Mercurial > forge
changeset 9722:394c375af7a9 octave-forge
Fixed typos
| author | mmarzolla |
|---|---|
| date | Thu, 15 Mar 2012 21:34:38 +0000 |
| parents | 0a73915a01c4 |
| children | b2ed3689e9a4 |
| files | main/queueing/doc/markovchains.txi main/queueing/doc/queueing.html main/queueing/doc/queueing.pdf main/queueing/inst/ctmc_taexps.m |
| diffstat | 4 files changed, 162 insertions(+), 98 deletions(-) [+] |
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--- a/main/queueing/doc/markovchains.txi Thu Mar 15 21:26:25 2012 +0000 +++ b/main/queueing/doc/markovchains.txi Thu Mar 15 21:34:38 2012 +0000 @@ -53,10 +53,10 @@ The evolution of a Markov chain with finite state space @math{@{1, 2, @dots{}, N@}} can be fully described by a stochastic matrix @math{{\bf P}(n) = P_{i,j}(n)} such that @math{P_{i, j}(n) = P( X_{n+1} = j\ |\ -X_n = j )}. If the Markov chain is homogeneous (that is, the +X_n = i )}. If the Markov chain is homogeneous (that is, the transition probability matrix @math{{\bf P}(n)} is time-independent), we can simply write @math{{\bf P} = P_{i, j}}, where @math{P_{i, j} = -P( X_{n+1} = j\ |\ X_n = j )} for all @math{n=0, 1, 2, @dots{}}. +P( X_{n+1} = j\ |\ X_n = i )} for all @math{n=0, 1, 2, @dots{}}. The transition probability matrix @math{\bf P} must satisfy the following two properties: (1) @math{P_{i, j} @geq{} 0} for all @@ -65,9 +65,17 @@ @c @DOCSTRING(dtmc_check_P) +@menu +* State occupancy probabilities (DTMC):: +* Birth-death process (DTMC):: +* First passage times (DTMC):: +* Mean time to absorption (DTMC):: +@end menu + @c @c @c +@node State occupancy probabilities (DTMC) @subsection State occupancy probabilities We denote with @math{{\bf \pi}(n) = (\pi_1(n), \pi_2(n), @dots{}, @@ -112,12 +120,15 @@ @end group @end example -@subsection Birth-Death process +@c +@node Birth-death process (DTMC) +@subsection Birth-death process @c @DOCSTRING(dtmc_bd) -@subsection First passage times +@node First passage times (DTMC) +@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 @@ -145,6 +156,7 @@ @DOCSTRING(dtmc_fpt) @c +@node Mean time to absorption (DTMC) @subsection Mean Time to Absorption @DOCSTRING(dtmc_mtta) @@ -179,15 +191,15 @@ @DOCSTRING(ctmc_check_Q) @menu -* State occupancy probabilities:: -* Birth-Death process:: -* Expected Sojourn Time:: -* Time-Averaged Expected Sojourn Time:: -* Mean Time to Absorption:: -* First Passage Times:: +* State occupancy probabilities (CTMC):: +* Birth-death process (CTMC):: +* Expected sojourn times (CTMC):: +* Time-averaged expected sojourn times (CTMC):: +* Mean time to absorption (CTMC):: +* First passage times (CTMC):: @end menu -@node State occupancy probabilities +@node State occupancy probabilities (CTMC) @subsection State occupancy probabilities Similarly to the discrete case, we denote with @math{{\bf \pi}(t) = @@ -238,16 +250,16 @@ @c @c @c -@node Birth-Death process -@subsection Birth-Death process +@node Birth-death process (CTMC) +@subsection Birth-Death Process @DOCSTRING(ctmc_bd) @c @c @c -@node Expected Sojourn Time -@subsection Expected Sojourn Time +@node Expected sojourn times (CTMC) +@subsection Expected Sojourn Times Given a @math{N} state continuous-time Markov Chain with infinitesimal generator matrix @math{\bf Q}, we define the vector @math{{\bf L}(t) = @@ -277,7 +289,7 @@ @iftex @tex -$$ {\bf L}(t) = \int_{u=0}^t {\bf \pi}(u) du$$ +$$ {\bf L}(t) = \int_0^t {\bf \pi}(u) du$$ @end tex @end iftex @ifnottex @@ -312,8 +324,8 @@ @c @c @c -@node Time-Averaged Expected Sojourn Time -@subsection Time-Averaged Expected Sojourn Time +@node Time-averaged expected sojourn times (CTMC) +@subsection Time-Averaged Expected Sojourn Times @DOCSTRING(ctmc_taexps) @@ -328,7 +340,7 @@ @c @c @c -@node Mean Time to Absorption +@node Mean time to absorption (CTMC) @subsection Mean Time to Absorption If we consider a Markov Chain with absorbing states, it is possible to @@ -400,7 +412,7 @@ @c @c @c -@node First Passage Times +@node First passage times (CTMC) @subsection First Passage Times @DOCSTRING(ctmc_fpt)
