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[Octave-cvsupdate] SF.net SVN: octave:[9813] trunk/octave-forge/main/queueing
|
From: <mma...@us...> - 2012-03-11 11:20:42
|
Revision: 9813
http://octave.svn.sourceforge.net/octave/?rev=9813&view=rev
Author: mmarzolla
Date: 2012-03-11 11:20:31 +0000 (Sun, 11 Mar 2012)
Log Message:
-----------
New function dtmc_bd() added; misc fixes and improvements
Modified Paths:
--------------
trunk/octave-forge/main/queueing/ChangeLog
trunk/octave-forge/main/queueing/DESCRIPTION
trunk/octave-forge/main/queueing/Makefile
trunk/octave-forge/main/queueing/NEWS
trunk/octave-forge/main/queueing/doc/INSTALL
trunk/octave-forge/main/queueing/doc/markovchains.txi
trunk/octave-forge/main/queueing/doc/queueing.html
trunk/octave-forge/main/queueing/doc/queueing.pdf
trunk/octave-forge/main/queueing/doc/summary.txi
trunk/octave-forge/main/queueing/inst/ctmc_bd.m
trunk/octave-forge/main/queueing/inst/ctmc_exps.m
trunk/octave-forge/main/queueing/inst/qnmm1k.m
trunk/octave-forge/main/queueing/inst/qnmmmk.m
trunk/octave-forge/main/queueing/inst/qnvisits.m
Added Paths:
-----------
trunk/octave-forge/main/queueing/inst/dtmc_bd.m
Removed Paths:
-------------
trunk/octave-forge/main/queueing/inst/ctmc_bd_solve.m
Modified: trunk/octave-forge/main/queueing/ChangeLog
===================================================================
--- trunk/octave-forge/main/queueing/ChangeLog 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/ChangeLog 2012-03-11 11:20:31 UTC (rev 9813)
@@ -1,6 +1,6 @@
2012-02-XX Moreno Marzolla <mar...@cs...>
- * Version 1.0.X released
+ * Version 1.X.0 released
* Fixed bug in qnvisits() which made the function behave incorrectly
under particular degenerate cases.
* Fixed bug in ctmc_exps() (wrong initial value in call to lsode)
@@ -9,6 +9,10 @@
* Function ctmc_exps() and ctmc_taexps() accept a scalar as second
argument; the old syntax is still supported, but may be deprecated
in future releases.
+ * Function ctmc_bd() now returns the infinitesimal generator matrix
+ of the birth-death process, not the steady-state solution.
+ * Function ctmc_bd_solve() has been removed
+ * New function dtmc_bd()
* Miscellaneous fixes/improvements to the documentation
2012-02-04 Moreno Marzolla <mar...@cs...>
Modified: trunk/octave-forge/main/queueing/DESCRIPTION
===================================================================
--- trunk/octave-forge/main/queueing/DESCRIPTION 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/DESCRIPTION 2012-03-11 11:20:31 UTC (rev 9813)
@@ -1,6 +1,6 @@
Name: queueing
-Version: 1.0.0
-Date: 2012-02-04
+Version: 1.X.0
+Date: 2012-02-XX
Author: Moreno Marzolla <mar...@cs...>
Maintainer: Moreno Marzolla <mar...@cs...>
Title: Queueing networks and Markov chains analysis package for GNU Octave
Modified: trunk/octave-forge/main/queueing/Makefile
===================================================================
--- trunk/octave-forge/main/queueing/Makefile 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/Makefile 2012-03-11 11:20:31 UTC (rev 9813)
@@ -1,5 +1,5 @@
-VERSIONNUM=1.0.0
-VERSIONDATE="2012-02-04"
+VERSIONNUM=1.X.0
+VERSIONDATE="2012-02-XX"
PROGNAME=queueing
DISTNAME=$(PROGNAME)-$(VERSIONNUM)
Modified: trunk/octave-forge/main/queueing/NEWS
===================================================================
--- trunk/octave-forge/main/queueing/NEWS 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/NEWS 2012-03-11 11:20:31 UTC (rev 9813)
@@ -1,4 +1,4 @@
-Summary of important user-visible changes for queueing-1.0.X
+Summary of important user-visible changes for queueing-1.X.0
** Function ctmc_exps() can now compute the expected sojourn time
until absorption for absorming CTMC
@@ -7,6 +7,14 @@
the second argument (time). The old syntax is still supported,
but may be deprecated in the future.
+** Function ctmc_bd() now returns the infinitesimal generator matrix Q
+ of the birth-death process with given rates, not the steady-state
+ solution.
