Copyright © 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2020 Moreno Marzolla.
Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies.
Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one.
Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions.
This manual documents how to install and run the Queueing package. It corresponds to version 1.2.7 of the package.
• Summary: | ||
• Installation and Getting Started: | Installation of the queueing package. | |
• Markov Chains: | Functions for Markov chains analysis. | |
• Single Station Queueing Systems: | Functions for single-station queueing systems. | |
• Queueing Networks: | Functions for queueing networks analysis. | |
• References: | References. | |
• Copying: | The GNU General Public License. | |
• Concept Index: | An item for each concept. | |
• Function Index: | An item for each function. |
Next: Installation and Getting Started, Previous: Top, Up: Top [Contents][Index]
• About the Queueing Package: | What is the Queueing package. | |
• Contributing Guidelines: | How to contribute. | |
• Acknowledgments: |
Next: Contributing Guidelines, Up: Summary [Contents][Index]
This document describes the queueing
package for GNU Octave
(queueing
in short). The queueing
package, previously
known as qnetworks
toolbox, is a collection of functions for
analyzing queueing networks and Markov chains written for GNU
Octave. Specifically, queueing
contains functions for analyzing
Jackson networks, open, closed or mixed product-form BCMP networks,
and computing performance bounds. The following algorithms are
available
queueing
provides functions for analyzing the following types of single-station
queueing systems:
Functions for Markov chain analysis are also provided (discrete- and continuous-time chains are supported):
The queueing
package 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 improve this package by contributing additional functions
and reporting bugs. See Contributing Guidelines.
If you use the queueing
package 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.
Next: Acknowledgments, Previous: About the Queueing Package, Up: Summary [Contents][Index]
Contributions and bug reports are always welcome. If you want
to contribute to the queueing
package, here are some
guidelines:
texinfo
format, so that it can be extracted and included into
the printable manual. See the existing functions for the documentation
style.
Send your contribution to Moreno Marzolla (moreno.marzolla@unibo.it). If you are a user of this package and find it useful, let me know by dropping me a line. Thanks.
Previous: Contributing Guidelines, Up: Summary [Contents][Index]
The following people (listed alphabetically) contributed to the
queueing
package, either by providing feedback, reporting bugs
or contributing code: Philip Carinhas, Phil Colbourn, Diego Didona,
Yves Durand, Marco Guazzone, Dmitry Kolesnikov, Michele Mazzucco,
Marco Paolieri.
Next: Markov Chains, Previous: Summary, Up: Top [Contents][Index]
• Installation through Octave package management system: | ||
• Manual installation: | ||
• Development sources: | ||
• Naming Conventions: | ||
• Quick start Guide: |
Next: Manual installation, Up: Installation and Getting Started [Contents][Index]
The most recent version of queueing
is 1.2.7 and can
be downloaded from Octave-Forge
https://octave.sourceforge.io/queueing/
Additional information can be found at
http://www.moreno.marzolla.name/software/queueing/
To install queueing
, follow these steps:
queueing
from Octave command prompt using this
command:
octave:1> pkg install -forge queueing
The command above will download and install the latest version of the
queueing
package from Octave Forge, and install it on your
machine.
If you do not have root access, you can perform a local install with:
octave:1> pkg install -local -forge queueing
This will install queueing
in your home directory, and the
package will be available to the current user only.
queueing
tarball from
Octave-Forge; to install the package in the system-wide location
issue this command at the Octave prompt:
octave:1> pkg install queueing-1.2.7.tar.gz
(you may need to start Octave as root in order to allow the installation to copy the files to the target locations). After this, all functions will be available each time Octave starts, without the need to tweak the search path.
If you do not have root access, you can do a local install using:
octave:1> pkg install -local queueing-1.2.7.tar.gz
queueing
package has been succesfully installed; you should see
something like:
octave:1>pkg list queueing Package Name | Version | Installation directory --------------+---------+----------------------- queueing | 1.2.7 | /home/moreno/octave/queueing-1.2.7
queueing
is no longer
automatically loaded on Octave start. To make the functions
available for use, you need to issue the command
octave:1>pkg load queueing
at the Octave prompt. To automatically load queueing
each time
Octave starts, you can add the command above to the startup script
(usually, ~/.octaverc on Unix systems).
queueing
from your system, use the
pkg uninstall command:
octave:1> pkg uninstall queueing
Next: Development sources, Previous: Installation through Octave package management system, Up: Installation and Getting Started [Contents][Index]
If you want to manually install queueing
in a custom location,
you can download the tarball and unpack it somewhere:
tar xvfz queueing-1.2.7.tar.gz cd queueing-1.2.7/queueing/
Copy all .m
files from the inst/ directory to some
target location. Then, start Octave with the -p option to add
the target location to the search path, so that Octave will find all
queueing
functions automatically:
octave -p /path/to/queueing
For example, if all queueing
m-files are in
/usr/local/queueing, you can start Octave as follows:
octave -p /usr/local/queueing
If you want, you can add the following line to ~/.octaverc:
addpath("/path/to/queueing");
so that the path /path/to/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.
