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queueing

This manual documents how to install and run the Queueing Toolbox. It corresponds to version 1.0.0 of the package.

• Summary
• Installation: Installation of the queueing toolbox.
• Getting Started: Getting started with 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.
• Contributing Guidelines: How to contribute.
• Acknowledgements: People who contributed to the queueing toolbox.
• 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

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);
• Computation of 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 (discrete and continuous time Markov chains are supported):

• Birth-death process;
• Computation of transient and steady-state occupancy probabilities;
• Computation of mean time to absorption;
• Computation of time-averages sojourn time.
• Computation of mean 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.

2 Installing the queueing toolbox

• Installation through Octave package management system
• Manual installation
• Content of the source distribution
• Using the queueing toolbox

2.1 Installation through Octave package management system

The most recent version of queueing is 1.0.0 and can be downloaded from Octave-forge

http://octave.sourceforge.net/

The package Web page is

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

To install queueing in the system-wide location, such that all functions are automatically available when Octave starts, you can use ‘pkg install’ command. At the Octave prompt, type the following:

     octave:1> pkg install queueing-1.0.0.tar.gz

(Note: 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. To uninstall queueing, use the ‘pkg uninstall queueing’ command.

If you do not have root access, you can do a local installation by issuing the following command at the Octave prompt:

     octave:1> pkg install -local queueing-1.0.0.tar.gz

This will install queueing within the user's home directory, and the package will be available to that 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.

2.2 Manual installation

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

     tar xfz queueing-1.0.0.tar.gz
     cd queueing-1.0.0/

Copy all .m files from the inst/ directory to some target location. Then, you can 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 Content of the source distribution

The queueing package source code in the Subversion repository contains the following subdirectories (some of which are not included in the installation archive):

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 Introduction and Getting Started

• Analysis of Closed Networks
• Analysis of Open Networks

In this chapter we give some usage examples of the queueing package. The reader is assumed to be familiar with Queueing Networks (although some basic terminology and notation will be given here). Additional usage examples are embedded in most of the function files; to display and execute the demos associated with function fname you can type demo fname at the Octave prompt. For example

     demo qnclosed

executes all demos (if any) for the qnclosed function.

3.1 Analysis of Closed Networks

Let us consider a simple closed network with K=3 service centers. Each center is of type M/M/1–FCFS. We denote with S_i the average service time at center i, i=1, 2, 3. Let 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 P. Specifically, a request completing service at center i is enqueued at center j with probability P_ij. Let us assume the following routing probability matrix:

         [ 0  0.3  0.7 ]
     P = [ 1  0    0   ]
         [ 1  0    0   ]

For example, according to matric P a job completing service at center 1 is routed to center 2 with probability 0.3, and is routed to center 3 with probability 0.7.

The network above can be analyzed with the qnclosed function; if there is just a single class of requests, as in the example above, qnclosed calls qnclosedsinglemva which implements the Mean Value Analysys (MVA) algorithm for single-class, product-form network.

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.

As can be seen, we must compute the visit ratios (or visit counts) V_k for each center k. The visit counts satisfy the following equations:

     V_j = sum_i V_i P_ij

We can compute V_k from the routing probability matrix P_ij using the qnvisits function:

     P = [0 0.3 0.7; 1 0 0; 1 0 0];
     V = qnvisits(P)
        ⇒ V = 1.00000 0.30000 0.70000

We can check that the computed values satisfy the above equation by evaluating the following expression:

     V*P
          ⇒ ans = 1.00000 0.30000 0.70000

which is equal to V. Hence, 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. 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.

3.2 Analysis of Open Networks

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. The visit counts V_k for open networks satisfy the following equation:

     V_j = sum_i V_i P_ij

where P_0j is the probability of an external arrival to center j. This can be computed as:

Assuming the same service times as in the previous example, the network can be analyzed with the qnopen function, 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

4 Markov Chains

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

4.1 Discrete-Time Markov Chains

• DTMC Stationary Probability
• DTMC First Passage Times

4.1.1 Stationary Probability

4.1.2 First Passage Times

The First Passage Time M_i j is defined as the average number of transitions needed to visit state j for the first time, starting from state i. Matrix \bf M satisfies the property that

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

4.2 Continuous-Time Markov Chains

• CTMC Stationary Probability
• Birth-Death process
• Expected Sojourn Time
• Time-Averaged Expected Sojourn Time
• Expected Time to Absorption
• CTMC First Passage Times

4.2.1 Stationary Probability

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

4.2.2 Birth-Death process

4.2.3 Expected Sojourn Time

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). Then, \bf L(t) is the solution of the following differential equation:

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

The function ctmc_exps can be used to compute \bf L(t), by using the lsode Octave function to solve the above linear differential equation.