--- a/main/queueing/doc/queueing.html Thu Mar 15 21:26:25 2012 +0000 +++ b/main/queueing/doc/queueing.html Thu Mar 15 21:34:38 2012 +0000 @@ -56,19 +56,19 @@ <ul> <li><a href="#Discrete_002dTime-Markov-Chains">4.1 Discrete-Time Markov Chains</a> <ul> -<li><a href="#Discrete_002dTime-Markov-Chains">4.1.1 State occupancy probabilities</a> -<li><a href="#Discrete_002dTime-Markov-Chains">4.1.2 Birth-Death process</a> -<li><a href="#Discrete_002dTime-Markov-Chains">4.1.3 First passage times</a> -<li><a href="#Discrete_002dTime-Markov-Chains">4.1.4 Mean Time to Absorption</a> +<li><a href="#State-occupancy-probabilities-_0028DTMC_0029">4.1.1 State occupancy probabilities</a> +<li><a href="#Birth_002ddeath-process-_0028DTMC_0029">4.1.2 Birth-death process</a> +<li><a href="#First-passage-times-_0028DTMC_0029">4.1.3 First Passage Times</a> +<li><a href="#Mean-time-to-absorption-_0028DTMC_0029">4.1.4 Mean Time to Absorption</a> </li></ul> <li><a href="#Continuous_002dTime-Markov-Chains">4.2 Continuous-Time Markov Chains</a> <ul> -<li><a href="#State-occupancy-probabilities">4.2.1 State occupancy probabilities</a> -<li><a href="#Birth_002dDeath-process">4.2.2 Birth-Death process</a> -<li><a href="#Expected-Sojourn-Time">4.2.3 Expected Sojourn Time</a> -<li><a href="#Time_002dAveraged-Expected-Sojourn-Time">4.2.4 Time-Averaged Expected Sojourn Time</a> -<li><a href="#Mean-Time-to-Absorption">4.2.5 Mean Time to Absorption</a> -<li><a href="#First-Passage-Times">4.2.6 First Passage Times</a> +<li><a href="#State-occupancy-probabilities-_0028CTMC_0029">4.2.1 State occupancy probabilities</a> +<li><a href="#Birth_002ddeath-process-_0028CTMC_0029">4.2.2 Birth-Death Process</a> +<li><a href="#Expected-sojourn-times-_0028CTMC_0029">4.2.3 Expected Sojourn Times</a> +<li><a href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">4.2.4 Time-Averaged Expected Sojourn Times</a> +<li><a href="#Mean-time-to-absorption-_0028CTMC_0029">4.2.5 Mean Time to Absorption</a> +<li><a href="#First-passage-times-_0028CTMC_0029">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> @@ -805,10 +805,10 @@ <p>The evolution of a Markov chain with finite state space {1, 2, <small class="dots">...</small>, N} can be fully described by a stochastic matrix \bf P(n) = P_i,j(n) such that P_i, j(n) = P( X_n+1 = j\ |\ -X_n = j ). If the Markov chain is homogeneous (that is, the +X_n = i ). If the Markov chain is homogeneous (that is, the transition probability matrix \bf P(n) is time-independent), we can simply write \bf P = P_i, j, where P_i, j = -P( X_n+1 = j\ |\ X_n = j ) for all n=0, 1, 2, <small class="dots">...</small>. +P( X_n+1 = j\ |\ X_n = i ) for all n=0, 1, 2, <small class="dots">...</small>. <p>The transition probability matrix \bf P must satisfy the following two properties: (1) P_i, j ≥ 0 for all @@ -827,6 +827,22 @@ </blockquote></div> +<ul class="menu"> +<li><a accesskey="1" href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a> +<li><a accesskey="2" href="#Birth_002ddeath-process-_0028DTMC_0029">Birth-death process (DTMC)</a> +<li><a accesskey="3" href="#First-passage-times-_0028DTMC_0029">First passage times (DTMC)</a> +<li><a accesskey="4" href="#Mean-time-to-absorption-_0028DTMC_0029">Mean time to absorption (DTMC)</a> +</ul> + +<div class="node"> +<a name="State-occupancy-probabilities-(DTMC)"></a> +<a name="State-occupancy-probabilities-_0028DTMC_0029"></a> +<p><hr> +Next: <a rel="next" accesskey="n" href="#Birth_002ddeath-process-_0028DTMC_0029">Birth-death process (DTMC)</a>, +Up: <a rel="up" accesskey="u" href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a> + +</div> + <h4 class="subsection">4.1.1 State occupancy probabilities</h4> <p>We denote with \bf \pi(n) = (\pi_1(n), \pi_2(n), <small class="dots">...