Next: Naming Conventions, Previous: Manual installation, Up: Installation and Getting Started [Contents][Index]
The source code of the queueing
package can be found in the
Mercurial repository at the URL:
https://sourceforge.net/p/octave/queueing/ci/default/tree/
The source distribution contains additional development files that are not present in the installation tarball. This section briefly describes the content of the source tree. This is only relevant for developers who want to modify the code or the documentation.
The source distribution contains the following directories:
Documentation sources. Most of the documentation is extracted from the comment blocks of function files from the inst/ directory.
This directory contains the m-files which implement the
various algorithms provided by queueing
. As a notational
convention, the names of functions for Queueing Networks begin with
the ‘qn’ prefix; the name of functions for Continuous-Time Markov
Chains (CTMCs) begin with the ‘ctmc’ prefix, and the names of
functions for Discrete-Time Markov Chains (DTMCs) begin with the
‘dtmc’ prefix.
This directory contains the test scripts used to run all function tests.
This directory contains functions that 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. 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 that 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.
Next: Quick start Guide, Previous: Development sources, Up: Installation and Getting Started [Contents][Index]
Most of the functions in the queueing
package obey a common
naming convention. Function names are made of several parts; the first
part is a prefix which indicates the class of problems the function
addresses:
Functions for continuous-time Markov chains
Functions for discrete-time Markov chains
Functions for analyzing single-station queueing systems (individual service centers)
Functions for analyzing queueing networks
Functions dealing with Markov chains start with either the ctmc
or dtmc
prefix; the prefix is optionally followed by an
additional string which hints at what the function does:
Birth-Death process
Mean Time to Absorption
First Passage Times
Expected Sojourn Times
Time-Averaged Expected Sojourn Times
For example, function ctmcbd
returns the infinitesimal
generator matrix for a continuous birth-death process, while
dtmcbd
returns the transition probability matrix for a discrete
birth-death process. Note that there exist functions ctmc
and
dtmc
(without any suffix) that compute steady-state and
transient state occupancy probabilities for CTMCs and DTMCs,
respectively. See Markov Chains.
Functions whose name starts with qs-
deal with single station
queueing systems. The suffix describes the type of system, e.g.,
qsmm1
for M/M/1, qnmmm
for M/M/m and so
on. See Single Station Queueing Systems.
Finally, functions whose name starts with qn-
deal with
queueing networks. The character that follows indicates whether the
function handles open ('o'
) or closed ('c'
) networks,
and whether there is a single customer class ('s'
) or multiple
classes ('m'
). The string mix
indicates that the
function supports mixed networks with both open and closed customer
classes.
Open, single-class network: open network with a single class of customers
Open, multiclass network: open network with multiple job classes
Closed, single-class network
Closed, multiclass network
Mixed network with open and closed classes of customers
The last part of the function name indicates the algorithm implemented by the function. See Queueing Networks.
Asymptotic Bounds Analysis
Balanced System Bounds
Geometric Bounds
PB Bounds
Composite Bounds (CB)
Mean Value Analysis (MVA) algorithm
Conditional MVA
MVA with general load-dependent servers
Approximate MVA
MVABLO approximation for blocking queueing networks
Convolution algorithm
Convolution algorithm with general load-dependent servers
Some deprecated functions may be present in the queueing
package; generally, these are functions that have been renamed, and
the old name is kept for a while for backward compatibility.
Deprecated functions are not documented and will be removed in future
releases. Calling a deprecated functions displays a warning message
that appears only once per session. The warning message can be turned
off with the command:
octave:1> warning ("off", "qn:deprecated-function");
However, you are strongly recommended to update your code to the new API. To help catching usages of deprecated functions, you can transform warnings into errors so that your application stops immediately:
octave:1> warning ("error", "qn:deprecated-function");
Previous: Naming Conventions, Up: Installation and Getting Started [Contents][Index]
You can use all functions by simply invoking their name with the
appropriate parameters; an error is shown in case of missing/wrong
parameters. Extensive documentation is provided for each function, and
can be displayed with the help
command. For example:
octave:2> help qncsmvablo
shows the documentation for the qncsmvablo
function.
Additional information can be found in the queueing
manual,
that is available in PDF format in doc/queueing.pdf and in HTML
format in doc/queueing.html.
Many functions have demo blocks showing usage examples. To execute the
demos for the qnclosed
function, use the demo
command:
octave:4> demo qnclosed
We now illustrate a few examples of how the queueing
package
can be used. More examples are provided in the manual.
