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queueing

This manual documents how to install and run the Queueing Toolbox. It corresponds to version 1.1.1 of the package.

• Summary
• Installation: Installation of the queueing toolbox.
• Markov Chains: Functions for Markov Chains.
• Single Station Queueing Systems: Functions for single-station queueing systems.
• Queueing Networks: Functions for queueing networks.
• References: References
• Copying: The GNU General Public License.
• Concept Index: An item for each concept.
• Function Index: An item for each function.
• Author Index: An item for each author.

1 Summary

• About the Queueing Toolbox: What is the Queueing Toolbox
• Contributing Guidelines: How to contribute
• Acknowledgements

1.1 About the Queueing Toolbox

This document describes the queueing toolbox for GNU Octave (queueing in short). The queueing toolbox, previously known as qnetworks, is a collection of functions written in GNU Octave for analyzing queueing networks and Markov chains. Specifically, queueing contains functions for analyzing Jackson networks, open, closed or mixed product-form BCMP networks, and computation of performance bounds. The following algorithms have been implemented

• Convolution for closed, single-class product-form networks with load-dependent service centers;
• Exact and approximate Mean Value Analysis (MVA) for single and multiple class product-form closed networks;
• MVA for mixed, multiple class product-form networks with load-independent service centers;
• Approximate MVA for closed, single-class networks with blocking (MVABLO algorithm by F. Akyildiz);
• Asymptotic Bounds, Balanced System Bounds and Geometric Bounds;

queueing provides functions for analyzing the following kind of single-station queueing systems:

• M/M/1
• M/M/m
• M/M/\infty
• M/M/1/k single-server, finite capacity system
• M/M/m/k multiple-server, finite capacity system
• Asymmetric M/M/m
• M/G/1 (general service time distribution)
• M/H_m/1 (Hyperexponential service time distribution)

Functions for Markov chain analysis are also provided:

• Birth-death process;
• Transient and steady-state occupancy probabilities;
• Mean times to absorption;
• Expected sojourn times and time-averaged sojourn times;
• Mean first passage times;

The queueing toolbox is distributed under the terms of the GNU General Public License (GPL), version 3 or later (see Copying). You are encouraged to share this software with others, and make this package more useful by contributing additional functions and reporting problems. See Contributing Guidelines.

If you use the queueing toolbox in a technical paper, please cite it as:

Moreno Marzolla, The qnetworks Toolbox: A Software Package for Queueing Networks Analysis. Khalid Al-Begain, Dieter Fiems and William J. Knottenbelt, Editors, Proceedings 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications (ASMTA 2010) Cardiff, UK, June 14–16, 2010, volume 6148 of Lecture Notes in Computer Science, Springer, pp. 102–116, ISBN 978-3-642-13567-5

If you use BibTeX, this is the citation block:

@inproceedings{queueing,
  author    = {Moreno Marzolla},
  title     = {The qnetworks Toolbox: A Software Package for Queueing 
               Networks Analysis},
  booktitle = {Analytical and Stochastic Modeling Techniques and 
               Applications, 17th International Conference, 
               ASMTA 2010, Cardiff, UK, June 14-16, 2010. Proceedings},
  editor    = {Khalid Al-Begain and Dieter Fiems and William J. Knottenbelt},
  year      = {2010},
  publisher = {Springer},
  series    = {Lecture Notes in Computer Science},
  volume    = {6148},
  pages     = {102--116},
  ee        = {http://dx.doi.org/10.1007/978-3-642-13568-2_8},
  isbn      = {978-3-642-13567-5}
}

An early draft of the paper above is available as Technical Report UBLCS-2010-04, February 2010, Department of Computer Science, University of Bologna, Italy.

1.2 Contributing Guidelines

Contributions and bug reports are always welcome. If you want to contribute to the queueing package, here are some guidelines:

• If you are contributing a new function, please embed proper documentation within the function itself. The documentation must be in texinfo format, so that it can be extracted and formatted into the printable manual. See the existing functions of the queueing package for the documentation style.
• Make sure that each new function properly checks the validity of its input parameters. For example, each function accepting vectors should check whether the dimensions match.
• Provide bibliographic references for each new algorithm you contribute. If your implementation differs in some way from the reference you give, please describe how and why your implementation differs. Add references to the doc/references.txi file.
• Include test and demo blocks with your code. Test blocks are particularly important, since most algorithms tend to be quite tricky to implement correctly. If appropriate, test blocks should also verify that the function fails on incorrect input parameters.

Send your contribution to Moreno Marzolla (marzolla@cs.unibo.it). If you are just a user of this package and find it useful, let me know by dropping me a line. Thanks.

1.3 Acknowledgements

The following people (listed in alphabetical order) contributed to the queueing package, either by providing feedback, reporting bugs or contributing code: Philip Carinhas, Phil Colbourn, Yves Durand, Marco Guazzone, Dmitry Kolesnikov.

2 Installing the queueing toolbox

• Installation through Octave package management system
• Manual installation
• Development sources
• Using the queueing toolbox

2.1 Installation through Octave package management system

The most recent version of queueing is 1.1.1 and can be downloaded from Octave-Forge

http://octave.sourceforge.net/queueing/

Additional information can be found at

http://www.moreno.marzolla.name/software/queueing/

If you have a recent version of GNU Octave and a network connection, you can install queueing directly from Octave command prompt using this command:

     octave:1> pkg install -forge queueing

The command above will automaticall download and install the latest version of the queueing toolbox from Octave Forge, and install it on your machine. You can verify that the package is indeed installed:

     octave:1>pkg list queueing
     Package Name  | Version | Installation directory
     --------------+---------+-----------------------
         queueing  |   1.1.1 | /home/moreno/octave/queueing-1.1.1

Note: Starting from version 1.1.1, queueing is no longer automatically loaded on startup. To load the package you need to issue the command pkg load queueing at the Octave prompt. To automatically load queueing each time Octave starts, you can add the command above to the startup script (.octaverc on Unix systems).

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-1.1.1.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, all functions will be readily available each time Octave starts, without the need to tweak the search path.

If you do not have root access, you can do a local install using:

     octave:1> pkg install -local queueing-1.1.1.tar.gz

This will install queueing within your home directory, and the package will be available to your user only.

Note: Octave version 3.2.3 as shipped with Ubuntu 10.04 seems to ignore -local and always tries to install the package on the system directory.

To remove queueing simply use

     octave:1> pkg uninstall queueing

2.2 Manual installation

If you want to manually install queueing in a custom location, you can download the tarball and unpack it somewhere:

     tar xvfz queueing-1.1.1.tar.gz
     cd queueing-1.1.1/queueing/

Copy all .m files from the inst/ directory to some target location. Then, start Octave with the -p option to add the target location to the search path, so that Octave will find all queueing functions automatically:

     octave -p /path/to/queueing

For example, if all queueing m-files are in /usr/local/queueing, you can start Octave as follows:

     octave -p /usr/local/queueing

If you want, you can add the following line to ~/.octaverc:

     addpath("/path/to/queueing");

so that the path /usr/local/queueing is automatically added to the search path each time Octave is started, and you no longer need to specify the -p option on the command line.

2.3 Development sources

The source code of the queueing package can be found in the Subversion repository at the URL:

http://octave.svn.sourceforge.net/viewvc/octave/trunk/octave-forge/main/queueing/

The source distribution contains additional development files which are not present in the installation tarball. This section briefly describes the content of the source tree. This is only relevant for developers who want to modify the code or documentation; normal users of the queueing package don't need

The source distribution contains the following directories:

doc/

Documentation source. Most of the documentation is extracted from the comment blocks of individual function files from the inst/ directory.

inst/

This directory contains the m-files which implement the various Queueing Network algorithms provided by queueing. As a notational convention, the names of source files containing functions for Queueing Networks start with the ‘qn’ prefix; the name of source files containing functions for Continuous-Time Markov Chains (CTMSs) start with the ‘ctmc’ prefix, and the names of files containing functions for Discrete-Time Markov Chains (DTMCs) start with the ‘dtmc’ prefix.

test/

This directory contains the test functions used to invoke all tests on all function files.

scripts/

This directory contains some utility scripts mostly from GNU Octave, which extract the documentation from the specially-formatted comments in the m-files.

examples/

This directory contains examples which are automatically extracted from the ‘demo’ blocks of the function files.

devel/

This directory contains function files which are either not working properly, or need additional testing before they are moved to the inst/ directory.