+
+** New function dtmc_bd() added
+
+** Function ctmc_bd_solve() has been removed
+
Summary of important user-visible changes for queueing-1.0.0
------------------------------------------------------------------------------
Modified: trunk/octave-forge/main/queueing/doc/INSTALL
===================================================================
--- trunk/octave-forge/main/queueing/doc/INSTALL 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/doc/INSTALL 2012-03-11 11:20:31 UTC (rev 9813)
@@ -16,7 +16,7 @@
1.1 Installation through Octave package management system
=========================================================
-The most recent version of `queueing' is 1.0.0 and can be downloaded
+The most recent version of `queueing' is 1.X.0 and can be downloaded
from Octave-Forge
`http://octave.sourceforge.net/queueing/'
@@ -37,13 +37,13 @@
octave:1>pkg list queueing
Package Name | Version | Installation directory
--------------+---------+-----------------------
- queueing *| 1.0.0 | /home/moreno/octave/queueing-1.0.0
+ queueing *| 1.X.0 | /home/moreno/octave/queueing-1.X.0
Alternatively, you can first download `queueing' from Octave-Forge;
then, to install the package in the system-wide location issue this
command at the Octave prompt:
- octave:1> pkg install _queueing-1.0.0.tar.gz_
+ octave:1> pkg install _queueing-1.X.0.tar.gz_
(you may need to start Octave as root in order to allow the
installation to copy the files to the target locations). After this,
@@ -52,7 +52,7 @@
If you do not have root access, you can do a local install using:
- octave:1> pkg install -local queueing-1.0.0.tar.gz
+ octave:1> pkg install -local queueing-1.X.0.tar.gz
This will install `queueing' within your home directory, and the
package will be available to your user only. *Note:* Octave version
@@ -69,8 +69,8 @@
If you want to manually install `queueing' in a custom location, you
can download the tarball and unpack it somewhere:
- tar xvfz queueing-1.0.0.tar.gz
- cd queueing-1.0.0/queueing/
+ tar xvfz queueing-1.X.0.tar.gz
+ cd queueing-1.X.0/queueing/
Copy all `.m' files from the `inst/' directory to some target
location. Then, start Octave with the `-p' option to add the target
Modified: trunk/octave-forge/main/queueing/doc/markovchains.txi
===================================================================
--- trunk/octave-forge/main/queueing/doc/markovchains.txi 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/doc/markovchains.txi 2012-03-11 11:20:31 UTC (rev 9813)
@@ -66,6 +66,9 @@
@math{n}. @math{\pi_i(n)} denotes the probability that the system is
in state @math{i} at step @math{n}.
+@c
+@subsection State occupancy probabilities
+
Given the transition probability matrix @math{\bf P} and the initial
state occupancy probability vector @math{{\bf \pi}(0) = (\pi_1(0),
\pi_2(0), @dots{}, \pi_N(0))} at step 0, the state occupancy
@@ -91,6 +94,7 @@
@math{{\bf \pi}(0)}. The stationary state occupancy probability vector
@math{\bf \pi} satisfies @math{{\bf \pi} = {\bf \pi} {\bf P}}.
+@c
@DOCSTRING(dtmc)
@noindent @strong{EXAMPLE}
@@ -101,6 +105,13 @@
@end group
@end example
+@subsection Birth-Death process
+
+@c
+@DOCSTRING(dtmc_bd)
+
+@subsection First passage times
+
The First Passage Time @math{M_{i j}} is defined as the average
number of transitions needed to visit state @math{j} for the first
time, starting from state @math{i}. Matrix @math{\bf M} satisfies the
@@ -123,6 +134,7 @@
@end example
@end ifnottex
+@c
@DOCSTRING(dtmc_fpt)
@c
@@ -132,16 +144,16 @@
@section Continuous-Time Markov Chains