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 p_0=(1,0,0,0).

      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));
      tt = linspace(0,10,100);
      p0 = zeros(1,N); p0(1)=1;
      L = ctmc_exps(Q,tt,p0);
      plot( tt, L(:,1), ";State 1;", "linewidth", 2, \
            tt, L(:,2), ";State 2;", "linewidth", 2, \
            tt, L(:,3), ";State 3;", "linewidth", 2, \
            tt, L(:,4), ";State 4 (absorbing);", "linewidth", 2);
      legend("location","northwest");
      xlabel("Time");
      ylabel("Expected sojourn time");

4.2.4 Time-Averaged Expected Sojourn Time

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-3,50,500);
      p = zeros(1,N); p(1)=1;
      M = ctmc_taexps(Q,t,p);
      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");

4.2.5 Expected Time to Absorption

If we consider a Markov Chain with absorbing states, it is possible to define the expected time to absorption as the expected time until the system goes into an absorbing state. More specifically, let us suppose that A is the set of transient (i.e., non-absorbing) states of a CTMC with N states and infinitesimal generator matrix \bf Q. The expected time to absorption \bf L_A(\infty) is defined as the solution of the following equation:

     L_A( inf ) Q_A = -pi_A(0)

where \bf Q_A is the restriction of matrix \bf Q to only states in A, and \bf \pi_A(0) is the initial state occupancy probability at time 0, restricted to states in A.

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;
      Q = diag(death,-1);
      Q -= diag(sum(Q,2));
      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.

4.2.6 First Passage Times

5 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.

5.1 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.

5.2 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.

5.3 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.

5.4 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.

5.5 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.

5.6 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

5.7 The M/G/1 System

5.8 The M/H_m/1 System

6 Queueing Networks

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

6.1 Introduction to QNs

Queueing Networks (QN) are a very simple yet powerful modeling tool which is used to analyze many kind of systems. In its simplest form, a QN is made of K service centers. Each service center i has a queue, which is connected to m_i (generally identical) servers. Customers (or requests) arrive at the service center, and join the queue if there is a slot available. Then, requests are served according to a (de)queueing policy. After service completes, the requests leave the service center.

The service centers for which m_i = \infty are called delay centers or infinite servers. If a service center has infinite servers, of course each new request will find one server available, so there will never be queueing.

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, there is an infinite number of identical servers so that each request always finds a server available, and there is no queueing

A population of requests or customers arrives to the system system, requesting service to the service centers. The request population may be open or closed. In open systems there is an infinite population of requests. New customers arrive from outside the system, and eventually leave the system. In closed systems there is a fixed population of request which continuously interacts with the system.

There might be a single class of requests, meaning that all requests behave in the same way (e.g., they spend the same average time on each particular server), or there might be multiple classes of requests.

6.1.1 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_i

External arrival rate to service center i.

\lambda

Overall external arrival rate to the whole system: \lambda = \sum_i \lambda_i.

S_i

Average service time. S_i is the average service time on service center i. In other words, S_i 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_ij

Routing probability matrix. \bf P = P_ij is a K \times K matrix such that P_ij 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_ij.

V_i

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

Model Outputs

U_i

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

R_i

Average response time. R_i is the average response time of service center i. 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_i

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

X_i

Throughput. X_i is the throughput of service center i. 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:

     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 models, the visit ratios satisfy the following equation:

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

6.1.2 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_ck in service at service center k. For open models, we denote with \bf \lambda = \lambda_ck the arrival rates, where \lambda_ck 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_risj such that P_risj 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_ci

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

\lambda

Overall external arrival rate to the whole system: \lambda = \sum_c \sum_i \lambda_ci

S_ci

Average service time. S_ci is the average service time on service center i for class c requests.

P_risj

Routing probability matrix. \bf P = P_risj is a C \times K \times C \times K matrix such that P_risj 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_ci

Average number of visits. V_ci is the average number of visits of class c requests to the service center i.

