well. <pre class="sp"> </pre> <strong>See also:</strong> qnmm1,qnmmm,qnmmmk. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p class="noindent">G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998, Section 6.4. <p><a name="index-Bolch_002c-G_002e-66"></a><a name="index-Greiner_002c-S_002e-67"></a><a name="index-de-Meer_002c-H_002e-68"></a><a name="index-Trivedi_002c-K_002e-69"></a> <!-- M/M/1/k --> <div class="node"> <a name="The-M%2fM%2f1%2fK-System"></a> <a name="The-M_002fM_002f1_002fK-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a>, Previous: <a rel="previous" accesskey="p" href="#The-M_002fM_002finf-System">The M/M/inf System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.4 The M/M/1/K System</h3> <p>In a M/M/1/K finite capacity system there can be at most k jobs at any time. If a new request tries to join the system when there are already K other requests, the arriving 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 \lambda and \mu. <p><a name="doc_002dqnmm1k"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>, <var>pK</var>] = <b>qnmm1k</b> (<var>lambda, mu, K</var>)<var><a name="index-qnmm1k-70"></a></var><br> <blockquote> <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-71"></a> 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; the maximum number of requests in the system is K, and the maximum queue length is K-1. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate (<var>lambda</var><code>>0</code>). <br><dt><var>mu</var><dd>Service rate (<var>mu</var><code>>0</code>). <br><dt><var>K</var><dd>Maximum number of requests allowed in the system (<var>K</var><code> ≥ 1</code>). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Service center utilization, which is defined as <var>U</var><code> = 1-</code><var>p0</var> <br><dt><var>R</var><dd>Service center response time <br><dt><var>Q</var><dd>Average number of requests in the system <br><dt><var>X</var><dd>Service center throughput <br><dt><var>p0</var><dd>Steady-state probability that there are no requests in the system <br><dt><var>pK</var><dd>Steady-state probability that there are K requests in the system (i.e., that the system is full) </dl> <p><var>lambda</var>, <var>mu</var> and <var>K</var> can be vectors of the same size. In this case, the results will be vectors as well. <pre class="sp"> </pre> <strong>See also:</strong> qnmm1,qnmminf,qnmmm. </blockquote></div> <!-- M/M/m/k --> <div class="node"> <a name="The-M%2fM%2fm%2fK-System"></a> <a name="The-M_002fM_002fm_002fK-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a>, Previous: <a rel="previous" accesskey="p" href="#The-M_002fM_002f1_002fK-System">The M/M/1/K System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.5 The M/M/m/K System</h3> <p>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 is made of K-m slots. The M/M/m/K system is always stable. <p><a name="doc_002dqnmmmk"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>, <var>pK</var>] = <b>qnmmmk</b> (<var>lambda, mu, m, K</var>)<var><a name="index-qnmmmk-72"></a></var><br> <blockquote> <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-73"></a> 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. The maximum queue length is K-m. This function generates and solves the underlying CTMC. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate (<var>lambda</var><code>>0</code>). <br><dt><var>mu</var><dd>Service rate (<var>mu</var><code>>0</code>). <br><dt><var>m</var><dd>Number of servers (<var>m</var><code> ≥ 1</code>). <br><dt><var>K</var><dd>Maximum number of requests allowed in the system, including those inside the service centers (<var>K</var><code> ≥ </code><var>m</var>). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Service center utilization <br><dt><var>R</var><dd>Service center response time <br><dt><var>Q</var><dd>Average number of requests in the system <br><dt><var>X</var><dd>Service center throughput <br><dt><var>p0</var><dd>Steady-state probability that there are no requests in the system. <br><dt><var>pK</var><dd>Steady-state probability that there are <var>K</var> requests in the system (i.e., probability that the system is full). </dl> <p><var>lambda</var>, <var>mu</var>, <var>m</var> and <var>K</var> can be either scalars, or vectors of the same size. In this case, the results will be vectors as well. <pre class="sp"> </pre> <strong>See also:</strong> qnmm1,qnmminf,qnmmm. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p class="noindent">G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998, Section 6.6. <p><a name="index-Bolch_002c-G_002e-74"></a><a name="index-Greiner_002c-S_002e-75"></a><a name="index-de-Meer_002c-H_002e-76"></a><a name="index-Trivedi_002c-K_002e-77"></a> <!-- Approximate M/M/m --> <div class="node"> <a name="The-Asymmetric-M%2fM%2fm-System"></a> <a name="The-Asymmetric-M_002fM_002fm-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-M_002fG_002f1-System">The M/G/1 System</a>, Previous: <a rel="previous" accesskey="p" href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.6 The Asymmetric M/M/m System</h3> <p>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. <p><a name="doc_002dqnammm"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnammm</b> (<var>lambda, mu</var>)<var><a name="index-qnammm-78"></a></var><br> <blockquote> <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-79"></a> Compute <em>approximate</em> utilization, response time, average number of requests in service and throughput for an asymmetric M/M/m queue. In this system there are m different service centers 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. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate (<var>lambda</var><code>>0</code>). <br><dt><var>mu</var><dd><var>mu</var><code>(i)</code> is the service rate of server i, 1 ≤ i ≤ m. The system must be ergodic (<var>lambda</var><code> < sum(</code><var>mu</var><code>)</code>). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Approximate service center utilization, U = \lambda / ( \sum_i \mu_i ). <br><dt><var>R</var><dd>Approximate service center response time <br><dt><var>Q</var><dd>Approximate number of requests in the system <br><dt><var>X</var><dd>Approximate service center throughput. If the system is ergodic, we will always have <var>X</var><code> = </code><var>lambda</var> </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnmmm. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p class="noindent">G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998 <p><a name="index-Bolch_002c-G_002e-80"></a><a name="index-Greiner_002c-S_002e-81"></a><a name="index-de-Meer_002c-H_002e-82"></a><a name="index-Trivedi_002c-K_002e-83"></a> <div class="node"> <a name="The-M%2fG%2f1-System"></a> <a name="The-M_002fG_002f1-System"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#The-M_002fHm_002f1-System">The M/Hm/1 System</a>, Previous: <a rel="previous" accesskey="p" href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.7 The M/G/1 System</h3> <p><a name="doc_002dqnmg1"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>] = <b>qnmg1</b> (<var>lambda, xavg, x2nd</var>)<var><a name="index-qnmg1-84"></a></var><br> <blockquote> <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-85"></a> 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 <var>xavg</var>, and by its second moment <var>x2nd</var>. The computations are based on results from L. Kleinrock, <cite>Queuing Systems</cite>, Wiley, Vol 2, and Pollaczek-Khinchine formula. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate. <br><dt><var>xavg</var><dd>Average service time <br><dt><var>x2nd</var><dd>Second moment of service time distribution </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Service center utilization <br><dt><var>R</var><dd>Service center response time <br><dt><var>Q</var><dd>Average number of requests in the system <br><dt><var>X</var><dd>Service center throughput <br><dt><var>p0</var><dd>probability that there is not any request at system </dl> <p><var>lambda</var>, <var>xavg</var>, <var>t2nd</var> can be vectors of the same size. In this case, the results will be vectors as well. <pre class="sp"> </pre> <strong>See also:</strong> qnmh1. </blockquote></div> <div class="node"> <a name="The-M%2fHm%2f1-System"></a> <a name="The-M_002fHm_002f1-System"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#The-M_002fG_002f1-System">The M/G/1 System</a>, Up: <a rel="up" accesskey="u" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a> </div> <h3 class="section">5.8 The M/H_m/1 System</h3> <p><a name="doc_002dqnmh1"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>p0</var>] = <b>qnmh1</b> (<var>lambda, mu, alpha</var>)<var><a name="index-qnmh1-86"></a></var><br> <blockquote> <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-87"></a> 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: <pre class="example"> ___ m \ B(x) = > alpha(j) * (1-exp(-mu(j)*x)) x>0 /__ j=1 </pre> <p>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. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Arrival rate. <br><dt><var>mu</var><dd><var>mu</var><code>(j)</code> is the phase j service rate. The total number of phases m is <code>length(</code><var>mu</var><code>)</code>. <br><dt><var>alpha</var><dd><var>alpha</var><code>(j)</code> is the probability that a request is served at phase j. <var>alpha</var> must have the same size as <var>mu</var>. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>Service center utilization <br><dt><var>R</var><dd>Service center response time <br><dt><var>Q</var><dd>Average number of requests in the system <br><dt><var>X</var><dd>Service center throughput </dl> <!-- @seealso{qnmhr1} --> </blockquote></div> <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Queueing-Networks"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Contributing-Guidelines">Contributing Guidelines</a>, Previous: <a rel="previous" accesskey="p" href="#Single-Station-Queueing-Systems">Single Station Queueing Systems</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="chapter">6 Queueing Networks</h2> <ul class="menu"> <li><a accesskey="1" href="#Introduction-to-QNs">Introduction to QNs</a>: A brief introduction to Queueing Networks. <li><a accesskey="2" href="#Generic-Algorithms">Generic Algorithms</a>: High-level functions for QN analysis <li><a accesskey="3" href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a>: Functions to analyze product-form QNs <li><a accesskey="4" href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a>: Functions to analyze non product-form QNs <li><a accesskey="5" href="#Bounds-on-performance">Bounds on performance</a>: Functions to compute performance bounds <li><a accesskey="6" href="#Utility-functions">Utility functions</a>: Utility functions to compute miscellaneous quantities </ul> <p><a name="index-queueing-networks-88"></a> <!-- INTRODUCTION --> <div class="node"> <a name="Introduction-to-QNs"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Generic-Algorithms">Generic Algorithms</a>, Up: <a rel="up" accesskey="u" href="#Queueing-Networks">Queueing Networks</a> </div> <h3 class="section">6.1 Introduction to QNs</h3> <p>Queueing Networks (QN) are a very simple yet powerful modeling tool which is used to analyze many kind of systems. In its simplest form, a QN is made of K service centers. Each service center i has a queue, which is connected to m_i (generally identical) <em>servers</em>. Customers (or requests) arrive at the service center, and join the queue if there is a slot available. Then, requests are served according to a (de)queueing policy. After service completes, the requests leave the service center. <p>The service centers for which m_i = \infty are called <em>delay centers</em> or <em>infinite servers</em>. If a service center has infinite servers, of course each new request will find one server available, so there will never be queueing. <p>Requests join the queue according to a <em>queueing policy</em>, such as: <dl> <dt><strong>FCFS</strong><dd>First-Come-First-Served <br><dt><strong>LCFS-PR</strong><dd>Last-Come-First-Served, Preemptive Resume <br><dt><strong>PS</strong><dd>Processor Sharing <br><dt><strong>IS</strong><dd>Infinite Server, there is an infinite number of identical servers so that each request always finds a server available, and there is no queueing </dl> <p>A population of <em>requests</em> or <em>customers</em> arrives to the system system, requesting service to the service centers. The request population may be <em>open</em> or <em>closed</em>. In open systems there is an infinite population of requests. New customers arrive from outside the system, and eventually leave the system. In closed systems there is a fixed population of request which continuously interacts with the system. <p>There might be a single class of requests, meaning that all requests behave in the same way (e.g., they spend the same average time on each particular server), or there might be multiple classes of requests. <h4 class="subsection">6.1.1 Single class models</h4> <p>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. <p class="noindent"><strong>Model Inputs</strong> <dl> <dt>\lambda_i<dd>External arrival rate to service center i. <br><dt>\lambda<dd>Overall external arrival rate to the whole system: \lambda = \sum_i \lambda_i. <br><dt>S_i<dd>Average service time. S_i is the average service time on service center i. In other words, S_i is the average time from the instant in which a request is extracted from the queue and starts being service, and the instant at which service finishes and the request moves to another queue (or exits the system). <br><dt>P_ij<dd>Routing probability matrix. \bf P = P_ij is a K \times K matrix such that P_ij is the probability that a request completing service at server i will move directly to server j, The probability that a request leaves the system after service at service center i is 1-\sum_j=1^K P_ij. <br><dt>V_i<dd>Average number of visits. V_i is the average number of visits to the service center i. This quantity will be described shortly. </dl> <p class="noindent"><strong>Model Outputs</strong> <dl> <dt>U_i<dd>Service center utilization. U_i is the utilization of service center i. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). <br><dt>R_i<dd>Average response time. R_i is the average response time of service center i. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service. <br><dt>Q_i<dd>Average number of customers. Q_i is the average number of requests in service center i. This includes both the requests in the queue, and the request being served. <br><dt>X_i<dd>Throughput. X_i is the throughput of service center i. The throughput is defined as the ratio of job completions (i.e., average number of jobs completed over a fixed interval of time). </dl> <p class="noindent">Given these output parameters, additional performance measures can be computed as follows: <dl> <dt>X<dd>System throughput, X = X_1 / V_1 <br><dt>R<dd>System response time, R = \sum_k=1^K R_k V_k <br><dt>Q<dd>Average number of requests in the system, Q = N-XZ </dl> <p>For open, single-class models, the scalar \lambda denotes the external arrival rate of requests to the system. The average number of visits satisfy the following equation: <pre class="example"> V == P0 + V*P; </pre> <p class="noindent">where P_0 j is the probability that an external arrival goes to service center j. If \lambda_j is the external arrival rate to service center j, and \lambda = \sum_j \lambda_j is the overall external arrival rate, then P_0 j = \lambda_j / \lambda. <p>For closed models, the visit ratios satisfy the following equation: <pre class="example"> V(1) == 1 && V == V*P; </pre> <h4 class="subsection">6.1.2 Multiple class models</h4> <p>In multiple class QN models, we assume that there exist C different classes of requests. Each request from class c spends on average time S_ck in service at service center k. For open models, we denote with \bf \lambda = \lambda_ck the arrival rates, where \lambda_ck is the external arrival rate of class c customers at service center k. For closed models, we denote with \bf N = (N_1, N_2, \ldots N_C) the population vector, where N_c is the number of class c requests in the system. <p>The transition probability matrix for these kind of networks will be a C \times K \times C \times K matrix \bf P = P_risj such that P_risj is the probability that a class r request which completes service at center i will join server j as a class s request. <p>Model input and outputs can be adjusted by adding additional indexes for the customer classes. <p class="noindent"><strong>Model Inputs</strong> <dl> <dt>\lambda_ci<dd>External arrival rate of class-c requests to service center i <br><dt>\lambda<dd>Overall external arrival rate to the whole system: \lambda = \sum_c \sum_i \lambda_ci <br><dt>S_ci<dd>Average service time. S_ci is the average service time on service center i for class c requests. <br><dt>P_risj<dd>Routing probability matrix. \bf P = P_risj is a C \times K \times C \times K matrix such that P_risj is the probability that a class r request which completes service at server i will move to server j as a class s request. <br><dt>V_ci<dd>Average number of visits. V_ci is the average number of visits of class c requests to the service center i. </dl> <p class="noindent"><strong>Model Outputs</strong> <dl> <dt>U_ci<dd>Utilization of service center i 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). <br><dt>R_ci<dd>Average response time experienced by class c requests on service center i. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service. <br><dt>Q_ci<dd>Average number of class c requests on service center i. This includes both the requests in the queue, and the request being served. <br><dt>X_ci<dd>Throughput of service center i for class c requests. The throughput is defined as the rate of completion of class c requests. </dl> <p class="noindent">It is possible to define aggregate performance measures as follows: <dl> <dt>U_i<dd>Utilization of service center i: <code>Ui = sum(U,1);</code> <br><dt>R_c<dd>System response time for class c requests: <code>Rc = sum( V.*R, 1 );</code> <br><dt>Q_c<dd>Average number of class c requests in the system: <code>Qc = sum( Q, 2 );</code> <br><dt>X_c<dd>Class c throughput: <code>Xc = X(:,1) ./ V(:,1);</code> </dl> <p>We can define the visit ratios V_sj for class s customers at service center j as follows: <p>V_sj = sum_r sum_i V_ri P_risj, for all s,j <p class="noindent">while for open networks: <p>V_sj = P_0sj + sum_r sum_i V_ri P_risj, for all s,j <p class="noindent">where P_0sj is the probability that an external arrival goes to service center j as a class-s request. If \lambda_sj is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_sj is the overall external arrival rate to the whole system, then P_0sj = \lambda_sj / \lambda. <div class="node"> <a name="Generic-Algorithms"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a>, Previous: <a rel="previous" accesskey="p" href="#Introduction-to-QNs">Introduction to QNs</a>, Up: <a rel="up" accesskey="u" href="#Queueing-Networks">Queueing Networks</a> </div> <h3 class="section">6.2 Generic Algorithms</h3> <p>The <code>queueing</code> package provides a couple of high-level functions for defining and solving QN models. These functions can be used to define a open or closed QN model (with single or multiple job classes), with arbitrary configuration and queueing disciplines. At the moment only product-form networks can be solved, See <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a>. <p>The network is defined by two parameters. The first one is the list of nodes, encoded as an Octave <em>cell array</em>. The second parameter is the visit ration <var>V</var>, which can be either a vector (for single-class models) or a two-dimensional matrix (for multiple-class models). <p>Individual nodes in the network are structures build using the <code>qnmknode</code> function. <p><a name="doc_002dqnmknode"></a> <div class="defun"> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S</var>)<var><a name="index-qnmknode-89"></a></var><br> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S, m</var>)<var><a name="index-qnmknode-90"></a></var><br> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/1-lcfs-pr", S</var>)<var><a name="index-qnmknode-91"></a></var><br> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S</var>)<var><a name="index-qnmknode-92"></a></var><br> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S, s2</var>)<var><a name="index-qnmknode-93"></a></var><br> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S</var>)<var><a name="index-qnmknode-94"></a></var><br> — Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S, s2</var>)<var><a name="index-qnmknode-95"></a></var><br> <blockquote> <p>Creates a node; this function can be used together with <code>qnsolve</code>. 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. <p><strong>INPUTS</strong> <dl> <dt><var>S</var><dd>Average service time. S can be either a scalar, a row vector, a column vector or a two-dimensional matrix. <ul> <li>If S is a scalar, it is assumed to be a load-independent, class-independent service time. <li>If S is a column vector, then <var>S</var><code>(c)</code> is assumed to the the load-independent service time for class c customers. <li>If S is a row vector, then <var>S</var><code>(n)</code> is assumed to be the class-independent service time at the node, when there are n requests. <li>Finally, if <var>S</var> is a two-dimensional matrix, then <var>S</var><code>(c,n)</code> is assumed to be the class c service time when there are n requests at the node. </ul> <br><dt><var>m</var><dd>Number of identical servers at the node. Default is <var>m</var><code>=1</code>. <br><dt><var>s2</var><dd>Squared coefficient of variation for the service time. Default is 1.0. </dl> <p>The returned struct <var>Q</var> should be considered opaque to the client. <!-- The returned struct @var{Q} has the following fields: --> <!-- @table @var --> <!-- @item Q.node --> <!-- (String) type of the node; valid values are @code{"m/m/m-fcfs"}, --> <!-- @code{"-/g/1-lcfs-pr"}, @code{"-/g/1-ps"} (Processor-Sharing) --> <!-- and @code{"-/g/inf"} (Infinite Server, or delay center). --> <!-- @item Q.S --> <!-- Average service time. If @code{@var{Q}.S} is a vector, then --> <!-- @code{@var{Q}.S(i)} is the average service time at that node --> <!-- if there are @math{i} requests. --> <!-- @item Q.m --> <!-- Number of identical servers at a @code{"m/m/m-fcfs"}. Default is 1. --> <!-- @item Q.c --> <!-- Number of customer classes. Default is 1. --> <!-- @end table --> <pre class="sp"> </pre> <strong>See also:</strong> qnsolve. </blockquote></div> <p>After the network has been defined, it is possible to solve it using the <code>qnsolve</code> function. Note that this function is somewhat less efficient than those described in later sections, but generally easier to use. <p><a name="doc_002dqnsolve"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnsolve</b> (<var>"closed", N, QQ, V</var>)<var><a name="index-qnsolve-96"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnsolve</b> (<var>"closed", N, QQ, V, Z</var>)<var><a name="index-qnsolve-97"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnsolve</b> (<var>"open", lambda, QQ, V</var>)<var><a name="index-qnsolve-98"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnsolve</b> (<var>"mixed", lambda, N, QQ, V</var>)<var><a name="index-qnsolve-99"></a></var><br> <blockquote> <p>General evaluator of QN models. Networks can be open, closed or mixed; single as well as multiclass networks are supported. <ul> <li>For <strong>closed</strong> networks, the following server types are supported: M/M/m–FCFS, -/G/\infty, -/G/1–LCFS-PR, -/G/1–PS and load-dependent variants. <li>For <strong>open</strong> networks, the following server types are supported: M/M/m–FCFS, -/G/\infty and -/G/1–PS. General load-dependent nodes are <em>not</em> supported. Multiclass open networks do not support multiple server M/M/m nodes, but only single server M/M/1–FCFS. <li>For <strong>mixed</strong> networks, the following server types are supported: M/M/1–FCFS, -/G/\infty and -/G/1–PS. General load-dependent nodes are <em>not</em> supported. </ul> <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Number of requests in the system for closed networks. For single-class networks, <var>N</var> must be a scalar. For multiclass networks, <var>N</var><code>(c)</code> is the population size of closed class c. <br><dt><var>lambda</var><dd>External arrival rate (scalar) for open networks. For single-class networks, <var>lambda</var> must be a scalar. For multiclass networks, <var>lambda</var><code>(c)</code> is the class c overall arrival rate. <br><dt><var>QQ</var><dd>List of queues in the network. This must be a cell array with N elements, such that <var>QQ</var><code>{i}</code> is a struct produced by the <code>qnmknode</code> function. <br><dt><var>Z</var><dd>External delay ("think time") for closed networks. Default 0. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If i is a FCFS node, then <var>U</var><code>(i)</code> is the utilization of service center i. If i is an IS node, then <var>U</var><code>(i)</code> is the <em>traffic intensity</em> defined as <var>X</var><code>(i)*</code><var>S</var><code>(i)</code>. <br><dt><var>R</var><dd><var>R</var><code>(i)</code> is the average response time of service center i. <br><dt><var>Q</var><dd><var>Q</var><code>(i)</code> is the average number of customers in service center i. <br><dt><var>X</var><dd><var>X</var><code>(i)</code> is the throughput of service center i. </dl> <p>Note that for multiclass networks, the computed results are per-class utilization, response time, number of customers and throughput: <var>U</var><code>(c,k)</code>, <var>R</var><code>(c,k)</code>, <var>Q</var><code>(c,k)</code>, <var>X</var><code>(c,k)</code>, </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <p>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: <ul> <li>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 <code>[0.2 0.1 0.1; 0.2 0.1 0.1]</code>. Thus, <var>S</var><code>(1,2) = 0.2</code> means that service time for class 1 customers where there are 2 requests in 0.2. Note that service times are class-independent; <li>Node 2 is a -/G/1–PS node, with service times S_12 = 0.4 for class 1, and S_22 = 0.6 for class 2 requests; <li>Node 3 is a -/G/\infty node (delay center), with service times S_13=1 and S_23=2 for class 1 and 2 respectively. </ul> <p>After defining the per-class visit count <var>V</var> such that <var>V</var><code>(c,k)</code> is the visit count of class c requests to service center k. We can define and solve the model as follows: <pre class="example"><pre class="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 );</pre></pre> <div class="node"> <a name="Algorithms-for-Product-Form-QNs"></a> <a name="Algorithms-for-Product_002dForm-QNs"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a>, Previous: <a rel="previous" accesskey="p" href="#Generic-Algorithms">Generic Algorithms</a>, Up: <a rel="up" accesskey="u" href="#Queueing-Networks">Queueing Networks</a> </div> <h3 class="section">6.3 Algorithms for Product-Form QNs</h3> <p>Product-form queueing networks fulfill the following assumptions: <ul> <li>The network can consist of open and closed job classes. <li>The following queueing disciplines are allowed: FCFS, PS, LCFS-PR and IS. <li>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. <li>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. <li>In open networks two kinds of arrival processes are allowed: i) the arrival process is Poisson, with arrival rate \lambda which can depend on the number of jobs in the network. ii) the arrival process consists of U independent Poisson arrival streams where the U job sources are assigned to the U chains; the arrival rate can be load dependent. </ul> <!-- Jackson Networks --> <h4 class="subsection">6.3.1 Jackson Networks</h4> <p>Jackson networks satisfy the following conditions: <ul> <li>There is only one job class in the network; the overall number of jobs in the system is unlimited. <li>There are N 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. <li>Arrival rates as well as routing probabilities are independent from the number of nodes in the network. <li>External arrivals and service times at the service centers are exponentially distributed, and in general can be load-dependent. <li>Service discipline at each node is FCFS </ul> <p>We define the <em>joint probability vector</em> \pi(k_1, k_2, \ldots k_N) as the steady-state probability that there are k_i requests at service center i, for all i=1,2, \ldots N. Jackson networks have the property that the joint probability is the product of the marginal probabilities \pi_i: <pre class="example"> <var>joint_prob</var> = prod( <var>pi</var> ) </pre> <p class="noindent">where \pi_i(k_i) is the steady-state probability that there are k_i requests at service center i. <p><a name="doc_002dqnjackson"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnjackson</b> (<var>lambda, S, P </var>)<var><a name="index-qnjackson-100"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnjackson</b> (<var>lambda, S, P, m </var>)<var><a name="index-qnjackson-101"></a></var><br> — Function File: <var>pr</var> = <b>qnjackson</b> (<var>lambda, S, P, m, k</var>)<var><a name="index-qnjackson-102"></a></var><br> <blockquote> <p><a name="index-open-network_002c-single-class-103"></a><a name="index-Jackson-network-104"></a> With three or four input parameters, this function computes the steady-state occupancy probabilities for a Jackson network. With five input parameters, this function computes the steady-state probability <var>pi</var><code>(j)</code> that there are <var>k</var><code>(j)</code> requests at service center j. <p>This function solves a subset of Jackson networks, with the following constraints: <ul> <li>External arrival rates are load-independent. <li>Service center i consists either of <var>m</var><code>(i) ≥ 1</code> identical servers with individual average service time <var>S</var><code>(i)</code>, or of an Infinite Server (IS) node. </ul> <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd><var>lambda</var><code>(i)</code> is the external arrival rate to service center i. <var>lambda</var> must be a vector of length N, <var>lambda</var><code>(i) ≥ 0</code>. <br><dt><var>S</var><dd><var>S</var><code>(i)</code> is the average service time on service center i <var>S</var> must be a vector of length N, <var>S</var><code>(i)>0</code>. <br><dt><var>P</var><dd><var>P</var><code>(i,j)</code> is the probability that a job which completes service at service center i proceeds to service center j. <var>P</var> must be a matrix of size N \times N. <br><dt><var>m</var><dd><var>m</var><code>(i)</code> is the number of servers at service center i. If <var>m</var><code>(i) < 1</code>, service center i is an infinite-server node. Otherwise, it is a regular FCFS queueing center with <var>m</var><code>(i)</code> servers. If this parameter is omitted, default is <var>m</var><code>(i) = 1</code> for all i. If this parameter is a scalar, it will be promoted to a vector with the same size as <var>lambda</var>. Otherwise, <var>m</var> must be a vector of length N. <br><dt><var>k</var><dd>Compute the steady-state probability that there are <var>k</var><code>(i)</code> requests at service center i. <var>k</var> must have the same length as <var>lambda</var>, with <var>k</var><code>(i) ≥ 0</code>. </dl> <p><strong>OUTPUT</strong> <dl> <dt><var>U</var><dd>If i is a FCFS node, then <var>U</var><code>(i)</code> is the utilization of service center i. If i is an IS node, then <var>U</var><code>(i)</code> is the <em>traffic intensity</em> defined as <var>X</var><code>(i)*</code><var>S</var><code>(i)</code>. <br><dt><var>R</var><dd><var>R</var><code>(i)</code> is the average response time of service center i. <br><dt><var>Q</var><dd><var>Q</var><code>(i)</code> is the average number of customers in service center i. <br><dt><var>X</var><dd><var>X</var><code>(i)</code> is the throughput of service center i. <br><dt><var>pr</var><dd><var>pr</var><code>(i)</code> is the steady state probability that there are <var>k</var><code>(i)</code> requests at service center i. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnopen. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>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</cite>, Wiley, 1998, pp. 284–287. <p><a name="index-Bolch_002c-G_002e-105"></a><a name="index-Greiner_002c-S_002e-106"></a><a name="index-de-Meer_002c-H_002e-107"></a><a name="index-Trivedi_002c-K_002e-108"></a> <h4 class="subsection">6.3.2 The Convolution Algorithm</h4> <p>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: <pre class="example"> k = [k1, k2, ... kn]; <span class="roman">population vector</span> p = 1/G(N+1) \prod F(i,k); </pre> <p>Here \pi(k_1, k_2, \ldots k_K) is the joint probability of having k_i requests at node i, for all i=1,2, \ldots K. <p>The <em>convolution algorithms</em> computes the normalization constants G = (G(0), G(1), \ldots G(N)) for single-class, closed networks with N requests. The normalization constants are returned as vector <var>G</var><code>=[</code><var>G</var><code>(1), </code><var>G</var><code>(2), ... </code><var>G</var><code>(N+1)]</code> where <var>G</var><code>(i+1)</code> 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). <p><code>queueing</code> implements the convolution algorithm, in the function <code>qnconvolution</code> and <code>qnconvolutionld</code>. The first one supports single-station nodes, multiple-station nodes and IS nodes. The second one supports networks with general load-dependent service centers. <!-- The Convolution Algorithm --> <p><a name="doc_002dqnconvolution"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>G</var>] = <b>qnconvolution</b> (<var>N, S, V</var>)<var><a name="index-qnconvolution-109"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>G</var>] = <b>qnconvolution</b> (<var>N, S, V, m</var>)<var><a name="index-qnconvolution-110"></a></var><br> <blockquote> <p><a name="index-closed-network-111"></a><a name="index-normalization-constant-112"></a><a name="index-convolution-algorithm-113"></a> This function implements the <em>convolution algorithm</em> for computing steady-state performance measures of product-form, single-class closed queueing networks. Load-independent service centers, multiple servers (M/M/m queues) and IS nodes are supported. For general load-dependent service centers, use the <code>qnconvolutionld</code> function instead. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Number of requests in the system (<var>N</var><code>>0</code>). <br><dt><var>S</var><dd><var>S</var><code>(k)</code> is the average service time on center k (<var>S</var><code>(k) ≥ 0</code>). <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the visit count of service center k (<var>V</var><code>(k) ≥ 0</code>). <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at center k. If <var>m</var><code>(k) < 1</code>, center k is a delay center (IS); if <var>m</var><code>(k) ≥ 1</code>, center k it is a regular M/M/m queueing center with <var>m</var><code>(k)</code> identical servers. Default is <var>m</var><code>(k) = 1</code> for all k. </dl> <p><strong>OUTPUT</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(k)</code> is the utilization of center k. For IS nodes, <var>U</var><code>(k)</code> is the <em>traffic intensity</em>. <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the average response time of center k. <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of customers at center k. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of center k. <br><dt><var>G</var><dd>Vector of normalization constants. <var>G</var><code>(n+1)</code> contains the value of the normalization constant with n requests G(n), n=0, <small class="dots">...</small>, N. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnconvolutionld. </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <p>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. Let <var>k</var><code>=[k_1, k_2, ... k_K]</code> be a valid population vector. Then, the steady-state probability <var>p</var><code>(i)</code> to have <var>k</var><code>(i)</code> requests at service center i can be computed as: <pre class="example"><pre class="verbatim"> k = [1 2 0]; K = sum(k); # 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] = qnconvolution( K, S, V, m ); p = [0 0 0]; # initialize p # Compute the probability to have k(i) jobs at service center i for i=1:3 p(i) = (V(i)*S(i))^k(i) / G(K+1) * \ (G(K-k(i)+1) - V(i)*S(i)*G(K-k(i)) ); printf("k(%d)=%d prob=%f\n", i, k(i), p(i) ); endfor</pre>-| k(1)=1 prob=0.17975 -| k(2)=2 prob=0.48404 -| k(3)=0 prob=0.52779 </pre> <p class="noindent"><strong>NOTE</strong> <p>For a network with K service centers and N requests, this implementation of the convolution algorithm has time and space complexity O(NK). <p class="noindent"><strong>REFERENCES</strong> <p>Jeffrey P. Buzen, <cite>Computational Algorithms for Closed Queueing Networks with Exponential Servers</cite>, Communications of the ACM, volume 16, number 9, september 1973, pp. 527–531. <a href="http://doi.acm.org/10.1145/362342.362345">http://doi.acm.org/10.1145/362342.362345</a> <p><a name="index-Buzen_002c-J_002e-P_002e-114"></a> 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</cite>, Wiley, 1998, pp. 313–317. <p><a name="index-Bolch_002c-G_002e-115"></a><a name="index-Greiner_002c-S_002e-116"></a><a name="index-de-Meer_002c-H_002e-117"></a><a name="index-Trivedi_002c-K_002e-118"></a> <!-- Convolution for load-dependent service centers --> <a name="doc_002dqnconvolutionld"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>G</var>] = <b>qnconvolutionld</b> (<var>N, S, V</var>)<var><a name="index-qnconvolutionld-119"></a></var><br> <blockquote> <p><a name="index-closed-network-120"></a><a name="index-normalization-constant-121"></a><a name="index-convolution-algorithm-122"></a><a name="index-load_002ddependent-service-center-123"></a> This function implements the <em>convolution algorithm</em> for product-form, single-class closed queueing networks with general load-dependent service centers. <p>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 <var>G</var><code>=[</code><var>G</var><code>(1), ..., </code><var>G</var><code>(N+1)]</code> where <var>G</var><code>(i+1)</code> is the value of G(i). <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Number of requests in the system (<var>N</var><code>>0</code>). <br><dt><var>S</var><dd><var>S</var><code>(k,n)</code> is the mean service time at center k where there are n requests, 1 ≤ n ≤ N. <var>S</var><code>(k,n)</code> = 1 / \mu_k,n, where \mu_k,n is the service rate of center k when there are n requests. <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the visit count of service center k (<var>V</var><code>(k) ≥ 0</code>). The length of <var>V</var> is the number of servers K in the network. </dl> <p><strong>OUTPUT</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(k)</code> is the utilization of center k. <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the average response time at center k. <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of customers in center k. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of center k. <br><dt><var>G</var><dd>Normalization constants (vector). <var>G</var><code>(n+1)</code> corresponds to G(n), as array indexes in Octave start from 1. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnconvolution. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Herb Schwetman, <cite>Some Computational Aspects of Queueing Network Models</cite>, Technical Report CSD-TR-354, Department of Computer Sciences, Purdue University, feb, 1981 (revised). <a href="http://www.cs.purdue.edu/research/technical_reports/1980/TR%2080-354.pdf">http://www.cs.purdue.edu/research/technical_reports/1980/TR%2080-354.pdf</a> <p><a name="index-Schwetman_002c-H_002e-124"></a> M. Reiser, H. Kobayashi, <cite>On The Convolution Algorithm for Separable Queueing Networks</cite>, 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. <a href="http://doi.acm.org/10.1145/800200.806187">http://doi.acm.org/10.1145/800200.806187</a> <p><a name="index-Reiser_002c-M_002e-125"></a><a name="index-Kobayashi_002c-H_002e-126"></a> 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</cite>, Wiley, 1998, pp. 313–317. Function <code>qnconvolutionld</code> 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()</code> in <code>qnconvolutionld</code> which has been made similar to function f_i defined in Schwetman, <code>Some Computational Aspects of Queueing Network Models</code>. <p><a name="index-Bolch_002c-G_002e-127"></a><a name="index-Greiner_002c-S_002e-128"></a><a name="index-de-Meer_002c-H_002e-129"></a><a name="index-Trivedi_002c-K_002e-130"></a> <h4 class="subsection">6.3.3 Open networks</h4> <!-- Open networks with single class --> <p><a name="doc_002dqnopensingle"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnopensingle</b> (<var>lambda, S, V</var>)<var><a name="index-qnopensingle-131"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnopensingle</b> (<var>lambda, S, V, m</var>)<var><a name="index-qnopensingle-132"></a></var><br> <blockquote> <p><a name="index-open-network_002c-single-class-133"></a><a name="index-BCMP-network-134"></a> Analyze open, single class BCMP queueing networks. <p>This function works for a subset of BCMP single-class open networks satisfying the following properties: <ul> <li>The allowed service disciplines at network nodes are: FCFS, PS, LCFS-PR, IS (infinite server); <li>Service times are exponentially distributed and load-independent; <li>Service center i can consist of <var>m</var><code>(i) ≥ 1</code> identical servers. <li>Routing is load-independent </ul> <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>Overall external arrival rate (<var>lambda</var><code>>0</code>). <br><dt><var>S</var><dd><var>S</var><code>(k)</code> is the average service time at center i (<var>S</var><code>(k)>0</code>). <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the average number of visits to center k (<var>V</var><code>(k) ≥ 0</code>). <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at center i. If <var>m</var><code>(k) < 1</code>, then service center k is a delay center (IS); otherwise it is a regular queueing center with <var>m</var><code>(k)</code> servers. Default is <var>m</var><code>(k) = 1</code> for each k. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If k is a queueing center, <var>U</var><code>(k)</code> is the utilization of center k. If k is an IS node, then <var>U</var><code>(k)</code> is the <em>traffic intensity</em> defined as <var>X</var><code>(k)*</code><var>S</var><code>(k)</code>. <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the average response time of center k. <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of requests at center k. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of center k. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnopen,qnclosed,qnvisits. </blockquote></div> <p>From the results computed by this function, it is possible to derive other quantities of interest as follows: <ul> <li><strong>System Response Time</strong>: The overall system response time can be computed as <code>R_s = dot(V,R);</code> <li><strong>Average number of requests</strong>: The average number of requests in the system can be computed as: <code>Q_s = sum(Q)</code> </ul> <p class="noindent"><strong>EXAMPLE</strong> <pre class="example"><pre class="verbatim"> lambda = 3; V = [16 7 8]; S = [0.01 0.02 0.03]; [U R Q X] = qnopensingle( lambda, S, V ); R_s = dot(R,V) # System response time N = sum(Q) # Average number in system</pre>-| R_s = 1.4062 -| N = 4.2186 </pre> <p class="noindent"><strong>REFERENCES</strong> <p>G. Bolch, S. Greiner, H. de Meer and K. Trivedi, <cite>Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications</cite>, Wiley, 1998. <p><a name="index-Bolch_002c-G_002e-135"></a><a name="index-Greiner_002c-S_002e-136"></a><a name="index-de-Meer_002c-H_002e-137"></a><a name="index-Trivedi_002c-K_002e-138"></a> <!-- Open network with multiple classes --> <p><a name="doc_002dqnopenmulti"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnopenmulti</b> (<var>lambda, S, V</var>)<var><a name="index-qnopenmulti-139"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnopenmulti</b> (<var>lambda, S, V, m</var>)<var><a name="index-qnopenmulti-140"></a></var><br> <blockquote> <p><a name="index-open-network_002c-multiple-classes-141"></a> Exact analysis of open, multiple-class BCMP networks. The network can be made of <em>single-server</em> queueing centers (FCFS, LCFS-PR or PS) or delay centers (IS). This function assumes a network with K service centers and C customer classes. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd><var>lambda</var><code>(c)</code> is the external arrival rate of class c customers (<var>lambda</var><code>(c)>0</code>). <br><dt><var>S</var><dd><var>S</var><code>(c,k)</code> is the mean service time of class c customers on the service center k (<var>S</var><code>(c,k)>0</code>). For FCFS nodes, average service times must be class-independent. <br><dt><var>V</var><dd><var>V</var><code>(c,k)</code> is the average number of visits of class c customers to service center k (<var>V</var><code>(c,k) ≥ 0 </code>). <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at service center k. Valid values are <var>m</var><code>(k) < 1</code> to denote a delay center (-/G/\infty), and <var>m</var><code>(k)==1</code> to denote a single server queueing center (M/M/1–FCFS, -/G/1–LCFS-PR or -/G/1–PS). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If k is a queueing center, then <var>U</var><code>(c,k)</code> is the class c utilization of center k. If k is an IS node, then <var>U</var><code>(c,k)</code> is the class c <em>traffic intensity</em> defined as <var>X</var><code>(c,k)*</code><var>S</var><code>(c,k)</code>. <br><dt><var>R</var><dd><var>R</var><code>(c,k)</code> is the class c response time at center k. The system response time for class c requests can be computed as <code>dot(</code><var>R</var><code>, </code><var>V</var><code>, 2)</code>. <br><dt><var>Q</var><dd><var>Q</var><code>(c,k)</code> is the average number of class c requests at center k. The average number of class c requests in the system <var>Qc</var> can be computed as <code>Qc = sum(</code><var>Q</var><code>, 2)</code> <br><dt><var>X</var><dd><var>X</var><code>(c,k)</code> is the class c throughput at center k. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnopen,qnopensingle,qnvisits. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, <cite>Quantitative System Performance: Computer System Analysis Using Queueing Network Models</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. In particular, see section 7.4.1 ("Open Model Solution Techniques"). <p><a name="index-Lazowska_002c-E_002e-D_002e-142"></a><a name="index-Zahorjan_002c-J_002e-143"></a><a name="index-Graham_002c-G_002e-S_002e-144"></a><a name="index-Sevcik_002c-K_002e-C_002e-145"></a> <h4 class="subsection">6.3.4 Closed Networks</h4> <!-- MVA for single class, closed networks --> <p><a name="doc_002dqnclosedsinglemva"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>G</var>] = <b>qnclosedsinglemva</b> (<var>N, S, V</var>)<var><a name="index-qnclosedsinglemva-146"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>G</var>] = <b>qnclosedsinglemva</b> (<var>N, S, V, m</var>)<var><a name="index-qnclosedsinglemva-147"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>, <var>G</var>] = <b>qnclosedsinglemva</b> (<var>N, S, V, m, Z</var>)<var><a name="index-qnclosedsinglemva-148"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-149"></a><a name="index-closed-network_002c-single-class-150"></a><a name="index-normalization-constant-151"></a> 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>qnclosedsinglemvald</code> instead. <p>Additionally, the normalization constant G(n), n=0, <small class="dots">...