Example 1 Compute the stationary state occupancy probabilities of a continuous-time Markov chain with infinitesimal generator matrix
/ -0.8 0.6 0.2 \ Q = | 0.3 -0.7 0.4 | \ 0.2 0.2 -0.4 /
Q = [ -0.8 0.6 0.2; \ 0.3 -0.7 0.4; \ 0.2 0.2 -0.4 ]; q = ctmc(Q) ⇒ q = 0.23256 0.32558 0.44186
Example 2 Compute the transient state occupancy probability after n=3 transitions of a three state discrete-time birth-death process, with birth probabilities \lambda_{01} = 0.3 and \lambda_{12} = 0.5 and death probabilities \mu_{10} = 0.5 and \mu_{21} = 0.7, assuming that the system is initially in state zero (i.e., the initial state occupancy probabilities are [1, 0, 0]).
n = 3; p0 = [1 0 0]; P = dtmcbd( [0.3 0.5], [0.5 0.7] ); p = dtmc(P,n,p0) ⇒ p = 0.55300 0.29700 0.15000
Example 3 Compute server utilization, response time, mean number of requests and throughput of a closed queueing network with N=4 requests and three M/M/1–FCFS queues with mean service times S = [1.0, 0.8, 1.4] and average number of visits V = [1.0, 0.8, 0.8]
S = [1.0 0.8 1.4]; V = [1.0 0.8 0.8]; N = 4; [U R Q X] = qncsmva(N, S, V) ⇒ U = 0.70064 0.44841 0.78471 R = 2.1030 1.2642 3.2433 Q = 1.47346 0.70862 1.81792 X = 0.70064 0.56051 0.56051
Example 4 Compute server utilization, response time, mean number of requests and throughput of an open queueing network with three M/M/1–FCFS queues with mean service times S = [1.0, 0.8, 1.4] and average number of visits V = [1.0, 0.8, 0.8]. The overall arrival rate is \lambda = 0.8 requests/second.
S = [1.0 0.8 1.4]; V = [1.0 0.8 0.8]; lambda = 0.8; [U R Q X] = qnos(lambda, S, V) ⇒ U = 0.80000 0.51200 0.89600 R = 5.0000 1.6393 13.4615 Q = 4.0000 1.0492 8.6154 X = 0.80000 0.64000 0.64000
Next: Single Station Queueing Systems, Previous: Installation and Getting Started, Up: Top [Contents][Index]
• Discrete-Time Markov Chains: | ||
• Continuous-Time Markov Chains: |
Next: Continuous-Time Markov Chains, Up: Markov Chains [Contents][Index]
Let X_0, X_1, …, X_n, … be a sequence of random variables defined over the discrete state space 1, 2, …. The sequence X_0, X_1, …, X_n, … is a stochastic process with discrete time 0, 1, 2, …. A Markov chain is a stochastic process {X_n, n=0, 1, …} which satisfies the following Markov property:
P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, …, X_0 = x_0) = P(X_{n+1} = x_{n+1} | X_n = x_n)
which basically means that the probability that the system is in a particular state at time n+1 only depends on the state the system was at time n.
The evolution of a Markov chain with finite state space {1, …, N} can be fully described by a stochastic matrix {\bf P}(n) = [ P_{i,j}(n) ] where P_{i, j}(n) = P( X_{n+1} = j\ |\ X_n = i ). If the Markov chain is homogeneous (that is, the transition probability matrix {\bf P}(n) is time-independent), we can write {\bf P} = [P_{i, j}], where P_{i, j} = P( X_{n+1} = j\ |\ X_n = i ) for all n=0, 1, ….
The transition probability matrix \bf P must be a stochastic matrix, meaning that it must satisfy the following two properties:
Property 1 requires that all probabilities are nonnegative; property 2 requires that the outgoing transition probabilities from any state i sum to one.
Check whether P is a valid transition probability matrix.
If P is valid, r is the size (number of rows or columns) of P. If P is not a transition probability matrix, r is set to zero, and err to an appropriate error string.
A DTMC is irreducible if every state can be reached with non-zero probability starting from every other state.
Check if P is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with transition matrix P.
INPUTS
P(i,j)
transition probability from state i to state j. P must be an N \times N stochastic matrix.
OUTPUTS
r
1 if P is irreducible, 0 otherwise (scalar)
s(i)
strongly connected component (SCC) that state i belongs to
(vector of length N). SCCs are numbered 1, 2, ….
The number of SCCs is max(s)
. If the graph is
strongly connected, then there is a single SCC and the predicate
all(s == 1)
evaluates to true
Next: Birth-death process (DTMC), Up: Discrete-Time Markov Chains [Contents][Index]
Given a discrete-time Markov chain with state space {1, …, N}, we denote with {\bf \pi}(n) = \left[\pi_1(n), … \pi_N(n) \right] the state occupancy probability vector at step n = 0, 1, …. \pi_i(n) is the probability that the system is in state i after n transitions.
Given the transition probability matrix \bf P and the initial state occupancy probability vector {\bf \pi}(0) = \left[\pi_1(0), …, \pi_N(0)\right], {\bf \pi}(n) can be computed as:
\pi(n) = \pi(0) P^n
Under certain conditions, there exists a stationary state occupancy probability {\bf \pi} = \lim_{n \rightarrow +\infty} {\bf \pi}(n), which is independent from {\bf \pi}(0). The vector \bf \pi is the solution of the following linear system:
/ | \pi P = \pi | \pi 1^T = 1 \
where \bf 1 is the row vector of ones, and ( \cdot )^T the transpose operator.
Compute stationary or transient state occupancy probabilities for a discrete-time Markov chain.