</small>, @@ -911,7 +927,17 @@ T, ss(2)*ones(size(T)), "r;Steady State;" ); xlabel("Time Step");</pre> </pre> - <h4 class="subsection">4.1.2 Birth-Death process</h4> + <div class="node"> +<a name="Birth-death-process-(DTMC)"></a> +<a name="Birth_002ddeath-process-_0028DTMC_0029"></a> +<p><hr> +Next: <a rel="next" accesskey="n" href="#First-passage-times-_0028DTMC_0029">First passage times (DTMC)</a>, +Previous: <a rel="previous" accesskey="p" href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a>, +Up: <a rel="up" accesskey="u" href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a> + +</div> + +<h4 class="subsection">4.1.2 Birth-death process</h4> <p><a name="doc_002ddtmc_005fbd"></a> @@ -942,7 +968,17 @@ </blockquote></div> -<h4 class="subsection">4.1.3 First passage times</h4> +<div class="node"> +<a name="First-passage-times-(DTMC)"></a> +<a name="First-passage-times-_0028DTMC_0029"></a> +<p><hr> +Next: <a rel="next" accesskey="n" href="#Mean-time-to-absorption-_0028DTMC_0029">Mean time to absorption (DTMC)</a>, +Previous: <a rel="previous" accesskey="p" href="#Birth_002ddeath-process-_0028DTMC_0029">Birth-death process (DTMC)</a>, +Up: <a rel="up" accesskey="u" href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a> + +</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 @@ -1001,6 +1037,15 @@ </blockquote></div> +<div class="node"> +<a name="Mean-time-to-absorption-(DTMC)"></a> +<a name="Mean-time-to-absorption-_0028DTMC_0029"></a> +<p><hr> +Previous: <a rel="previous" accesskey="p" href="#First-passage-times-_0028DTMC_0029">First passage times (DTMC)</a>, +Up: <a rel="up" accesskey="u" href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a> + +</div> + <h4 class="subsection">4.1.4 Mean Time to Absorption</h4> <p><a name="doc_002ddtmc_005fmtta"></a> @@ -1084,18 +1129,19 @@ </blockquote></div> <ul class="menu"> -<li><a accesskey="1" href="#State-occupancy-probabilities">State occupancy probabilities</a> -<li><a accesskey="2" href="#Birth_002dDeath-process">Birth-Death process</a> -<li><a accesskey="3" href="#Expected-Sojourn-Time">Expected Sojourn Time</a> -<li><a accesskey="4" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a> -<li><a accesskey="5" href="#Mean-Time-to-Absorption">Mean Time to Absorption</a> -<li><a accesskey="6" href="#First-Passage-Times">First Passage Times</a> +<li><a accesskey="1" href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a> +<li><a accesskey="2" href="#Birth_002ddeath-process-_0028CTMC_0029">Birth-death process (CTMC)</a> +<li><a accesskey="3" href="#Expected-sojourn-times-_0028CTMC_0029">Expected sojourn times (CTMC)</a> +<li><a accesskey="4" href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">Time-averaged expected sojourn times (CTMC)</a> +<li><a accesskey="5" href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a> +<li><a accesskey="6" href="#First-passage-times-_0028CTMC_0029">First passage times (CTMC)</a> </ul> <div class="node"> -<a name="State-occupancy-probabilities"></a> +<a name="State-occupancy-probabilities-(CTMC)"></a> +<a name="State-occupancy-probabilities-_0028CTMC_0029"></a> <p><hr> -Next: <a rel="next" accesskey="n" href="#Birth_002dDeath-process">Birth-Death process</a>, +Next: <a rel="next" accesskey="n" href="#Birth_002ddeath-process-_0028CTMC_0029">Birth-death process (CTMC)</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> @@ -1177,16 +1223,16 @@ q = ctmc(Q)</pre> ⇒ q = 0.50000 0.50000 </pre> <div class="node"> -<a name="Birth-Death-process"></a> -<a name="Birth_002dDeath-process"></a> +<a name="Birth-death-process-(CTMC)"></a> +<a name="Birth_002ddeath-process-_0028CTMC_0029"></a> <p><hr> -Next: <a rel="next" accesskey="n" href="#Expected-Sojourn-Time">Expected Sojourn Time</a>, -Previous: <a rel="previous" accesskey="p" href="#State-occupancy-probabilities">State occupancy probabilities</a>, +Next: <a rel="next" accesskey="n" href="#Expected-sojourn-times-_0028CTMC_0029">Expected sojourn times (CTMC)</a>, +Previous: <a rel="previous" accesskey="p" href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> -<h4 class="subsection">4.2.2 Birth-Death process</h4> +<h4 class="subsection">4.2.2 Birth-Death Process</h4> <p><a name="doc_002dctmc_005fbd"></a> @@ -1218,15 +1264,16 @@ </blockquote></div> <div class="node"> -<a name="Expected-Sojourn-Time"></a> +<a name="Expected-sojourn-times-(CTMC)"></a> +<a name="Expected-sojourn-times-_0028CTMC_0029"></a> <p><hr> -Next: <a rel="next" accesskey="n" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a>, -Previous: <a rel="previous" accesskey="p" href="#Birth_002dDeath-process">Birth-Death process</a>, +Next: <a rel="next" accesskey="n" href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">Time-averaged expected sojourn times (CTMC)</a>, +Previous: <a