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[Octave-cvsupdate] SF.net SVN: octave:[9808] trunk/octave-forge/main/queueing
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From: <mma...@us...> - 2012-03-10 16:03:56
|
Revision: 9808
http://octave.svn.sourceforge.net/octave/?rev=9808&view=rev
Author: mmarzolla
Date: 2012-03-10 16:03:47 +0000 (Sat, 10 Mar 2012)
Log Message:
-----------
fixed bug in texinfo documentation
Modified Paths:
--------------
trunk/octave-forge/main/queueing/doc/queueing.html
trunk/octave-forge/main/queueing/doc/queueing.pdf
trunk/octave-forge/main/queueing/inst/ctmc_exps.m
Modified: trunk/octave-forge/main/queueing/doc/queueing.html
===================================================================
--- trunk/octave-forge/main/queueing/doc/queueing.html 2012-03-10 16:00:45 UTC (rev 9807)
+++ trunk/octave-forge/main/queueing/doc/queueing.html 2012-03-10 16:03:47 UTC (rev 9808)
@@ -1100,36 +1100,46 @@
<p><a name="doc_002dctmc_005fexps"></a>
<div class="defun">
-— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, tt, p</var>)<var><a name="index-ctmc_005fexps-20"></a></var><br>
+— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, tt, p </var>)<var><a name="index-ctmc_005fexps-20"></a></var><br>
+— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fexps-21"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-21"></a><a name="index-Expected-sojourn-time-22"></a>
-Compute the expected total time <var>L</var><code>(t,j)</code> spent in state
-j during the time interval <code>[0,</code><var>tt</var><code>(t))</code>, assuming
-that at time 0 the state occupancy probability was <var>p</var>.
+ <p><a name="index-Markov-chain_002c-continuous-time-22"></a><a name="index-Expected-sojourn-time-23"></a>
+With three arguments, compute the expected time <var>L</var><code>(t,j)</code>
+spent in each state j during the time interval
+<code>[0,</code><var>tt</var><code>(t))</code>, assuming that at time 0 the state occupancy
+probability was <var>p</var>. With two arguments, compute the expected
+time <var>L</var><code>(j)</code> spent in each state j until absorption.
<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 <code>sum(</code><var>Q</var><code>,2) == 0</code>
+<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>tt</var><dd>This parameter is a vector used for numerical integration. The first
element <var>tt</var><code>(1)</code> must be 0, and the last element
<var>tt</var><code>(end)</code> must be the upper bound of the interval
[0,t) of interest (<var>tt</var><code>(end) == t</code>).
- <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
+ <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><var>L</var><code>(t,j)</code> is the expected time spent in state j
-during the interval <code>[0,</code><var>tt</var><code>(t))</code>. <code>1 ≤ </code><var>t</var><code> ≤ length(</code><var>tt</var><code>)</code>
+<dt><var>L</var><dd>If this function is called with three arguments, <var>L</var> is a matrix
+of size <code>[length(</code><var>tt</var><code>), N]</code> where <var>L</var><code>(t,j)</code> is the
+expected time spent in state j during the interval
+<code>[0,</code><var>tt</var><code>(t)]</code>. If this function is called with two
+arguments, <var>L</var> is a vector with N elements where
+<var>L</var><code>(j)</code> is the expected time spent in state j until
+absorption, if j is a transient state. If j
+is an absorbing state, <var>L</var><code>(j) = 0</code>.
</dl>
@@ -1175,9 +1185,9 @@
<p><a name="doc_002dctmc_005ftaexps"></a>
<div class="defun">
-— Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, tt, p</var>)<var><a name="index-ctmc_005ftaexps-23"></a></var><br>
+— Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, tt, p</var>)<var><a name="index-ctmc_005ftaexps-24"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-24"></a><a name="index-Time_002dalveraged-sojourn-time-25"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-25"></a><a name="index-Time_002dalveraged-sojourn-time-26"></a>
Compute the <em>time-averaged sojourn time</em> <var>M</var><code>(t,j)</code>,
defined as the fraction of the time interval <code>[0,</code><var>tt</var><code>(t))</code> spent in
state j, assuming that at time 0 the state occupancy
@@ -1264,12 +1274,13 @@
<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-26"></a></var><br>
+— Function File: <var>t</var> = <b>ctmc_mtta</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fmtta-27"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-27"></a><a name="index-Mean-time-to-absorption-28"></a>
-Compute the Mean-Time to Absorption (MTTA) starting from initial
-occupancy probability <var>p</var> at time 0. If there are no absorbing
-states, this function fails with an error.