With a single argument, compute the stationary state occupancy
probabilities p(1), …, p(N)
for a
discrete-time Markov chain with finite state space {1, …,
N} and with N \times N transition matrix
P. With three arguments, compute the transient state occupancy
probabilities p(1), …, p(N)
that the system is in
state i after n steps, given initial occupancy
probabilities p0(1), …, p0(N).
INPUTS
P(i,j)
transition probabilities from state i to state j. P must be an N \times N irreducible stochastic matrix, meaning that the sum of each row must be 1 (\sum_{j=1}^N P_{i, j} = 1), and the rank of P must be N.
n
Number of transitions after which state occupancy probabilities are computed (scalar, n ≥ 0)
p0(i)
probability that at step 0 the system is in state i (vector of length N).
OUTPUTS
p(i)
If this function is called with a single argument, p(i)
is the steady-state probability that the system is in state i.
If this function is called with three arguments, p(i)
is the probability that the system is in state i
after n transitions, given the probabilities
p0(i)
that the initial state is i.
See also: ctmc.
EXAMPLE
The following example is from GrSn97. Let us consider a maze with nine rooms, as shown in the following figure
+-----+-----+-----+ | | | | | 1 2 3 | | | | | +- -+- -+- -+ | | | | | 4 5 6 | | | | | +- -+- -+- -+ | | | | | 7 8 9 | | | | | +-----+-----+-----+
A mouse is placed in one of the rooms and can wander around. At each step, the mouse moves from the current room to a neighboring one with equal probability. For example, if it is in room 1, it can move to room 2 and 4 with probability 1/2, respectively; if the mouse is in room 8, it can move to either 7, 5 or 9 with probability 1/3.
The transition probabilities P_{i, j} from room i to room j can be summarized in the following matrix:
/ 0 1/2 0 1/2 0 0 0 0 0 \ | 1/3 0 1/3 0 1/3 0 0 0 0 | | 0 1/2 0 0 0 1/2 0 0 0 | | 1/3 0 0 0 1/3 0 1/3 0 0 | P = | 0 1/4 0 1/4 0 1/4 0 1/4 0 | | 0 0 1/3 0 1/3 0 0 0 1/3 | | 0 0 0 1/2 0 0 0 1/2 0 | | 0 0 0 0 1/3 0 1/3 0 1/3 | \ 0 0 0 0 0 1/2 0 1/2 0 /
The stationary state occupancy probabilities can then be computed with the following code:
P = zeros(9,9); P(1,[2 4] ) = 1/2; P(2,[1 5 3] ) = 1/3; P(3,[2 6] ) = 1/2; P(4,[1 5 7] ) = 1/3; P(5,[2 4 6 8]) = 1/4; P(6,[3 5 9] ) = 1/3; P(7,[4 8] ) = 1/2; P(8,[7 5 9] ) = 1/3; P(9,[6 8] ) = 1/2; p = dtmc(P); disp(p)
⇒ 0.083333 0.125000 0.083333 0.125000 0.166667 0.125000 0.083333 0.125000 0.083333
Next: Expected number of visits (DTMC), Previous: State occupancy probabilities (DTMC), Up: Discrete-Time Markov Chains [Contents][Index]
Returns the transition probability matrix P for a discrete
birth-death process over state space {1, …, N}.
For each i=1, …, (N-1),
b(i)
is the transition probability from state
i to (i+1), and d(i)
is the transition
probability from state (i+1) to i.
Matrix \bf P is defined as:
/ \ | 1-b(1) b(1) | | d(1) (1-d(1)-b(2)) b(2) | | d(2) (1-d(2)-b(3)) b(3) | | | | ... ... ... | | | | d(N-2) (1-d(N-2)-b(N-1)) b(N-1) | | d(N-1) 1-d(N-1) | \ /
where \lambda_i and \mu_i are the birth and death probabilities, respectively.
See also: ctmcbd.
Next: Time-averaged expected sojourn times (DTMC), Previous: Birth-death process (DTMC), Up: Discrete-Time Markov Chains [Contents][Index]
Given a N state discrete-time Markov chain with transition matrix \bf P and an integer n ≥ 0, we let L_i(n) be the the expected number of visits to state i during the first n transitions. The vector {\bf L}(n) = \left[ L_1(n), …, L_N(n) \right] is defined as
n n ___ ___ \ \ i L(n) = > pi(i) = > pi(0) P /___ /___ i=0 i=0
where {\bf \pi}(i) = {\bf \pi}(0){\bf P}^i is the state occupancy probability after i transitions, and {\bf \pi}(0) = \left[\pi_1(0), …, \pi_N(0) \right] are the initial state occupancy probabilities.
If \bf P is absorbing, i.e., the stochastic process eventually enters a state with no outgoing transitions, then we can compute the expected number of visits until absorption \bf L. To do so, we first rearrange the states by rewriting \bf P as
/ Q | R \ P = |---+---| \ 0 | I /
where the first t states are transient and the last r states are absorbing (t+r = N). The matrix {\bf N} = ({\bf I} - {\bf Q})^{-1} is called the fundamental matrix; N_{i,j} is the expected number of times the process is in the j-th transient state assuming it started in the i-th transient state. If we reshape \bf N to the size of \bf P (filling missing entries with zeros), we have that, for absorbing chains, {\bf L} = {\bf \pi}(0){\bf N}.