@menu
-* CTMC Stationary Probability::
+* State occupancy probabilities::
* Birth-Death process::
* Expected Sojourn Time::
* Time-Averaged Expected Sojourn Time::
* Expected Time to Absorption::
-* CTMC First Passage Times::
+* First Passage Times::
@end menu
-@node CTMC Stationary Probability
-@subsection Stationary Probability
+@node State occupancy probabilities
+@subsection State occupancy probabilities
@DOCSTRING(ctmc)
@@ -305,7 +317,7 @@
@c
@c
@c
-@node CTMC First Passage Times
+@node First Passage Times
@subsection First Passage Times
@DOCSTRING(ctmc_fpt)
Modified: trunk/octave-forge/main/queueing/doc/queueing.html
===================================================================
--- trunk/octave-forge/main/queueing/doc/queueing.html 2012-03-10 20:51:31 UTC (rev 9812)
+++ trunk/octave-forge/main/queueing/doc/queueing.html 2012-03-11 11:20:31 UTC (rev 9813)
@@ -55,14 +55,19 @@
<li><a name="toc_Markov-Chains" href="#Markov-Chains">4 Markov Chains</a>
<ul>
<li><a href="#Discrete_002dTime-Markov-Chains">4.1 Discrete-Time Markov Chains</a>
+<ul>
+<li><a href="#Discrete_002dTime-Markov-Chains">4.1.1 State occupancy probabilities</a>
+<li><a href="#Discrete_002dTime-Markov-Chains">4.1.2 Birth-Death process</a>
+<li><a href="#Discrete_002dTime-Markov-Chains">4.1.3 First passage times</a>
+</li></ul>
<li><a href="#Continuous_002dTime-Markov-Chains">4.2 Continuous-Time Markov Chains</a>
<ul>
-<li><a href="#CTMC-Stationary-Probability">4.2.1 Stationary Probability</a>
+<li><a href="#State-occupancy-probabilities">4.2.1 State occupancy probabilities</a>
<li><a href="#Birth_002dDeath-process">4.2.2 Birth-Death process</a>
<li><a href="#Expected-Sojourn-Time">4.2.3 Expected Sojourn Time</a>
<li><a href="#Time_002dAveraged-Expected-Sojourn-Time">4.2.4 Time-Averaged Expected Sojourn Time</a>
<li><a href="#Expected-Time-to-Absorption">4.2.5 Expected Time to Absorption</a>
-<li><a href="#CTMC-First-Passage-Times">4.2.6 First Passage Times</a>
+<li><a href="#First-Passage-Times">4.2.6 First Passage Times</a>
</li></ul>
</li></ul>
<li><a name="toc_Single-Station-Queueing-Systems" href="#Single-Station-Queueing-Systems">5 Single Station Queueing Systems</a>
@@ -121,7 +126,7 @@
<h2 class="unnumbered">queueing</h2>
<p>This manual documents how to install and run the Queueing Toolbox.
-It corresponds to version 1.0.0 of the package.
+It corresponds to version 1.X.0 of the package.
<!-- -->
<ul class="menu">
@@ -189,8 +194,7 @@
<li>Approximate MVA for closed, single-class networks with blocking
(MVABLO algorithm by F. Akyildiz);
- <li>Computation of Asymptotic Bounds, Balanced System Bounds
-and Geometric Bounds;
+ <li>Asymptotic Bounds, Balanced System Bounds and Geometric Bounds;
</ul>
@@ -209,15 +213,15 @@
<li>M/H_m/1 (Hyperexponential service time distribution)
</ul>
- <p>Functions for Markov chain analysis are also provided (discrete and
-continuous time Markov chains are supported):
+ <p>Functions for Markov chain analysis are also provided, for discrete-time
+chains (DTMC) or continuous-time chains (CTMC):
<ul>
-<li>Birth-death process;
-<li>Computation of transient and steady-state occupancy probabilities;
-<li>Computation of mean time to absorption;
-<li>Computation of time-averages sojourn time.
-<li>Computation of mean passage times
+<li>Birth-death process (DTMC and CTMC);
+<li>Transient and steady-state occupancy probabilities (DTMC and CTMC);
+<li>Mean time to absorption (CTMC);
+<li>Time-averaged sojourn times (CTMC);
+<li>First passage times (DTMC and CTMC).