rel="previous" accesskey="p" href="#Birth_002ddeath-process-_0028CTMC_0029">Birth-death process (CTMC)</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> -<h4 class="subsection">4.2.3 Expected Sojourn Time</h4> +<h4 class="subsection">4.2.3 Expected Sojourn Times</h4> <p>Given a N state continuous-time Markov Chain with infinitesimal generator matrix \bf Q, we define the vector \bf L(t) = @@ -1322,16 +1369,16 @@ ylabel("Expected sojourn time");</pre> </pre> <div class="node"> -<a name="Time-Averaged-Expected-Sojourn-Time"></a> -<a name="Time_002dAveraged-Expected-Sojourn-Time"></a> +<a name="Time-averaged-expected-sojourn-times-(CTMC)"></a> +<a name="Time_002daveraged-expected-sojourn-times-_0028CTMC_0029"></a> <p><hr> -Next: <a rel="next" accesskey="n" href="#Mean-Time-to-Absorption">Mean Time to Absorption</a>, -Previous: <a rel="previous" accesskey="p" href="#Expected-Sojourn-Time">Expected Sojourn Time</a>, +Next: <a rel="next" accesskey="n" href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a>, +Previous: <a rel="previous" accesskey="p" href="#Expected-sojourn-times-_0028CTMC_0029">Expected sojourn times (CTMC)</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> -<h4 class="subsection">4.2.4 Time-Averaged Expected Sojourn Time</h4> +<h4 class="subsection">4.2.4 Time-Averaged Expected Sojourn Times</h4> <p><a name="doc_002dctmc_005ftaexps"></a> @@ -1364,7 +1411,7 @@ <dl> <dt><var>M</var><dd>If this function is called with three parameters, <var>M</var><code>(i)</code> -is the expected fraction of the interval 0,t] spent in state +is the expected fraction of the interval [0,t] spent in state i assuming that the state occupancy probability at time zero is <var>p</var>. If this function is called with two parameters, <var>M</var><code>(i)</code> is the expected fraction of time until absorption @@ -1397,10 +1444,11 @@ ylabel("Time-averaged Expected sojourn time");</pre> </pre> <div class="node"> -<a name="Mean-Time-to-Absorption"></a> +<a name="Mean-time-to-absorption-(CTMC)"></a> +<a name="Mean-time-to-absorption-_0028CTMC_0029"></a> <p><hr> -Next: <a rel="next" accesskey="n" href="#First-Passage-Times">First Passage Times</a>, -Previous: <a rel="previous" accesskey="p" href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a>, +Next: <a rel="next" accesskey="n" href="#First-passage-times-_0028CTMC_0029">First passage times (CTMC)</a>, +Previous: <a rel="previous" accesskey="p" href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">Time-averaged expected sojourn times (CTMC)</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> @@ -1490,9 +1538,10 @@ <p><a name="index-Bolch_002c-G_002e-42"></a><a name="index-Greiner_002c-S_002e-43"></a><a name="index-de-Meer_002c-H_002e-44"></a><a name="index-Trivedi_002c-K_002e-45"></a> <div class="node"> -<a name="First-Passage-Times"></a> +<a name="First-passage-times-(CTMC)"></a> +<a name="First-passage-times-_0028CTMC_0029"></a> <p><hr> -Previous: <a rel="previous" accesskey="p" href="#Mean-Time-to-Absorption">Mean Time to Absorption</a>, +Previous: <a rel="previous" accesskey="p" href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a>, Up: <a rel="up" accesskey="u" href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a> </div> @@ -5516,8 +5565,8 @@ <li><a href="#index-Approximate-MVA-179">Approximate MVA</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Asymmetric-_0040math_007bM_002fM_002fm_007d-system-79">Asymmetric M/M/m system</a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-BCMP-network-134">BCMP network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> -<li><a href="#index-Birth_002ddeath-process-30">Birth-death process</a>: <a href="#Birth_002dDeath-process">Birth-Death process</a></li> -<li><a href="#index-Birth_002ddeath-process-11">Birth-death process</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> +<li><a href="#index-Birth_002ddeath-process-30">Birth-death process</a>: <a href="#Birth_002ddeath-process-_0028CTMC_0029">Birth-death process (CTMC)</a></li> +<li><a href="#index-Birth_002ddeath-process-11">Birth-death process</a>: <a href="#Birth_002ddeath-process-_0028DTMC_0029">Birth-death process (DTMC)</a></li> <li><a