Compute the expected number of visits to each state during the first n transitions, or until abrosption.
INPUTS
P(i,j)
N \times N transition matrix. P(i,j)
is the
transition probability from state i to state j.
n
Number of steps during which the expected number of visits are
computed (n ≥ 0). If n=0
, returns
p0. If n > 0
, returns the expected number of
visits after exactly n transitions.
p0(i)
Initial state occupancy probabilities; p0(i)
is
the probability that the system is in state i at step 0.
OUTPUTS
L(i)
When called with two arguments, L(i)
is the expected
number of visits to state i before absorption. When
called with three arguments, L(i)
is the expected number
of visits to state i during the first n transitions.
REFERENCES
See also: ctmcexps.
Next: Mean time to absorption (DTMC), Previous: Expected number of visits (DTMC), Up: Discrete-Time Markov Chains [Contents][Index]
Compute the time-averaged sojourn times M(i)
,
defined as the fraction of time spent in state i during the
first n transitions (or until absorption), assuming that the
state occupancy probabilities at time 0 are p0.
INPUTS
P(i,j)
N \times N transition probability matrix.
n
Number of transitions during which the time-averaged expected sojourn times are computed (scalar, n ≥ 0). if n = 0, returns p0.
p0(i)
Initial state occupancy probabilities (vector of length N).
OUTPUTS
M(i)
If this function is called with three arguments, M(i)
is
the expected fraction of steps {0, …, n} spent in
state i, assuming that the state occupancy probabilities at
time zero are p0. If this function is called with two
arguments, M(i)
is the expected fraction of steps spent
in state i until absorption. M is a vector of length
N.
See also: dtmcexps.
Next: First passage times (DTMC), Previous: Time-averaged expected sojourn times (DTMC), Up: Discrete-Time Markov Chains [Contents][Index]
The mean time to absorption is defined as the average number of transitions that are required to enter an absorbing state, starting from a transient state or given initial state occupancy probabilities {\bf \pi}(0).
Let t_i be the expected number of transitions before being absorbed in any absorbing state, starting from state i. The vector {\bf t} = [t_1, …, t_N] can be computed from the fundamental matrix \bf N (see Expected number of visits (DTMC)) as
t = N c
where \bf c is a column vector of 1’s.
Let {\bf B} = [ B_{i, j} ] be a matrix where B_{i, j} is the probability of being absorbed in state j, starting from transient state i. Again, using matrices \bf N and \bf R (see Expected number of visits (DTMC)) we can write
B = N R
Compute the expected number of steps before absorption for a DTMC with state space {1, …, N} and transition probability matrix P.
INPUTS
P(i,j)
N \times N transition probability matrix.
P(i,j)
is the transition probability from state
i to state j.
p0(i)
Initial state occupancy probabilities (vector of length N).
OUTPUTS
t
t(i)
When called with a single argument, t is a vector of length
N such that t(i)
is the expected number of steps
before being absorbed in any absorbing state, starting from state
i; if i is absorbing, t(i) = 0
. When
called with two arguments, t is a scalar, and represents the
expected number of steps before absorption, starting from the initial
state occupancy probability p0.
N(i)
N(i,j)
When called with a single argument, N is the N \times N
fundamental matrix for P. N(i,j)
is the expected
number of visits to transient state j before absorption, if the
system started in transient state i. The initial state is counted
if i = j. When called with two arguments, N is a vector
of length N such that N(j)
is the expected number
of visits to transient state j before absorption, given initial
state occupancy probability P0.
B(i)
B(i,j)
When called with a single argument, B is a N \times N
matrix where B(i,j)
is the probability of being
absorbed in state j, starting from transient state i;
if j is not absorbing, B(i,j) = 0
; if i
is absorbing, B(i,i) = 1
and B(i,j) = 0
for all i \neq j. When called with two arguments, B is
a vector of length N where B(j)
is the
probability of being absorbed in state j, given initial state
occupancy probabilities p0.
REFERENCES
See also: ctmcmtta.
Previous: Mean time to absorption (DTMC), Up: Discrete-Time Markov Chains [Contents][Index]
The First Passage Time M_{i, j} is the average number of transitions needed to enter state j for the first time, starting from state i. Matrix \bf M satisfies the property
___ \ M_ij = 1 + > P_ij * M_kj /___ k!=j
To compute {\bf M} = [ M_{i, j}] a different formulation is used. Let \bf W be the N \times N matrix having each row equal to the stationary state occupancy probability vector \bf \pi for \bf P; let \bf I be the N \times N identity matrix (i.e., the matrix of all ones). Define \bf Z as follows:
-1 Z = (I - P + W)
Then, we have that
Z_jj - Z_ij M_ij = ----------- \pi_j
According to the definition above, M_{i,i} = 0. We arbitrarily set M_{i,i} to the mean recurrence time r_i for state i, that is the average number of transitions needed to return to state i starting from it. r_i is:
1 r_i = ----- \pi_i
Compute mean first passage times and mean recurrence times for an irreducible discrete-time Markov chain over the state space {1, …, N}.