+ <p><a name="index-Markov-chain_002c-continuous-time-28"></a><a name="index-Mean-time-to-absorption-29"></a>
+Compute the Mean-Time to Absorption (MTTA) of the CTMC described by
+the infinitesimal generator matrix <var>Q</var>, starting from initial
+occupancy probability <var>p</var>. If there are no absorbing states, this
+function fails with an error.
<p><strong>INPUTS</strong>
@@ -1326,7 +1337,7 @@
Performance Evaluation with Computer Science Applications</cite>, Wiley,
1998.
- <p><a name="index-Bolch_002c-G_002e-29"></a><a name="index-Greiner_002c-S_002e-30"></a><a name="index-de-Meer_002c-H_002e-31"></a><a name="index-Trivedi_002c-K_002e-32"></a>
+ <p><a name="index-Bolch_002c-G_002e-30"></a><a name="index-Greiner_002c-S_002e-31"></a><a name="index-de-Meer_002c-H_002e-32"></a><a name="index-Trivedi_002c-K_002e-33"></a>
<div class="node">
<a name="CTMC-First-Passage-Times"></a>
<p><hr>
@@ -1340,10 +1351,10 @@
<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-33"></a></var><br>
-— Function File: <var>m</var> = <b>ctmc_fpt</b> (<var>Q, i, j</var>)<var><a name="index-ctmc_005ffpt-34"></a></var><br>
+— Function File: <var>M</var> = <b>ctmc_fpt</b> (<var>Q</var>)<var><a name="index-ctmc_005ffpt-34"></a></var><br>
+— Function File: <var>m</var> = <b>ctmc_fpt</b> (<var>Q, i, j</var>)<var><a name="index-ctmc_005ffpt-35"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-35"></a><a name="index-First-passage-times-36"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-36"></a><a name="index-First-passage-times-37"></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
@@ -1449,9 +1460,9 @@
<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-qnmm1-37"></a></var><br>
+— 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-qnmm1-38"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-38"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-39"></a>
Compute utilization, response time, average number of requests
and throughput for a M/M/1 queue.
@@ -1496,7 +1507,7 @@
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-39"></a><a name="index-Greiner_002c-S_002e-40"></a><a name="index-de-Meer_002c-H_002e-41"></a><a name="index-Trivedi_002c-K_002e-42"></a>
+ <p><a name="index-Bolch_002c-G_002e-40"></a><a name="index-Greiner_002c-S_002e-41"></a><a name="index-de-Meer_002c-H_002e-42"></a><a name="index-Trivedi_002c-K_002e-43"></a>
<!-- M/M/m -->
<div class="node">
<a name="The-M%2fM%2fm-System"></a>
@@ -1522,10 +1533,10 @@
<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-43"></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-44"></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</var>)<var><a name="index-qnmmm-44"></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-45"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-45"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-46"></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.
@@ -1577,7 +1588,7 @@
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-46"></a><a name="index-Greiner_002c-S_002e-47"></a><a name="index-de-Meer_002c-H_002e-48"></a><a name="index-Trivedi_002c-K_002e-49"></a>
+ <p><a name="index-Bolch_002c-G_002e-47"></a><a name="index-Greiner_002c-S_002e-48"></a><a name="index-de-Meer_002c-H_002e-49"></a><a name="index-Trivedi_002c-K_002e-50"></a>
<!-- M/M/inf -->
<div class="node">
<a name="The-M%2fM%2finf-System"></a>
@@ -1600,7 +1611,7 @@
<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-50"></a></var><br>
+— 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-51"></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
@@ -1608,7 +1619,7 @@
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-51"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f_007dinf-system-52"></a>
<p><strong>INPUTS</strong>
@@ -1626,7 +1637,7 @@
different from the utilization, which in the case of M/M/\infty
centers is always zero.
