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[Octave-cvsupdate] SF.net SVN: octave:[9928] trunk/octave-forge/main/queueing
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From: <mma...@us...> - 2012-03-16 22:29:26
|
Revision: 9928
http://octave.svn.sourceforge.net/octave/?rev=9928&view=rev
Author: mmarzolla
Date: 2012-03-16 22:29:17 +0000 (Fri, 16 Mar 2012)
Log Message:
-----------
fixed test cases
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_mtta.m
trunk/octave-forge/main/queueing/inst/dtmc_check_P.m
trunk/octave-forge/main/queueing/inst/dtmc_fpt.m
trunk/octave-forge/main/queueing/inst/dtmc_mtta.m
trunk/octave-forge/main/queueing/inst/qnvisits.m
Modified: trunk/octave-forge/main/queueing/doc/queueing.html
===================================================================
--- trunk/octave-forge/main/queueing/doc/queueing.html 2012-03-16 19:17:13 UTC (rev 9927)
+++ trunk/octave-forge/main/queueing/doc/queueing.html 2012-03-16 22:29:17 UTC (rev 9928)
@@ -817,13 +817,13 @@
<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>
+— Function File: [<var>r</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.
+Check if <var>P</var> is a valid transition probability matrix. If <var>P</var>
+is valid, <var>r</var> is the size (number of rows or columns) of <var>P</var>.
+If <var>P</var> is not a transition probability matrix, <var>r</var> is set to
+zero, and <var>err</var> to an appropriate error string.
</blockquote></div>
@@ -871,14 +871,14 @@
— 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><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><a name="index-Markov-chain_002c-transient-probabilities-9"></a><a name="index-Transient-probabilities-10"></a>
+Compute steady-state or transient state occupancy probabilities for a
+Discrete-Time Markov Chain. With a single argument, compute the
+steady-state occupancy probability vector <var>p</var><code>(1), ...,
+</code><var>p</var><code>(N)</code> given the N \times N transition probability matrix
+<var>P</var>. With three arguments, compute the state occupancy
+probabilities <var>p</var><code>(1), ..., </code><var>p</var><code>(N)</code> after <var>n</var>
+steps, given initial occupancy probability vector <var>p0</var>.
<p><strong>INPUTS</strong>
@@ -942,9 +942,9 @@
<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>
+— Function File: <var>P</var> = <b>dtmc_bd</b> (<var>birth, death</var>)<var><a name="index-dtmc_005fbd-11"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-discrete-time-10"></a><a name="index-Birth_002ddeath-process-11"></a>
+ <p><a name="index-Markov-chain_002c-discrete-time-12"></a><a name="index-Birth_002ddeath-process-13"></a>
Returns the N \times N transition probability matrix P
for a birth-death process with given rates.
@@ -994,15 +994,14 @@
<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>
+— Function File: <var>M</var> = <b>dtmc_fpt</b> (<var>P</var>)<var><a name="index-dtmc_005ffpt-14"></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><a name="index-First-passage-times-15"></a><a name="index-Mean-recurrence-times-16"></a>
+Compute the mean first passage times matrix \bf M, such that
+<var>M</var><code>(i,j)</code> is 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. Diagonal elements of <var>M</var> are the mean recurrence
+times.
<p><strong>INPUTS</strong>
@@ -1013,26 +1012,17 @@
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
+<dt><var>M</var><dd><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>.
+<var>M</var><code>(i,i)</code> is the <em>mean recurrence time</em> of state
+i, 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>
@@ -1051,10 +1041,10 @@
<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>
+— Function File: [<var>t</var> <var>B</var>] = <b>dtmc_mtta</b> (<var>P</var>)<var><a name="index-dtmc_005fmtta-17"></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-18"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-disctete-time-18"></a><a name="index-Mean-time-to-absorption-19"></a>
+ <p><a name="index-Markov-chain_002c-disctete-time-19"></a><a name="index-Mean-time-to-absorption-20"></a><a name="index-Absorption-probabilities-21"></a>
Compute the expected number of steps before absorption for the
DTMC with N \times N transition probability matrix <var>P</var>.