href="#index-blocking-queueing-network-229">blocking queueing network</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-bounds_002c-asymptotic-241">bounds, asymptotic</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-bounds_002c-balanced-system-256">bounds, balanced system</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> @@ -5534,13 +5583,13 @@ <li><a href="#index-Closed-network_002c-single-class-180">Closed network, single class</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-closed-network_002c-single-class-150">closed network, single class</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-CMVA-171">CMVA</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> -<li><a href="#index-Continuous-time-Markov-chain-25">Continuous time Markov chain</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> +<li><a href="#index-Continuous-time-Markov-chain-25">Continuous time Markov chain</a>: <a href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a></li> <li><a href="#index-convolution-algorithm-113">convolution algorithm</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-copyright-295">copyright</a>: <a href="#Copying">Copying</a></li> -<li><a href="#index-Discrete-time-Markov-chain-6">Discrete time Markov chain</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-Expected-sojourn-time-34">Expected sojourn time</a>: <a href="#Expected-Sojourn-Time">Expected Sojourn Time</a></li> -<li><a href="#index-First-passage-times-49">First passage times</a>: <a href="#First-Passage-Times">First Passage Times</a></li> -<li><a href="#index-First-passage-times-15">First passage times</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> +<li><a href="#index-Discrete-time-Markov-chain-6">Discrete time Markov chain</a>: <a href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a></li> +<li><a href="#index-Expected-sojourn-time-34">Expected sojourn time</a>: <a href="#Expected-sojourn-times-_0028CTMC_0029">Expected sojourn times (CTMC)</a></li> +<li><a href="#index-First-passage-times-49">First passage times</a>: <a href="#First-passage-times-_0028CTMC_0029">First passage times (CTMC)</a></li> +<li><a href="#index-First-passage-times-15">First passage times</a>: <a href="#First-passage-times-_0028DTMC_0029">First passage times (DTMC)</a></li> <li><a href="#index-Jackson-network-104">Jackson network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-load_002ddependent-service-center-123">load-dependent service center</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-g_t_0040math_007bM_002fG_002f1_007d-system-85">M/G/1 system</a>: <a href="#The-M_002fG_002f1-System">The M/G/1 System</a></li> @@ -5550,19 +5599,22 @@ <li><a href="#index-g_t_0040math_007bM_002fM_002f_007dinf-system-64">M/M/inf system</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002fm_007d-system-58">M/M/m system</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-g_t_0040math_007bM_002fM_002fm_002fK_007d-system-73">M/M/m/K system</a>: <a href="#The-M_002fM_002fm_002fK-System">The M/M/m/K System</a></li> -<li><a href="#index-Markov-chain_002c-continuous-time-48">Markov chain, continuous time</a>: <a href="#First-Passage-Times">First Passage Times</a></li> -<li><a href="#index-Markov-chain_002c-continuous-time-40">Markov chain, continuous time</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> -<li><a href="#index-Markov-chain_002c-continuous-time-37">Markov chain, continuous time</a>: <a href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a></li> -<li><a href="#index-Markov-chain_002c-continuous-time-33">Markov chain, continuous time</a>: <a href="#Expected-Sojourn-Time">Expected Sojourn Time</a></li> -<li><a href="#index-Markov-chain_002c-continuous-time-29">Markov chain, continuous time</a>: <a href="#Birth_002dDeath-process">Birth-Death process</a></li> -<li><a href="#index-Markov-chain_002c-continuous-time-24">Markov chain, continuous time</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> +<li><a href="#index-Markov-chain_002c-continuous-time-48">Markov chain, continuous time</a>: <a href="#First-passage-times-_0028CTMC_0029">First passage times (CTMC)</a></li> +<li><a href="#index-Markov-chain_002c-continuous-time-40">Markov chain, continuous