Model Outputs

U_ci

Utilization of service center i 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_ci

Average response time experienced by class c requests on service center i. 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_ci

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

X_ci

Throughput of service center i 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_i

Utilization of service center i: Ui = sum(U,1);

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_sj 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_0sj is the probability that an external arrival goes to service center j as a class-s request. If \lambda_sj is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_sj is the overall external arrival rate to the whole system, then P_0sj = \lambda_sj / \lambda.

6.2 Generic Algorithms

The queueing package provides a couple of high-level functions for defining and solving QN models. These functions can be used to define a open or closed QN model (with single or multiple job classes), with arbitrary configuration and queueing disciplines. At the moment only product-form networks can be solved, See Algorithms for Product-Form QNs.

The network is defined by two parameters. The first one is the list of nodes, encoded as an Octave cell array. The second parameter is the visit ration 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 the qnsolve function. Note that this function is somewhat less efficient than those described in later sections, but generally easier to use.

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_12 = 0.4 for class 1, and S_22 = 0.6 for class 2 requests;
• Node 3 is a -/G/\infty node (delay center), with service times S_13=1 and S_23=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 );

6.3 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.

6.3.1 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.

6.3.2 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 G = (G(0), G(1), \ldots G(N)) 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.

6.3.3 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").

6.3.4 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.

6.3.5 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

6.4 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

6.5 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.

6.6 Utility functions

6.6.1 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");

6.6.2 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_sj 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_0sj is the probability that an external arrival goes to service center j as a class-s request. If \lambda_sj is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_sj is the overall external arrival rate to the whole system, then P_0sj = \lambda_sj / \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 );

6.6.3 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

Appendix A 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 will be extracted and formatted into the printable manual. See the existing functions of the queueing package for the documentation style.
• The documentation should be as precise as possible. In particular, always state what the valid ranges of the parameters are.
• If you are contributing a new function, ensure that the function properly checks the validity of its input parameters. For example, each function accepting vectors should check whether the dimensions match.
• Always provide bibliographic references for each algorithm you contribute. If your implementation differs in some way from the reference you give, please describe how and why your implementation differs.
• Include Octave test and demo blocks with your code. Test blocks are particularly important, because Queueing Network algorithms tend to be quite complex to implement correctly, and we must ensure that the implementations provided with the queueing package are (mostly) correct.

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

Appendix B 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.

Appendix C GNU GENERAL PUBLIC LICENSE

     Copyright © 2007 Free Software Foundation, Inc. http://fsf.org/
     
     Everyone is permitted to copy and distribute verbatim copies of this
     license document, but changing it is not allowed.

Preamble

The GNU General Public License is a free, copyleft license for software and other kinds of works.

The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program—to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.

To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.

For the developers' and authors' protection, the GPL clearly explains that there is no warranty for this free software. For both users' and authors' sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.

Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.

Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS

0. Definitions.

   “This License” refers to version 3 of the GNU General Public License.

   “Copyright” also means copyright-like laws that apply to other kinds of works, such as semiconductor masks.

   “The Program” refers to any copyrightable work licensed under this License. Each licensee is addressed as “you”. “Licensees” and “recipients” may be individuals or organizations.

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   A “covered work” means either the unmodified Program or a work based on the Program.

   To “propagate” a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.

   To “convey” a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.

   An interactive user interface displays “Appropriate Legal Notices” to the extent that it includes a convenient and prominently visible feature that (1) displays an appropriate copyright notice, and (2) tells the user that there is no warranty for the work (except to the extent that warranties are provided), that licensees may convey the work under this License, and how to view a copy of this License. If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion.

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   The Corresponding Source for a work in source code form is that same work.

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   All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.

   You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.

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4. Conveying Verbatim Copies.

   You may convey verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program.

   You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee.

5. Conveying Modified Source Versions.

   You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:

   1. The work must carry prominent notices stating that you modified it, and giving a relevant date.
   2. The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to “keep intact all notices”.
   3. You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.
   4. If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.

   A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an “aggregate” if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

6. Conveying Non-Source Forms.

   You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:

   1. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.
   2. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.
   3. Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.
   4. Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.
   5. Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.

   A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work.

   A “User Product” is either (1) a “consumer product”, which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, “normally used” refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product.

   “Installation Information” for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.

   If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).

   The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network.

   Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.

7. Additional Terms.

   “Additional permissions” are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions.

   When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission.

   Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms:

   1. Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or
   2. Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or
   3. Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or
   4. Limiting the use for publicity purposes of names of licensors or authors of the material; or
   5. Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or
   6. Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.

   All other non-permissive additional terms are considered “further restrictions” within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying.

   If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms.

   Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.

8. Termination.

   You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).

   However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

   Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

   Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.

9. Acceptance Not Required for Having Copies.

   You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

10. Automatic Licensing of Downstream Recipients.

    Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.

    An “entity transaction” is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party's predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts.

    You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.

11. Patents.

    A “contributor” is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's “contributor version”.

    A contributor's “essential patent claims” are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, “control” includes the right to grant patent sublicenses in a manner consistent with the requirements of this License.

    Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor's essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version.

    In the following three paragraphs, a “patent license” is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To “grant” such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party.

    If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. “Knowingly relying” means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

    If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

    A patent license is “discriminatory” if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

    Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

12. No Surrender of Others' Freedom.

    If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

13. Use with the GNU Affero General Public License.

    Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

14. Revised Versions of this License.

    The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

    Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License “or any later version” applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

    If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

    Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version.

15. Disclaimer of Warranty.

    THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

16. Limitation of Liability.

    IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

17. Interpretation of Sections 15 and 16.

    If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

END OF TERMS AND CONDITIONS

How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

     one line to give the program's name and a brief idea of what it does.
     Copyright (C) year name of author
     
     This program is free software: you can redistribute it and/or modify
     it under the terms of the GNU General Public License as published by
     the Free Software Foundation, either version 3 of the License, or (at
     your option) any later version.
     
     This program is distributed in the hope that it will be useful, but
     WITHOUT ANY WARRANTY; without even the implied warranty of
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
     General Public License for more details.
     
     You should have received a copy of the GNU General Public License
     along with this program.  If not, see http://www.gnu.org/licenses/.

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

     program Copyright (C) year name of author
     This program comes with ABSOLUTELY NO WARRANTY; for details type ‘show w’.
     This is free software, and you are welcome to redistribute it
     under certain conditions; type ‘show c’ for details.

The hypothetical commands ‘show w’ and ‘show c’ should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read http://www.gnu.org/philosophy/why-not-lgpl.html.

Concept Index

• 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
• 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: CTMC Stationary Probability
• convolution algorithm: Algorithms for Product-Form QNs
• copyright: Copying
• Discrete time Markov chain: DTMC Stationary Probability
• Expected sojourn time: Expected Sojourn Time
• First passage times: CTMC First Passage Times
• First passage times: DTMC First Passage Times
• 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: CTMC First Passage Times
• Markov chain, continuous time: Expected Time to Absorption
• Markov chain, continuous time: Time-Averaged Expected Sojourn Time
• Markov chain, continuous time: Expected Sojourn Time
• Markov chain, continuous time: Birth-Death process
• Markov chain, continuous time: CTMC Stationary Probability
• Markov chain, discrete time: DTMC First Passage Times
• Markov chain, discrete time: DTMC Stationary Probability
• Markov chain, state occupancy probabilities: CTMC Stationary Probability
• Markov chain, stationary probabilities: DTMC Stationary Probability
• Mean time to absorption: Expected Time to Absorption
• 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: CTMC Stationary Probability
• Stationary probabilities: DTMC Stationary Probability
• Time-alveraged sojourn time: Time-Averaged Expected Sojourn Time
• traffic intensity: The M/M/inf System
• warranty: Copying

Function Index

• ctmc: CTMC Stationary Probability
• ctmc_bd: Birth-Death process
• ctmc_exps: Expected Sojourn Time
• ctmc_fpt: CTMC First Passage Times
• ctmc_mtta: Expected Time to Absorption
• ctmc_taexps: Time-Averaged Expected Sojourn Time
• dtmc: DTMC Stationary Probability
• dtmc_fpt: DTMC First Passage Times
• 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.: Expected Time to Absorption
• 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.: Expected Time to Absorption
• 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.: Expected Time to Absorption
• 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.: Expected Time to Absorption
• Wong, E.: Utility functions
• Zahorjan, J.: Utility functions
• Zahorjan, J.: Bounds on performance
• Zahorjan, J.: Algorithms for Product-Form QNs