The queueing package ships with a Makefile which can be used to produce the documentation (in PDF and HTML format), and automatically execute all function tests. Specifically, the following targets are defined:

all

Running ‘make’ (or ‘make all’) on the top-level directory builds the programs used to extract the documentation from the comments embedded in the m-files, and then produce the documentation in PDF and HTML format (doc/queueing.pdf and doc/queueing.html, respectively).

check

Running ‘make check’ will execute all tests contained in the m-files. If you modify the code of any function in the inst/ directory, you should run the tests to ensure that no errors have been introduced. You are also encouraged to contribute new tests, especially for functions which are not adequately validated.

clean

distclean

dist

The ‘make clean’, ‘make distclean’ and ‘make dist’ commands are used to clean up the source directory and prepare the distribution archive in compressed tar format.

2.4 Using the queueing toolbox

You can use all functions by simply invoking their name with the appropriate parameters; the queueing package should display an error message in case of missing/wrong parameters. You can display the help text for any function using the help command. For example:

     octave:2> help qnmvablo

prints the documentation for the qnmvablo function. Additional information can be found in the queueing manual, which is available in PDF format in doc/queueing.pdf and in HTML format in doc/queueing.html.

Within GNU Octave, you can also run the test and demo blocks associated to the functions, using the test and demo commands respectively. To run all the tests of, say, the qnmvablo function:

     octave:3> test qnmvablo
     -| PASSES 4 out of 4 tests

To execute the demos of the qnclosed function, use the following:

     octave:4> demo qnclosed

3 Markov Chains

• Discrete-Time Markov Chains
• Continuous-Time Markov Chains

3.1 Discrete-Time Markov Chains

Let X_0, X_1, ..., X_n, ... be a sequence of random variables defined over a discete state space 0, 1, 2, .... The sequence X_0, X_1, ..., X_n, ... is a stochastic process with discrete time 0, 1, 2, .... A Markov chain is a stochastic process {X_n, n=0, 1, 2, ...} which satisfies the following Markov property:

P(X_n+1 = x_n+1 | X_n = x_n, X_n-1 = x_n-1, ..., X_0 = x_0) = P(X_n+1 = x_n+1 | X_n = x_n)

which basically means that the probability that the system is in a particular state at time n+1 only depends on the state the system was at time n.

The evolution of a Markov chain with finite state space {1, 2, ..., 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 = i ). If the Markov chain is homogeneous (that is, the transition probability matrix \bf P(n) is time-independent), we can write \bf P = [P_i, j], where P_i, j = P( X_n+1 = j\ |\ X_n = i ) for all n=0, 1, ....

The transition probability matrix \bf P must satisfy the following two properties: (1) P_i, j ≥ 0 for all i, j, and (2) \sum_j=1^N P_i,j = 1 for all i

• State occupancy probabilities (DTMC)
• Birth-death process (DTMC)
• Expected number of visits (DTMC)
• Time-averaged expected sojourn times (DTMC)
• Mean time to absorption (DTMC)
• First passage times (DTMC)

3.1.1 State occupancy probabilities

We denote with \bf \pi(n) = \left(\pi_1(n), \pi_2(n), ..., \pi_N(n) \right) the state occupancy probability vector at step n. \pi_i(n) denotes the probability that the system is in state i after n transitions.

Given the transition probability matrix \bf P and the initial state occupancy probability vector \bf \pi(0) = \left(\pi_1(0), \pi_2(0), ..., \pi_N(0)\right), \bf \pi(n) can be computed as:

     \pi(n) = \pi(0) P^n

Under certain conditions, there exists a stationary state occupancy probability \bf \pi = \lim_n \rightarrow +\infty \bf \pi(n), which is independent from \bf \pi(0). The stationary vector \bf \pi is the solution of the following linear system:

     /
     | \pi P   = \pi
     | \pi 1^T = 1
     \

where \bf 1 is the row vector of ones, and ( \cdot )^T the transpose operator.

EXAMPLE

This example is from GrSn97. Let us consider a maze with nine rooms, as shown in the following figure

     +-----+-----+-----+
     |     |     |     |
     |  1     2     3  |
     |     |     |     |
     +-   -+-   -+-   -+
     |     |     |     |
     |  4     5     6  |
     |     |     |     |
     +-   -+-   -+-   -+
     |     |     |     |
     |  7     8     9  |
     |     |     |     |
     +-----+-----+-----+

A mouse is placed in one of the rooms and can wander around. At each step, the mouse moves from the current room to a neighboring one with equal probability: if it is in room 1, it can move to room 2 and 4 with probability 1/2, respectively. If the mouse is in room 8, it can move to either 7, 5 or 9 with probability 1/3.

The transition probability \bf P from room i to room j is the following:

             / 0     1/2   0     1/2   0     0     0     0     0   \
             | 1/3   0     1/3   0     1/3   0     0     0     0   |
             | 0     1/2   0     0     0     1/2   0     0     0   |
             | 1/3   0     0     0     1/3   0     1/3   0     0   |
         P = | 0     1/4   0     1/4   0     1/4   0     1/4   0   |
             | 0     0     1/3   0     1/3   0     0     0     1/3 |
             | 0     0     0     1/2   0     0     0     1/2   0   |
             | 0     0     0     0     1/3   0     1/3   0     1/3 |
             \ 0     0     0     0     0     1/2   0     1/2   0   /

The stationary state occupancy probability vector can be computed using the following code:

      P = zeros(9,9);
      P(1,[2 4]    ) = 1/2;
      P(2,[1 5 3]  ) = 1/3;
      P(3,[2 6]    ) = 1/2;
      P(4,[1 5 7]  ) = 1/3;
      P(5,[2 4 6 8]) = 1/4;
      P(6,[3 5 9]  ) = 1/3;
      P(7,[4 8]    ) = 1/2;
      P(8,[7 5 9]  ) = 1/3;
      P(9,[6 8]    ) = 1/2;
      p = dtmc(P);
      disp(p)

         ⇒ 0.083333   0.125000   0.083333   0.125000
            0.166667   0.125000   0.083333   0.125000
            0.083333

3.1.2 Birth-death process

3.1.3 Expected Number of Visits

Given a N state discrete-time Markov chain with transition matrix \bf P and an integer n ≥ 0, we let L_i(n) be the the expected number of visits to state i during the first n transitions. The vector \bf L(n) = ( L_1(n), L_2(n), ..., L_N(n) ) is defined as

              n            n
             ___          ___
            \            \           i
     L(n) =  >   pi(i) =  >   pi(0) P
            /___         /___
             i=0          i=0

where \bf \pi(i) = \bf \pi(0)\bf P^i is the state occupancy probability after i transitions.

If \bf P is absorbing, i.e., the stochastic process eventually reaches a state with no outgoing transitions with probability 1, then we can compute the expected number of visits until absorption \bf L. To do so, we first rearrange the states to rewrite matrix \bf P as:

         / Q | R \
     P = |---+---|
         \ 0 | I /

where the first t states are transient and the last r states are absorbing (t+r = N). The matrix \bf N = (\bf I - \bf Q)^-1 is called the fundamental matrix; N_i,j is the expected number of times that the process is in the j-th transient state if it started in the i-th transient state. If we reshape \bf N to the size of \bf P (filling missing entries with zeros), we have that, for absorbing chains \bf L = \bf \pi(0)\bf N.

3.1.4 Time-averaged expected sojourn times

3.1.5 Mean Time to Absorption

The mean time to absorption is defined as the average number of transitions which are required to reach an absorbing state, starting from a transient state (or given an initial state occupancy probability vector \bf \pi(0)).