- <p><a name="index-traffic-intensity-52"></a>
+ <p><a name="index-traffic-intensity-53"></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
@@ -1654,7 +1665,7 @@
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-53"></a><a name="index-Greiner_002c-S_002e-54"></a><a name="index-de-Meer_002c-H_002e-55"></a><a name="index-Trivedi_002c-K_002e-56"></a>
+ <p><a name="index-Bolch_002c-G_002e-54"></a><a name="index-Greiner_002c-S_002e-55"></a><a name="index-de-Meer_002c-H_002e-56"></a><a name="index-Trivedi_002c-K_002e-57"></a>
<!-- M/M/1/k -->
<div class="node">
<a name="The-M%2fM%2f1%2fK-System"></a>
@@ -1678,9 +1689,9 @@
<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-57"></a></var><br>
+— 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-58"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-58"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-59"></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
@@ -1747,9 +1758,9 @@
<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-59"></a></var><br>
+— 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-60"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-60"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-61"></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
@@ -1807,7 +1818,7 @@
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-61"></a><a name="index-Greiner_002c-S_002e-62"></a><a name="index-de-Meer_002c-H_002e-63"></a><a name="index-Trivedi_002c-K_002e-64"></a>
+ <p><a name="index-Bolch_002c-G_002e-62"></a><a name="index-Greiner_002c-S_002e-63"></a><a name="index-de-Meer_002c-H_002e-64"></a><a name="index-Trivedi_002c-K_002e-65"></a>
<!-- Approximate M/M/m -->
<div class="node">
@@ -1829,9 +1840,9 @@
<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-65"></a></var><br>
+— 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-66"></a></var><br>
<blockquote>
- <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-66"></a>
+ <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-67"></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
@@ -1878,7 +1889,7 @@
and Markov Chains: Modeling and Performance Evaluation with Computer
Science Applications</cite>, Wiley, 1998
- <p><a name="index-Bolch_002c-G_002e-67"></a><a name="index-Greiner_002c-S_002e-68"></a><a name="index-de-Meer_002c-H_002e-69"></a><a name="index-Trivedi_002c-K_002e-70"></a>
+ <p><a name="index-Bolch_002c-G_002e-68"></a><a name="index-Greiner_002c-S_002e-69"></a><a name="index-de-Meer_002c-H_002e-70"></a><a name="index-Trivedi_002c-K_002e-71"></a>
<div class="node">
<a name="The-M%2fG%2f1-System"></a>
<a name="The-M_002fG_002f1-System"></a>
@@ -1894,9 +1905,9 @@
<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-qnmg1-71"></a></var><br>
+— 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-qnmg1-72"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-72"></a>
+ <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-73"></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
@@ -1953,9 +1964,9 @@
<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-qnmh1-73"></a></var><br>
+— 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-qnmh1-74"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-74"></a>
+ <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-75"></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:
@@ -2037,7 +2048,7 @@
<li><a accesskey="6" href="#Utility-functions">Utility functions</a>: Utility functions to compute miscellaneous quantities
</ul>
-<p><a name="index-queueing-networks-75"></a>
+<p><a name="index-queueing-networks-76"></a>
<!-- INTRODUCTION -->
<div class="node">
<a name="Introduction-to-QNs"></a>
@@ -2298,13 +2309,13 @@
<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-76"></a></var><br>
-— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S, m</var>)<var><a name="index-qnmknode-77"></a></var><br>
-— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/1-lcfs-pr", S</var>)<var><a name="index-qnmknode-78"></a></var><br>
-— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S</var>)<var><a name="index-qnmknode-79"></a></var><br>
-— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S, s2</var>)<var><a name="index-qnmknode-80"></a></var><br>
-— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S</var>)<var><a name="index-qnmknode-81"></a></var><br>
-— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S, s2</var>)<var><a name="index-qnmknode-82"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S</var>)<var><a name="index-qnmknode-77"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S, m</var>)<var><a name="index-qnmknode-78"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/1-lcfs-pr", S</var>)<var><a name="index-qnmknode-79"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S</var>)<var><a name="index-qnmknode-80"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S, s2</var>)<var><a name="index-qnmknode-81"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S</var>)<var><a name="index-qnmknode-82"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S, s2</var>)<var><a name="index-qnmknode-83"></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
@@ -2373,10 +2384,10 @@
<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-83"></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-84"></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-85"></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-86"></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</var>)<var><a name="index-qnsolve-84"></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-85"></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-86"></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-87"></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.