</ul>
@@ -308,7 +312,7 @@
<h3 class="section">2.1 Installation through Octave package management system</h3>
-<p>The most recent version of <code>queueing</code> is 1.0.0 and can
+<p>The most recent version of <code>queueing</code> is 1.X.0 and can
be downloaded from Octave-Forge
<p><a href="http://octave.sourceforge.net/queueing/">http://octave.sourceforge.net/queueing/</a>
@@ -330,13 +334,13 @@
<pre class="example"> octave:1><kbd>pkg list queueing</kbd>
Package Name | Version | Installation directory
--------------+---------+-----------------------
- queueing *| 1.0.0 | /home/moreno/octave/queueing-1.0.0
+ queueing *| 1.X.0 | /home/moreno/octave/queueing-1.X.0
</pre>
<p>Alternatively, you can first download <code>queueing</code> from
Octave-Forge; then, to install the package in the system-wide
location issue this command at the Octave prompt:
-<pre class="example"> octave:1> <kbd>pkg install </kbd><em>queueing-1.0.0.tar.gz</em>
+<pre class="example"> octave:1> <kbd>pkg install </kbd><em>queueing-1.X.0.tar.gz</em>
</pre>
<p class="noindent">(you may need to start Octave as root in order to allow the
installation to copy the files to the target locations). After this,
@@ -345,7 +349,7 @@
<p>If you do not have root access, you can do a local install using:
-<pre class="example"> octave:1> <kbd>pkg install -local queueing-1.0.0.tar.gz</kbd>
+<pre class="example"> octave:1> <kbd>pkg install -local queueing-1.X.0.tar.gz</kbd>
</pre>
<p>This will install <code>queueing</code> within your home directory, and the
package will be available to your user only. <strong>Note:</strong> Octave
@@ -371,8 +375,8 @@
<p>If you want to manually install <code>queueing</code> in a custom location,
you can download the tarball and unpack it somewhere:
-<pre class="example"> <kbd>tar xvfz queueing-1.0.0.tar.gz</kbd>
- <kbd>cd queueing-1.0.0/queueing/</kbd>
+<pre class="example"> <kbd>tar xvfz queueing-1.X.0.tar.gz</kbd>
+ <kbd>cd queueing-1.X.0/queueing/</kbd>
</pre>
<p>Copy all <code>.m</code> files from the <samp><span class="file">inst/</span></samp> directory to some
target location. Then, start Octave with the <samp><span class="option">-p</span></samp> option to add
@@ -813,7 +817,9 @@
n. \pi_i(n) denotes the probability that the system is
in state i at step n.
- <p>Given the transition probability matrix \bf P and the initial
+<h4 class="subsection">4.1.1 State occupancy probabilities</h4>
+
+<p>Given the transition probability matrix \bf P and the initial
state occupancy probability vector \bf \pi(0) = (\pi_1(0),
\pi_2(0), <small class="dots">...</small>, \pi_N(0)) at step 0, the state occupancy
probability vector \bf \pi(n) at step n can be
@@ -889,7 +895,40 @@
T, ss(2)*ones(size(T)), "r;Steady State;" );
xlabel("Time Step");</pre>
</pre>
- <p>The First Passage Time M_i j is defined as the average
+ <h4 class="subsection">4.1.2 Birth-Death process</h4>
+
+<p><a name="doc_002ddtmc_005fbd"></a>
+
+<div class="defun">
+— Function File: <var>P</var> = <b>dtmc_bd</b> (<var>birth, death</var>)<var><a name="index-dtmc_005fbd-7"></a></var><br>
+<blockquote>
+ <p><a name="index-Markov-chain_002c-discrete-time-8"></a><a name="index-Birth_002ddeath-process-9"></a>
+Returns the N \times N transition probability matrix P
+for a birth-death process with given rates.
+
+ <p><strong>INPUTS</strong>
+
+ <dl>
+<dt><var>birth</var><dd>Vector with N-1 elements, where <var>birth</var><code>(i)</code> is the
+transition probability from state i to state i+1.
+
+ <br><dt><var>death</var><dd>Vector with N-1 elements, where <var>death</var><code>(i)</code> is the
+transition probability from state i+1 to state i.
+
+ </dl>
+
+ <p><strong>OUTPUTS</strong>
+
+ <dl>
+<dt><var>P</var><dd>Transition probability matrix for the birth-death process.
+
+ </dl>
+
+ </blockquote></div>
+
+<h4 class="subsection">4.1.3 First passage times</h4>
+
+<p>The First Passage Time M_i j is defined as the average
number of transitions needed to visit state j for the first
time, starting from state i. Matrix \bf M satisfies the
property that
@@ -903,10 +942,10 @@
<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-7"></a></var><br>
-— Function File: <var>m</var> = <b>dtmc_fpt</b> (<var>P, i, j</var>)<var><a name="index-dtmc_005ffpt-8"></a></var><br>
+— Function File: <var>M</var> = <b>dtmc_fpt</b> (<var>P</var>)<var><a name="index-dtmc_005ffpt-10"></a></var><br>
+— Function File: <var>m</var> = <b>dtmc_fpt</b> (<var>P, i, j</var>)<var><a name="index-dtmc_005ffpt-11"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-discrete-time-9"></a><a name="index-First-passage-times-10"></a>
+ <p><a name="index-Markov-chain_002c-discrete-time-12"></a><a name="index-First-passage-times-13"></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