Let \bf t_i be the expected number of transitions before being absorbed in any absorbing state, starting from state i. Vector \bf t can be computed from the fundamental matrix \bf N (see Expected number of visits (DTMC)) as

     t = 1 N

Let \bf B = [ B_i, j ] be a matrix where B_i, j is the probability of being absorbed in state j, starting from transient state i. Again, using matrices \bf N and \bf R (see Expected number of visits (DTMC)) we can write

     B = N R

3.1.6 First Passage Times

The First Passage Time M_i, j is 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

                ___
               \
     M_ij = 1 + >   P_ij * M_kj
               /___
               k!=j

To compute \bf M = [ M_i, j] a different formulation is used. Let \bf W be the N \times N matrix having each row equal to the steady-state probability vector \bf \pi for \bf P; let \bf I be the N \times N identity matrix. Define \bf Z as follows:

                    -1
     Z = (I - P + W)

Then, we have that

            Z_jj - Z_ij
     M_ij = -----------
               \pi_j

According to the definition above, M_i,i = 0. We arbitrarily let M_i,i to be the mean recurrence time r_i for state i, that is the average number of transitions needed to return to state i starting from it. r_i is:

             1
     r_i = -----
           \pi_i

3.2 Continuous-Time Markov Chains

A stochastic process {X(t), t ≥ 0} is a continuous-time Markov chain if, for all integers n, and for any sequence t_0, t_1 , \ldots, t_n, t_n+1 such that t_0 < t_1 < \ldots < t_n < t_n+1, we have

P(X_n+1 = x_n+1 | X_n = x_n, X_n-1 = x_n-1, ..., X_0 = x_0) = P(X_n+1 = x_n+1 | X_n = x_n)

A continuous-time Markov chain is defined according to an infinitesimal generator matrix \bf Q = [Q_i,j], where for each i \neq j, Q_i, j is the transition rate from state i to state j. The matrix \bf Q must satisfy the property that, for all i, \sum_j=1^N Q_i, j = 0.

• 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)

3.2.1 State occupancy probabilities

Similarly to the discrete case, we denote with \bf \pi(t) = (\pi_1(t), \pi_2(t), ..., \pi_N(t) ) the state occupancy probability vector at time t. \pi_i(t) is the probability that the system is in state i at time t ≥ 0.

Given the infinitesimal generator matrix \bf Q and the initial state occupancy probabilities \bf \pi(0) = (\pi_1(0), \pi_2(0), ..., \pi_N(0)), the state occupancy probabilities \bf \pi(t) at time t can be computed as:

     \pi(t) = \pi(0) exp(Qt)

where \exp( \bf Q t ) is the matrix exponential of \bf Q t. Under certain conditions, there exists a stationary state occupancy probability \bf \pi = \lim_t \rightarrow +\infty \bf \pi(t), which is independent from \bf \pi(0). \bf \pi is the solution of the following linear system:

     /
     | \pi Q   = 0
     | \pi 1^T = 1
     \

EXAMPLE

Consider a two-state CTMC such that transition rates between states are equal to 1. This can be solved as follows:

      Q = [ -1  1; \
             1 -1  ];
      q = ctmc(Q)

         ⇒ q = 0.50000   0.50000

3.2.2 Birth-Death Process

3.2.3 Expected Sojourn Times

Given a N state continuous-time Markov Chain with infinitesimal generator matrix \bf Q, we define the vector \bf L(t) = (L_1(t), L_2(t), \ldots, L_N(t)) such that L_i(t) is the expected sojourn time in state i during the interval [0,t), assuming that the initial occupancy probability at time 0 was \bf \pi(0). \bf L(t) can be expressed as the solution of the following differential equation:

      dL
      --(t) = L(t) Q + pi(0),    L(0) = 0
      dt

Alternatively, \bf L(t) can also be expressed in integral form as:

            / t
     L(t) = |   pi(u) du
            / 0

where \bf \pi(t) = \bf \pi(0) \exp(\bf Qt) is the state occupancy probability at time t; \exp(\bf Qt) is the matrix exponential of \bf Qt.

If there are absorbing states, we can define the vector expected sojourn times until absorption \bf L(\infty), where for each transient state i, L_i(\infty) is the expected total time spent in state i until absorption, assuming that the system started with a given state occupancy probability vector \bf \pi(0). Let \tau be the set of transient (i.e., non absorbing) states; let \bf Q_\tau be the restriction of \bf Q to the transient substates only. Similarly, let \bf \pi_\tau(0) be the restriction of the initial probability vector \bf \pi(0) to transient states \tau.

The expected time to absorption \bf L_\tau(\infty) is defined as the solution of the following equation:

     L_T( inf ) Q_T = -pi_T(0)

EXAMPLE

Let us consider a pure-birth, 4-states CTMC such that the transition rate from state i to state i+1 is \lambda_i = i \lambda (i=1, 2, 3), with \lambda = 0.5. The following code computes the expected sojourn time in state i, given the initial occupancy probability \bf \pi_0=(1,0,0,0).

      lambda = 0.5;
      N = 4;
      b = lambda*[1:N-1];
      d = zeros(size(b));
      Q = ctmc_bd(b,d);
      t = linspace(0,10,100);
      p0 = zeros(1,N); p0(1)=1;
      L = zeros(length(t),N);
      for i=1:length(t)
        L(i,:) = ctmc_exps(Q,t(i),p0);
      endfor
      plot( t, L(:,1), ";State 1;", "linewidth", 2, \
            t, L(:,2), ";State 2;", "linewidth", 2, \
            t, L(:,3), ";State 3;", "linewidth", 2, \
            t, L(:,4), ";State 4;", "linewidth", 2 );
      legend("location","northwest");
      xlabel("Time");
      ylabel("Expected sojourn time");

3.2.4 Time-Averaged Expected Sojourn Times

EXAMPLE

      lambda = 0.5;
      N = 4;
      birth = lambda*linspace(1,N-1,N-1);
      death = zeros(1,N-1);
      Q = diag(birth,1)+diag(death,-1);
      Q -= diag(sum(Q,2));
      t = linspace(1e-5,30,100);
      p = zeros(1,N); p(1)=1;
      M = zeros(length(t),N);
      for i=1:length(t)
        M(i,:) = ctmc_taexps(Q,t(i),p);
      endfor
      clf;
      plot(t, M(:,1), ";State 1;", "linewidth", 2, \
           t, M(:,2), ";State 2;", "linewidth", 2, \
           t, M(:,3), ";State 3;", "linewidth", 2, \
           t, M(:,4), ";State 4 (absorbing);", "linewidth", 2 );
      legend("location","east");
      xlabel("Time");
      ylabel("Time-averaged Expected sojourn time");

3.2.5 Mean Time to Absorption

EXAMPLE

Let us consider a simple model of a redundant disk array. We assume that the array is made of 5 independent disks, such that the array can tolerate up to 2 disk failures without losing data. If three or more disks break, the array is dead and unrecoverable. We want to estimate the Mean-Time-To-Failure (MTTF) of the disk array.

We model this system as a 4 states Markov chain with state space \ 2, 3, 4, 5 \. State i denotes the fact that exactly i disks are active; state 2 is absorbing. Let \mu be the failure rate of a single disk. The system starts in state 5 (all disks are operational). We use a pure death process, with death rate from state i to state i-1 is \mu i, for i = 3, 4, 5).

The MTTF of the disk array is the MTTA of the Markov Chain, and can be computed with the following expression:

      mu = 0.01;
      death = [ 3 4 5 ] * mu;
      birth = 0*death;
      Q = ctmc_bd(birth,death);
      t = ctmc_mtta(Q,[0 0 0 1])

         ⇒ t = 78.333

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998.

3.2.6 First Passage Times

4 Single Station Queueing Systems

Single Station Queueing Systems contain a single station, and are thus quite easy to analyze. The queueing package contains functions for handling the following types of queues:

• The M/M/1 System: Single-server queueing station.
• The M/M/m System: Multiple-server queueing station.
• The M/M/inf System: Infinite-server (delay center) station.
• The M/M/1/K System: Single-server, finite-capacity queueing station.
• The M/M/m/K System: Multiple-server, finite-capacity queueing station.
• The Asymmetric M/M/m System: Asymmetric multiple-server queueing station.
• The M/G/1 System: Single-server with general service time distribution.
• The M/Hm/1 System: Single-server with hyperexponential service time distribution.

The functions which analyze the queues above can be used as building blocks for analyzing Queueing Networks. For example, Jackson networks can be solved by computing the aggregate arrival rates to each node, and then solving each node in isolation as if it were a single station queueing system.