@@ -2554,11 +2565,11 @@
<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-87"></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-88"></a></var><br>
-— Function File: <var>pr</var> = <b>qnjackson</b> (<var>lambda, S, P, m, k</var>)<var><a name="index-qnjackson-89"></a></var><br>
+— 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-88"></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-89"></a></var><br>
+— Function File: <var>pr</var> = <b>qnjackson</b> (<var>lambda, S, P, m, k</var>)<var><a name="index-qnjackson-90"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-single-class-90"></a><a name="index-Jackson-network-91"></a>
+ <p><a name="index-open-network_002c-single-class-91"></a><a name="index-Jackson-network-92"></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
@@ -2640,7 +2651,7 @@
Performance Evaluation with Computer Science Applications</cite>, Wiley,
1998, pp. 284–287.
- <p><a name="index-Bolch_002c-G_002e-92"></a><a name="index-Greiner_002c-S_002e-93"></a><a name="index-de-Meer_002c-H_002e-94"></a><a name="index-Trivedi_002c-K_002e-95"></a>
+ <p><a name="index-Bolch_002c-G_002e-93"></a><a name="index-Greiner_002c-S_002e-94"></a><a name="index-de-Meer_002c-H_002e-95"></a><a name="index-Trivedi_002c-K_002e-96"></a>
<h4 class="subsection">6.3.2 The Convolution Algorithm</h4>
@@ -2674,10 +2685,10 @@
<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-96"></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-97"></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</var>)<var><a name="index-qnconvolution-97"></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-98"></a></var><br>
<blockquote>
- <p><a name="index-closed-network-98"></a><a name="index-normalization-constant-99"></a><a name="index-convolution-algorithm-100"></a>
+ <p><a name="index-closed-network-99"></a><a name="index-normalization-constant-100"></a><a name="index-convolution-algorithm-101"></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
@@ -2768,20 +2779,20 @@
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-101"></a>
+ <p><a name="index-Buzen_002c-J_002e-P_002e-102"></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-102"></a><a name="index-Greiner_002c-S_002e-103"></a><a name="index-de-Meer_002c-H_002e-104"></a><a name="index-Trivedi_002c-K_002e-105"></a>
+ <p><a name="index-Bolch_002c-G_002e-103"></a><a name="index-Greiner_002c-S_002e-104"></a><a name="index-de-Meer_002c-H_002e-105"></a><a name="index-Trivedi_002c-K_002e-106"></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-106"></a></var><br>
+— 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-107"></a></var><br>
<blockquote>
- <p><a name="index-closed-network-107"></a><a name="index-normalization-constant-108"></a><a name="index-convolution-algorithm-109"></a><a name="index-load_002ddependent-service-center-110"></a>
+ <p><a name="index-closed-network-108"></a><a name="index-normalization-constant-109"></a><a name="index-convolution-algorithm-110"></a><a name="index-load_002ddependent-service-center-111"></a>
This function implements the <em>convolution algorithm</em> for
product-form, single-class closed queueing networks with general
load-dependent service centers.
@@ -2841,7 +2852,7 @@
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-111"></a>
+ <p><a name="index-Schwetman_002c-H_002e-112"></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
@@ -2849,7 +2860,7 @@
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-112"></a><a name="index-Kobayashi_002c-H_002e-113"></a>
+ <p><a name="index-Reiser_002c-M_002e-113"></a><a name="index-Kobayashi_002c-H_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,
@@ -2861,7 +2872,7 @@
function f_i defined in Schwetman, <code>Some Computational
Aspects of Queueing Network Models</code>.
- <p><a name="index-Bolch_002c-G_002e-114"></a><a name="index-Greiner_002c-S_002e-115"></a><a name="index-de-Meer_002c-H_002e-116"></a><a name="index-Trivedi_002c-K_002e-117"></a>
+ <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>
<h4 class="subsection">6.3.3 Open networks</h4>
@@ -2869,10 +2880,10 @@
<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-118"></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-119"></a></var><br>
+— 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-119"></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-120"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-single-class-120"></a><a name="index-BCMP-network-121"></a>
+ <p><a name="index-open-network_002c-single-class-121"></a><a name="index-BCMP-network-122"></a>
Analyze open, single class BCMP queueing networks.