INPUTS
P(i,j)
transition probability from state i to state j. P must be an irreducible stochastic matrix, which means that the sum of each row must be 1 (\sum_{j=1}^N P_{i j} = 1), and the rank of P must be N.
OUTPUTS
M(i,j)
For all 1 ≤ i, j ≤ N, i \neq j, M(i,j)
is
the average number of transitions before state j is entered
for the first time, starting from state i.
M(i,i)
is the mean recurrence time of state
i, and represents the average time needed to return to state
i.
REFERENCES
See also: ctmcfpt.
Previous: Discrete-Time Markov Chains, Up: Markov Chains [Contents][Index]
A stochastic process {X(t), t ≥ 0} is a continuous-time Markov chain if, for all integers n, and for any sequence t_0, t_1 , …, t_n, t_{n+1} such that t_0 < t_1 < … < t_n < t_{n+1}, we have
P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, ..., X_0 = x_0) = P(X_{n+1} = x_{n+1} | X_n = x_n)
A continuous-time Markov chain is defined according to an infinitesimal generator matrix {\bf Q} = [Q_{i,j}], where for each i \neq j, Q_{i, j} is the transition rate from state i to state j. The matrix \bf Q must satisfy the property that, for all i, \sum_{j=1}^N Q_{i, j} = 0.
If Q is a valid infinitesimal generator matrix, return the size (number of rows or columns) of Q. If Q is not an infinitesimal generator matrix, set result to zero, and err to an appropriate error string.
Similarly to the DTMC case, a CTMC is irreducible if every state is eventually reachable from every other state in finite time.
Check if Q is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with infinitesimal generator matrix Q.
INPUTS
Q(i,j)
Infinitesimal generator matrix. Q is a N \times N square
matrix where Q(i,j)
is the transition rate from state
i to state j, for 1 ≤ i, j ≤ N,
i \neq j.
OUTPUTS
r
1 if Q is irreducible, 0 otherwise.
s(i)
strongly connected component (SCC) that state i belongs to.
SCCs are numbered 1, 2, …. If the graph is strongly
connected, then there is a single SCC and the predicate all(s == 1)
evaluates to true.
Next: Birth-death process (CTMC), Up: Continuous-Time Markov Chains [Contents][Index]
Similarly to the discrete case, we denote with {\bf \pi}(t) = \left[\pi_1(t), …, \pi_N(t) \right] the state occupancy probability vector at time t. \pi_i(t) is the probability that the system is in state i at time t ≥ 0.
Given the infinitesimal generator matrix \bf Q and initial state occupancy probabilities {\bf \pi}(0) = \left[\pi_1(0), …, \pi_N(0)\right], the occupancy probabilities {\bf \pi}(t) at time t can be computed as:
\pi(t) = \pi(0) exp(Qt)
where \exp( {\bf Q} t ) is the matrix exponential of {\bf Q} t. Under certain conditions, there exists a stationary state occupancy probability {\bf \pi} = \lim_{t \rightarrow +\infty} {\bf \pi}(t) that is independent from {\bf \pi}(0). \bf \pi is the solution of the following linear system:
/ | \pi Q = 0 | \pi 1^T = 1 \
Compute stationary or transient state occupancy probabilities for a continuous-time Markov chain.
With a single argument, compute the stationary state occupancy probabilities p(1), …, p(N) for a continuous-time Markov chain with finite state space {1, …, N} and N \times N infinitesimal generator matrix Q. With three arguments, compute the state occupancy probabilities p(1), …, p(N) that the system is in state i at time t, given initial state occupancy probabilities p0(1), …, p0(N) at time 0.
INPUTS
Q(i,j)
Infinitesimal generator matrix. Q is a N \times N square
matrix where Q(i,j)
is the transition rate from state
i to state j, for 1 ≤ i \neq j ≤ N.
Q must satisfy the property that \sum_{j=1}^N Q_{i, j} =
0
t
Time at which to compute the transient probability (t ≥ 0). If omitted, the function computes the steady state occupancy probability vector.
p0(i)
probability that the system is in state i at time 0.
OUTPUTS
p(i)
If this function is invoked with a single argument, p(i)
is the steady-state probability that the system is in state i,
i = 1, …, N. If this function is invoked with three
arguments, p(i)
is the probability that the system is in
state i at time t, given the initial occupancy
probabilities p0(1), …, p0(N).
See also: dtmc.
EXAMPLE
Consider a two-state CTMC where all transition rates between states are equal to 1. The stationary state occupancy probabilities can be computed as follows:
Q = [ -1 1; ... 1 -1 ]; q = ctmc(Q)
⇒ q = 0.50000 0.50000
Next: Expected sojourn times (CTMC), Previous: State occupancy probabilities (CTMC), Up: Continuous-Time Markov Chains [Contents][Index]
Returns the infinitesimal generator matrix Q for a
continuous birth-death process over the finite state space
{1, …, N}. For each i=1, …, (N-1),
b(i)
is the transition rate from state i to
state (i+1), and d(i)
is the transition rate from state
(i+1) to state i.