4.1 The M/M/1 System

The M/M/1 system is made of a single server connected to an unlimited FCFS queue. The mean arrival rate is Poisson with arrival rate \lambda; the service time is exponentially distributed with average service rate \mu. The system is stable if \lambda < \mu.

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.3.

4.2 The M/M/m System

The M/M/m system is similar to the M/M/1 system, except that there are m \geq 1 identical servers connected to a single queue. Thus, at most m requests can be served at the same time. The M/M/m system can be seen as a single server with load-dependent service rate \mu(n), which is a function of the number n of nodes in the center:

     mu(n) = min(m,n)*mu

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.5.

4.3 The M/M/inf System

The M/M/\infty system is similar to the M/M/m system, except that there are infinitely many identical servers (that is, m = \infty). Each new request is assigned to a new server, so that queueing never occurs. The M/M/\infty system is always stable.

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.4.

4.4 The M/M/1/K System

In a M/M/1/K finite capacity system there can be at most k jobs at any time. If a new request tries to join the system when there are already K other requests, the arriving request is lost. The queue has K-1 slots. The M/M/1/K system is always stable, regardless of the arrival and service rates \lambda and \mu.

4.5 The M/M/m/K System

The M/M/m/K finite capacity system is similar to the M/M/1/k system except that the number of servers is m, where 1 \leq m \leq K. The queue is made of K-m slots. The M/M/m/K system is always stable.

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 6.6.

4.6 The Asymmetric M/M/m System

The Asymmetric M/M/m system contains m servers connected to a single queue. Differently from the M/M/m system, in the asymmetric M/M/m each server may have a different service time.

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998

4.7 The M/G/1 System

4.8 The M/H_m/1 System

5 Queueing Networks

• Introduction to QNs: A brief introduction to Queueing Networks.
• Algorithms for Product-Form QNs: Functions to analyze product-form QNs
• Algorithms for non Product-Form QNs: Functions to analyze non product-form QNs
• Generic Algorithms: High-level functions for QN analysis
• Bounds on performance: Functions to compute performance bounds
• Utility functions: Utility functions to compute miscellaneous quantities

5.1 Introduction to QNs

Queueing Networks (QN) are a very simple yet powerful modeling tool which can be used to analyze many kind of systems. In its simplest form, a QN is made of K service centers. Each center k has a queue, which is connected to m_k (generally identical) servers. Arriving customers (requests) join the queue if there is a slot available. Then, requests are served according to a (de)queueing policy (e.g., FIFO). After service completes, requests leave the server and can join another queue or exit from the system.

Service centers for which m_k = \infty are called delay centers or infinite servers. In this kind of centers, every request always finds one available server, so queueing never occurs.

Requests join the queue according to a queueing policy, such as:

FCFS

First-Come-First-Served

LCFS-PR

Last-Come-First-Served, Preemptive Resume

PS

Processor Sharing

IS

Infinite Server (for which m_k = \infty).

Queueing models can be open or closed. In open models there is an infinite population of requests; new customers are generated outside the system, and eventually leave the system. In closed systems there is a fixed population of request.

Queueing models can have a single request class, meaning that all requests behave in the same way (e.g., they spend the same average time on each particular server). In multiclass models there are multiple request classes, each one with its own parameters. Furthermore, in multiclass models there can be open and closed classes of requests within the same system.

5.1.1 Single class models

In single class models, all requests are indistinguishable and belong to the same class. This means that every request has the same average service time, and all requests move through the system with the same routing probabilities.

Model Inputs

\lambda_k

External arrival rate to service center k.

\lambda

Overall external arrival rate to the whole system: \lambda = \sum_k \lambda_k.

S_k

Average service time. S_k is the average service time on service center k. In other words, S_k is the average time from the instant in which a request is extracted from the queue and starts being service, and the instant at which service finishes and the request moves to another queue (or exits the system).

P_i, j

Routing probability matrix. \bf P = [P_i, j] is a K \times K matrix such that P_i, j is the probability that a request completing service at server i will move directly to server j, The probability that a request leaves the system after service at service center i is 1-\sum_j=1^K P_i, j.

V_k

Average number of visits. V_k is the average number of visits to the service center k. This quantity will be described shortly.

Model Outputs

U_k

Service center utilization. U_k is the utilization of service center k. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests).

R_k

Average response time. R_k is the average response time of service center k. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service.

Q_k

Average number of customers. Q_k is the average number of requests in service center k. This includes both the requests in the queue, and the request being served.

X_k

Throughput. X_k is the throughput of service center k. The throughput is defined as the ratio of job completions (i.e., average number of jobs completed over a fixed interval of time).

Given these output parameters, additional performance measures can be computed as follows:

X

System throughput, X = X_1 / V_1

R

System response time, R = \sum_k=1^K R_k V_k

Q

Average number of requests in the system, Q = N-XZ

For open, single-class models, the scalar \lambda denotes the external arrival rate of requests to the system. The average number of visits satisfy the following equation:

                       K
                      ___
                     \
     V_j = P_(0, j) + >   V_i P_(i, j)
                     /___
                      i=1

where P_0, j is the probability that an external arrival goes to service center j. If \lambda_j is the external arrival rate to service center j, and \lambda = \sum_j \lambda_j is the overall external arrival rate, then P_0, j = \lambda_j / \lambda.

For closed models, the visit ratios satisfy the following equation:

     
     V_1 = 1
     
             K
            ___
           \
     V_j =  >   V_i P_(i, j)
           /___
            i=1
     

5.1.2 Multiple class models

In multiple class QN models, we assume that there exist C different classes of requests. Each request from class c spends on average time S_c, k in service at service center k. For open models, we denote with \bf \lambda = \lambda_ck the arrival rates, where \lambda_c, k is the external arrival rate of class c customers at service center k. For closed models, we denote with \bf N = (N_1, N_2, \ldots, N_C) the population vector, where N_c is the number of class c requests in the system.

The transition probability matrix for these kind of networks will be a C \times K \times C \times K matrix \bf P = [P_r, i, s, j] such that P_r, i, s, j is the probability that a class r request which completes service at center i will join server j as a class s request.

Model input and outputs can be adjusted by adding additional indexes for the customer classes.

Model Inputs

\lambda_c, k

External arrival rate of class-c requests to service center k

\lambda

Overall external arrival rate to the whole system: \lambda = \sum_c \sum_k \lambda_c, k

S_c, k

Average service time. S_c, k is the average service time on service center k for class c requests.

P_r, i, s, j

Routing probability matrix. \bf P = [P_r, i, s, j] is a C \times K \times C \times K matrix such that P_r, i, s, j is the probability that a class r request which completes service at server i will move to server j as a class s request.

V_c, k

Average number of visits. V_c, k is the average number of visits of class c requests to center k.

Model Outputs

U_c, k

Utilization of service center k by class c requests. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests).

R_c, k

Average response time experienced by class c requests on service center k. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service.

Q_c, k

Average number of class c requests on service center k. This includes both the requests in the queue, and the request being served.

X_c, k

Throughput of service center k for class c requests. The throughput is defined as the rate of completion of class c requests.

It is possible to define aggregate performance measures as follows:

U_k

Utilization of service center k: Uk = sum(U,k);

R_c

System response time for class c requests: Rc = sum( V.*R, 1 );

Q_c

Average number of class c requests in the system: Qc = sum( Q, 2 );

X_c

Class c throughput: Xc = X(:,1) ./ V(:,1);

We can define the visit ratios V_s, j for class s customers at service center j as follows:

V_sj = sum_r sum_i V_ri P_risj, for all s,j

while for open networks:

V_sj = P_0sj + sum_r sum_i V_ri P_risj, for all s,j

where P_0, s, j is the probability that an external arrival goes to service center j as a class-s request. If \lambda_s, j is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_s, j is the overall external arrival rate to the whole system, then P_0, s, j = \lambda_s, j / \lambda.