</small>, N is computed; G(n) can be used in conjunction with the BCMP theorem to compute steady-state probabilities. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Population size (number of requests in the system, <var>N</var><code> ≥ 0</code>). If <var>N</var><code> == 0</code>, this function returns <var>U</var><code> = </code><var>R</var><code> = </code><var>Q</var><code> = </code><var>X</var><code> = 0</code> <br><dt><var>S</var><dd><var>S</var><code>(k)</code> is the mean service time on server k (<var>S</var><code>(k)>0</code>). <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the average number of visits to service center k (<var>V</var><code>(k) ≥ 0</code>). <br><dt><var>Z</var><dd>External delay for customers (<var>Z</var><code> ≥ 0</code>). Default is 0. <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at center k (if <var>m</var> is a scalar, all centers have that number of servers). If <var>m</var><code>(k) < 1</code>, center k is a delay center (IS); otherwise it is a regular queueing center (FCFS, LCFS-PR or PS) with <var>m</var><code>(k)</code> servers. Default is <var>m</var><code>(k) = 1</code> for all k (each service center has a single server). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If k is a FCFS, LCFS-PR or PS node (<var>m</var><code>(k) == 1</code>), then <var>U</var><code>(k)</code> is the utilization of center k. If k is an IS node (<var>m</var><code>(k) < 1</code>), then <var>U</var><code>(k)</code> is the <em>traffic intensity</em> defined as <var>X</var><code>(k)*</code><var>S</var><code>(k)</code>. <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the response time at center k. The <em>Residence Time</em> at center k is <var>R</var><code>(k) * </code><var>V</var><code>(k)</code>. The system response time <var>Rsys</var> can be computed either as <var>Rsys</var><code> = </code><var>N</var><code>/</code><var>Xsys</var><code> - Z</code> or as <var>Rsys</var><code> = dot(</code><var>R</var><code>,</code><var>V</var><code>)</code> <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of requests at center k. The number of requests in the system can be computed either as <code>sum(</code><var>Q</var><code>)</code>, or using the formula <var>N</var><code>-</code><var>Xsys</var><code>*</code><var>Z</var>. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of center k. The system throughput <var>Xsys</var> can be computed as <var>Xsys</var><code> = </code><var>X</var><code>(1) / </code><var>V</var><code>(1)</code> <br><dt><var>G</var><dd>Normalization constants. <var>G</var><code>(n+1)</code> corresponds to the value of the normalization constant G(n), n=0, <small class="dots">...</small>, 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. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedsinglemvald. </blockquote></div> <p>From the results provided by this function, it is possible to derive other quantities of interest as follows: <p class="noindent"><strong>EXAMPLE</strong> <pre class="example"><pre class="verbatim"> S = [ 0.125 0.3 0.2 ]; V = [ 16 10 5 ]; N = 20; m = ones(1,3); Z = 4; [U R Q X] = qnclosedsinglemva(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 );</pre></pre> <p class="noindent"><strong>REFERENCES</strong> <p>M. Reiser and S. S. Lavenberg, <cite>Mean-Value Analysis of Closed Multichain Queuing Networks</cite>, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. <a href="http://doi.acm.org/10.1145/322186.322195">http://doi.acm.org/10.1145/322186.322195</a> <p><a name="index-Reiser_002c-M_002e-152"></a><a name="index-Lavenberg_002c-S_002e-S_002e-153"></a> This implementation is described in R. Jain , <cite>The Art of Computer Systems Performance Analysis</cite>, Wiley, 1991, p. 577. Multi-server nodes <!-- and the computation of @math{G(N)}, --> 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</cite>, Wiley, 1998, Section 8.2.1, "Single Class Queueing Networks". <p><a name="index-Jain_002c-R_002e-154"></a><a name="index-Bolch_002c-G_002e-155"></a><a name="index-Greiner_002c-S_002e-156"></a><a name="index-de-Meer_002c-H_002e-157"></a><a name="index-Trivedi_002c-K_002e-158"></a> <!-- MVA for single class, closed networks with load dependent servers --> <a name="doc_002dqnclosedsinglemvald"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvald</b> (<var>N, S, V</var>)<var><a name="index-qnclosedsinglemvald-159"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvald</b> (<var>N, S, V, Z</var>)<var><a name="index-qnclosedsinglemvald-160"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-161"></a><a name="index-closed-network_002c-single-class-162"></a><a name="index-load_002ddependent-service-center-163"></a> Exact MVA algorithm for closed, single class queueing networks with load-dependent service centers. This function supports FCFS, LCFS-PR, PS and IS nodes. For networks with only fixed-rate service centers and multiple-server nodes, the function <code>qnclosedsinglemva</code> is more efficient. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Population size (number of requests in the system, <var>N</var><code> ≥ 0</code>). If <var>N</var><code> == 0</code>, this function returns <var>U</var><code> = </code><var>R</var><code> = </code><var>Q</var><code> = </code><var>X</var><code> = 0</code> <br><dt><var>S</var><dd><var>S</var><code>(k,n)</code> is the mean service time at center k where there are n requests, 1 ≤ n ≤ N. <var>S</var><code>(k,n)</code> = 1 / \mu_k,n, where \mu_k,n is the service rate of center k when there are n requests. <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the average number of visits to service center k (<var>V</var><code>(k) ≥ 0</code>). <br><dt><var>Z</var><dd>external delay ("think time", <var>Z</var><code> ≥ 0</code>); default 0. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(k)</code> is the 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. <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the response time on service center k. <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of requests in service center k. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of service center k. </dl> </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>M. Reiser and S. S. Lavenberg, <cite>Mean-Value Analysis of Closed Multichain Queuing Networks</cite>, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. <a href="http://doi.acm.org/10.1145/322186.322195">http://doi.acm.org/10.1145/322186.322195</a> <p>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</cite>, Wiley, 1998, Section 8.2.4.1, “Networks with Load-Deèpendent Service: Closed Networks”. <p><a name="index-Bolch_002c-G_002e-164"></a><a name="index-Greiner_002c-S_002e-165"></a><a name="index-de-Meer_002c-H_002e-166"></a><a name="index-Trivedi_002c-K_002e-167"></a> <!-- CMVA for single class, closed networks with a single load dependent servers --> <a name="doc_002dqncmva"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qncmva</b> (<var>N, S, Sld, V</var>)<var><a name="index-qncmva-168"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qncmva</b> (<var>N, S, Sld, V, Z</var>)<var><a name="index-qncmva-169"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-170"></a><a name="index-CMVA-171"></a> Implementation of the Conditional MVA (CMVA) algorithm, a numerically stable variant of MVA for load-dependent servers. CMVA is described in G. Casale, <cite>A Note on Stable Flow-Equivalent Aggregation in Closed Networks</cite>. The network is made of M service centers and a delay center. Servers 1, \ldots, M-1 are load-independent; server M is load-dependent. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Population size (number of requests in the system, <var>N</var><code> ≥ 0</code>). If <var>N</var><code> == 0</code>, this function returns <var>U</var><code> = </code><var>R</var><code> = </code><var>Q</var><code> = </code><var>X</var><code> = 0</code> <br><dt><var>S</var><dd><var>S</var><code>(k)</code> is the mean service time on server k = 1, <small class="dots">...</small>, M-1 (<var>S</var><code>(k) > 0</code>). <br><dt><var>Sld</var><dd><var>Sld</var><code>(n)</code> is the mean service time on server M when there are n requests, n=1, <small class="dots">...</small>, N. <var>Sld</var><code>(n) = </code> 1 / \mu(n), where \mu(n) is the service rate at center N when there are n requests. <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the average number of visits to service center k= 1, <small class="dots">...</small>, M (<var>V</var><code>(k) ≥ 0</code>). <br><dt><var>Z</var><dd>External delay for customers (<var>Z</var><code> ≥ 0</code>). Default is 0. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(k)</code> is the utilization of center k=1, <small class="dots">...</small>, M <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the response time at center k=1, <small class="dots">...</small>, M. The system response time <var>Rsys</var> can be computed as <var>Rsys</var><code> = </code><var>N</var><code>/</code><var>Xsys</var><code> - Z</code> <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of requests at center k=1, <small class="dots">...</small>, M. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of center k=1, <small class="dots">...</small>, M. </dl> </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>G. Casale. <cite>A note on stable flow-equivalent aggregation in closed networks</cite>. Queueing Syst. Theory Appl., 60:193–202, December 2008. <p><a name="index-Casale_002c-G_002e-172"></a> <!-- Approximate MVA for single class, closed networks --> <p><a name="doc_002dqnclosedsinglemvaapprox"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvaapprox</b> (<var>N, S, V</var>)<var><a name="index-qnclosedsinglemvaapprox-173"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvaapprox</b> (<var>N, S, V, m</var>)<var><a name="index-qnclosedsinglemvaapprox-174"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvaapprox</b> (<var>N, S, V, m, Z</var>)<var><a name="index-qnclosedsinglemvaapprox-175"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvaapprox</b> (<var>N, S, V, m, Z, tol</var>)<var><a name="index-qnclosedsinglemvaapprox-176"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedsinglemvaapprox</b> (<var>N, S, V, m, Z, tol, iter_max</var>)<var><a name="index-qnclosedsinglemvaapprox-177"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-178"></a><a name="index-Approximate-MVA-179"></a><a name="index-Closed-network_002c-single-class-180"></a><a name="index-Closed-network_002c-approximate-analysis-181"></a> 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 <code>qnclosedsinglemvald</code> instead. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Population size (number of requests in the system, <var>N</var><code> > 0</code>). <br><dt><var>S</var><dd><var>S</var><code>(k)</code> is the mean service time on server k (<var>S</var><code>(k)>0</code>). <br><dt><var>V</var><dd><var>V</var><code>(k)</code> is the average number of visits to service center k (<var>V</var><code>(k) ≥ 0</code>). <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at center k (if <var>m</var> is a scalar, all centers have that number of servers). If <var>m</var><code>(k) < 1</code>, center k is a delay center (IS); if <var>m</var><code>(k) == 1</code>, center k is a regular queueing center (FCFS, LCFS-PR or PS) with one server (default). This function does not support multiple server nodes (<var>m</var><code>(k) > 1</code>). <br><dt><var>Z</var><dd>External delay for customers (<var>Z</var><code> ≥ 0</code>). Default is 0. <br><dt><var>tol</var><dd>Stopping tolerance. The algorithm stops when the maximum relative difference between the new and old value of the queue lengths <var>Q</var> becomes less than the tolerance. Default is 10^-5. <br><dt><var>iter_max</var><dd>Maximum number of iterations (<var>iter_max</var><code>>0</code>. The function aborts if convergenge is not reached within the maximum number of iterations. Default is 100. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If k is a FCFS, LCFS-PR or PS node (<var>m</var><code>(k) == 1</code>), then <var>U</var><code>(k)</code> is the utilization of center k. If k is an IS node (<var>m</var><code>(k) < 1</code>), then <var>U</var><code>(k)</code> is the <em>traffic intensity</em> defined as <var>X</var><code>(k)*</code><var>S</var><code>(k)</code>. <br><dt><var>R</var><dd><var>R</var><code>(k)</code> is the response time at center k. The system response time <var>Rsys</var> can be computed as <var>Rsys</var><code> = </code><var>N</var><code>/</code><var>Xsys</var><code> - Z</code> <br><dt><var>Q</var><dd><var>Q</var><code>(k)</code> is the average number of requests at center k. The number of requests in the system can be computed either as <code>sum(</code><var>Q</var><code>)</code>, or using the formula <var>N</var><code>-</code><var>Xsys</var><code>*</code><var>Z</var>. <br><dt><var>X</var><dd><var>X</var><code>(k)</code> is the throughput of center k. The system throughput <var>Xsys</var> can be computed as <var>Xsys</var><code> = </code><var>X</var><code>(1) / </code><var>V</var><code>(1)</code> </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedsinglemva,qnclosedsinglemvald. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>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</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. In particular, see section 6.4.2.2 ("Approximate Solution Techniques"). <p><a name="index-Lazowska_002c-E_002e-D_002e-182"></a><a name="index-Zahorjan_002c-J_002e-183"></a><a name="index-Graham_002c-G_002e-S_002e-184"></a><a name="index-Sevcik_002c-K_002e-C_002e-185"></a> <!-- MVA for multiple class, closed networks --> <p><a name="doc_002dqnclosedmultimva"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimva</b> (<var>N, S </var>)<var><a name="index-qnclosedmultimva-186"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimva</b> (<var>N, S, V</var>)<var><a name="index-qnclosedmultimva-187"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimva</b> (<var>N, S, V, m</var>)<var><a name="index-qnclosedmultimva-188"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimva</b> (<var>N, S, V, m, Z</var>)<var><a name="index-qnclosedmultimva-189"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimva</b> (<var>N, S, P</var>)<var><a name="index-qnclosedmultimva-190"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimva</b> (<var>N, S, P, m</var>)<var><a name="index-qnclosedmultimva-191"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-192"></a><a name="index-closed-network_002c-multiple-classes-193"></a> Analyze closed, multiclass queueing networks with K service centers and C independent customer classes (chains) using the Mean Value Analysys (MVA) algorithm. <p>Queueing policies at service centers can be any of the following: <dl> <dt><strong>FCFS</strong><dd>(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. <br><dt><strong>PS</strong><dd>(Processor Sharing) customers are served in parallel by a single server, each customer receiving an equal share of the service rate. <br><dt><strong>LCFS-PR</strong><dd>(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. <br><dt><strong>IS</strong><dd>(Infinite Server) customers are delayed independently of other customers at the service center (there is effectively an infinite number of servers). </dl> <blockquote> <b>Note:</b> If this function is called specifying the visit ratios <var>V</var>, class switching is <strong>not</strong> allowed. <p>If this function is called specifying the routing probability matrix <var>P</var>, then class switching <strong>is</strong> allowed; however, in this case all nodes are restricted to be fixed rate service centers or delay centers: multiple-server and general load-dependent centers are not supported.