@@ -956,31 +995,31 @@
<h3 class="section">4.2 Continuous-Time Markov Chains</h3>
<ul class="menu">
-<li><a accesskey="1" href="#CTMC-Stationary-Probability">CTMC Stationary Probability</a>
+<li><a accesskey="1" href="#State-occupancy-probabilities">State occupancy probabilities</a>
<li><a accesskey="2" href="#Birth_002dDeath-process">Birth-Death process</a>
<li><a accesskey="3" href="#Expected-Sojourn-Time">Expected Sojourn Time</a>
<li><a accesskey="4" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a>
<li><a accesskey="5" href="#Expected-Time-to-Absorption">Expected Time to Absorption</a>
-<li><a accesskey="6" href="#CTMC-First-Passage-Times">CTMC First Passage Times</a>
+<li><a accesskey="6" href="#First-Passage-Times">First Passage Times</a>
</ul>
<div class="node">
-<a name="CTMC-Stationary-Probability"></a>
+<a name="State-occupancy-probabilities"></a>
<p><hr>
Next: <a rel="next" accesskey="n" href="#Birth_002dDeath-process">Birth-Death process</a>,
Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a>
</div>
-<h4 class="subsection">4.2.1 Stationary Probability</h4>
+<h4 class="subsection">4.2.1 State occupancy probabilities</h4>
<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-11"></a></var><br>
-— Function File: <var>p</var> = <b>ctmc</b> (<var>Q, t. q0</var>)<var><a name="index-ctmc-12"></a></var><br>
+— Function File: <var>p</var> = <b>ctmc</b> (<var>Q</var>)<var><a name="index-ctmc-14"></a></var><br>
+— Function File: <var>p</var> = <b>ctmc</b> (<var>Q, t. q0</var>)<var><a name="index-ctmc-15"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-13"></a><a name="index-Continuous-time-Markov-chain-14"></a><a name="index-Markov-chain_002c-state-occupancy-probabilities-15"></a><a name="index-Stationary-probabilities-16"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-16"></a><a name="index-Continuous-time-Markov-chain-17"></a><a name="index-Markov-chain_002c-state-occupancy-probabilities-18"></a><a name="index-Stationary-probabilities-19"></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
@@ -1033,7 +1072,7 @@
<a name="Birth_002dDeath-process"></a>
<p><hr>
Next: <a rel="next" accesskey="n" href="#Expected-Sojourn-Time">Expected Sojourn Time</a>,
-Previous: <a rel="previous" accesskey="p" href="#CTMC-Stationary-Probability">CTMC Stationary Probability</a>,
+Previous: <a rel="previous" accesskey="p" href="#State-occupancy-probabilities">State occupancy probabilities</a>,
Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a>
</div>
@@ -1043,11 +1082,11 @@
<p><a name="doc_002dctmc_005fbd"></a>
<div class="defun">
-— Function File: <var>p</var> = <b>ctmc_bd</b> (<var>birth, death</var>)<var><a name="index-ctmc_005fbd-17"></a></var><br>
+— Function File: <var>Q</var> = <b>ctmc_bd</b> (<var>birth, death</var>)<var><a name="index-ctmc_005fbd-20"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-18"></a><a name="index-Birth_002ddeath-process-19"></a>
-Compute the steady-state solution of a birth-death process with state
-space (1, <small class="dots">...</small>, N).
+ <p><a name="index-Markov-chain_002c-continuous-time-21"></a><a name="index-Birth_002ddeath-process-22"></a>
+Returns the N \times N infinitesimal generator matrix Q
+for a birth-death process with given rates.
<p><strong>INPUTS</strong>
@@ -1063,8 +1102,7 @@
<p><strong>OUTPUTS</strong>
<dl>
-<dt><var>p</var><dd><var>p</var><code>(i)</code> is the steady-state probability that the system is
-in state i, i=1, <small class="dots">...</small>, N.
+<dt><var>Q</var><dd>Infinitesimal generator matrix for the birth-death process.
</dl>
@@ -1100,10 +1138,10 @@
<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-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>
+— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, t, p </var>)<var><a name="index-ctmc_005fexps-23"></a></var><br>
+— Function File: <var>L</var> = <b>ctmc_exps</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fexps-24"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-22"></a><a name="index-Expected-sojourn-time-23"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-25"></a><a name="index-Expected-sojourn-time-26"></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
@@ -1130,9 +1168,9 @@
<dl>
<dt><var>L</var><dd>If this function is called with three arguments, <var>L</var><code>(i)</code> is
-the expected time spent in state j during the interval
+the expected time spent in state i during the interval
[0,t]. If this function is called with two arguments
-<var>L</var><code>(i)</code> is the expected time spent in state i until
+<var>L</var><code>(i)</code> is either the expected time spent in state i until
absorption (if i is a transient state), or zero
(if <var>i</var> is an absorbing state).