5.1.3 Closed Network Example

We now give a simple example on how the queueing toolbox can be used to analyze a closed network. Let us consider a simple closed network with K=3 M/M/1–FCFS centers. We denote with S_k the average service time at center k, k=1, 2, 3. We use S_1 = 1.0, S_2 = 2.0 and S_3 = 0.8. The routing of jobs within the network is described with a routing probability matrix \bf P. Specifically, a request completing service at center i is enqueued at center j with probability P_i, j. We use the following routing matrix:

         / 0  0.3  0.7 \
     P = | 1  0    0   |
         \ 1  0    0   /

The network above can be analyzed with the qnclosed function see doc-qnclosed. qnclosed requires the following parameters:

N

Number of requests in the network (since we are considering a closed network, the number of requests is fixed)

S

Array of average service times at the centers: S(k) is the average service time at center k.

V

Array of visit ratios: V(k) is the average number of visits to center k.

We can compute V_k from the routing probability matrix P_i, j using the qnvisits function see doc-qnvisits. We can analyze the network for a given population size N (for example, N=10) as follows:

     N = 10;
     S = [1 2 0.8];
     P = [0 0.3 0.7; 1 0 0; 1 0 0];
     V = qnvisits(P);
     [U R Q X] = qnclosed( N, S, V )
        ⇒ U = 0.99139 0.59483 0.55518
        ⇒ R = 7.4360  4.7531  1.7500
        ⇒ Q = 7.3719  1.4136  1.2144
        ⇒ X = 0.99139 0.29742 0.69397

The output of qnclosed includes the vector of utilizations U_k at center k, response time R_k, average number of customers Q_k and throughput X_k. In our example, the throughput of center 1 is X_1 = 0.99139, and the average number of requests in center 3 is Q_3 = 1.2144. The utilization of center 1 is U_1 = 0.99139, which is the higher value among the service centers. Tus, center 1 is the bottleneck device.

This network can also be analyzed with the qnsolve function see doc-qnsolve. qnsolve can handle open, closed or mixed networks, and allows the network to be described in a very flexible way. First, let Q1, Q2 and Q3 be the variables describing the service centers. Each variable is instantiated with the qnmknode function.

     Q1 = qnmknode( "m/m/m-fcfs", 1 );
     Q2 = qnmknode( "m/m/m-fcfs", 2 );
     Q3 = qnmknode( "m/m/m-fcfs", 0.8 );

The first parameter of qnmknode is a string describing the type of the node. Here we use "m/m/m-fcfs" to denote a M/M/m–FCFS center. The second parameter gives the average service time. An optional third parameter can be used to specify the number m of service centers. If omitted, it is assumed m=1 (single-server node).

Now, the network can be analyzed as follows:

     N = 10;
     V = [1 0.3 0.7];
     [U R Q X] = qnsolve( "closed", N, { Q1, Q2, Q3 }, V )
        ⇒ U = 0.99139 0.59483 0.55518
        ⇒ R = 7.4360  4.7531  1.7500
        ⇒ Q = 7.3719  1.4136  1.2144
        ⇒ X = 0.99139 0.29742 0.69397

Of course, we get exactly the same results. Other functions can be used for closed networks, see Algorithms for Product-Form QNs.

5.1.4 Open Network Example

Open networks can be analyzed in a similar way. Let us consider an open network with K=3 service centers, and routing probability matrix as follows:

         / 0  0.3  0.5 \
     P = ! 1  0    0   |
         \ 1  0    0   /

In this network, requests can leave the system from center 1 with probability 1-(0.3+0.5) = 0.2. We suppose that external jobs arrive at center 1 with rate \lambda_1 = 0.15; there are no arrivals at centers 2 and 3.

Similarly to closed networks, we first need to compute the visit counts V_k to center k. Again, we use the qnvisits function as follows:

     P = [0 0.3 0.5; 1 0 0; 1 0 0];
     lambda = [0.15 0 0];
     V = qnvisits(P, lambda)
        ⇒ V = 5.00000 1.50000 2.50000

where lambda(k) is the arrival rate at center k, and P is the routing matrix. Assuming the same service times as in the previous example, the network can be analyzed with the qnopen function see doc-qnopen, as follows:

     S = [1 2 0.8];
     [U R Q X] = qnopen( sum(lambda), S, V )
        ⇒ U = 0.75000 0.45000 0.30000
        ⇒ R = 4.0000  3.6364  1.1429
        ⇒ Q = 3.00000 0.81818 0.42857
        ⇒ X = 0.75000 0.22500 0.37500

The first parameter of the qnopen function is the (scalar) aggregate arrival rate.

Again, it is possible to use the qnsolve high-level function:

     Q1 = qnmknode( "m/m/m-fcfs", 1 );
     Q2 = qnmknode( "m/m/m-fcfs", 2 );
     Q3 = qnmknode( "m/m/m-fcfs", 0.8 );
     lambda = [0.15 0 0];
     [U R Q X] = qnsolve( "open", sum(lambda), { Q1, Q2, Q3 }, V )
        ⇒ U = 0.75000 0.45000 0.30000
        ⇒ R = 4.0000  3.6364  1.1429
        ⇒ Q = 3.00000 0.81818 0.42857
        ⇒ X = 0.75000 0.22500 0.37500

5.2 Algorithms for Product-Form QNs

Product-form queueing networks fulfill the following assumptions:

• The network can consist of open and closed job classes.
• The following queueing disciplines are allowed: FCFS, PS, LCFS-PR and IS.
• Service times for FCFS nodes must be exponentially distributed and class-independent. Service centers at PS, LCFS-PR and IS nodes can have any kind of service time distribution with a rational Laplace transform. Furthermore, for PS, LCFS-PR and IS nodes, different classes of customers can have different service times.
• The service rate of an FCFS node is only allowed to depend on the number of jobs at this node; in a PS, LCFS-PR and IS node the service rate for a particular job class can also depend on the number of jobs of that class at the node.
• In open networks two kinds of arrival processes are allowed: i) the arrival process is Poisson, with arrival rate \lambda which can depend on the number of jobs in the network. ii) the arrival process consists of U independent Poisson arrival streams where the U job sources are assigned to the U chains; the arrival rate can be load dependent.

5.2.1 Jackson Networks

Jackson networks satisfy the following conditions:

• There is only one job class in the network; the overall number of jobs in the system is unlimited.
• There are N service centers in the network. Each service center may have Poisson arrivals from outside the system. A job can leave the system from any node.
• Arrival rates as well as routing probabilities are independent from the number of nodes in the network.
• External arrivals and service times at the service centers are exponentially distributed, and in general can be load-dependent.
• Service discipline at each node is FCFS

We define the joint probability vector \pi(k_1, k_2, \ldots, k_N) as the steady-state probability that there are k_i requests at service center i, for all i=1, 2, \ldots, N. Jackson networks have the property that the joint probability is the product of the marginal probabilities \pi_i:

     joint_prob = prod( pi )

where \pi_i(k_i) is the steady-state probability that there are k_i requests at service center i.

REFERENCES

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, pp. 284–287.

5.2.2 The Convolution Algorithm

According to the BCMP theorem, the state probability of a closed single class queueing network with K nodes and N requests can be expressed as:

     k = [k1, k2, ... kn]; population vector
     p = 1/G(N+1) \prod F(i,k);

Here \pi(k_1, k_2, \ldots, k_K) is the joint probability of having k_i requests at node i, for all i=1, 2, \ldots, K.

The convolution algorithms computes the normalization constants \bf G = \left(G(0), G(1), \ldots, G(N)\right) for single-class, closed networks with N requests. The normalization constants are returned as vector G=[G(1), G(2), ... G(N+1)] where G(i+1) is the value of G(i) (remember that Octave uses 1-base vectors). The normalization constant can be used to compute all performance measures of interest (utilization, average response time and so on).

queueing implements the convolution algorithm, in the function qnconvolution and qnconvolutionld. The first one supports single-station nodes, multiple-station nodes and IS nodes. The second one supports networks with general load-dependent service centers.