time</a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> +<li><a href="#index-Markov-chain_002c-continuous-time-37">Markov chain, continuous time</a>: <a href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">Time-averaged expected sojourn times (CTMC)</a></li> +<li><a href="#index-Markov-chain_002c-continuous-time-33">Markov chain, continuous time</a>: <a href="#Expected-sojourn-times-_0028CTMC_0029">Expected sojourn times (CTMC)</a></li> +<li><a href="#index-Markov-chain_002c-continuous-time-29">Markov chain, continuous time</a>: <a href="#Birth_002ddeath-process-_0028CTMC_0029">Birth-death process (CTMC)</a></li> +<li><a href="#index-Markov-chain_002c-continuous-time-24">Markov chain, continuous time</a>: <a href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a></li> <li><a href="#index-Markov-chain_002c-continuous-time-21">Markov chain, continuous time</a>: <a href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a></li> +<li><a href="#index-Markov-chain_002c-discrete-time-14">Markov chain, discrete time</a>: <a href="#First-passage-times-_0028DTMC_0029">First passage times (DTMC)</a></li> +<li><a href="#index-Markov-chain_002c-discrete-time-10">Markov chain, discrete time</a>: <a href="#Birth_002ddeath-process-_0028DTMC_0029">Birth-death process (DTMC)</a></li> +<li><a href="#index-Markov-chain_002c-discrete-time-5">Markov chain, discrete time</a>: <a href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a></li> <li><a href="#index-Markov-chain_002c-discrete-time-2">Markov chain, discrete time</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-Markov-chain_002c-disctete-time-18">Markov chain, disctete time</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-Markov-chain_002c-state-occupancy-probabilities-26">Markov chain, state occupancy probabilities</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> -<li><a href="#index-Markov-chain_002c-stationary-probabilities-7">Markov chain, stationary probabilities</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-Mean-time-to-absorption-41">Mean time to absorption</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> -<li><a href="#index-Mean-time-to-absorption-19">Mean time to absorption</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> +<li><a href="#index-Markov-chain_002c-disctete-time-18">Markov chain, disctete time</a>: <a href="#Mean-time-to-absorption-_0028DTMC_0029">Mean time to absorption (DTMC)</a></li> +<li><a href="#index-Markov-chain_002c-state-occupancy-probabilities-26">Markov chain, state occupancy probabilities</a>: <a href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a></li> +<li><a href="#index-Markov-chain_002c-stationary-probabilities-7">Markov chain, stationary probabilities</a>: <a href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a></li> +<li><a href="#index-Mean-time-to-absorption-41">Mean time to absorption</a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> +<li><a href="#index-Mean-time-to-absorption-19">Mean time to absorption</a>: <a href="#Mean-time-to-absorption-_0028DTMC_0029">Mean time to absorption (DTMC)</a></li> <li><a href="#index-Mean-Value-Analysys-_0028MVA_0029-149">Mean Value Analysys (MVA)</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Mean-Value-Analysys-_0028MVA_0029_002c-approximate-178">Mean Value Analysys (MVA), approximate</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-mixed-network-221">mixed network</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> @@ -5575,9 +5627,9 @@ <li><a href="#index-queueing-network-with-blocking-228">queueing network with blocking</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> <li><a href="#index-queueing-networks-88">queueing networks</a>: <a href="#Queueing-Networks">Queueing Networks</a></li> <li><a href="#index-RS-blocking-239">RS blocking</a>: <a href="#Algorithms-for-non-Product_002dform-QNs">Algorithms for non Product-form QNs</a></li> -<li><a href="#index-Stationary-probabilities-27">Stationary probabilities</a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> -<li><a href="#index-Stationary-probabilities-8">Stationary probabilities</a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-Time_002dalveraged-sojourn-time-38">Time-alveraged