@@ -1183,9 +1221,9 @@
<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-24"></a></var><br>
+— Function File: <var>M</var> = <b>ctmc_taexps</b> (<var>Q, t, p</var>)<var><a name="index-ctmc_005ftaexps-27"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-25"></a><a name="index-Time_002dalveraged-sojourn-time-26"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-28"></a><a name="index-Time_002dalveraged-sojourn-time-29"></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] spent in
state i, assuming that the state occupancy probabilities at
@@ -1242,7 +1280,7 @@
<div class="node">
<a name="Expected-Time-to-Absorption"></a>
<p><hr>
-Next: <a rel="next" accesskey="n" href="#CTMC-First-Passage-Times">CTMC First Passage Times</a>,
+Next: <a rel="next" accesskey="n" href="#First-Passage-Times">First Passage Times</a>,
Previous: <a rel="previous" accesskey="p" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a>,
Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a>
@@ -1268,9 +1306,9 @@
<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-27"></a></var><br>
+— Function File: <var>t</var> = <b>ctmc_mtta</b> (<var>Q, p</var>)<var><a name="index-ctmc_005fmtta-30"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-28"></a><a name="index-Mean-time-to-absorption-29"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-31"></a><a name="index-Mean-time-to-absorption-32"></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
@@ -1331,9 +1369,9 @@
Performance Evaluation with Computer Science Applications</cite>, Wiley,
1998.
- <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>
+ <p><a name="index-Bolch_002c-G_002e-33"></a><a name="index-Greiner_002c-S_002e-34"></a><a name="index-de-Meer_002c-H_002e-35"></a><a name="index-Trivedi_002c-K_002e-36"></a>
<div class="node">
-<a name="CTMC-First-Passage-Times"></a>
+<a name="First-Passage-Times"></a>
<p><hr>
Previous: <a rel="previous" accesskey="p" href="#Expected-Time-to-Absorption">Expected Time to Absorption</a>,
Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a>
@@ -1345,10 +1383,10 @@
<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-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>
+— Function File: <var>M</var> = <b>ctmc_fpt</b> (<var>Q</var>)<var><a name="index-ctmc_005ffpt-37"></a></var><br>
+— Function File: <var>m</var> = <b>ctmc_fpt</b> (<var>Q, i, j</var>)<var><a name="index-ctmc_005ffpt-38"></a></var><br>
<blockquote>
- <p><a name="index-Markov-chain_002c-continuous-time-36"></a><a name="index-First-passage-times-37"></a>
+ <p><a name="index-Markov-chain_002c-continuous-time-39"></a><a name="index-First-passage-times-40"></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
@@ -1454,9 +1492,9 @@
<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-qnmm1-38"></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-qnmm1-41"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-39"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f1_007d-system-42"></a>
Compute utilization, response time, average number of requests
and throughput for a M/M/1 queue.
@@ -1501,7 +1539,7 @@
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-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>
+ <p><a name="index-Bolch_002c-G_002e-43"></a><a name="index-Greiner_002c-S_002e-44"></a><a name="index-de-Meer_002c-H_002e-45"></a><a name="index-Trivedi_002c-K_002e-46"></a>
<!-- M/M/m -->
<div class="node">
<a name="The-M%2fM%2fm-System"></a>
@@ -1527,10 +1565,10 @@
<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-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>
+— 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-47"></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-48"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-46"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002fm_007d-system-49"></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.
@@ -1582,7 +1620,7 @@
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-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>
+ <p><a name="index-Bolch_002c-G_002e-50"></a><a name="index-Greiner_002c-S_002e-51"></a><a name="index-de-Meer_002c-H_002e-52"></a><a name="index-Trivedi_002c-K_002e-53"></a>
<!-- M/M/inf -->
<div class="node">
<a name="The-M%2fM%2finf-System"></a>
@@ -1605,7 +1643,7 @@
<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-51"></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-54"></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
@@ -1613,7 +1651,7 @@
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-52"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f_007dinf-system-55"></a>
<p><strong>INPUTS</strong>
@@ -1631,7 +1669,7 @@
different from the utilization, which in the case of M/M/\infty
centers is always zero.