</blockquote> <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd><var>N</var><code>(c)</code> is the number of class c requests in the system; <var>N</var><code>(c) ≥ 0</code>. If class c has no requests (<var>N</var><code>(c) = 0</code>), then <var>U</var><code>(c,k) = </code><var>R</var><code>(c,k) = </code><var>Q</var><code>(c,k) = </code><var>X</var><code>(c,k) = 0</code> for all <var>k</var>. <br><dt><var>S</var><dd><var>S</var><code>(c,k)</code> is the mean service time for class c customers at center k (<var>S</var><code>(c,k) ≥ 0</code>). If service time at center k is class-dependent, then center #mathk is assumed to be of type -/G/1–PS (Processor Sharing). If center k is a FCFS node (<var>m</var><code>(k)>1</code>), then the service times <strong>must</strong> be class-independent. <br><dt><var>V</var><dd><var>V</var><code>(c,k)</code> is the average number of visits of class c customers to service center k; <var>V</var><code>(c,k) ≥ 0</code>, default is 1. <strong>If you pass this parameter, class switching is not allowed</strong> <br><dt><var>P</var><dd><var>P</var><code>(r,i,s,j)</code> is the probability that a class r job completing service at center i is routed to center j as a class s job. <strong>If you pass this parameter, class switching is allowed</strong>. <br><dt><var>m</var><dd>If <var>m</var><code>(k)<1</code>, then center k is assumed to be a delay center (IS node -/G/\infty). If <var>m</var><code>(k)==1</code>, then service center k is a regular queueing center (M/M/1–FCFS, -/G/1–LCFS-PR or -/G/1–PS). Finally, if <var>m</var><code>(k)>1</code>, center k is a M/M/m–FCFS center with <var>m</var><code>(k)</code> identical servers. Default is <var>m</var><code>(k)=1</code> for each k. <br><dt><var>Z</var><dd><var>Z</var><code>(c)</code> is the class c external delay (think time); <var>Z</var><code>(c) ≥ 0</code>. Default is 0. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If k is a FCFS, LCFS-PR or PS node, then <var>U</var><code>(c,k)</code> is the class c utilization at center k. If k is an IS node, then <var>U</var><code>(c,k)</code> is the class c <em>traffic intensity</em> at center k, defined as <var>U</var><code>(c,k) = </code><var>X</var><code>(c,k)*</code><var>S</var><code>(c,k)</code>. <br><dt><var>R</var><dd><var>R</var><code>(c,k)</code> is the class c response time at center k. The class c <em>residence time</em> at center k is <var>R</var><code>(c,k) * </code><var>C</var><code>(c,k)</code>. The total class c system response time is <code>dot(</code><var>R</var><code>, </code><var>V</var><code>, 2)</code>. <br><dt><var>Q</var><dd><var>Q</var><code>(c,k)</code> is the average number of class c requests at center k. The total number of requests at center k is <code>sum(</code><var>Q</var><code>(:,k))</code>. The total number of class c requests in the system is <code>sum(</code><var>Q</var><code>(c,:))</code>. <br><dt><var>X</var><dd><var>X</var><code>(c,k)</code> is the class c throughput at center k. The class c system throughput can be computed as <var>X</var><code>(c,1) / </code><var>V</var><code>(c,1)</code>. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosed, qnclosedmultimvaapprox. </blockquote></div> <p class="noindent"><strong>NOTE</strong> <p>Given a network with K service centers, C job classes and population vector \bf N=(N_1, N_2, \ldots N_C), 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 <samp><span class="command">qnclosedmultimvaapprox</span></samp> function, which implements the approximate MVA algorithm. Note however that <samp><span class="command">qnclosedmultimvaapprox</span></samp> will only provide approximate results. <p class="noindent"><strong>REFERENCES</strong> <p>M. Reiser and S. S. Lavenberg, <cite>Mean-Value Analysis of Closed Multichain Queuing Networks</cite>, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. <a href="http://doi.acm.org/10.1145/322186.322195">http://doi.acm.org/10.1145/322186.322195</a> <p><a name="index-Reiser_002c-M_002e-194"></a><a name="index-Lavenberg_002c-S_002e-S_002e-195"></a> 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</cite>, 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</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. In particular, see section 7.4.2.1 ("Exact Solution Techniques"). <p><a name="index-Bolch_002c-G_002e-196"></a><a name="index-Greiner_002c-S_002e-197"></a><a name="index-de-Meer_002c-H_002e-198"></a><a name="index-Trivedi_002c-K_002e-199"></a><a name="index-Lazowska_002c-E_002e-D_002e-200"></a><a name="index-Zahorjan_002c-J_002e-201"></a><a name="index-Graham_002c-G_002e-S_002e-202"></a><a name="index-Sevcik_002c-K_002e-C_002e-203"></a> <!-- Approximate MVA, with Bard-Schweitzer approximation --> <a name="doc_002dqnclosedmultimvaapprox"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimvaapprox</b> (<var>N, S, V</var>)<var><a name="index-qnclosedmultimvaapprox-204"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimvaapprox</b> (<var>N, S, V, m</var>)<var><a name="index-qnclosedmultimvaapprox-205"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimvaapprox</b> (<var>N, S, V, m, Z</var>)<var><a name="index-qnclosedmultimvaapprox-206"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimvaapprox</b> (<var>N, S, V, m, Z, tol</var>)<var><a name="index-qnclosedmultimvaapprox-207"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosedmultimvaapprox</b> (<var>N, S, V, m, Z, tol, iter_max</var>)<var><a name="index-qnclosedmultimvaapprox-208"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-209"></a><a name="index-Approximate-MVA-210"></a><a name="index-Closed-network_002c-multiple-classes-211"></a><a name="index-Closed-network_002c-approximate-analysis-212"></a> Analyze closed, multiclass queueing networks with K service centers and C customer classes using the approximate Mean Value Analysys (MVA) algorithm. <p>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 approximately equal to the queue length with population set \bf N, times (n-1)/n: <pre class="example"> Q_i(N-1c) ~ (n-1)/n Q_i(N) </pre> <p>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. <p>This implementation works for networks made of infinite server (IS) nodes and single-server nodes only. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd><var>N</var><code>(c)</code> is the number of class c requests in the system (<var>N</var><code>(c)>0</code>). <br><dt><var>S</var><dd><var>S</var><code>(c,k)</code> is the mean service time for class c customers at center k (<var>S</var><code>(c,k) ≥ 0</code>). <br><dt><var>V</var><dd><var>V</var><code>(c,k)</code> is the average number of visits of class c requests to center k (<var>V</var><code>(c,k) ≥ 0</code>). <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at service center k. If <var>m</var><code>(k) < 1</code>, then the service center k is assumed to be a delay center (IS). If <var>m</var><code>(k) == 1</code>, 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. <br><dt><var>Z</var><dd><var>Z</var><code>(c)</code> is the class c external delay. Default is 0. <br><dt><var>tol</var><dd>Stopping tolerance (<var>tol</var><code>>0</code>). The algorithm stops if the queue length computed on two subsequent iterations are less than <var>tol</var>. Default is 10^-5. <br><dt><var>iter_max</var><dd>Maximum number of iterations (<var>iter_max</var><code>>0</code>. The function aborts if convergenge is not reached within the maximum number of iterations. Default is 100. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd>If k is a FCFS, LCFS-PR or PS node, then <var>U</var><code>(c,k)</code> is the utilization of class c requests on service center k. If k is an IS node, then <var>U</var><code>(c,k)</code> is the class c <em>traffic intensity</em> at device k, defined as <var>U</var><code>(c,k) = </code><var>X</var><code>(c)*</code><var>S</var><code>(c,k)</code> <br><dt><var>R</var><dd><var>R</var><code>(c,k)</code> is the response time of class c requests at service center k. <br><dt><var>Q</var><dd><var>Q</var><code>(c,k)</code> is the average number of class c requests at service center k. <br><dt><var>X</var><dd><var>X</var><code>(c,k)</code> is the class c throughput at service center k. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosed. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Y. Bard, <cite>Some Extensions to Multiclass Queueing Network Analysis</cite>, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, feb. 1979, pp. 51–62. <p><a name="index-Bard_002c-Y_002e-213"></a> P. Schweitzer, <cite>Approximate Analysis of Multiclass Closed Networks of Queues</cite>, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25–29. <p><a name="index-Schweitzer_002c-P_002e-214"></a> 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</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. 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. <p><a name="index-Lazowska_002c-E_002e-D_002e-215"></a><a name="index-Zahorjan_002c-J_002e-216"></a><a name="index-Graham_002c-G_002e-S_002e-217"></a><a name="index-Sevcik_002c-K_002e-C_002e-218"></a> <h4 class="subsection">6.3.5 Mixed Networks</h4> <!-- MVA for mixed networks --> <p><a name="doc_002dqnmix"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnmix</b> (<var>lambda, N, S, V, m</var>)<var><a name="index-qnmix-219"></a></var><br> <blockquote> <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-220"></a><a name="index-mixed-network-221"></a> Solution of mixed queueing networks through MVA. The network consists of K service centers (single-server or delay centers) and C independent customer chains. Both open and closed chains are possible. <var>lambda</var> is the vector of per-chain arrival rates (open classes); <var>N</var> is the vector of populations for closed chains. <blockquote> <b>Note:</b> In this implementation class switching is <strong>not</strong> allowed. Each customer class <em>must</em> correspond to an independent chain. </blockquote> <p>If the network is made of open or closed classes only, then this function calls <code>qnopenmulti</code> or <code>qnclosedmultimva</code> respectively, and prints a warning message. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dt><var>N</var><dd>For each customer chain c: <ul> <li>if c is a closed chain, then <var>N</var><code>(c)>0</code> is the number of class c requests and <var>lambda</var><code>(c)</code> must be zero; <li>If c is an open chain, <var>lambda</var><code>(c)>0</code> is the arrival rate of class c requests and <var>N</var><code>(c)</code> must be zero; </ul> <p class="noindent">For each c, the following must hold: <pre class="example"> (<var>lambda</var>(c)>0 && <var>N</var>(c)==0) || (<var>lambda</var>(c)==0 && <var>N</var>(c)>0) </pre> <p>which means that either <var>lambda</var><code>(c)</code> is nonzero and <var>N</var><code>(n)</code> is zero, or the other way around. If for some c, <var>lambda</var>(c) \neq 0 and <var>N</var>(c) \neq 0, an error is reported and this function aborts. <br><dt><var>S</var><dd><var>S</var><code>(c,k)</code> is the mean service time for class c customers on service center k, <var>S</var><code>(c,k) ≥ 0</code>. For FCFS nodes, service times must be class-independent. <br><dt><var>V</var><dd><var>V</var><code>(c,k)</code> is the average number of visits of class c customers to service center k (<var>V</var><code>(c,k) ≥ 0</code>). <br><dt><var>m</var><dd><var>m</var><code>(k)</code> is the number of servers at service center k. Only single-server (<var>m</var><code>(k)==1</code>) or IS (Infinite Server) nodes (<var>m</var><code>(k)<1</code>) are supported. If omitted, each service center is assumed to have a single server. Queueing discipline for single-server nodes can be FCFS, PS or LCFS-PR. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(c,k)</code> is the utilization of class c requests on service center k. <br><dt><var>R</var><dd><var>R</var><code>(c,k)</code> is the response time of class c requests on service center k. <br><dt><var>Q</var><dd><var>Q</var><code>(c,k)</code> is the average number of class c requests on service center k. <br><dt><var>X</var><dd><var>X</var><code>(c,k)</code> is the class c throughput on service center k. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedmultimva, qnopenmulti. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, <cite>Quantitative System Performance: Computer System Analysis Using Queueing Network Models</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. 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. <p><a name="index-Lazowska_002c-E_002e-D_002e-222"></a><a name="index-Zahorjan_002c-J_002e-223"></a><a name="index-Graham_002c-G_002e-S_002e-224"></a><a name="index-Sevcik_002c-K_002e-C_002e-225"></a> Herb Schwetman, <cite>Implementing the Mean Value Algorithm for the Solution of Queueing Network Models</cite>, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, feb 15, 1982, available at <a href="http://www.cs.purdue.edu/research/technical_reports/1980/TR%2080-355.pdf">http://www.cs.purdue.edu/research/technical_reports/1980/TR%2080-355.pdf</a> <p><a name="index-Schwetman_002c-H_002e-226"></a> <div class="node"> <a name="Algorithms-for-non-Product-form-QNs"></a> <a name="Algorithms-for-non-Product_002dform-QNs"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Bounds-on-performance">Bounds on performance</a>, Previous: <a rel="previous" accesskey="p" href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a>, Up: <a rel="up" accesskey="u" href="#Queueing-Networks">Queueing Networks</a> </div> <h3 class="section">6.4 Algorithms for non Product-Form QNs</h3> <!-- MVABLO algorithm for approximate analysis of closed, single class --> <!-- QN with blocking --> <p><a name="doc_002dqnmvablo"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnmvablo</b> (<var>N, S, M, P</var>)<var><a name="index-qnmvablo-227"></a></var><br> <blockquote> <p><a name="index-queueing-network-with-blocking-228"></a><a name="index-blocking-queueing-network-229"></a><a name="index-closed-network_002c-finite-capacity-230"></a> MVA algorithm for closed queueing networks with blocking. <samp><span class="command">qnmvablo</span></samp> computes approximate utilization, response time and mean queue length for closed, single class queueing networks with blocking. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>population size, i.e., number of requests in the system. <var>N</var> must be strictly greater than zero, and less than the overall network capacity: <code>0 < </code><var>N</var><code> < sum(</code><var>M</var><code>)</code>. <br><dt><var>S</var><dd>Average service time. <var>S</var><code>(i)</code> is the average service time requested on server i (<var>S</var><code>(i) > 0</code>). <br><dt><var>M</var><dd>Server capacity. <var>M</var><code>(i)</code> is the capacity of service center i. The capacity is the maximum number of requests in a service center, including the request currently in service (<var>M</var><code>(i) ≥ 1</code>). <br><dt><var>P</var><dd><var>P</var><code>(i,j)</code> is the probability that a request which completes service at server i will be transferred to server j. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(i)</code> is the utilization of service center i. <br><dt><var>R</var><dd><var>R</var><code>(i)</code> is the average response time of service center i. <br><dt><var>Q</var><dd><var>Q</var><code>(i)</code> is the average number of requests in service center i (including the request in service). <br><dt><var>X</var><dd><var>X</var><code>(i)</code> is the throughput of service center i. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnopen, qnclosed. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Ian F. Akyildiz, <cite>Mean Value Analysis for Blocking Queueing Networks</cite>, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418–428. <a href="http://dx.doi.org/10.1109/32.4663">http://dx.doi.org/10.1109/32.4663</a> <p><a name="index-Akyildiz_002c-I_002e-F_002e-231"></a> <a name="doc_002dqnmarkov"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnmarkov</b> (<var>lambda, S, C, P</var>)<var><a name="index-qnmarkov-232"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnmarkov</b> (<var>lambda, S, C, P, m</var>)<var><a name="index-qnmarkov-233"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnmarkov</b> (<var>N, S, C, P</var>)<var><a name="index-qnmarkov-234"></a></var><br> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnmarkov</b> (<var>N, S, C, P, m</var>)<var><a name="index-qnmarkov-235"></a></var><br> <blockquote> <p><a name="index-closed-network_002c-multiple-classes-236"></a><a name="index-closed-network_002c-finite-capacity-237"></a><a name="index-blocking-queueing-network-238"></a><a name="index-RS-blocking-239"></a> Compute utilization, response time, average queue length and throughput for open or closed queueing networks with finite capacity. 