@@ -1079,9 +1069,10 @@
<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
+in state j, starting from transient 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>.
@@ -1118,9 +1109,9 @@
<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>
+— Function File: [<var>result</var> <var>err</var>] = <b>ctmc_check_Q</b> (<var>Q</var>)<var><a name="index-ctmc_005fcheck_005fQ-22"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-21"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-23"></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
@@ -1171,10 +1162,10 @@
<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>
+— Function File: <var>p</var> = <b>ctmc</b> (<var>Q</var>)<var><a name="index-ctmc-24"></a></var><br>
+— Function File: <var>p</var> = <b>ctmc</b> (<var>Q, t. p0</var>)<var><a name="index-ctmc-25"></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>
+ <p><a name="index-Markov-chain_002c-continuous-time-26"></a><a name="index-Continuous-time-Markov-chain-27"></a><a name="index-Markov-chain_002c-state-occupancy-probabilities-28"></a><a name="index-Stationary-probabilities-29"></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
@@ -1237,9 +1228,9 @@
<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>
+— Function File: <var>Q</var> = <b>ctmc_bd</b> (<var>birth, death</var>)<var><a name="index-ctmc_005fbd-30"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-29"></a><a name="index-Birth_002ddeath-process-30"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-31"></a><a name="index-Birth_002ddeath-process-32"></a>
Returns the N \times N infinitesimal generator matrix Q
for a birth-death process with given rates.
@@ -1300,10 +1291,10 @@
<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>
+— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, t, p </var>)<var><a name="index-ctmc_005fexps-33"></a></var><br>
+— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fexps-34"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-33"></a><a name="index-Expected-sojourn-time-34"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-35"></a><a name="index-Expected-sojourn-time-36"></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
@@ -1383,10 +1374,10 @@
<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>
+— Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, t, p</var>)<var><a name="index-ctmc_005ftaexps-37"></a></var><br>
+— Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, p</var>)<var><a name="index-ctmc_005ftaexps-38"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-37"></a><a name="index-Time_002dalveraged-sojourn-time-38"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-39"></a><a name="index-Time_002dalveraged-sojourn-time-40"></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
@@ -1473,9 +1464,9 @@
<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>
+— Function File: <var>t</var> = <b>ctmc_mtta</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fmtta-41"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-40"></a><a name="index-Mean-time-to-absorption-41"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-42"></a><a name="index-Mean-time-to-absorption-43"></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
@@ -1536,7 +1527,7 @@
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>
+ <p><a name="index-Bolch_002c-G_002e-44"></a><a name="index-Greiner_002c-S_002e-45"></a><a name="index-de-Meer_002c-H_002e-46"></a><a name="index-Trivedi_002c-K_002e-47"></a>
<div class="node">
<a name="First-passage-times-(CTMC)"></a>
<a name="First-passage-times-_0028CTMC_0029"></a>
@@ -1551,10 +1542,10 @@
<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-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>
+— Function File: <var>M</var> = <b>ctmc_fpt</b> (<var>Q</var>)<var><a name="index-ctmc_005ffpt-48"></a></var><br>
+— Function File: <var>m</var> = <b>ctmc_fpt</b> (<var>Q, i, j</var>)<var><a name="index-ctmc_005ffpt-49"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-48"></a><a name="index-First-passage-times-49"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-50"></a><a name="index-First-passage-times-51"></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
@@ -1660,9 +1651,9 @@
<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-qnmm1-50"></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-qnmm1-52"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-51"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-53"></a>
Compute utilization, response time, average number of requests
and throughput for a M/M/1 queue.