<p>This function works for a subset of BCMP single-class open networks
@@ -2965,16 +2976,16 @@
Networks and Markov Chains: Modeling and Performance Evaluation with
Computer Science Applications</cite>, Wiley, 1998.
- <p><a name="index-Bolch_002c-G_002e-122"></a><a name="index-Greiner_002c-S_002e-123"></a><a name="index-de-Meer_002c-H_002e-124"></a><a name="index-Trivedi_002c-K_002e-125"></a>
+ <p><a name="index-Bolch_002c-G_002e-123"></a><a name="index-Greiner_002c-S_002e-124"></a><a name="index-de-Meer_002c-H_002e-125"></a><a name="index-Trivedi_002c-K_002e-126"></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-126"></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-127"></a></var><br>
+— 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-127"></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-128"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-multiple-classes-128"></a>
+ <p><a name="index-open-network_002c-multiple-classes-129"></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
@@ -3039,7 +3050,7 @@
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-129"></a><a name="index-Zahorjan_002c-J_002e-130"></a><a name="index-Graham_002c-G_002e-S_002e-131"></a><a name="index-Sevcik_002c-K_002e-C_002e-132"></a>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-130"></a><a name="index-Zahorjan_002c-J_002e-131"></a><a name="index-Graham_002c-G_002e-S_002e-132"></a><a name="index-Sevcik_002c-K_002e-C_002e-133"></a>
<h4 class="subsection">6.3.4 Closed Networks</h4>
@@ -3047,11 +3058,11 @@
<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-133"></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-134"></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-135"></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</var>)<var><a name="index-qnclosedsinglemva-134"></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-135"></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-136"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-136"></a><a name="index-closed-network_002c-single-class-137"></a><a name="index-normalization-constant-138"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-137"></a><a name="index-closed-network_002c-single-class-138"></a><a name="index-normalization-constant-139"></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
@@ -3097,8 +3108,11 @@
<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 as <var>Rsys</var><code> = </code><var>N</var><code>/</code><var>Xsys</var><code> - Z</code>
+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
@@ -3149,7 +3163,7 @@
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-139"></a><a name="index-Lavenberg_002c-S_002e-S_002e-140"></a>
+ <p><a name="index-Reiser_002c-M_002e-140"></a><a name="index-Lavenberg_002c-S_002e-S_002e-141"></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)}, -->
@@ -3158,15 +3172,15 @@
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-141"></a><a name="index-Bolch_002c-G_002e-142"></a><a name="index-Greiner_002c-S_002e-143"></a><a name="index-de-Meer_002c-H_002e-144"></a><a name="index-Trivedi_002c-K_002e-145"></a>
+ <p><a name="index-Jain_002c-R_002e-142"></a><a name="index-Bolch_002c-G_002e-143"></a><a name="index-Greiner_002c-S_002e-144"></a><a name="index-de-Meer_002c-H_002e-145"></a><a name="index-Trivedi_002c-K_002e-146"></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-146"></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-147"></a></var><br>
+— 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-147"></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-148"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-148"></a><a name="index-closed-network_002c-single-class-149"></a><a name="index-load_002ddependent-service-center-150"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-149"></a><a name="index-closed-network_002c-single-class-150"></a><a name="index-load_002ddependent-service-center-151"></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
@@ -3224,15 +3238,15 @@
1998, Section 8.2.4.1, “Networks with Load-Deèpendent Service: Closed
Networks”.
- <p><a name="index-Bolch_002c-G_002e-151"></a><a name="index-Greiner_002c-S_002e-152"></a><a name="index-de-Meer_002c-H_002e-153"></a><a name="index-Trivedi_002c-K_002e-154"></a>
+ <p><a name="index-Bolch_002c-G_002e-152"></a><a name="index-Greiner_002c-S_002e-153"></a><a name="index-de-Meer_002c-H_002e-154"></a><a name="index-Trivedi_002c-K_002e-155"></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-155"></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-156"></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</var>)<var><a name="index-qncmva-156"></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-157"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-157"></a><a name="index-CMVA-158"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-158"></a><a name="index-CMVA-159"></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
@@ -3286,19 +3300,19 @@
closed networks</cite>. Queueing Syst. Theory Appl., 60:193–202, December
2008.