sojourn time</a>: <a href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a></li> +<li><a href="#index-Stationary-probabilities-27">Stationary probabilities</a>: <a href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a></li> +<li><a href="#index-Stationary-probabilities-8">Stationary probabilities</a>: <a href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a></li> +<li><a href="#index-Time_002dalveraged-sojourn-time-38">Time-alveraged sojourn time</a>: <a href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">Time-averaged expected sojourn times (CTMC)</a></li> <li><a href="#index-traffic-intensity-65">traffic intensity</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-warranty-294">warranty</a>: <a href="#Copying">Copying</a></li> </ul><div class="node"> @@ -5594,18 +5646,18 @@ <ul class="index-fn" compact> -<li><a href="#index-ctmc-22"><code>ctmc</code></a>: <a href="#State-occupancy-probabilities">State occupancy probabilities</a></li> -<li><a href="#index-ctmc_005fbd-28"><code>ctmc_bd</code></a>: <a href="#Birth_002dDeath-process">Birth-Death process</a></li> +<li><a href="#index-ctmc-22"><code>ctmc</code></a>: <a href="#State-occupancy-probabilities-_0028CTMC_0029">State occupancy probabilities (CTMC)</a></li> +<li><a href="#index-ctmc_005fbd-28"><code>ctmc_bd</code></a>: <a href="#Birth_002ddeath-process-_0028CTMC_0029">Birth-death process (CTMC)</a></li> <li><a href="#index-ctmc_005fcheck_005fQ-20"><code>ctmc_check_Q</code></a>: <a href="#Continuous_002dTime-Markov-Chains">Continuous-Time Markov Chains</a></li> -<li><a href="#index-ctmc_005fexps-31"><code>ctmc_exps</code></a>: <a href="#Expected-Sojourn-Time">Expected Sojourn Time</a></li> -<li><a href="#index-ctmc_005ffpt-46"><code>ctmc_fpt</code></a>: <a href="#First-Passage-Times">First Passage Times</a></li> -<li><a href="#index-ctmc_005fmtta-39"><code>ctmc_mtta</code></a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> -<li><a href="#index-ctmc_005ftaexps-35"><code>ctmc_taexps</code></a>: <a href="#Time_002dAveraged-Expected-Sojourn-Time">Time-Averaged Expected Sojourn Time</a></li> -<li><a href="#index-dtmc-3"><code>dtmc</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-dtmc_005fbd-9"><code>dtmc_bd</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> +<li><a href="#index-ctmc_005fexps-31"><code>ctmc_exps</code></a>: <a href="#Expected-sojourn-times-_0028CTMC_0029">Expected sojourn times (CTMC)</a></li> +<li><a href="#index-ctmc_005ffpt-46"><code>ctmc_fpt</code></a>: <a href="#First-passage-times-_0028CTMC_0029">First passage times (CTMC)</a></li> +<li><a href="#index-ctmc_005fmtta-39"><code>ctmc_mtta</code></a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> +<li><a href="#index-ctmc_005ftaexps-35"><code>ctmc_taexps</code></a>: <a href="#Time_002daveraged-expected-sojourn-times-_0028CTMC_0029">Time-averaged expected sojourn times (CTMC)</a></li> +<li><a href="#index-dtmc-3"><code>dtmc</code></a>: <a href="#State-occupancy-probabilities-_0028DTMC_0029">State occupancy probabilities (DTMC)</a></li> +<li><a href="#index-dtmc_005fbd-9"><code>dtmc_bd</code></a>: <a href="#Birth_002ddeath-process-_0028DTMC_0029">Birth-death process (DTMC)</a></li> <li><a href="#index-dtmc_005fcheck_005fP-1"><code>dtmc_check_P</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-dtmc_005ffpt-12"><code>dtmc_fpt</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> -<li><a href="#index-dtmc_005fmtta-16"><code>dtmc_mtta</code></a>: <a href="#Discrete_002dTime-Markov-Chains">Discrete-Time Markov Chains</a></li> +<li><a href="#index-dtmc_005ffpt-12"><code>dtmc_fpt</code></a>: <a href="#First-passage-times-_0028DTMC_0029">First passage times (DTMC)</a></li> +<li><a href="#index-dtmc_005fmtta-16"><code>dtmc_mtta</code></a>: <a href="#Mean-time-to-absorption-_0028DTMC_0029">Mean time to absorption (DTMC)</a></li> <li><a href="#index-population_005fmix-284"><code>population_mix</code></a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-qnammm-78"><code>qnammm</code></a>: <a href="#The-Asymmetric-M_002fM_002fm-System">The Asymmetric M/M/m System</a></li> <li><a href="#index-qnclosed-278"><code>qnclosed</code></a>: <a href="#Utility-functions">Utility functions</a></li> @@ -5662,7 +5714,7 @@ <li><a href="#index-Bolch_002c-G_002e-66">Bolch, G.