@@ -1707,7 +1698,7 @@
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-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>
+ <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/m -->
<div class="node">
<a name="The-M%2fM%2fm-System"></a>
@@ -1733,10 +1724,10 @@
<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-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>
+— 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-58"></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-59"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-58"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-60"></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.
@@ -1788,7 +1779,7 @@
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-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>
+ <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>
<!-- M/M/inf -->
<div class="node">
<a name="The-M%2fM%2finf-System"></a>
@@ -1811,7 +1802,7 @@
<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-63"></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-65"></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
@@ -1819,7 +1810,7 @@
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-64"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f_007dinf-system-66"></a>
<p><strong>INPUTS</strong>
@@ -1837,7 +1828,7 @@
different from the utilization, which in the case of M/M/\infty
centers is always zero.
- <p><a name="index-traffic-intensity-65"></a>
+ <p><a name="index-traffic-intensity-67"></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
@@ -1865,7 +1856,7 @@
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-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>
+ <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>
<!-- M/M/1/k -->
<div class="node">
<a name="The-M%2fM%2f1%2fK-System"></a>
@@ -1889,9 +1880,9 @@
<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-70"></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-72"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-71"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-73"></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
@@ -1958,9 +1949,9 @@
<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-72"></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-74"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-73"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-75"></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
@@ -2018,7 +2009,7 @@
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-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>
+ <p><a name="index-Bolch_002c-G_002e-76"></a><a name="index-Greiner_002c-S_002e-77"></a><a name="index-de-Meer_002c-H_002e-78"></a><a name="index-Trivedi_002c-K_002e-79"></a>
<!-- Approximate M/M/m -->
<div class="node">
@@ -2040,9 +2031,9 @@
<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-78"></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-80"></a></var><br>
<blockquote>
- <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-79"></a>
+ <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-81"></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
@@ -2089,7 +2080,7 @@
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>
+ <p><a name="index-Bolch_002c-G_002e-82"></a><a name="index-Greiner_002c-S_002e-83"></a><a name="index-de-Meer_002c-H_002e-84"></a><a name="index-Trivedi_002c-K_002e-85"></a>
<div class="node">
<a name="The-M%2fG%2f1-System"></a>
<a name="The-M_002fG_002f1-System"></a>
@@ -2105,9 +2096,9 @@
<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-qnmg1-84"></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-qnmg1-86"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-85"></a>
+ <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-87"></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
@@ -2164,9 +2155,9 @@
<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-qnmh1-86"></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-qnmh1-88"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-87"></a>
+ <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-89"></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:
@@ -2248,7 +2239,7 @@
<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>
+<p><a name="index-queueing-networks-90"></a>
<!-- INTRODUCTION -->
<div class="node">
<a name="Introduction-to-QNs"></a>
@@ -2509,13 +2500,13 @@
<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-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>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S</var>)<var><a name="index-qnmknode-91"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S, m</var>)<var><a name="index-qnmknode-92"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/1-lcfs-pr", S</var>)<var><a name="index-qnmknode-93"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S</var>)<var><a name="index-qnmknode-94"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S, s2</var>)<var><a name="index-qnmknode-95"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S</var>)<var><a name="index-qnmknode-96"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S, s2</var>)<var><a name="index-qnmknode-97"></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
@@ -2584,10 +2575,10 @@
<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-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>
+— 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-98"></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-99"></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-100"></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-101"></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.