- <p><a name="index-Casale_002c-G_002e-159"></a>
+ <p><a name="index-Casale_002c-G_002e-160"></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-160"></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-161"></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-162"></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-163"></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-164"></a></var><br>
+— 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-161"></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-162"></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-163"></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-164"></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-165"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-165"></a><a name="index-Approximate-MVA-166"></a><a name="index-Closed-network_002c-single-class-167"></a><a name="index-Closed-network_002c-approximate-analysis-168"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-166"></a><a name="index-Approximate-MVA-167"></a><a name="index-Closed-network_002c-single-class-168"></a><a name="index-Closed-network_002c-approximate-analysis-169"></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
@@ -3377,20 +3391,20 @@
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-169"></a><a name="index-Zahorjan_002c-J_002e-170"></a><a name="index-Graham_002c-G_002e-S_002e-171"></a><a name="index-Sevcik_002c-K_002e-C_002e-172"></a>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-170"></a><a name="index-Zahorjan_002c-J_002e-171"></a><a name="index-Graham_002c-G_002e-S_002e-172"></a><a name="index-Sevcik_002c-K_002e-C_002e-173"></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-173"></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-174"></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-175"></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-176"></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-177"></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-178"></a></var><br>
+— 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-174"></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-175"></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-176"></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-177"></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-178"></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-179"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-179"></a><a name="index-closed-network_002c-multiple-classes-180"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-180"></a><a name="index-closed-network_002c-multiple-classes-181"></a>
Analyze closed, multiclass queueing networks with K service
centers and C independent customer classes (chains) using the
Mean Value Analysys (MVA) algorithm.
@@ -3477,8 +3491,10 @@
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 total class c system response time
-can be computed as <code>dot(</code><var>R</var><code>, </code><var>V</var><code>, 2)</code>.
+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
@@ -3518,7 +3534,7 @@
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-181"></a><a name="index-Lavenberg_002c-S_002e-S_002e-182"></a>
+ <p><a name="index-Reiser_002c-M_002e-182"></a><a name="index-Lavenberg_002c-S_002e-S_002e-183"></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,
@@ -3528,18 +3544,18 @@
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-183"></a><a name="index-Greiner_002c-S_002e-184"></a><a name="index-de-Meer_002c-H_002e-185"></a><a name="index-Trivedi_002c-K_002e-186"></a><a name="index-Lazowska_002c-E_002e-D_002e-187"></a><a name="index-Zahorjan_002c-J_002e-188"></a><a name="index-Graham_002c-G_002e-S_002e-189"></a><a name="index-Sevcik_002c-K_002e-C_002e-190"></a>
+ <p><a name="index-Bolch_002c-G_002e-184"></a><a name="index-Greiner_002c-S_002e-185"></a><a name="index-de-Meer_002c-H_002e-186"></a><a name="index-Trivedi_002c-K_002e-187"></a><a name="index-Lazowska_002c-E_002e-D_002e-188"></a><a name="index-Zahorjan_002c-J_002e-189"></a><a name="index-Graham_002c-G_002e-S_002e-190"></a><a name="index-Sevcik_002c-K_002e-C_002e-191"></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-191"></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-192"></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-193"></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-194"></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-195"></a></var><br>
+— 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-192"></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-193"></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-194"></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-195"></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-196"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-196"></a><a name="index-Approximate-MVA-197"></a><a name="index-Closed-network_002c-multiple-classes-198"></a><a name="index-Closed-network_002c-approximate-analysis-199"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-197"></a><a name="index-Approximate-MVA-198"></a><a name="index-Closed-network_002c-multiple-classes-199"></a><a name="index-Closed-network_002c-approximate-analysis-200"></a>
Analyze closed, multiclass queueing networks with K service
centers and C customer classes using the approximate Mean
Value Analysys (MVA) algorithm.