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. <p>More specifically, networks which can me analyzed by this function have the following properties: <ul> <li>There exists only a single class of customers. <li>The network has K service centers. Center i has m_i > 0 servers, and has a total (finite) capacity of C_i \geq m_i which includes both buffer space and servers. The buffer space at service center i is therefore C_i - m_i. <li>The network can be open, with external arrival rate to center i equal to \lambda_i, or closed with fixed population size N. For closed networks, the population size N must be strictly less than the network capacity: N < \sum_i C_i. <li>Average service times are load-independent. <li>P_ij 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 P_ij. <li>Blocking type is Repetitive-Service (RS). Service center j is <em>saturated</em> if the number of requests is equal to its capacity <code>C_j</code>. 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. </ul> <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dt><var>N</var><dd>If the first argument is a vector <var>lambda</var>, it is considered to be the external arrival rate <var>lambda</var><code>(i) ≥ 0</code> to service center i of an open network. If the first argument is a scalar, it is considered as the population size <var>N</var> of a closed network; in this case <var>N</var> must be strictly less than the network capacity: <var>N</var><code> < sum(</code><var>C</var><code>)</code>. <br><dt><var>S</var><dd><var>S</var><code>(i)</code> is the average service time at service center i <br><dt><var>C</var><dd><var>C</var><code>(i)</code> is the Capacity of service center i. The capacity includes both the buffer and server space <var>m</var><code>(i)</code>. Thus the buffer space is <var>C</var><code>(i)-</code><var>m</var><code>(i)</code>. <br><dt><var>P</var><dd><var>P</var><code>(i,j)</code> is the transition probability from service center i to service center j. <br><dt><var>m</var><dd><var>m</var><code>(i)</code> is the number of servers at service center i. Note that <var>m</var><code>(i) ≥ </code><var>C</var><code>(i)</code> for each <var>i</var>. If <var>m</var> is omitted, all service centers are assumed to have a single server (<var>m</var><code>(i) = 1</code> for all i). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>U</var><dd><var>U</var><code>(i)</code> is the utilization of service center i. <br><dt><var>R</var><dd><var>R</var><code>(i)</code> is the response time on service center i. <br><dt><var>Q</var><dd><var>Q</var><code>(i)</code> is the average number of customers in the service center i, <em>including</em> the request in service. <br><dt><var>X</var><dd><var>X</var><code>(i)</code> is the throughput of service center i. </dl> <blockquote> <b>Note:</b> The space complexity of this implementation is O( \prod_i=1^K (C_i + 1)^2). The time complexity is dominated by the time needed to solve a linear system with \prod_i=1^K (C_i + 1) unknowns. </blockquote> </blockquote></div> <div class="node"> <a name="Bounds-on-performance"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Utility-functions">Utility functions</a>, Previous: <a rel="previous" accesskey="p" href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a>, Up: <a rel="up" accesskey="u" href="#Queueing-Networks">Queueing Networks</a> </div> <h3 class="section">6.5 Bounds on performance</h3> <p><a name="doc_002dqnopenab"></a> <div class="defun"> — Function File: [<var>Xu</var>, <var>Rl</var>] = <b>qnopenab</b> (<var>lambda, D</var>)<var><a name="index-qnopenab-240"></a></var><br> <blockquote> <p><a name="index-bounds_002c-asymptotic-241"></a><a name="index-open-network-242"></a> Compute Asymptotic Bounds for single-class, open Queueing Networks with K service centers. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>overall arrival rate to the system (scalar). Abort if <var>lambda</var><code> ≤ 0</code> <br><dt><var>D</var><dd><var>D</var><code>(k)</code> is the service demand at center k. The service demand vector <var>D</var> must be nonempty, and all demands must be nonnegative (<var>D</var><code>(k) ≥ 0</code> for all k). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Xu</var><dd>Upper bound on the system throughput. <br><dt><var>Rl</var><dd>Lower bound on the system response time. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnopenbsb. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, <cite>Quantitative System Performance: Computer System Analysis Using Queueing Network Models</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. In particular, see section 5.2 ("Asymptotic Bounds"). <p><a name="index-Lazowska_002c-E_002e-D_002e-243"></a><a name="index-Zahorjan_002c-J_002e-244"></a><a name="index-Graham_002c-G_002e-S_002e-245"></a><a name="index-Sevcik_002c-K_002e-C_002e-246"></a> <a name="doc_002dqnclosedab"></a> <div class="defun"> — Function File: [<var>Xl</var>, <var>Xu</var>, <var>Rl</var>, <var>Ru</var>] = <b>qnclosedab</b> (<var>N, D</var>)<var><a name="index-qnclosedab-247"></a></var><br> — Function File: [<var>Xl</var>, <var>Xu</var>, <var>Rl</var>, <var>Ru</var>] = <b>qnclosedab</b> (<var>N, D, Z</var>)<var><a name="index-qnclosedab-248"></a></var><br> <blockquote> <p><a name="index-bounds_002c-asymptotic-249"></a><a name="index-closed-network-250"></a> Compute Asymptotic Bounds for single-class, closed Queueing Networks with K service centers. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>number of requests in the system (scalar, <var>N</var><code>>0</code>). <br><dt><var>D</var><dd><var>D</var><code>(k)</code> is the service demand of service center k, <var>D</var><code>(k) ≥ 0</code>. <br><dt><var>Z</var><dd>external delay (think time, scalar, <var>Z</var><code> ≥ 0</code>). If omitted, it is assumed to be zero. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Xl</var><dt><var>Xu</var><dd>Lower and upper bound on the system throughput. <br><dt><var>Rl</var><dt><var>Ru</var><dd>Lower and upper bound on the system response time. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedbsb, qnclosedgb, qnclosedpb. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p class="noindent">Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, <cite>Quantitative System Performance: Computer System Analysis Using Queueing Network Models</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. In particular, see section 5.2 ("Asymptotic Bounds"). <p><a name="index-Lazowska_002c-E_002e-D_002e-251"></a><a name="index-Zahorjan_002c-J_002e-252"></a><a name="index-Graham_002c-G_002e-S_002e-253"></a><a name="index-Sevcik_002c-K_002e-C_002e-254"></a> <p><a name="doc_002dqnopenbsb"></a> <div class="defun"> — Function File: [<var>Xu</var>, <var>Rl</var>, <var>Ru</var>] = <b>qnopenbsb</b> (<var>lambda, D</var>)<var><a name="index-qnopenbsb-255"></a></var><br> <blockquote> <p><a name="index-bounds_002c-balanced-system-256"></a><a name="index-open-network-257"></a> Compute Balanced System Bounds for single-class, open Queueing Networks with K service centers. <p><strong>INPUTS</strong> <dl> <dt><var>lambda</var><dd>overall arrival rate to the system (scalar). Abort if <var>lambda</var><code> < 0 </code> <br><dt><var>D</var><dd><var>D</var><code>(k)</code> is the service demand at center k. The service demand vector <var>D</var> must be nonempty, and all demands must be nonnegative (<var>D</var><code>(k) ≥ 0</code> for all k). </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Xl</var><dd>Lower bound on the system throughput. <br><dt><var>Rl</var><dt><var>Ru</var><dd>Lower and upper bound on the system response time. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnopenab. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, <cite>Quantitative System Performance: Computer System Analysis Using Queueing Network Models</cite>, Prentice Hall, 1984. <a href="http://www.cs.washington.edu/homes/lazowska/qsp/">http://www.cs.washington.edu/homes/lazowska/qsp/</a>. In particular, see section 5.4 ("Balanced Systems Bounds"). <p><a name="index-Lazowska_002c-E_002e-D_002e-258"></a><a name="index-Zahorjan_002c-J_002e-259"></a><a name="index-Graham_002c-G_002e-S_002e-260"></a><a name="index-Sevcik_002c-K_002e-C_002e-261"></a> <a name="doc_002dqnclosedbsb"></a> <div class="defun"> — Function File: [<var>Xl</var>, <var>Xu</var>, <var>Rl</var>, <var>Ru</var>] = <b>qnclosedbsb</b> (<var>N, D</var>)<var><a name="index-qnclosedbsb-262"></a></var><br> — Function File: [<var>Xl</var>, <var>Xu</var>, <var>Rl</var>, <var>Ru</var>] = <b>qnclosedbsb</b> (<var>N, D, Z</var>)<var><a name="index-qnclosedbsb-263"></a></var><br> <blockquote> <p><a name="index-bounds_002c-balanced-system-264"></a><a name="index-closed-network-265"></a> Compute Balanced System Bounds for single-class, closed Queueing Networks with K service centers. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>number of requests in the system (scalar). <br><dt><var>D</var><dd><var>D</var><code>(k)</code> is the service demand at center k; <var>K</var><code>(k) ≥ 0</code>. <br><dt><var>Z</var><dd>external delay (think time, scalar, <var>Z</var><code> ≥ 0</code>). If omitted, it is assumed to be zero. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Xl</var><dt><var>Xu</var><dd>Lower and upper bound on the system throughput. <br><dt><var>Rl</var><dt><var>Ru</var><dd>Lower and upper bound on the system response time. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedab, qnclosedgb, qnclosedpb. </blockquote></div> <p><a name="doc_002dqnclosedpb"></a> <div class="defun"> — Function File: [<var>Xl</var>, <var>Xu</var>] = <b>qnclosedpb</b> (<var>N, D </var>)<var><a name="index-qnclosedpb-266"></a></var><br> <blockquote> <p>Compute PB Bounds (C. H. Hsieh and S. Lam, 1987) for single-class, closed Queueing Networks with K service centers. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>number of requests in the system (scalar). Must be <var>N</var><code> > 0</code>. <br><dt><var>D</var><dd><var>D</var><code>(k)</code> is the service demand of service center k. Must be <var>D</var><code>(k) ≥ 0</code> for all k. <br><dt><var>Z</var><dd>external delay (think time, scalar). If omitted, it is assumed to be zero. Must be <var>Z</var><code> ≥ 0</code>. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Xl</var><dt><var>Xu</var><dd>Lower and upper bounds on the system throughput. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedab, qbclosedbsb, qnclosedgb. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>The original paper describing PB Bounds is C. H. Hsieh and S. Lam, <cite>Two classes of performance bounds for closed queueing networks</cite>, PEVA, vol. 7, n. 1, pp. 3–30, 1987 <p>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</cite>, IEEE Transactions on Computers, 57(6):780-794, June 2008. <p><a name="index-Hsieh_002c-C_002e-H-267"></a><a name="index-Lam_002c-S_002e-268"></a><a name="index-Casale_002c-G_002e-269"></a><a name="index-Muntz_002c-R_002e-R_002e-270"></a><a name="index-Serazzi_002c-G_002e-271"></a> <a name="doc_002dqnclosedgb"></a> <div class="defun"> — Function File: [<var>Xl</var>, <var>Xu</var>, <var>Ql</var>, <var>Qu</var>] = <b>qnclosedgb</b> (<var>N, D, Z</var>)<var><a name="index-qnclosedgb-272"></a></var><br> <blockquote> <p><a name="index-bounds_002c-geometric-273"></a><a name="index-closed-network-274"></a> Compute Geometric Bounds (GB) for single-class, closed Queueing Networks. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>number of requests in the system (scalar, <var>N</var><code> > 0</code>). <br><dt><var>D</var><dd><var>D</var><code>(k)</code> is the service demand of service center k (<var>D</var><code>(k) ≥ 0</code>). <br><dt><var>Z</var><dd>external delay (think time, scalar). If omitted, it is assumed to be zero. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>Xl</var><dt><var>Xu</var><dd>Lower and upper bound on the system throughput. If <var>Z</var><code>>0</code>, these bounds are computed using <em>Geometric Square-root Bounds</em> (GSB). If <var>Z</var><code>==0</code>, these bounds are computed using <em>Geometric Bounds</em> (GB) <br><dt><var>Ql</var><dt><var>Qu</var><dd><var>Ql</var><code>(i)</code> and <var>Qu</var><code>(i)</code> are the lower and upper bounds respectively of the queue length for service center i. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedab. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>G. Casale, R. R. Muntz, G. Serazzi, <cite>Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks</cite>, IEEE Transactions on Computers, 57(6):780-794, June 2008. <a href="http://doi.ieeecomputersociety.org/10.1109/TC.2008.37">http://doi.ieeecomputersociety.org/10.1109/TC.2008.37</a> <p><a name="index-Casale_002c-G_002e-275"></a><a name="index-Muntz_002c-R_002e-R_002e-276"></a><a name="index-Serazzi_002c-G_002e-277"></a> In this implementation we set X^+ and X^- as the upper and lower Asymptotic Bounds as computed by the <code>qnclosedab</code> function, respectively. <div class="node"> <a name="Utility-functions"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#Bounds-on-performance">Bounds on performance</a>, Up: <a rel="up" accesskey="u" href="#Queueing-Networks">Queueing Networks</a> </div> <h3 class="section">6.6 Utility functions</h3> <h4 class="subsection">6.6.1 Open or closed networks</h4> <p><a name="doc_002dqnclosed"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnclosed</b> (<var>N, S, V, <small class="dots">...</small></var>)<var><a name="index-qnclosed-278"></a></var><br> <blockquote> <p><a name="index-closed-network-279"></a> 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. <p>This function dispatches the computation to one of <code>qnclosedsinglemva</code>, <code>qnclosedsinglemvald</code> or <code>qnclosedmultimva</code>. <ul> <li>If <var>N</var> 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</var> is a vector, then <var>S</var><code>(k)</code> is the average service time of center k, and this function calls <code>qnclosedsinglemva</code> which supports load-independent service centers. If <var>S</var> is a matrix, <var>S</var><code>(k,i)</code> is the average service time at service center k when i ≥ 1 jobs are present; in this case, the network is analyzed with the <code>qnclosedsinglemvald</code> function. <li>If <var>N</var> 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>qnclosedmultimva</code> function. </ul> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedsinglemva, qnclosedsinglemvald, qnclosedmultimva. </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <pre class="example"><pre class="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(1,3); # All centers are single-server Z = 2; # External delay N = 15; # Maximum population to consider V = qnvisits(P); # Compute number of visits from P D = V .* S; # Compute service demand from S and V X_bsb_lower = X_bsb_upper = zeros(1,N); X_ab_lower = X_ab_upper = zeros(1,N); X_mva = zeros(1,N); for n=1:N [X_bsb_lower(n) X_bsb_upper(n)] = qnclosedbsb(n, D, Z); [X_ab_lower(n) X_ab_upper(n)] = qnclosedab(n, D, 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]); xlabel("Number of Requests n"); ylabel("System Throughput X(n)"); legend("location","southeast");</pre></pre> <p><a name="doc_002dqnopen"></a> <div class="defun"> — Function File: [<var>U</var>, <var>R</var>, <var>Q</var>, <var>X</var>] = <b>qnopen</b> (<var>lambda, S, V, <small class="dots">...