- <p><a name="index-traffic-intensity-53"></a>
+ <p><a name="index-traffic-intensity-56"></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
@@ -1659,7 +1697,7 @@
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-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>
+ <p><a name="index-Bolch_002c-G_002e-57"></a><a name="index-Greiner_002c-S_002e-58"></a><a name="index-de-Meer_002c-H_002e-59"></a><a name="index-Trivedi_002c-K_002e-60"></a>
<!-- M/M/1/k -->
<div class="node">
<a name="The-M%2fM%2f1%2fK-System"></a>
@@ -1683,9 +1721,9 @@
<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-58"></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-61"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-59"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002f1_002fK_007d-system-62"></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
@@ -1752,9 +1790,9 @@
<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-60"></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-63"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-61"></a>
+ <p><a name="index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-64"></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
@@ -1812,7 +1850,7 @@
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-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>
+ <p><a name="index-Bolch_002c-G_002e-65"></a><a name="index-Greiner_002c-S_002e-66"></a><a name="index-de-Meer_002c-H_002e-67"></a><a name="index-Trivedi_002c-K_002e-68"></a>
<!-- Approximate M/M/m -->
<div class="node">
@@ -1834,9 +1872,9 @@
<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-66"></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-69"></a></var><br>
<blockquote>
- <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-67"></a>
+ <p><a name="index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-70"></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
@@ -1883,7 +1921,7 @@
and Markov Chains: Modeling and Performance Evaluation with Computer
Science Applications</cite>, Wiley, 1998
- <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>
+ <p><a name="index-Bolch_002c-G_002e-71"></a><a name="index-Greiner_002c-S_002e-72"></a><a name="index-de-Meer_002c-H_002e-73"></a><a name="index-Trivedi_002c-K_002e-74"></a>
<div class="node">
<a name="The-M%2fG%2f1-System"></a>
<a name="The-M_002fG_002f1-System"></a>
@@ -1899,9 +1937,9 @@
<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-qnmg1-72"></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-qnmg1-75"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-73"></a>
+ <p><a name="index-g_t_0040math_007bM_002fG_002f1_007d-system-76"></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
@@ -1958,9 +1996,9 @@
<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-qnmh1-74"></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-qnmh1-77"></a></var><br>
<blockquote>
- <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-75"></a>
+ <p><a name="index-g_t_0040math_007bM_002fH_005fm_002f1_007d-system-78"></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:
@@ -2042,7 +2080,7 @@
<li><a accesskey="6" href="#Utility-functions">Utility functions</a>: Utility functions to compute miscellaneous quantities
</ul>
-<p><a name="index-queueing-networks-76"></a>
+<p><a name="index-queueing-networks-79"></a>
<!-- INTRODUCTION -->
<div class="node">
<a name="Introduction-to-QNs"></a>
@@ -2303,13 +2341,13 @@
<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-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>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S</var>)<var><a name="index-qnmknode-80"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/m-fcfs", S, m</var>)<var><a name="index-qnmknode-81"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"m/m/1-lcfs-pr", S</var>)<var><a name="index-qnmknode-82"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S</var>)<var><a name="index-qnmknode-83"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/1-ps", S, s2</var>)<var><a name="index-qnmknode-84"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S</var>)<var><a name="index-qnmknode-85"></a></var><br>
+— Function File: <var>Q</var> = <b>qnmknode</b> (<var>"-/g/inf", S, s2</var>)<var><a name="index-qnmknode-86"></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
@@ -2378,10 +2416,10 @@
<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-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>
+— 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-87"></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-88"></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-89"></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-90"></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.
@@ -2559,11 +2597,11 @@
<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-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>
+— 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-91"></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-92"></a></var><br>
+— Function File: <var>pr</var> = <b>qnjackson</b> (<var>lambda, S, P, m, k</var>)<var><a name="index-qnjackson-93"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-single-class-91"></a><a name="index-Jackson-network-92"></a>
+ <p><a name="index-open-network_002c-single-class-94"></a><a name="index-Jackson-network-95"></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
@@ -2645,7 +2683,7 @@
Performance Evaluation with Computer Science Applications</cite>, Wiley,
1998, pp. 284–287.
- <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>
+ <p><a name="index-Bolch_002c-G_002e-96"></a><a name="index-Greiner_002c-S_002e-97"></a><a name="index-de-Meer_002c-H_002e-98"></a><a name="index-Trivedi_002c-K_002e-99"></a>
<h4 class="subsection">6.3.2 The Convolution Algorithm</h4>
@@ -2679,10 +2717,10 @@
<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-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>
+— 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-100"></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-101"></a></var><br>
<blockquote>
- <p><a name="index-closed-network-99"></a><a name="index-normalization-constant-100"></a><a name="index-convolution-algorithm-101"></a>
+ <p><a name="index-closed-network-102"></a><a name="index-normalization-constant-103"></a><a name="index-convolution-algorithm-104"></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
@@ -2773,20 +2811,20 @@
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-102"></a>
+ <p><a name="index-Buzen_002c-J_002e-P_002e-105"></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-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>
+ <p><a name="index-Bolch_002c-G_002e-106"></a><a name="index-Greiner_002c-S_002e-107"></a><a name="index-de-Meer_002c-H_002e-108"></a><a name="index-Trivedi_002c-K_002e-109"></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-107"></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-110"></a></var><br>
<blockquote>
- <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>
+ <p><a name="index-closed-network-111"></a><a name="index-normalization-constant-112"></a><a name="index-convolution-algorithm-113"></a><a name="index-load_002ddependent-service-center-114"></a>
This function implements the <em>convolution algorithm</em> for
product-form, single-class closed queueing networks with general
load-dependent service centers.