@@ -3624,12 +3640,12 @@
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-200"></a>
+ <p><a name="index-Bard_002c-Y_002e-201"></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-201"></a>
+ <p><a name="index-Schweitzer_002c-P_002e-202"></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>,
@@ -3640,7 +3656,7 @@
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-202"></a><a name="index-Zahorjan_002c-J_002e-203"></a><a name="index-Graham_002c-G_002e-S_002e-204"></a><a name="index-Sevcik_002c-K_002e-C_002e-205"></a>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-203"></a><a name="index-Zahorjan_002c-J_002e-204"></a><a name="index-Graham_002c-G_002e-S_002e-205"></a><a name="index-Sevcik_002c-K_002e-C_002e-206"></a>
<h4 class="subsection">6.3.5 Mixed Networks</h4>
@@ -3648,9 +3664,9 @@
<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-206"></a></var><br>
+— 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-207"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-207"></a><a name="index-mixed-network-208"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-208"></a><a name="index-mixed-network-209"></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
@@ -3741,14 +3757,14 @@
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-209"></a><a name="index-Zahorjan_002c-J_002e-210"></a><a name="index-Graham_002c-G_002e-S_002e-211"></a><a name="index-Sevcik_002c-K_002e-C_002e-212"></a>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-210"></a><a name="index-Zahorjan_002c-J_002e-211"></a><a name="index-Graham_002c-G_002e-S_002e-212"></a><a name="index-Sevcik_002c-K_002e-C_002e-213"></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-213"></a>
+ <p><a name="index-Schwetman_002c-H_002e-214"></a>
<div class="node">
<a name="Algorithms-for-non-Product-form-QNs"></a>
@@ -3767,9 +3783,9 @@
<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-214"></a></var><br>
+— 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-215"></a></var><br>
<blockquote>
- <p><a name="index-queueing-network-with-blocking-215"></a><a name="index-blocking-queueing-network-216"></a><a name="index-closed-network_002c-finite-capacity-217"></a>
+ <p><a name="index-queueing-network-with-blocking-216"></a><a name="index-blocking-queueing-network-217"></a><a name="index-closed-network_002c-finite-capacity-218"></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.
@@ -3824,16 +3840,16 @@
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-218"></a>
+ <p><a name="index-Akyildiz_002c-I_002e-F_002e-219"></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-219"></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-220"></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-221"></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-222"></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</var>)<var><a name="index-qnmarkov-220"></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-221"></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-222"></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-223"></a></var><br>
<blockquote>
- <p><a name="index-closed-network_002c-multiple-classes-223"></a><a name="index-closed-network_002c-finite-capacity-224"></a><a name="index-blocking-queueing-network-225"></a><a name="index-RS-blocking-226"></a>
+ <p><a name="index-closed-network_002c-multiple-classes-224"></a><a name="index-closed-network_002c-finite-capacity-225"></a><a name="index-blocking-queueing-network-226"></a><a name="index-RS-blocking-227"></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
@@ -3943,9 +3959,9 @@
<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-227"></a></var><br>
+— Function File: [<var>Xu</var>, <var>Rl</var>] = <b>qnopenab</b> (<var>lambda, D</var>)<var><a name="index-qnopenab-228"></a></var><br>
<blockquote>
- <p><a name="index-bounds_002c-asymptotic-228"></a><a name="index-open-network-229"></a>
+ <p><a name="index-bounds_002c-asymptotic-229"></a><a name="index-open-network-230"></a>
Compute Asymptotic Bounds for single-class, open Queueing Networks
with K service centers.
@@ -3985,14 +4001,14 @@
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-230"></a><a name="index-Zahorjan_002c-J_002e-231"></a><a name="index-Graham_002c-G_002e-S_002e-232"></a><a name="index-Sevcik_002c-K_002e-C_002e-233"></a>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-231"></a><a name="index-Zahorjan_002c-J_002e-232"></a><a name="index-Graham_002c-G_002e-S_002e-233"></a><a name="index-Sevcik_002c-K_002e-C_002e-234"></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-234"></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-235"></a></var><br>
+— 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-235"></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-236"></a></var><br>
<blockquote>
- <p><a name="index-bounds_002c-asymptotic-236"></a><a name="index-closed-network-237"></a>
+ <p><a name="index-bounds_002c-asymptotic-237"></a><a name="index-closed-network-238"></a>
Compute Asymptotic Bounds for single-class, closed Queueing Networks
wit...
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