EXAMPLE

The normalization constant G can be used to compute the steady-state probabilities for a closed single class product-form Queueing Network with K nodes. Let k=[k_1, k_2, ..., k_K] be a valid population vector. Then, the steady-state probability p(i) to have k(i) requests at service center i can be computed as:

      k = [1 2 0];
      K = sum(k); # Total population size
      S = [ 1/0.8 1/0.6 1/0.4 ];
      m = [ 2 3 1 ];
      V = [ 1 .667 .2 ];
      [U R Q X G] = qnconvolution( K, S, V, m );
      p = [0 0 0]; # initialize p
      # Compute the probability to have k(i) jobs at service center i
      for i=1:3
        p(i) = (V(i)*S(i))^k(i) / G(K+1) * \
               (G(K-k(i)+1) - V(i)*S(i)*G(K-k(i)) );
        printf("k(%d)=%d prob=%f\n", i, k(i), p(i) );
      endfor

     -| k(1)=1 prob=0.17975
     -| k(2)=2 prob=0.48404
     -| k(3)=0 prob=0.52779

NOTE

For a network with K service centers and N requests, this implementation of the convolution algorithm has time and space complexity O(NK).

REFERENCES

Jeffrey P. Buzen, Computational Algorithms for Closed Queueing Networks with Exponential Servers, Communications of the ACM, volume 16, number 9, september 1973, pp. 527–531. http://doi.acm.org/10.1145/362342.362345

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, pp. 313–317.

REFERENCES

Herb Schwetman, Some Computational Aspects of Queueing Network Models, Technical Report CSD-TR-354, Department of Computer Sciences, Purdue University, feb, 1981 (revised). http://www.cs.purdue.edu/research/technical_reports/1980/TR%2080-354.pdf

M. Reiser, H. Kobayashi, On The Convolution Algorithm for Separable Queueing Networks, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29–31, 1976). SIGMETRICS '76. ACM, New York, NY, pp. 109–117. http://doi.acm.org/10.1145/800200.806187

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, pp. 313–317. Function qnconvolutionld is slightly different from the version described in Bolch et al. because it supports general load-dependent centers (while the version in the book does not). The modification is in the definition of function F() in qnconvolutionld which has been made similar to function f_i defined in Schwetman, Some Computational Aspects of Queueing Network Models.

5.2.3 Open networks

From the results computed by this function, it is possible to derive other quantities of interest as follows:

• System Response Time: The overall system response time can be computed as R_s = dot(V,R);
• Average number of requests: The average number of requests in the system can be computed as: Q_s = sum(Q)

EXAMPLE

      lambda = 3;
      V = [16 7 8];
      S = [0.01 0.02 0.03];
      [U R Q X] = qnopensingle( lambda, S, V );
      R_s = dot(R,V) # System response time
      N = sum(Q) # Average number in system

     -| R_s =  1.4062
     -| N =  4.2186

REFERENCES

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998.

REFERENCES

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.1 ("Open Model Solution Techniques").

5.2.4 Closed Networks

From the results provided by this function, it is possible to derive other quantities of interest as follows:

EXAMPLE

      S = [ 0.125 0.3 0.2 ];
      V = [ 16 10 5 ];
      N = 20;
      m = ones(1,3);
      Z = 4;
      [U R Q X] = qnclosedsinglemva(N,S,V,m,Z);
      X_s = X(1)/V(1); # System throughput
      R_s = dot(R,V); # System response time
      printf("\t    Util      Qlen     RespT      Tput\n");
      printf("\t--------  --------  --------  --------\n");
      for k=1:length(S)
        printf("Dev%d\t%8.4f  %8.4f  %8.4f  %8.4f\n", k, U(k), Q(k), R(k), X(k) );
      endfor
      printf("\nSystem\t          %8.4f  %8.4f  %8.4f\n\n", N-X_s*Z, R_s, X_s );

REFERENCES

M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. http://doi.acm.org/10.1145/322186.322195

This implementation is described in R. Jain , The Art of Computer Systems Performance Analysis, Wiley, 1991, p. 577. Multi-server nodes are treated according to G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 8.2.1, "Single Class Queueing Networks".

REFERENCES

M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. http://doi.acm.org/10.1145/322186.322195

This implementation is described in G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 8.2.4.1, “Networks with Load-Deèpendent Service: Closed Networks”.

REFERENCES

G. Casale. A note on stable flow-equivalent aggregation in closed networks. Queueing Syst. Theory Appl., 60:193–202, December 2008.

REFERENCES

This implementation is based on Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 6.4.2.2 ("Approximate Solution Techniques").

NOTE

Given a network with K service centers, C job classes and population vector \bf N=(N_1, N_2, \ldots, N_C), the MVA algorithm requires space O(C \prod_i (N_i + 1)). The time complexity is O(CK\prod_i (N_i + 1)). This implementation is slightly more space-efficient (see details in the code). While the space requirement can be mitigated by using some optimizations, the time complexity can not. If you need to analyze large closed networks you should consider the qnclosedmultimvaapprox function, which implements the approximate MVA algorithm. Note however that qnclosedmultimvaapprox will only provide approximate results.

REFERENCES

M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. http://doi.acm.org/10.1145/322186.322195

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998 and Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.2.1 ("Exact Solution Techniques").

REFERENCES

Y. Bard, Some Extensions to Multiclass Queueing Network Analysis, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, feb. 1979, pp. 51–62.

P. Schweitzer, Approximate Analysis of Multiclass Closed Networks of Queues, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25–29.

This implementation is based on Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.2.2 ("Approximate Solution Techniques"). This implementation is slightly different from the one described above, as it computes the average response times R instead of the residence times.

5.2.5 Mixed Networks

REFERENCES

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.3 ("Mixed Model Solution Techniques"). Note that in this function we compute the mean response time R instead of the mean residence time as in the reference.

Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, feb 15, 1982, available at http://www.cs.purdue.edu/research/technical_reports/1980/TR%2080-355.pdf

5.3 Algorithms for non Product-Form QNs

REFERENCES

Ian F. Akyildiz, Mean Value Analysis for Blocking Queueing Networks, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418–428. http://dx.doi.org/10.1109/32.4663

5.4 Generic Algorithms

The queueing package provides a high-level function qnsolve for analyzing QN models. qnsolve takes as input a high-level description of the queueing model, and delegates the actual solution of the model to one of the lower-level function. qnsolve supports single or multiclass models, but at the moment only product-form networks can be analyzed. For non product-form networks See Algorithms for non Product-Form QNs.

qnsolve accepts two input parameters. The first one is the list of nodes, encoded as an Octave cell array. The second parameter is the vector of visit ratios V, which can be either a vector (for single-class models) or a two-dimensional matrix (for multiple-class models).

Individual nodes in the network are structures build using the qnmknode function.

After the network has been defined, it is possible to solve it using qnsolve.

EXAMPLE

Let us consider a closed, multiclass network with C=2 classes and K=3 service center. Let the population be M=(2, 1) (class 1 has 2 requests, and class 2 has 1 request). The nodes are as follows:

• Node 1 is a M/M/1–FCFS node, with load-dependent service times. Service times are class-independent, and are defined by the matrix [0.2 0.1 0.1; 0.2 0.1 0.1]. Thus, S(1,2) = 0.2 means that service time for class 1 customers where there are 2 requests in 0.2. Note that service times are class-independent;
• Node 2 is a -/G/1–PS node, with service times S_1, 2 = 0.4 for class 1, and S_2, 2 = 0.6 for class 2 requests;
• Node 3 is a -/G/\infty node (delay center), with service times S_1, 3=1 and S_2, 3=2 for class 1 and 2 respectively.

After defining the per-class visit count V such that V(c,k) is the visit count of class c requests to service center k. We can define and solve the model as follows:

      QQ = { qnmknode( "m/m/m-fcfs", [0.2 0.1 0.1; 0.2 0.1 0.1] ), \
             qnmknode( "-/g/1-ps", [0.4; 0.6] ), \
             qnmknode( "-/g/inf", [1; 2] ) };
      V = [ 1 0.6 0.4; \
            1 0.3 0.7 ];
      N = [ 2 1 ];
      [U R Q X] = qnsolve( "closed", N, QQ, V );

5.5 Bounds on performance

REFERENCES

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 5.2 ("Asymptotic Bounds").

REFERENCES

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 5.2 ("Asymptotic Bounds").

REFERENCES

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 5.4 ("Balanced Systems Bounds").

REFERENCES

The original paper describing PB Bounds is C. H. Hsieh and S. Lam, Two classes of performance bounds for closed queueing networks, PEVA, vol. 7, n. 1, pp. 3–30, 1987

This function implements the non-iterative variant described in G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008.

