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qnetworks

This manual documents how to install and run the qnetworks package. It corresponds to qnetworks version 0.8.4.

• Preface
• Installation: Installation of the qnetworks package.
• Getting Started: Getting started with qnetworks.
• 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 to qnetworks.
• Acknowledgements: People who contributed to qnetworks.
• 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 Preface

This document describes the qnetworks package; qnetworks is a collection of functions written in GNU Octave which implement numerical algorithms for analyzing Queueing Network models. In particular, the qnetworks package contains functions for analyzing Jackson networks, open, closed or mixed product-form BCMP networks, and computation of performance bounds. More specifically, the following algorithms have been implemented

• Convolution for closed, single-class product-form networks with load-dependent service centers;
• Mean Value Analysis (MVA) algorithms for single as well as multiple-class closed product-form networks.
• MVA for mixed, multiple class product-form networks with load-independent service centers;
• Closed, single-class networks with blocking using the approximate MVABLO algorithm by F. Akyildiz;
• Closed, multiple class product-form networks using approximate MVA (Bard-Schweitzer approximation);
• Computation of Asymptotic Bounds, Balanced System Bounds as well as Geometric Bounds;

Furthermore, qnetworks 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)

Finally, functions for Markov Chain analysis are provided as well:

• Birth-death process;
• Computation of mean time to absorption;
• Computation of time-averages sojourn time.

qnetworks 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.

You can cite qnetworks in a technical paper as:

Moreno Marzolla, The qnetworks Toolbox: A Software Package for Queueing Networks Analysis. Khalid Al-Begain, Dieter Fiems and William J. Knottenbelt, Editors, Proceedings Analytical and Stochastic Modeling Techniques and Applications, 17th International Conference, 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

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

If you use BibTeX, this is the citation block:

@inproceedings{qnetworks,
  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}
}

2 Installing the qnetworks package

The most recent version of the qnetworks package can be downloaded from

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

Currently, the most recent version is 0.8.4, and is distributed in source form in the qnetworks-0.8.4.tar.gz archive. Download the source tarball from the URL above, unpack it somewhere:

     tar xfz qnetworks-0.8.4.tar.gz
     cd qnetworks-0.8.4/

The qnetworks source distribution includes the following subdirectories:

doc/

Documentation source code

inst/

This directory contains the actual m-files which implement the various Queueing Network algorithms. Note that, as a notational convention, most of the source files (and the functions defined therein) start with the ‘qn’ prefix.

test/

This directory contains the automated test function, which is the one used by GNU Octave.

scripts/

This directory contains some utility scripts taken 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 in the source code. Some of these examples are included into the documentation.

broken/

This directory contains scripts which are not working properly, or need additional testing before they can be included in the inst/ directory.

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

all

Running ‘make’ (or ‘make all’) on the top-level qnetworks directory will build the necessary scripts used to extract the documentation from the comments embedded in the m-files. Then, it will produce the doc/qnetworks.pdf and doc/qnetworks.html files which contain the documentation in PDF and HTML format, respectively.

check

Running ‘make check’ will execute all the test functions contained in the m-files. If you modify the code of any function from the inst/ directory, please run the test to ensure that no errors are introduced. Furthermore, you are encouraged to contribute new test functions to qnetworks, to help verifying that the algorithms are implemented correctly.

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.

If you want to define a new function, simply create the corresponding m-file in the inst/ directory. Each file should have the exact name of the function it defines. You should also add test blocks to ensure that your function computes the correct value; test blocks are extremely useful to help spot errors when a function is modified.

To install the qnetworks package, you can copy all .m files from the inst/ directory to a location where Octave can find them. You can also start Octave with the -p option to tell it to add a path to its search path, so that it will find the functions automatically without the need to move them:

     octave -p /path/to/qnetworks

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

     octave -p /usr/local/qnetworks

Alternatively, you can use the ‘pkg install’ Octave command to install the tarball directly. Start GNU Octave, and type the following command at the prompt:

     octave:1> pkg install qnetworks-0.8.4.tar.gz

In this way, the qnetworks package can be uninstalled at any time with the ‘pkg uninstall qnetworks’ Octave command.