Matrix \bf Q is therefore defined as:
/ \ | -b(1) b(1) | | d(1) -(d(1)+b(2)) b(2) | | d(2) -(d(2)+b(3)) b(3) | | | | ... ... ... | | | | d(N-2) -(d(N-2)+b(N-1)) b(N-1) | | d(N-1) -d(N-1) | \ /
where \lambda_i and \mu_i are the birth and death rates, respectively.
See also: dtmcbd.
Next: Time-averaged expected sojourn times (CTMC), Previous: Birth-death process (CTMC), Up: Continuous-Time Markov Chains [Contents][Index]
Given a N state continuous-time Markov Chain with infinitesimal generator matrix \bf Q, we define the vector {\bf L}(t) = \left[L_1(t), …, L_N(t)\right] such that L_i(t) is the expected sojourn time in state i during the interval [0,t), assuming that the initial occupancy probabilities at time 0 were {\bf \pi}(0). {\bf L}(t) can be expressed as the solution of the following differential equation:
dL --(t) = L(t) Q + pi(0), L(0) = 0 dt
Alternatively, {\bf L}(t) can also be expressed in integral form as:
/ t L(t) = | pi(u) du / 0
where {\bf \pi}(t) = {\bf \pi}(0) \exp({\bf Q}t) is the state occupancy probability at time t; \exp({\bf Q}t) is the matrix exponential of {\bf Q}t.
If there are absorbing states, we can define the vector of expected sojourn times until absorption {\bf L}(\infty), where for each transient state i, L_i(\infty) is the expected total time spent in state i until absorption, assuming that the system started with given state occupancy probabilities {\bf \pi}(0). Let \tau be the set of transient (i.e., non absorbing) states; let {\bf Q}_\tau be the restriction of \bf Q to the transient sub-states only. Similarly, let {\bf \pi}_\tau(0) be the restriction of the initial state occupancy probability vector {\bf \pi}(0) to transient states \tau.
The expected time to absorption {\bf L}_\tau(\infty) is defined as the solution of the following equation:
L_T( inf ) Q_T = -pi_T(0)
With three arguments, compute the expected times L(i)
spent in each state i during the time interval [0,t],
assuming that the initial occupancy vector is p. With two
arguments, compute the expected time L(i)
spent in each
transient state i until absorption.
Note: In its current implementation, this function requires that an absorbing state is reachable from any non-absorbing state of Q.
INPUTS
Q(i,j)
N \times N infinitesimal generator matrix. Q(i,j)
is the transition rate from state i to state j,
1 ≤ i, j ≤ N, i \neq j.
The matrix Q must also satisfy the
condition \sum_{j=1}^N Q_{i,j} = 0 for every i=1, …, N.
t
If given, compute the expected sojourn times in [0,t]
p(i)
Initial occupancy probability vector; p(i)
is the
probability the system is in state i at time 0, i = 1,
…, N
OUTPUTS
L(i)
If this function is called with three arguments, L(i)
is
the expected time spent in state i during the interval
[0,t]. If this function is called with two arguments
L(i)
is the expected time spent in transient state
i until absorption; if state i is absorbing,
L(i)
is zero.
See also: dtmcexps.
EXAMPLE
Let us consider a 4-states pure birth continuous process where 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 for each state i, given initial occupancy probabilities {\bf \pi}_0=[1, 0, 0, 0].
lambda = 0.5; N = 4; b = lambda*[1:N-1]; d = zeros(size(b)); Q = ctmcbd(b,d); t = linspace(0,10,100); p0 = zeros(1,N); p0(1)=1; L = zeros(length(t),N); for i=1:length(t) L(i,:) = ctmcexps(Q,t(i),p0); endfor plot( t, L(:,1), ";State 1;", "linewidth", 2, ... t, L(:,2), ";State 2;", "linewidth", 2, ... t, L(:,3), ";State 3;", "linewidth", 2, ... t, L(:,4), ";State 4;", "linewidth", 2 ); legend("location","northwest"); legend("boxoff"); xlabel("Time"); ylabel("Expected sojourn time");
Next: Mean time to absorption (CTMC), Previous: Expected sojourn times (CTMC), Up: Continuous-Time Markov Chains [Contents][Index]
Compute the time-averaged sojourn time M(i)
,
defined as the fraction of the time interval [0,t] (or until
absorption) spent in state i, assuming that the state
occupancy probabilities at time 0 are p.
INPUTS
Q(i,j)
Infinitesimal generator matrix. Q(i,j)
is the transition
rate from state i to state j,
1 ≤ i,j ≤ N, i \neq j. The
matrix Q must also satisfy the condition \sum_{j=1}^N Q_{i,j} = 0
t
Time. If omitted, the results are computed until absorption.
p0(i)
initial state occupancy probabilities. p0(i)
is the
probability that the system is in state i at time 0, i
= 1, …, N
OUTPUTS
M(i)
When called with three arguments, M(i)
is the expected
fraction of the interval [0,t] spent in state i
assuming that the state occupancy probability at time zero is
p. When called with two arguments, M(i)
is the
expected fraction of time until absorption spent in state i;
in this case the mean time to absorption is sum(M)
.