REFERENCES

G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008. http://doi.ieeecomputersociety.org/10.1109/TC.2008.37

In this implementation we set X^+ and X^- as the upper and lower Asymptotic Bounds as computed by the qnclosedab function, respectively.

5.6 Utility functions

5.6.1 Open or closed networks

EXAMPLE

      P = [0 0.3 0.7; 1 0 0; 1 0 0]; # Transition probability matrix
      S = [1 0.6 0.2]; # Average service times
      m = ones(1,3); # All centers are single-server
      Z = 2; # External delay
      N = 15; # Maximum population to consider
     
      V = qnvisits(P); # Compute number of visits from P
      D = V .* S; # Compute service demand from S and V
      X_bsb_lower = X_bsb_upper = zeros(1,N);
      X_ab_lower = X_ab_upper = zeros(1,N);
      X_mva = zeros(1,N);
      for n=1:N
        [X_bsb_lower(n) X_bsb_upper(n)] = qnclosedbsb(n, D, Z);
        [X_ab_lower(n) X_ab_upper(n)] = qnclosedab(n, D, Z);
        [U R Q X] = qnclosed( n, S, V, m, Z );
        X_mva(n) = X(1)/V(1);
      endfor
      close all;
      plot(1:N, X_ab_lower,"g;Asymptotic Bounds;", \
           1:N, X_bsb_lower,"k;Balanced System Bounds;", \
           1:N, X_mva,"b;MVA;", "linewidth", 2, \
           1:N, X_bsb_upper,"k", \
           1:N, X_ab_upper,"g" );
      axis([1,N,0,1]);
      xlabel("Number of Requests n");
      ylabel("System Throughput X(n)");
      legend("location","southeast");

5.6.2 Computation of the visit counts

For single-class networks the average number of visits satisfy the following equation:

     V == P0 + V*P;

where P_0, j is the probability that an external arrival goes to service center j. If \lambda_j is the external arrival rate to service center j, and \lambda = \sum_j \lambda_j is the overall external arrival rate, then P_0, j = \lambda_j / \lambda.

For closed networks, the visit ratios satisfy the following equation:

     V(1) == 1 && V == V*P;

The definitions above can be extended to multiple class networks as follows. We define the visit ratios V_s, j for class s customers at service center j as follows:

V_sj = sum_r sum_i V_ri P_risj, for all s,j V_s1 = 1, for all s

while for open networks:

V_sj = P_0sj + sum_r sum_i V_ri P_risj, for all s,j

where P_0, s, j is the probability that an external arrival goes to service center j as a class-s request. If \lambda_s, j is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_s, j is the overall external arrival rate to the whole system, then P_0, s, j = \lambda_s, j / \lambda.

EXAMPLE

      P = [ 0 0.4 0.6 0; \
            0.2 0 0.2 0.6; \
            0 0 0 1; \
            0 0 0 0 ];
      lambda = [0.1 0 0 0.3];
      V = qnvisits(P,lambda);
      S = [2 1 2 1.8];
      m = [3 1 1 2];
      [U R Q X] = qnopensingle( sum(lambda), S, V, m );

5.6.3 Other utility functions

REFERENCES

Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, feb 15, 1982, available at http://www.cs.purdue.edu/research/technical_reports/1980/TR 80-355.pdf

Note that the slightly different problem of generating all tuples k_1, k_2, \ldots, k_N such that \sum_i k_i = k and k_i are nonnegative integers, for some fixed integer k ≥ 0 has been described in S. Santini, Computing the Indices for a Complex Summation, unpublished report, available at http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf

REFERENCES

Zahorjan, J. and Wong, E. The solution of separable queueing network models using mean value analysis. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI http://doi.acm.org/10.1145/1010629.805477

6 References

[Aky88]

Ian F. Akyildiz, Mean Value Analysis for Blocking Queueing Networks, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418–428. DOI 10.1109/32.4663

[Bar79]

Y. Bard, Some Extensions to Multiclass Queueing Network Analysis, proc. 4th Int. Symp. on Modelling and Performance Evaluation of Computer Systems, feb. 1979, pp. 51–62.

[BCMP75]

Forest Baskett, K. Mani Chandy, Richard R. Muntz, and Fernando G. Palacios. 1975. Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. J. ACM 22, 2 (April 1975), 248—260, DOI 10.1145/321879.321887

[BGMT98]

G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998.

[Buz73]

Jeffrey P. Buzen, Computational Algorithms for Closed Queueing Networks with Exponential Servers, Communications of the ACM, volume 16, number 9, september 1973, pp. 527–531. DOI 10.1145/362342.362345

[CMS08]

G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008. DOI 10.1109/TC.2008.37

[GrSn97]

Charles M. Grinstead, J. Laurie Snell, (July 1997). Introduction to Probability. American Mathematical Society. ISBN 978-0821807491; this excellent textbook is available in PDF format and can be used under the terms of the GNU Free Documentation License (FDL)

[Jac04]

James R. Jackson, Jobshop-Like Queueing Systems, Vol. 50, No. 12, Ten Most Influential Titles of "Management Science's" First Fifty Years (Dec., 2004), pp. 1796-1802, available online

[Jai91]

R. Jain, The Art of Computer Systems Performance Analysis, Wiley, 1991, p. 577.

[HsLa87]

C. H. Hsieh and S. Lam, Two classes of performance bounds for closed queueing networks, PEVA, vol. 7, n. 1, pp. 3–30, 1987

[LZGS84]

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. available online.

[ReKo76]

M. Reiser, H. Kobayashi, On The Convolution Algorithm for Separable Queueing Networks, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29–31, 1976). SIGMETRICS '76. ACM, New York, NY, pp. 109–117. DOI 10.1145/800200.806187

[ReLa80]

M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. DOI 10.1145/322186.322195

[Sch79]

P. Schweitzer, Approximate Analysis of Multiclass Closed Networks of Queues, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25—29

[Sch81]

Herb Schwetman, Some Computational Aspects of Queueing Network Models, Technical Report CSD-TR-354, Department of Computer Sciences, Purdue University, feb, 1981 (revised).

[Sch82]

Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, feb 15, 1982.

[Tij03]

H. C. Tijms, A first course in stochastic models, John Wiley and Sons, 2003, ISBN 0471498807, ISBN 9780471498803, DOI 10.1002/047001363X

[ZaWo81]

Zahorjan, J. and Wong, E. The solution of separable queueing network models using mean value analysis. SIGMETRICS Perform. Eval. Rev. 10, 3 (Sep. 1981), 80-85. DOI DOI 10.1145/1010629.805477