You should now be able to use all qnetworks functions by simply invoking their name. You can also get on-line help about each function using the help Octave command. For example:

     octave:2> help qnmvablo

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

Within GNU Octave, you can also run the tests and demo blocks associated to the various functions, using the test and demo commands. For example, to run all the tests blocks of the qnmvablo function, use the following:

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

To execute the demo blocks of the qnclosed function, use the following:

     octave:4> demo qnclosed

3 Getting Started

• Analysis of Closed Networks
• Analysis of Open Networks
• Markov Chains Analysis

In this chapter we illustrate some usage examples of the qnetworks package. Additional examples are embedded in some 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 probability matrix is defined as follows:

This network can be analyzed with the qnclosed function; in the case of single-class closed networks, as in the example above, the qnclosed function is a front-end to qnclosedsinglemva which implements the Mean Value Analysys (MVA) algorithm for single-class Queueing Networks.

qnclosed requires the following parameters:

N

Number of requests in the network

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.

The visit count V_k to center k can be computed with 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

Note that the visit counts V_k satisfy the following relation:

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

Now, we can analyze the network for a given population size N (for example, N=10) as follows:

     N = 10;
     S = [1 2 0.8];
     V = [1 0.3 0.7];
     [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 the qnclosed function includes the center k utilization U_k, response time R_k, average number of customers Q_k and throughput X_k. In this particular case, for 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: note that center 1 has the higher utilization among the service centers, so it is the bottleneck device.

You can also use the qnsolve function to analyze this network. qnsolve can be applied to open, closed or mixed networks, and allows the network to be described in a very flexible way.

First, we let Q1, Q2 and Q3 to 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 QN.

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:

Note that in this network, requests can leave the system with probability (1-0.3-0.5 = 0.2 after completing service at center 1. Let us consider external arrivals to center 1 with rate \lambda_1 = 0.15; arrival rates to center 2 and 3 are zero.

Similarly with closed networks, we first need to compute the visit counts V_k to center k. This can be done with 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

The visit counts V_k open networks satisfy the following equation:

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

Note that 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

3.3 Markov Chains Analysis

(TBD)

4 Markov Chains

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

4.1 Discrete-Time Markov Chains

4.2 Continuous-Time Markov Chains

• Birth-Death process
• Expected Sojourn Time
• Time-Averaged Expected Sojourn Time
• Expected Time to Absorption

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_solve(Q)
    ⇒ q = 0.50000   0.50000

4.2.1 Birth-Death process

4.2.2 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:

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, i=1,2,3 given initial occupancy probability p_0=(1,0,0,0).

      lambda = 0.5;
      N = 4;
      birth = lambda*linspace(N-1,1,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");

4.2.3 Time-Averaged Expected Sojourn Time

EXAMPLE

      lambda = 0.5;
      N = 4;
      birth = lambda*linspace(N-1,1,N-1);
      death = zeros(1,N-1);
      Q = diag(birth,1)+diag(death,-1);
      Q -= diag(sum(Q,2));
      t = linspace(0.001,10,100);
      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);
      legend("location","northeast");
      xlabel("Time");
      ylabel("Time-averaged Expected sojourn time");

4.2.4 Expected Time to Absorption

If we have 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 the the CTMC with N states and infinitesimal generator matrix \bf Q. Then, we define \bf L_A(\infty) as the expected time to absorption 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, also restricted only 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. 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 \ such that state i denotes that exactly i disks are active. State 2 is absorbing. Let \lambda denotes the failure rate of a single disk. The system starts in state 5 (all disks are operational). This is a pure death process, where the death rate from state i to state i-1 (for all i>2) is \lambda i.

It turns out that the MTTF in this case is the MTTA. We can compute the MTTF (MTTA) as follows:

      lambda = 0.01;
      death = [ 3*lambda 4*lambda 5*lambda ];
      Q = diag(death,-1);
      Q -= diag(sum(Q,2));
      t = ctmc_mtta(Q,[0 0 0 1])

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 Single Station Queueing Systems

Single Station Queueing Systems, as the name suggests, are special queueing systems which are made by a single station. Thus, such systems are quite easy to analyze. The qnetworks package contains functions for handling the following kind of systems:

• 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.