@@ -2765,11 +2756,11 @@
<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-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>
+— 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-102"></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-103"></a></var><br>
+— Function File: <var>pr</var> = <b>qnjackson</b> (<var>lambda, S, P, m, k</var>)<var><a name="index-qnjackson-104"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-single-class-103"></a><a name="index-Jackson-network-104"></a>
+ <p><a name="index-open-network_002c-single-class-105"></a><a name="index-Jackson-network-106"></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
@@ -2851,7 +2842,7 @@
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>
+ <p><a name="index-Bolch_002c-G_002e-107"></a><a name="index-Greiner_002c-S_002e-108"></a><a name="index-de-Meer_002c-H_002e-109"></a><a name="index-Trivedi_002c-K_002e-110"></a>
<h4 class="subsection">6.3.2 The Convolution Algorithm</h4>
@@ -2885,10 +2876,10 @@
<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-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>
+— 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-111"></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-112"></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>
+ <p><a name="index-closed-network-113"></a><a name="index-normalization-constant-114"></a><a name="index-convolution-algorithm-115"></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
@@ -2979,20 +2970,20 @@
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-114"></a>
+ <p><a name="index-Buzen_002c-J_002e-P_002e-116"></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>
+ <p><a name="index-Bolch_002c-G_002e-117"></a><a name="index-Greiner_002c-S_002e-118"></a><a name="index-de-Meer_002c-H_002e-119"></a><a name="index-Trivedi_002c-K_002e-120"></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>
+— 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-121"></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>
+ <p><a name="index-closed-network-122"></a><a name="index-normalization-constant-123"></a><a name="index-convolution-algorithm-124"></a><a name="index-load_002ddependent-service-center-125"></a>
This function implements the <em>convolution algorithm</em> for
product-form, single-class closed queueing networks with general
load-dependent service centers.
@@ -3052,7 +3043,7 @@
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-124"></a>
+ <p><a name="index-Schwetman_002c-H_002e-126"></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
@@ -3060,7 +3051,7 @@
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-125"></a><a name="index-Kobayashi_002c-H_002e-126"></a>
+ <p><a name="index-Reiser_002c-M_002e-127"></a><a name="index-Kobayashi_002c-H_002e-128"></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,
@@ -3072,7 +3063,7 @@
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>
+ <p><a name="index-Bolch_002c-G_002e-129"></a><a name="index-Greiner_002c-S_002e-130"></a><a name="index-de-Meer_002c-H_002e-131"></a><a name="index-Trivedi_002c-K_002e-132"></a>
<h4 class="subsection">6.3.3 Open networks</h4>
@@ -3080,10 +3071,10 @@
<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-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>
+— 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-133"></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-134"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-single-class-133"></a><a name="index-BCMP-network-134"></a>
+ <p><a name="index-open-network_002c-single-class-135"></a><a name="index-BCMP-network-136"></a>
Analyze open, single class BCMP queueing networks.
<p>This function works for a subset of BCMP single-class open networks
@@ -3176,16 +3167,16 @@
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>
+ <p><a name="index-Bolch_002c-G_002e-137"></a><a name="index-Greiner_002c-S_002e-138"></a><a name="index-de-Meer_002c-H_002e-139"></a><a name="index-Trivedi_002c-K_002e-140"></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>
+— 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-141"></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-142"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-multiple-classes-141"></a>
+ <p><a name="index-open-network_002c-multiple-classes-143"></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
@@ -3250,7 +3241,7 @@
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-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>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-144"></a><a name="index-Zahorjan_002c-J_002e-145"></a><a name="index-Graham_002c-G_002e-S_002e-146"></a><a name="index-Sevcik_002c-K_002e-C_002e-147"></a>
<h4 class="subsection">6.3.4 Closed Networks</h4>
@@ -3258,11 +3249,11 @@
<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-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>
+— 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-148"></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-149"></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-150"></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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-151"></a><a name="index-closed-network_002c-single-class-152"></a><a name="index-normalization-constant-153"></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
@@ -3363,7 +3354,7 @@
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-152"></a><a name="index-Lavenberg_002c-S_002e-S_002e-153"></a>
+ <p><a name="index-Reiser_002c-M_002e-154"></a><a name="index-Lavenberg_002c-S_002e-S_002e-155"></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)}, -->
@@ -3372,15 +3363,15 @@
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-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>
+ <p><a name="index-Jain_002c-R_002e-156"></a><a name="index-Bolch_002c-G_002e-157"></a><a name="index-Greiner_002c-S_002e-158"></a><a name="index-de-Meer_002c-H_002e-159"></a><a name="index-Trivedi_002c-K_002e-160"></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>
+— 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-161"></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-162"></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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-163"></a><a name="index-closed-network_002c-single-class-164"></a><a name="index-load_002ddependent-service-center-165"></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
@@ -3438,15 +3429,15 @@
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>
+ <p><a name="index-Bolch_002c-G_002e-166"></a><a name="index-Greiner_002c-S_002e-167"></a><a name="index-de-Meer_002c-H_002e-168"></a><a name="index-Trivedi_002c-K_002e-169"></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>
+— 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-170"></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-171"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-170"></a><a name="index-CMVA-171"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-172"></a><a name="index-CMVA-173"></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
@@ -3500,19 +3491,19 @@
closed networks</cite>. Queueing Syst. Theory Appl., 60:193–202, December
2008.