Appendix A GNU GENERAL PUBLIC LICENSE

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Concept Index

• Absorption probabilities: Mean time to absorption (DTMC)
• Approximate MVA: Algorithms for Product-Form QNs
• Asymmetric M/M/m system: The Asymmetric M/M/m System
• BCMP network: Algorithms for Product-Form QNs
• Birth-death process: Birth-death process (CTMC)
• Birth-death process: Birth-death process (DTMC)
• blocking queueing network: Algorithms for non Product-Form QNs
• bounds, asymptotic: Bounds on performance
• bounds, balanced system: Bounds on performance
• bounds, geometric: Bounds on performance
• closed network: Utility functions
• closed network: Bounds on performance
• closed network: Algorithms for Product-Form QNs
• Closed network, approximate analysis: Algorithms for Product-Form QNs
• closed network, finite capacity: Algorithms for non Product-Form QNs
• closed network, multiple classes: Utility functions
• closed network, multiple classes: Algorithms for non Product-Form QNs
• Closed network, multiple classes: Algorithms for Product-Form QNs
• closed network, multiple classes: Algorithms for Product-Form QNs
• Closed network, single class: Algorithms for Product-Form QNs
• closed network, single class: Algorithms for Product-Form QNs
• CMVA: Algorithms for Product-Form QNs
• Continuous time Markov chain: State occupancy probabilities (CTMC)
• convolution algorithm: Algorithms for Product-Form QNs
• copyright: Copying
• Discrete time Markov chain: State occupancy probabilities (DTMC)
• Expected sojourn time: Expected sojourn times (CTMC)
• Expected sojourn times: Expected number of visits (DTMC)
• First passage times: First passage times (CTMC)
• First passage times: First passage times (DTMC)
• Fundamental matrix: Mean time to absorption (DTMC)
• Jackson network: Algorithms for Product-Form QNs
• load-dependent service center: Algorithms for Product-Form QNs
• M/G/1 system: The M/G/1 System
• M/H_m/1 system: The M/Hm/1 System
• M/M/1 system: The M/M/1 System
• M/M/1/K system: The M/M/1/K System
• M/M/inf system: The M/M/inf System
• M/M/m system: The M/M/m System
• M/M/m/K system: The M/M/m/K System
• Markov chain, continuous time: Mean time to absorption (CTMC)
• Markov chain, continuous time: Time-averaged expected sojourn times (CTMC)
• Markov chain, continuous time: Expected sojourn times (CTMC)
• Markov chain, continuous time: Birth-death process (CTMC)
• Markov chain, continuous time: State occupancy probabilities (CTMC)
• Markov chain, continuous time: Continuous-Time Markov Chains
• Markov chain, discrete time: Birth-death process (DTMC)
• Markov chain, discrete time: State occupancy probabilities (DTMC)
• Markov chain, discrete time: Discrete-Time Markov Chains
• Markov chain, state occupancy probabilities: State occupancy probabilities (CTMC)
• Markov chain, stationary probabilities: State occupancy probabilities (DTMC)
• Markov chain, transient probabilities: State occupancy probabilities (DTMC)
• Mean recurrence times: First passage times (DTMC)
• Mean time to absorption: Mean time to absorption (CTMC)
• Mean time to absorption: Mean time to absorption (DTMC)
• Mean Value Analysys (MVA): Algorithms for Product-Form QNs
• Mean Value Analysys (MVA), approximate: Algorithms for Product-Form QNs
• mixed network: Algorithms for Product-Form QNs
• normalization constant: Algorithms for Product-Form QNs
• open network: Utility functions
• open network: Bounds on performance
• open network, multiple classes: Algorithms for Product-Form QNs
• open network, single class: Algorithms for Product-Form QNs
• population mix: Utility functions
• queueing network with blocking: Algorithms for non Product-Form QNs
• queueing networks: Queueing Networks
• RS blocking: Algorithms for non Product-Form QNs
• Stationary probabilities: State occupancy probabilities (CTMC)
• Stationary probabilities: State occupancy probabilities (DTMC)
• Time-alveraged sojourn time: Time-averaged expected sojourn times (CTMC)
• Time-alveraged sojourn time: Time-averaged expected sojourn times (DTMC)
• traffic intensity: The M/M/inf System
• Transient probabilities: State occupancy probabilities (DTMC)
• warranty: Copying

Function Index

• ctmc: State occupancy probabilities (CTMC)
• ctmc_bd: Birth-death process (CTMC)
• ctmc_check_Q: Continuous-Time Markov Chains
• ctmc_exps: Expected sojourn times (CTMC)
• ctmc_fpt: First passage times (CTMC)
• ctmc_mtta: Mean time to absorption (CTMC)
• ctmc_taexps: Time-averaged expected sojourn times (CTMC)
• dtmc: State occupancy probabilities (DTMC)
• dtmc_bd: Birth-death process (DTMC)
• dtmc_check_P: Discrete-Time Markov Chains
• dtmc_exps: Time-averaged expected sojourn times (DTMC)
• dtmc_exps: Expected number of visits (DTMC)
• dtmc_fpt: First passage times (DTMC)
• dtmc_mtta: Mean time to absorption (DTMC)
• population_mix: Utility functions
• qnammm: The Asymmetric M/M/m System
• qnclosed: Utility functions
• qnclosedab: Bounds on performance
• qnclosedbsb: Bounds on performance
• qnclosedgb: Bounds on performance
• qnclosedmultimva: Algorithms for Product-Form QNs
• qnclosedmultimvaapprox: Algorithms for Product-Form QNs
• qnclosedpb: Bounds on performance
• qnclosedsinglemva: Algorithms for Product-Form QNs
• qnclosedsinglemvaapprox: Algorithms for Product-Form QNs
• qnclosedsinglemvald: Algorithms for Product-Form QNs
• qncmva: Algorithms for Product-Form QNs
• qnconvolution: Algorithms for Product-Form QNs
• qnconvolutionld: Algorithms for Product-Form QNs
• qnjackson: Algorithms for Product-Form QNs
• qnmarkov: Algorithms for non Product-Form QNs
• qnmg1: The M/G/1 System
• qnmh1: The M/Hm/1 System
• qnmix: Algorithms for Product-Form QNs
• qnmknode: Generic Algorithms
• qnmm1: The M/M/1 System
• qnmm1k: The M/M/1/K System
• qnmminf: The M/M/inf System
• qnmmm: The M/M/m System
• qnmmmk: The M/M/m/K System
• qnmvablo: Algorithms for non Product-Form QNs
• qnmvapop: Utility functions
• qnopen: Utility functions
• qnopenab: Bounds on performance
• qnopenbsb: Bounds on performance
• qnopenmulti: Algorithms for Product-Form QNs
• qnopensingle: Algorithms for Product-Form QNs
• qnsolve: Generic Algorithms
• qnvisits: Utility functions

Author Index

• Akyildiz, I. F.: Algorithms for non Product-Form QNs
• Bard, Y.: Algorithms for Product-Form QNs
• Bolch, G.: Algorithms for Product-Form QNs
• Bolch, G.: The Asymmetric M/M/m System
• Bolch, G.: The M/M/m/K System
• Bolch, G.: The M/M/inf System
• Bolch, G.: The M/M/m System
• Bolch, G.: The M/M/1 System
• Bolch, G.: Mean time to absorption (CTMC)
• Buzen, J. P.: Algorithms for Product-Form QNs
• Casale, G.: Bounds on performance
• Casale, G.: Algorithms for Product-Form QNs
• de Meer, H.: Algorithms for Product-Form QNs
• de Meer, H.: The Asymmetric M/M/m System
• de Meer, H.: The M/M/m/K System
• de Meer, H.: The M/M/inf System
• de Meer, H.: The M/M/m System
• de Meer, H.: The M/M/1 System
• de Meer, H.: Mean time to absorption (CTMC)
• Graham, G. S.: Bounds on performance
• Graham, G. S.: Algorithms for Product-Form QNs
• Greiner, S.: Algorithms for Product-Form QNs
• Greiner, S.: The Asymmetric M/M/m System
• Greiner, S.: The M/M/m/K System
• Greiner, S.: The M/M/inf System
• Greiner, S.: The M/M/m System
• Greiner, S.: The M/M/1 System
• Greiner, S.: Mean time to absorption (CTMC)
• Hsieh, C. H: Bounds on performance
• Jain, R.: Algorithms for Product-Form QNs
• Kobayashi, H.: Algorithms for Product-Form QNs
• Lam, S.: Bounds on performance
• Lavenberg, S. S.: Algorithms for Product-Form QNs
• Lazowska, E. D.: Bounds on performance
• Lazowska, E. D.: Algorithms for Product-Form QNs
• Muntz, R. R.: Bounds on performance
• Reiser, M.: Algorithms for Product-Form QNs
• Santini, S.: Utility functions
• Schweitzer, P.: Algorithms for Product-Form QNs
• Schwetman, H.: Utility functions
• Schwetman, H.: Algorithms for Product-Form QNs
• Serazzi, G.: Bounds on performance
• Sevcik, K. C.: Bounds on performance
• Sevcik, K. C.: Algorithms for Product-Form QNs
• Trivedi, K.: Algorithms for Product-Form QNs
• Trivedi, K.: The Asymmetric M/M/m System
• Trivedi, K.: The M/M/m/K System
• Trivedi, K.: The M/M/inf System
• Trivedi, K.: The M/M/m System
• Trivedi, K.: The M/M/1 System
• Trivedi, K.: Mean time to absorption (CTMC)
• Wong, E.: Utility functions
• Zahorjan, J.: Utility functions
• Zahorjan, J.: Bounds on performance
• Zahorjan, J.: Algorithms for Product-Form QNs