Note that these functions can be used as building blocks for analyzing Queueing Networks. For example, Jackson networks can be analyzed 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, while 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 servers connected to a single queue. Thus, at most m requests can be in service at the same time. The M/M/m system can be seen as a single server with load-dependent service time \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 servers (that is, m = \infty). Thus, there is no queueing at this node, as each arriving job always finds a free server. The M/M/\infty system is always stable, regardless what the arrival and service rates \lambda and \mu are.

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: A brief introduction to Queueing Networks.
• Generic Queueing Network Analysis: High-level functions for QN analysis
• Algorithms for Product-Form QN: Functions to analyze product-form QNs
• Algorithms for non Product-form QN: 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

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 scalar N denotes the population size, i.e., the number of requests in the system. For closed networks, 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:

TBD

while for open networks:

TBD

where P_0 sj 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_0 sj = \lambda_sj / \lambda.

6.2 Generic Queueing Network Analysis

The qnetworks 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 QN.

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 QN

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

qnetworks 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("i=%d k(i)=%d prob=%f\n", i, k(i), p(i) );
      endfor
-| i=1 k(i)=1 prob=0.17975
     -| i=2 k(i)=2 prob=0.48404
     -| i=3 k(i)=0 prob=0.52779

NOTE

For a network with K service centers and population size N, 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("System throughput\t%f\n", X_s );
      printf("System Response Time\t%f\n", R_s );
      printf("Avg. number in system\t%f\n\n", N-X_s*Z );
      printf("\t\tUtil\t\tQlen\t\tRes. time\tTput\n");
      for k=1:length(S)
        printf("Device %d\t%f\t%f\t%f\t%f\n", k, U(k), Q(k), R(k), X(k) );
      endfor

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”.

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 QN

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, no. 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
      Nmax = 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,Nmax);
      X_ab_lower = X_ab_upper = zeros(1,Nmax);
      X_mva = zeros(1,Nmax);
      for n=1:Nmax
        [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:Nmax, X_ab_lower,"g;Asymptotic Bounds;", \
           1:Nmax, X_bsb_lower,"k;Balanced System Bounds;", \
           1:Nmax, X_mva,"b;MVA;", "linewidth", 2, \
           1:Nmax, X_bsb_upper,"k", \
           1:Nmax, X_ab_upper,"g" );
      title("MVA vs Balanced System Bounds");
      axis([1,Nmax,0,1]);
      xlabel("Request Population Size N");
      ylabel("System Throughput X(N)");
      legend("location","southeast");

6.6.2 Computation of the visit counts

For closed, single-class networks the number of visits satisfies the following conditions:

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

while for open networks with arrival rates lambda(i):

V == lambda + V*P

where N is the dimension (number of rows or columns) of P.

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 for some fixed k 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 qnetworks package, here are some guidelines you should consider:

• 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 qnetworks 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 qnetworks package are (mostly) correct.

Send your contribution to Moreno Marzolla (marzolla@cs.unibo.it).

Appendix B Acknowledgements

The following people (listed in alphabetical order) contributed to the qnetworks package, either by providing feedback, reporting bugs or contributing code: Philip Carinhas, 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.

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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.