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-Bolch_002c-G_002e-59">Bolch, G.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-Bolch_002c-G_002e-52">Bolch, G.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> -<li><a href="#index-Bolch_002c-G_002e-42">Bolch, G.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> +<li><a href="#index-Bolch_002c-G_002e-42">Bolch, G.</a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> <li><a href="#index-Buzen_002c-J_002e-P_002e-114">Buzen, J. P.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Casale_002c-G_002e-269">Casale, G.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Casale_002c-G_002e-172">Casale, G.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> @@ -5672,7 +5724,7 @@ <li><a href="#index-de-Meer_002c-H_002e-68">de Meer, H.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-de-Meer_002c-H_002e-61">de Meer, H.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-de-Meer_002c-H_002e-54">de Meer, H.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> -<li><a href="#index-de-Meer_002c-H_002e-44">de Meer, H.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> +<li><a href="#index-de-Meer_002c-H_002e-44">de Meer, H.</a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> <li><a href="#index-Graham_002c-G_002e-S_002e-245">Graham, G. S.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Graham_002c-G_002e-S_002e-144">Graham, G. S.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Greiner_002c-S_002e-106">Greiner, S.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> @@ -5681,7 +5733,7 @@ <li><a href="#index-Greiner_002c-S_002e-67">Greiner, S.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-Greiner_002c-S_002e-60">Greiner, S.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-Greiner_002c-S_002e-53">Greiner, S.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> -<li><a href="#index-Greiner_002c-S_002e-43">Greiner, S.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> +<li><a href="#index-Greiner_002c-S_002e-43">Greiner, S.</a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> <li><a href="#index-Hsieh_002c-C_002e-H-267">Hsieh, C. H</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li> <li><a href="#index-Jain_002c-R_002e-154">Jain, R.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> <li><a href="#index-Kobayashi_002c-H_002e-126">Kobayashi, H.</a>: <a href="#Algorithms-for-Product_002dForm-QNs">Algorithms for Product-Form QNs</a></li> @@ -5704,7 +5756,7 @@ <li><a href="#index-Trivedi_002c-K_002e-69">Trivedi, K.</a>: <a href="#The-M_002fM_002finf-System">The M/M/inf System</a></li> <li><a href="#index-Trivedi_002c-K_002e-62">Trivedi, K.</a>: <a href="#The-M_002fM_002fm-System">The M/M/m System</a></li> <li><a href="#index-Trivedi_002c-K_002e-55">Trivedi, K.</a>: <a href="#The-M_002fM_002f1-System">The M/M/1 System</a></li> -<li><a href="#index-Trivedi_002c-K_002e-45">Trivedi, K.</a>: <a href="#Mean-Time-to-Absorption">Mean Time to Absorption</a></li> +<li><a href="#index-Trivedi_002c-K_002e-45">Trivedi, K.</a>: <a href="#Mean-time-to-absorption-_0028CTMC_0029">Mean time to absorption (CTMC)</a></li> <li><a href="#index-Wong_002c-E_002e-293">Wong, E.</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-Zahorjan_002c-J_002e-292">Zahorjan, J.</a>: <a href="#Utility-functions">Utility functions</a></li> <li><a href="#index-Zahorjan_002c-J_002e-244">Zahorjan, J.</a>: <a href="#Bounds-on-performance">Bounds on performance</a></li>
--- a/main/queueing/inst/ctmc_taexps.m Thu Mar 15 21:26:25 2012 +0000 +++ b/main/queueing/inst/ctmc_taexps.m Thu Mar 15 21:34:38 2012 +0000 @@ -53,7 +53,7 @@ ## ## @item M ## If this function is called with three parameters, @code{@var{M}(i)} -## is the expected fraction of the interval @math{0,t]} spent in state +## is the expected fraction of the interval @math{[0,t]} spent in state ## @math{i} assuming that the state occupancy probability at time zero ## is @var{p}. If this function is called with two parameters, ## @code{@var{M}(i)} is the expected fraction of time until absorption