- <p><a name="index-Casale_002c-G_002e-172"></a>
+ <p><a name="index-Casale_002c-G_002e-174"></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>
+— 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-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</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</var>)<var><a name="index-qnclosedsinglemvaapprox-177"></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-178"></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-179"></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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-180"></a><a name="index-Approximate-MVA-181"></a><a name="index-Closed-network_002c-single-class-182"></a><a name="index-Closed-network_002c-approximate-analysis-183"></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
@@ -3591,20 +3582,20 @@
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-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>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-184"></a><a name="index-Zahorjan_002c-J_002e-185"></a><a name="index-Graham_002c-G_002e-S_002e-186"></a><a name="index-Sevcik_002c-K_002e-C_002e-187"></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>
+— 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-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</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, V, m</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, V, m, Z</var>)<var><a name="index-qnclosedmultimva-191"></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-192"></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-193"></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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-194"></a><a name="index-closed-network_002c-multiple-classes-195"></a>
Analyze closed, multiclass queueing networks with K service
centers and C independent customer classes (chains) using the
Mean Value Analysys (MVA) algorithm.
@@ -3734,7 +3725,7 @@
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-194"></a><a name="index-Lavenberg_002c-S_002e-S_002e-195"></a>
+ <p><a name="index-Reiser_002c-M_002e-196"></a><a name="index-Lavenberg_002c-S_002e-S_002e-197"></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,
@@ -3744,18 +3735,18 @@
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-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>
+ <p><a name="index-Bolch_002c-G_002e-198"></a><a name="index-Greiner_002c-S_002e-199"></a><a name="index-de-Meer_002c-H_002e-200"></a><a name="index-Trivedi_002c-K_002e-201"></a><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>
<!-- 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>
+— 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-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</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</var>)<var><a name="index-qnclosedmultimvaapprox-208"></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-209"></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-210"></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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-211"></a><a name="index-Approximate-MVA-212"></a><a name="index-Closed-network_002c-multiple-classes-213"></a><a name="index-Closed-network_002c-approximate-analysis-214"></a>
Analyze closed, multiclass queueing networks with K service
centers and C customer classes using the approximate Mean
Value Analysys (MVA) algorithm.
@@ -3840,12 +3831,12 @@
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-213"></a>
+ <p><a name="index-Bard_002c-Y_002e-215"></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>
+ <p><a name="index-Schweitzer_002c-P_002e-216"></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>,
@@ -3856,7 +3847,7 @@
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-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>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-217"></a><a name="index-Zahorjan_002c-J_002e-218"></a><a name="index-Graham_002c-G_002e-S_002e-219"></a><a name="index-Sevcik_002c-K_002e-C_002e-220"></a>
<h4 class="subsection">6.3.5 Mixed Networks</h4>
@@ -3864,9 +3855,9 @@
<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-219"></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-221"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-220"></a><a name="index-mixed-network-221"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-222"></a><a name="index-mixed-network-223"></a>
Solution of mixed queueing networks through MVA. ...
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