</small></var>)<var><a name="index-qnopen-280"></a></var><br> <blockquote> <p><a name="index-open-network-281"></a> Compute utilization, response time, average number of requests in the system, and throughput for open queueing networks. If <var>lambda</var> is a scalar, the network is considered a single-class QN and is solved using <code>qnopensingle</code>. If <var>lambda</var> is a vector, the network is considered as a multiclass QN and solved using <code>qnopenmulti</code>. <pre class="sp"> </pre> <strong>See also:</strong> qnopensingle, qnopenmulti. </blockquote></div> <!-- Compute the visit counts --> <h4 class="subsection">6.6.2 Computation of the visit counts</h4> <p>For single-class networks the average number of visits satisfy the following equation: <pre class="example"> V == P0 + V*P; </pre> <p class="noindent">where P_0 j is the probability that an external arrival goes to service center j. If \lambda_j is the external arrival rate to service center j, and \lambda = \sum_j \lambda_j is the overall external arrival rate, then P_0 j = \lambda_j / \lambda. <p>For closed networks, the visit ratios satisfy the following equation: <pre class="example"> V(1) == 1 && V == V*P; </pre> <p>The definitions above can be extended to multiple class networks as follows. We define the visit ratios V_sj for class s customers at service center j as follows: <p>V_sj = sum_r sum_i V_ri P_risj, for all s,j V_s1 = 1, for all s <p class="noindent">while for open networks: <p>V_sj = P_0sj + sum_r sum_i V_ri P_risj, for all s,j <p class="noindent">where P_0sj is the probability that an external arrival goes to service center j as a class-s request. If \lambda_sj is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_sj is the overall external arrival rate to the whole system, then P_0sj = \lambda_sj / \lambda. <p><a name="doc_002dqnvisits"></a> <div class="defun"> — Function File: [<var>V</var> <var>ch</var>] = <b>qnvisits</b> (<var>P</var>)<var><a name="index-qnvisits-282"></a></var><br> — Function File: <var>V</var> = <b>qnvisits</b> (<var>P, lambda</var>)<var><a name="index-qnvisits-283"></a></var><br> <blockquote> <p>Compute the average number of visits to the service centers of a single class, open or closed Queueing Network with N service centers. <p><strong>INPUTS</strong> <dl> <dt><var>P</var><dd>Routing probability matrix. For single class networks, <var>P</var><code>(i,j)</code> is the probability that a request which completed service at center i is routed to center j. For closed networks it must hold that <code>sum(</code><var>P</var><code>,2)==1</code>. The routing graph myst be strongly connected, meaning that it must be possible to eventually reach each node starting from each node. For multiple class networks, <var>P</var><code>(r,i,s,j)</code> is the 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. <br><dt><var>lambda</var><dd>(open networks only) vector of external arrivals. For single class networks, <var>lambda</var><code>(i)</code> is the external arrival rate to center i. For multiple class networks, <var>lambda</var><code>(r,i)</code> is the arrival rate of class r requests to center i. If this parameter is omitted, the network is assumed to be closed. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>V</var><dd>For single class networks, <var>V</var><code>(i)</code> is the average number of visits to server i. For multiple class networks, <var>V</var><code>(r,i)</code> is the class r visit ratio at center i. <br><dt><var>ch</var><dd>(For closed networks only). <var>ch</var><code>(c)</code> is the chain number that class c belongs to. Different classes can belong to the same chain. Chains are numbered 1, 2, <small class="dots">...</small>. The total number of chains is <code>max(</code><var>ch</var><code>)</code>. </dl> </blockquote></div> <p class="noindent"><strong>EXAMPLE</strong> <pre class="example"><pre class="verbatim"> 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 = qnvisits(P,lambda); S = [2 1 2 1.8]; m = [3 1 1 2]; [U R Q X] = qnopensingle( sum(lambda), S, V, m );</pre></pre> <h4 class="subsection">6.6.3 Other utility functions</h4> <p><a name="doc_002dpopulation_005fmix"></a> <div class="defun"> — Function File: pop_mix = <b>population_mix</b> (<var>k, N</var>)<var><a name="index-population_005fmix-284"></a></var><br> <blockquote> <p><a name="index-population-mix-285"></a><a name="index-closed-network_002c-multiple-classes-286"></a> Return the set of valid population mixes with exactly <var>k</var> customers, for a closed multiclass Queueing Network with population vector <var>N</var>. More specifically, given a multiclass Queueing Network with C customer classes, such that there are <var>N</var><code>(i)</code> requests of class i, a k-mix <var>mix</var> is a C-dimensional vector with the following properties: <pre class="example"> all( mix >= 0 ); all( mix <= N ); sum( mix ) == k; </pre> <p class="noindent">This function enumerates all valid k-mixes, such that <var>pop_mix</var><code>(i)</code> is a C dimensional row vector representing a valid population mix, for all i. <p><strong>INPUTS</strong> <dl> <dt><var>k</var><dd>Total population size of the requested mix. <var>k</var> must be a nonnegative integer <br><dt><var>N</var><dd><var>N</var><code>(i)</code> is the number of class i requests. The condition <var>k</var><code> ≤ sum(</code><var>N</var><code>)</code> must hold. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>pop_mix</var><dd><var>pop_mix</var><code>(i,j)</code> is the number of class j requests in the i-th population mix. The number of population mixes is <code>rows( </code><var>pop_mix</var><code> ) </code>. </dl> <p>Note that if you are interested in the number of k-mixes and you don't care to enumerate them, you can use the funcion <code>qnmvapop</code>. <pre class="sp"> </pre> <strong>See also:</strong> qnmvapop. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Herb Schwetman, <cite>Implementing the Mean Value Algorithm for the Solution of Queueing Network Models</cite>, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, feb 15, 1982, available at <a href="http://www.cs.purdue.edu/research/technical_reports/1980/TR 80-355.pdf">http://www.cs.purdue.edu/research/technical_reports/1980/TR 80-355.pdf</a> <p>Note that the slightly different problem of generating all tuples k_1, k_2, \ldots k_N such that \sum_i k_i = k and k_i are nonnegative integers, for some fixed integer k ≥ 0 has been described in S. Santini, <cite>Computing the Indices for a Complex Summation</cite>, unpublished report, available at <a href="http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf">http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf</a> <p><a name="index-Schwetman_002c-H_002e-287"></a><a name="index-Santini_002c-S_002e-288"></a> <a name="doc_002dqnmvapop"></a> <div class="defun"> — Function File: <var>H</var> = <b>qnmvapop</b> (<var>N</var>)<var><a name="index-qnmvapop-289"></a></var><br> <blockquote> <p><a name="index-population-mix-290"></a><a name="index-closed-network_002c-multiple-classes-291"></a> Given a network with C customer classes, this function computes the number of valid population mixes <var>H</var><code>(r,n)</code> that can be constructed by the multiclass MVA algorithm by allocating n customers to the first r classes. <p><strong>INPUTS</strong> <dl> <dt><var>N</var><dd>Population vector. <var>N</var><code>(c)</code> is the number of class-c requests in the system. The total number of requests in the network is <code>sum(</code><var>N</var><code>)</code>. </dl> <p><strong>OUTPUTS</strong> <dl> <dt><var>H</var><dd><var>H</var><code>(r,n)</code> is the number of valid populations that can be constructed allocating n customers to the first r classes. </dl> <pre class="sp"> </pre> <strong>See also:</strong> qnclosedmultimva,population_mix. </blockquote></div> <p class="noindent"><strong>REFERENCES</strong> <p>Zahorjan, J. and Wong, E. <cite>The solution of separable queueing network models using mean value analysis</cite>. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI <a href="http://doi.acm.org/10.1145/1010629.805477">http://doi.acm.org/10.1145/1010629.805477</a> <p><a name="index-Zahorjan_002c-J_002e-292"></a><a name="index-Wong_002c-E_002e-293"></a> <!-- Appendix starts here --> <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Contributing-Guidelines"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Acknowledgements">Acknowledgements</a>, Previous: <a rel="previous" accesskey="p" href="#Queueing-Networks">Queueing Networks</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="appendix">Appendix A Contributing Guidelines</h2> <p>Contributions and bug reports are <em>always</em> welcome. If you want to contribute to the <code>queueing</code> package, here are some guidelines: <ul> <li>If you are contributing a new function, please embed proper documentation within the function itself. The documentation must be in <code>texinfo</code> format, so that it will be extracted and formatted into the printable manual. See the existing functions of the <code>queueing</code> package for the documentation style. <li>The documentation should be as precise as possible. In particular, always state what the valid ranges of the parameters are. <li>If you are contributing a new function, ensure that the function properly checks the validity of its input parameters. For example, each function accepting vectors should check whether the dimensions match. <li>Always provide bibliographic references for each algorithm you contribute. If your implementation differs in some way from the reference you give, please describe how and why your implementation differs. <li>Include Octave test and demo blocks with your code. Test blocks are particularly important, because Queueing Network algorithms tend to be quite complex to implement correctly, and we must ensure that the implementations provided with the <code>queueing</code> package are (mostly) correct. </ul> <p>Send your contribution to Moreno Marzolla (<a href="mailto:marzolla@cs.unibo.it">marzolla@cs.unibo.it</a>). Even if you are just a user of <code>queueing</code>, and find this package useful, let me know by dropping me a line. Thanks. <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <!-- *- texinfo -*- --> <!-- Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla --> <!-- This file is part of the queueing toolbox, a Queueing Networks --> <!-- analysis package for GNU Octave. --> <!-- 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; see the file COPYING. If not, see --> <!-- <http://www.gnu.org/licenses/>. --> <div class="node"> <a name="Acknowledgements"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Copying">Copying</a>, Previous: <a rel="previous" accesskey="p" href="#Contributing-Guidelines">Contributing Guidelines</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="appendix">Appendix B Acknowledgements</h2> <p>The following people (listed in alphabetical order) contributed to the <code>queueing</code> package, either by providing feedback, reporting bugs or contributing code: Philip Carinhas, Phil Colbourn, Yves Durand, Marco Guazzone, Dmitry Kolesnikov. <!-- DO NOT EDIT! Generated automatically by munge-texi. --> <div class="node"> <a name="Copying"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Concept-Index">Concept Index</a>, Previous: <a rel="previous" accesskey="p" href="#Acknowledgements">Acknowledgements</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="appendix">Appendix C GNU GENERAL PUBLIC LICENSE</h2> <p><a name="index-warranty-294"></a><a name="index-copyright-295"></a> <div align="center">Version 3, 29 June 2007</div> <pre class="display"> Copyright © 2007 Free Software Foundation, Inc. <a href="http://fsf.org/">http://fsf.org/</a> Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. </pre> <h3 class="heading">Preamble</h3> <p>The GNU General Public License is a free, copyleft license for software and other kinds of works. <p>The licenses for most software and other practical works are designed to take away your freedom to share and change the works. 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SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. <li>Limitation of Liability. <p>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. <li>Interpretation of Sections 15 and 16. <p>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. </ol> <h3 class="heading">END OF TERMS AND CONDITIONS</h3> <h3 class="heading">How to Apply These Terms to Your New Programs</h3> <p>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. <p>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. <pre class="smallexample"> <var>one line to give the program's name and a brief idea of what it does.</var> Copyright (C) <var>year</var> <var>name of author</var> 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 <a href="http://www.gnu.org/licenses/">http://www.gnu.org/licenses/</a>. </pre> <p>Also add information on how to contact you by electronic and paper mail. <p>If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: <pre class="smallexample"> <var>program</var> Copyright (C) <var>year</var> <var>name of author</var> This program comes with ABSOLUTELY NO WARRANTY; for details type ‘<samp><span class="samp">show w</span></samp>’. This is free software, and you are welcome to redistribute it under certain conditions; type ‘<samp><span class="samp">show c</span></samp>’ for details. </pre> <p>The hypothetical commands ‘<samp><span class="samp">show w</span></samp>’ and ‘<samp><span class="samp">show c</span></samp>’ 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”. <p>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 <a href="http://www.gnu.org/licenses/">http://www.gnu.org/licenses/</a>. <p>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. But first, please read <a href="http://www.gnu.org/philosophy/why-not-lgpl.html">http://www.gnu.org/philosophy/why-not-lgpl.html</a>. <!