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Concept Index

• Asymmetric M/M/m system: The Asymmetric M/M/m System
• BCMP network: Algorithms for Product-Form QN
• blocking queueing network: Algorithms for non Product-form QN
• 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 QN
• closed network, approximate analysis: Algorithms for Product-Form QN
• closed network, finite capacity: Algorithms for non Product-form QN
• closed network, multiple classes: Utility functions
• closed network, multiple classes: Algorithms for non Product-form QN
• closed network, multiple classes: Algorithms for Product-Form QN
• closed network, single class: Algorithms for Product-Form QN
• convolution algorithm: Algorithms for Product-Form QN
• copyright: Copying
• expected sojourn time: Expected Sojourn Time
• Jackson network: Algorithms for Product-Form QN
• load-dependent service center: Algorithms for Product-Form QN
• 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: Expected Time to Absorption
• Markov chain, continuous-time: Time-Averaged Expected Sojourn Time
• Markov chain, continuous-time: Expected Sojourn Time
• Markov chain, continuous-time: Continuous-Time Markov Chains
• Markov chain, continuous-time, birth-death process: Birth-Death process
• Markov chain, discrete-time: Discrete-Time Markov Chains
• mean time to absorption: Expected Time to Absorption
• Mean Value Analysys (MVA): Algorithms for Product-Form QN
• Mean Value Analysys (MVA), approximate: Algorithms for Product-Form QN
• mixed network: Algorithms for Product-Form QN
• normalization constant: Algorithms for Product-Form QN
• open network: Utility functions
• open network: Bounds on performance
• open network, multiple classes: Algorithms for Product-Form QN
• open network, single class: Algorithms for Product-Form QN
• population mix: Utility functions
• queueing network with blocking: Algorithms for non Product-form QN
• queueing networks: Queueing Networks
• RS blocking: Algorithms for non Product-form QN
• time-alveraged sojourn time: Time-Averaged Expected Sojourn Time
• traffic intensity: The M/M/inf System
• warranty: Copying

Function Index

• ctmc_bd_solve: Birth-Death process
• ctmc_exps: Expected Sojourn Time
• ctmc_mtta: Expected Time to Absorption
• ctmc_solve: Continuous-Time Markov Chains
• ctmc_taexps: Time-Averaged Expected Sojourn Time
• dtmc_solve: Discrete-Time Markov Chains
• 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 QN
• qnclosedmultimvaapprox: Algorithms for Product-Form QN
• qnclosedpb: Bounds on performance
• qnclosedsinglemva: Algorithms for Product-Form QN
• qnclosedsinglemvald: Algorithms for Product-Form QN
• qnconvolution: Algorithms for Product-Form QN
• qnconvolutionld: Algorithms for Product-Form QN
• qnjackson: Algorithms for Product-Form QN
• qnmarkov: Algorithms for non Product-form QN
• qnmg1: The M/G/1 System
• qnmh1: The M/Hm/1 System
• qnmix: Algorithms for Product-Form QN
• qnmknode: Generic Queueing Network Analysis
• 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 QN
• qnmvapop: Utility functions
• qnopen: Utility functions
• qnopenab: Bounds on performance
• qnopenbsb: Bounds on performance
• qnopenmulti: Algorithms for Product-Form QN
• qnopensingle: Algorithms for Product-Form QN
• qnsolve: Generic Queueing Network Analysis
• qnvisits: Utility functions

Author Index

• Akyildiz, I. F.: Algorithms for non Product-form QN
• Bard, Y.: Algorithms for Product-Form QN
• Bolch, G.: Algorithms for Product-Form QN
• 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 QN
• Casale, G.: Bounds on performance
• de Meer, H.: Algorithms for Product-Form QN
• 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 QN
• Greiner, S.: Algorithms for Product-Form QN
• 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 QN
• Kobayashi, H.: Algorithms for Product-Form QN
• Lam, S.: Bounds on performance
• Lavenberg, S. S.: Algorithms for Product-Form QN
• Lazowska, E. D.: Bounds on performance
• Lazowska, E. D.: Algorithms for Product-Form QN
• Muntz, R. R.: Bounds on performance
• Reiser, M.: Algorithms for Product-Form QN
• Santini, S.: Utility functions
• Schweitzer, P.: Algorithms for Product-Form QN
• Schwetman, H.: Utility functions
• Schwetman, H.: Algorithms for Product-Form QN
• Serazzi, G.: Bounds on performance
• Sevcik, K. C.: Bounds on performance
• Sevcik, K. C.: Algorithms for Product-Form QN
• Trivedi, K.: Algorithms for Product-Form QN
• 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 QN