@@ -2846,7 +2884,7 @@
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-112"></a>
+ <p><a name="index-Schwetman_002c-H_002e-115"></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
@@ -2854,7 +2892,7 @@
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-113"></a><a name="index-Kobayashi_002c-H_002e-114"></a>
+ <p><a name="index-Reiser_002c-M_002e-116"></a><a name="index-Kobayashi_002c-H_002e-117"></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,
@@ -2866,7 +2904,7 @@
function f_i defined in Schwetman, <code>Some Computational
Aspects of Queueing Network Models</code>.
- <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-118"></a><a name="index-Greiner_002c-S_002e-119"></a><a name="index-de-Meer_002c-H_002e-120"></a><a name="index-Trivedi_002c-K_002e-121"></a>
<h4 class="subsection">6.3.3 Open networks</h4>
@@ -2874,10 +2912,10 @@
<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-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>
+— 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-122"></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-123"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-single-class-121"></a><a name="index-BCMP-network-122"></a>
+ <p><a name="index-open-network_002c-single-class-124"></a><a name="index-BCMP-network-125"></a>
Analyze open, single class BCMP queueing networks.
<p>This function works for a subset of BCMP single-class open networks
@@ -2970,16 +3008,16 @@
Networks and Markov Chains: Modeling and Performance Evaluation with
Computer Science Applications</cite>, Wiley, 1998.
- <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>
+ <p><a name="index-Bolch_002c-G_002e-126"></a><a name="index-Greiner_002c-S_002e-127"></a><a name="index-de-Meer_002c-H_002e-128"></a><a name="index-Trivedi_002c-K_002e-129"></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-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>
+— 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-130"></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-131"></a></var><br>
<blockquote>
- <p><a name="index-open-network_002c-multiple-classes-129"></a>
+ <p><a name="index-open-network_002c-multiple-classes-132"></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
@@ -3044,7 +3082,7 @@
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-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>
+ <p><a name="index-Lazowska_002c-E_002e-D_002e-133"></a><a name="index-Zahorjan_002c-J_002e-134"></a><a name="index-Graham_002c-G_002e-S_002e-135"></a><a name="index-Sevcik_002c-K_002e-C_002e-136"></a>
<h4 class="subsection">6.3.4 Closed Networks</h4>
@@ -3052,11 +3090,11 @@
<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-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>
+— 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-137"></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-138"></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-139"></a></var><br>
<blockquote>
- <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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-140"></a><a name="index-closed-network_002c-single-class-141"></a><a name="index-normalization-constant-142"></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
@@ -3157,7 +3195,7 @@
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-140"></a><a name="index-Lavenberg_002c-S_002e-S_002e-141"></a>
+ <p><a name="index-Reiser_002c-M_002e-143"></a><a name="index-Lavenberg_002c-S_002e-S_002e-144"></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)}, -->
@@ -3166,15 +3204,15 @@
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-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>
+ <p><a name="index-Jain_002c-R_002e-145"></a><a name="index-Bolch_002c-G_002e-146"></a><a name="index-Greiner_002c-S_002e-147"></a><a name="index-de-Meer_002c-H_002e-148"></a><a name="index-Trivedi_002c-K_002e-149"></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-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>
+— 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-150"></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-151"></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-load_002ddependent-service-center-151"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-152"></a><a name="index-closed-network_002c-single-class-153"></a><a name="index-load_002ddependent-service-center-154"></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
@@ -3232,15 +3270,15 @@
1998, Section 8.2.4.1, “Networks with Load-Deèpendent Service: Closed
Networks”.
- <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>
+ <p><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>
<!-- 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-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>
+— 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-159"></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-160"></a></var><br>
<blockquote>
- <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-158"></a><a name="index-CMVA-159"></a>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029-161"></a><a name="index-CMVA-162"></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
@@ -3294,19 +3332,19 @@
closed networks</cite>. Queueing Syst. Theory Appl., 60:193–202, December
2008.
- <p><a name="index-Casale_002c-G_002e-160"></a>
+ <p><a name="index-Casale_002c-G_002e-163"></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-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>
+— 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-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</var>)<var><a name="index-qnclosedsinglemvaapprox-165"></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-166"></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-167"></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-168"></a></var><br>
<blockquote>
- <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>
+ <p><a name="index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-169"></a><a name="index-Approximate-MVA-170"></a><a name="index-Closed-n...
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