-- INDEX --> <div class="node"> <a name="Concept-Index"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Function-Index">Function Index</a>, Previous: <a rel="previous" accesskey="p" href="#Copying">Copying</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="unnumbered">Concept Index</h2> <ul class="index-cp" compact> <li><a href="#index-Approximate-MVA-179">Approximate MVA</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-79">Asymmetric M/M/m system</a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-BCMP-network-134">BCMP network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Birth_002ddeath-process-30">Birth-death process</a>: <a href="#Birth_002dDeath-process">Birth-Death process</a></li> <li><a href="#index-Birth_002ddeath-process-11">Birth-death process</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-blocking-queueing-network-229">blocking queueing network</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-bounds_002c-asymptotic-241">bounds, asymptotic</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-bounds_002c-balanced-system-256">bounds, balanced system</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-bounds_002c-geometric-273">bounds, geometric</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-closed-network-279">closed network</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-closed-network-250">closed network</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-closed-network-111">closed network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Closed-network_002c-approximate-analysis-181">Closed network, approximate analysis</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-closed-network_002c-finite-capacity-230">closed network, finite capacity</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-closed-network_002c-multiple-classes-286">closed network, multiple classes</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-closed-network_002c-multiple-classes-236">closed network, multiple classes</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-Closed-network_002c-multiple-classes-211">Closed network, multiple classes</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-closed-network_002c-multiple-classes-193">closed network, multiple classes</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Closed-network_002c-single-class-180">Closed network, single class</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-closed-network_002c-single-class-150">closed network, single class</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-CMVA-171">CMVA</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Continuous-time-Markov-chain-25">Continuous time Markov chain</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> <li><a href="#index-convolution-algorithm-113">convolution algorithm</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-copyright-295">copyright</a>: <a href="#Copying">Copying</a></li> <li><a href="#index-Discrete-time-Markov-chain-6">Discrete time Markov chain</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Expected-sojourn-time-34">Expected sojourn time</a>: <a href="#Expected-Sojourn-Time">Expected Sojourn Time</a></li> <li><a href="#index-First-passage-times-49">First passage times</a>: <a href="#First-Passage-Times">First Passage Times</a></li> <li><a href="#index-First-passage-times-15">First passage times</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Jackson-network-104">Jackson network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-load_002ddependent-service-center-123">load-dependent service center</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-g_t_0040math_007bM_002fG_002f1_007d-system-85">M/G/1 system</a>: <a href="#The-M_002fG_002f1-System">The M/G/1 System</a></li> <li><a href="#index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-87">M/H_m/1 system</a>: <a href="#The-M_002fHm_002f1-System">The M/Hm/1 System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002f1_007d-system-51">M/M/1 system</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-71">M/M/1/K system</a>: <a href="#The-M_002fM_002f1_002fK-System">The M/M/1/K System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002f_007dinf-system-64">M/M/inf system</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002fm_007d-system-58">M/M/m system</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-73">M/M/m/K system</a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-48">Markov chain, continuous time</a>: <a href="#First-Passage-Times">First Passage Times</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-40">Markov chain, continuous time</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-37">Markov chain, continuous time</a>: <a href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-33">Markov chain, continuous time</a>: <a href="#Expected-Sojourn-Time">Expected Sojourn Time</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-29">Markov chain, continuous time</a>: <a href="#Birth_002dDeath-process">Birth-Death process</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-24">Markov chain, continuous time</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-21">Markov chain, continuous time</a>: <a href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a></li> <li><a href="#index-Markov-chain_002c-discrete-time-2">Markov chain, discrete time</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Markov-chain_002c-disctete-time-18">Markov chain, disctete time</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Markov-chain_002c-state-occupancy-probabilities-26">Markov chain, state occupancy probabilities</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> <li><a href="#index-Markov-chain_002c-stationary-probabilities-7">Markov chain, stationary probabilities</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Mean-time-to-absorption-41">Mean time to absorption</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-Mean-time-to-absorption-19">Mean time to absorption</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Mean-Value-Analysys-_0028MVA_0029-149">Mean Value Analysys (MVA)</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-178">Mean Value Analysys (MVA), approximate</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-mixed-network-221">mixed network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-normalization-constant-112">normalization constant</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-open-network-281">open network</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-open-network-242">open network</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-open-network_002c-multiple-classes-141">open network, multiple classes</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-open-network_002c-single-class-103">open network, single class</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-population-mix-285">population mix</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-queueing-network-with-blocking-228">queueing network with blocking</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-queueing-networks-88">queueing networks</a>: <a href="#Queueing-Networks">Queueing Networks</a></li> <li><a href="#index-RS-blocking-239">RS blocking</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-Stationary-probabilities-27">Stationary probabilities</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> <li><a href="#index-Stationary-probabilities-8">Stationary probabilities</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-Time_002dalveraged-sojourn-time-38">Time-alveraged sojourn time</a>: <a href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a></li> <li><a href="#index-traffic-intensity-65">traffic intensity</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-warranty-294">warranty</a>: <a href="#Copying">Copying</a></li> </ul><div class="node"> <a name="Function-Index"></a> <p><hr> Next: <a rel="next" accesskey="n" href="#Author-Index">Author Index</a>, Previous: <a rel="previous" accesskey="p" href="#Concept-Index">Concept Index</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="unnumbered">Function Index</h2> <ul class="index-fn" compact> <li><a href="#index-ctmc-22"><code>ctmc</code></a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> <li><a href="#index-ctmc_005fbd-28"><code>ctmc_bd</code></a>: <a href="#Birth_002dDeath-process">Birth-Death process</a></li> <li><a href="#index-ctmc_005fcheck_005fQ-20"><code>ctmc_check_Q</code></a>: <a href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a></li> <li><a href="#index-ctmc_005fexps-31"><code>ctmc_exps</code></a>: <a href="#Expected-Sojourn-Time">Expected Sojourn Time</a></li> <li><a href="#index-ctmc_005ffpt-46"><code>ctmc_fpt</code></a>: <a href="#First-Passage-Times">First Passage Times</a></li> <li><a href="#index-ctmc_005fmtta-39"><code>ctmc_mtta</code></a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-ctmc_005ftaexps-35"><code>ctmc_taexps</code></a>: <a href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a></li> <li><a href="#index-dtmc-3"><code>dtmc</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-dtmc_005fbd-9"><code>dtmc_bd</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-dtmc_005fcheck_005fP-1"><code>dtmc_check_P</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-dtmc_005ffpt-12"><code>dtmc_fpt</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-dtmc_005fmtta-16"><code>dtmc_mtta</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> <li><a href="#index-population_005fmix-284"><code>population_mix</code></a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-qnammm-78"><code>qnammm</code></a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-qnclosed-278"><code>qnclosed</code></a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-qnclosedab-247"><code>qnclosedab</code></a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-qnclosedbsb-262"><code>qnclosedbsb</code></a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-qnclosedgb-272"><code>qnclosedgb</code></a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-qnclosedmultimva-186"><code>qnclosedmultimva</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnclosedmultimvaapprox-204"><code>qnclosedmultimvaapprox</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnclosedpb-266"><code>qnclosedpb</code></a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-qnclosedsinglemva-146"><code>qnclosedsinglemva</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnclosedsinglemvaapprox-173"><code>qnclosedsinglemvaapprox</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnclosedsinglemvald-159"><code>qnclosedsinglemvald</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qncmva-168"><code>qncmva</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnconvolution-109"><code>qnconvolution</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnconvolutionld-119"><code>qnconvolutionld</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnjackson-100"><code>qnjackson</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnmarkov-232"><code>qnmarkov</code></a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-qnmg1-84"><code>qnmg1</code></a>: <a href="#The-M_002fG_002f1-System">The M/G/1 System</a></li> <li><a href="#index-qnmh1-86"><code>qnmh1</code></a>: <a href="#The-M_002fHm_002f1-System">The M/Hm/1 System</a></li> <li><a href="#index-qnmix-219"><code>qnmix</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnmknode-89"><code>qnmknode</code></a>: <a href="#Generic-Algorithms">Generic Algorithms</a></li> <li><a href="#index-qnmm1-50"><code>qnmm1</code></a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> <li><a href="#index-qnmm1k-70"><code>qnmm1k</code></a>: <a href="#The-M_002fM_002f1_002fK-System">The M/M/1/K System</a></li> <li><a href="#index-qnmminf-63"><code>qnmminf</code></a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-qnmmm-56"><code>qnmmm</code></a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-qnmmmk-72"><code>qnmmmk</code></a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> <li><a href="#index-qnmvablo-227"><code>qnmvablo</code></a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-qnmvapop-289"><code>qnmvapop</code></a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-qnopen-280"><code>qnopen</code></a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-qnopenab-240"><code>qnopenab</code></a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-qnopenbsb-255"><code>qnopenbsb</code></a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-qnopenmulti-139"><code>qnopenmulti</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnopensingle-131"><code>qnopensingle</code></a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-qnsolve-96"><code>qnsolve</code></a>: <a href="#Generic-Algorithms">Generic Algorithms</a></li> <li><a href="#index-qnvisits-282"><code>qnvisits</code></a>: <a href="#Utility-functions">Utility functions</a></li> </ul><div class="node"> <a name="Author-Index"></a> <p><hr> Previous: <a rel="previous" accesskey="p" href="#Function-Index">Function Index</a>, Up: <a rel="up" accesskey="u" href="#Top">Top</a> </div> <h2 class="unnumbered">Author Index</h2> <ul class="index-au" compact> <li><a href="#index-Akyildiz_002c-I_002e-F_002e-231">Akyildiz, I. F.</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-Bard_002c-Y_002e-213">Bard, Y.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Bolch_002c-G_002e-105">Bolch, G.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Bolch_002c-G_002e-80">Bolch, G.</a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-Bolch_002c-G_002e-74">Bolch, G.</a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> <li><a href="#index-Bolch_002c-G_002e-66">Bolch, G.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-Bolch_002c-G_002e-59">Bolch, G.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-Bolch_002c-G_002e-52">Bolch, G.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> <li><a href="#index-Bolch_002c-G_002e-42">Bolch, G.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-Buzen_002c-J_002e-P_002e-114">Buzen, J. P.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Casale_002c-G_002e-269">Casale, G.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Casale_002c-G_002e-172">Casale, G.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-de-Meer_002c-H_002e-107">de Meer, H.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-de-Meer_002c-H_002e-82">de Meer, H.</a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-de-Meer_002c-H_002e-76">de Meer, H.</a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> <li><a href="#index-de-Meer_002c-H_002e-68">de Meer, H.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-de-Meer_002c-H_002e-61">de Meer, H.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-de-Meer_002c-H_002e-54">de Meer, H.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> <li><a href="#index-de-Meer_002c-H_002e-44">de Meer, H.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-Graham_002c-G_002e-S_002e-245">Graham, G. S.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Graham_002c-G_002e-S_002e-144">Graham, G. S.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Greiner_002c-S_002e-106">Greiner, S.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Greiner_002c-S_002e-81">Greiner, S.</a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-Greiner_002c-S_002e-75">Greiner, S.</a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> <li><a href="#index-Greiner_002c-S_002e-67">Greiner, S.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-Greiner_002c-S_002e-60">Greiner, S.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-Greiner_002c-S_002e-53">Greiner, S.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> <li><a href="#index-Greiner_002c-S_002e-43">Greiner, S.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-Hsieh_002c-C_002e-H-267">Hsieh, C. H</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Jain_002c-R_002e-154">Jain, R.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Kobayashi_002c-H_002e-126">Kobayashi, H.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Lam_002c-S_002e-268">Lam, S.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Lavenberg_002c-S_002e-S_002e-153">Lavenberg, S. S.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Lazowska_002c-E_002e-D_002e-243">Lazowska, E. D.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Lazowska_002c-E_002e-D_002e-142">Lazowska, E. D.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Muntz_002c-R_002e-R_002e-270">Muntz, R. R.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Reiser_002c-M_002e-125">Reiser, M.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Santini_002c-S_002e-288">Santini, S.</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-Schweitzer_002c-P_002e-214">Schweitzer, P.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Schwetman_002c-H_002e-287">Schwetman, H.</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-Schwetman_002c-H_002e-124">Schwetman, H.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Serazzi_002c-G_002e-271">Serazzi, G.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Sevcik_002c-K_002e-C_002e-246">Sevcik, K. C.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Sevcik_002c-K_002e-C_002e-145">Sevcik, K. C.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Trivedi_002c-K_002e-108">Trivedi, K.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Trivedi_002c-K_002e-83">Trivedi, K.</a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-Trivedi_002c-K_002e-77">Trivedi, K.</a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> <li><a href="#index-Trivedi_002c-K_002e-69">Trivedi, K.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-Trivedi_002c-K_002e-62">Trivedi, K.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-Trivedi_002c-K_002e-55">Trivedi, K.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> <li><a href="#index-Trivedi_002c-K_002e-45">Trivedi, K.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> <li><a href="#index-Wong_002c-E_002e-293">Wong, E.</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-Zahorjan_002c-J_002e-292">Zahorjan, J.</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-Zahorjan_002c-J_002e-244">Zahorjan, J.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Zahorjan_002c-J_002e-143">Zahorjan, J.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> </ul></body></html>