See also: ctmcexps.
EXAMPLE
lambda = 0.5; N = 4; birth = lambda*linspace(1,N-1,N-1); death = zeros(1,N-1); Q = diag(birth,1)+diag(death,-1); Q -= diag(sum(Q,2)); t = linspace(1e-5,30,100); p = zeros(1,N); p(1)=1; M = zeros(length(t),N); for i=1:length(t) M(i,:) = ctmctaexps(Q,t(i),p); endfor clf; plot(t, M(:,1), ";State 1;", "linewidth", 2, ... t, M(:,2), ";State 2;", "linewidth", 2, ... t, M(:,3), ";State 3;", "linewidth", 2, ... t, M(:,4), ";State 4 (absorbing);", "linewidth", 2 ); legend("location","east"); legend("boxoff"); xlabel("Time"); ylabel("Time-averaged Expected sojourn time");
Next: First passage times (CTMC), Previous: Time-averaged expected sojourn times (CTMC), Up: Continuous-Time Markov Chains [Contents][Index]
Compute the Mean-Time to Absorption (MTTA) of the CTMC described by the infinitesimal generator matrix Q, starting from initial occupancy probabilities p. If there are no absorbing states, this function fails with an error.
INPUTS
Q(i,j)
N \times N infinitesimal generator matrix. Q(i,j)
is the transition rate from state i to state j, i
\neq j. The matrix Q must satisfy the condition
\sum_{j=1}^N Q_{i,j} = 0
p(i)
probability that the system is in state i at time 0, for each i=1, …, N
OUTPUTS
t
Mean time to absorption of the process represented by matrix Q. If there are no absorbing states, this function fails.
REFERENCES
See also: ctmcexps.
EXAMPLE
Let us consider a simple model of redundant disk array. We assume that the array is made of 5 independent disks and 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 continuous Markov chain with state space { 2, 3, 4, 5 }. In state i there are exactly i active (i.e., non failed) disks; 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, where the 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 as follows:
mu = 0.01; death = [ 3 4 5 ] * mu; birth = 0*death; Q = ctmcbd(birth,death); t = ctmcmtta(Q,[0 0 0 1])
⇒ t = 78.333
Previous: Mean time to absorption (CTMC), Up: Continuous-Time Markov Chains [Contents][Index]
Compute mean first passage times for an irreducible continuous-time Markov chain.
INPUTS
Q(i,j)
Infinitesimal generator matrix. Q is a N \times N
square matrix where Q(i,j)
is the transition rate from
state i to state j, for 1 ≤ i, j ≤ N,
i \neq j. Transition rates must be nonnegative, and
\sum_{j=1}^N Q_{i,j} = 0
i
Initial state.
j
Destination state.
OUTPUTS
M(i,j)
average time before state
j is visited for the first time, starting from state i.
We let M(i,i) = 0
.
m
m is the average time before state j is visited for the first time, starting from state i.
See also: ctmcmtta.
Next: Queueing Networks, Previous: Markov Chains, Up: Top [Contents][Index]
Single Station Queueing Systems contain a single station, and can
usually be analyzed easily. 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 Erlang-B Formula: | ||
• The Erlang-C Formula: | ||
• The Engset Formula: | ||
• 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. |
Next: The M/M/m System, Up: Single Station Queueing Systems [Contents][Index]
The M/M/1 system contains a single server connected to an unbounded FCFS queue. Requests arrive according to a Poisson process with rate \lambda; the service time is exponentially distributed with average service rate \mu. The system is stable if \lambda < \mu.
Compute utilization, response time, average number of requests and throughput for a M/M/1 queue.
INPUTS
lambda
Arrival rate (lambda ≥ 0
).
mu
Service rate (mu > lambda
).
k
Number of requests in the system (k ≥ 0
).
OUTPUTS
U
Server utilization
R
Server response time
Q
Average number of requests in the system
X
Server throughput. If the system is ergodic (mu >
lambda
), we always have X = lambda
p0
Steady-state probability that there are no requests in the system.
pk
Steady-state probability that there are k requests in the system. (including the one being served).
If this function is called with less than three input parameters, lambda and mu can be vectors of the same size. In this case, the results will be vectors as well.
REFERENCES
See also: qsmmm, qsmminf, qsmmmk.
Next: The Erlang-B Formula, Previous: The M/M/1 System, Up: Single Station Queueing Systems [Contents][Index]
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 shared FCFS 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 requests in the system:
mu(n) = min(m,n)*mu
where \mu is the service rate of each individual server.
Compute utilization, response time, average number of requests in service and throughput for a M/M/m queue, a queueing system with m identical servers connected to a single FCFS queue.
INPUTS
lambda
Arrival rate (lambda>0
).
mu
Service rate (mu>lambda
).
m
Number of servers (m ≥ 1
).
Default is m=1
.
k
Number of requests in the system (k ≥ 0
).
OUTPUTS
U
Service center utilization, U = \lambda / (m \mu).
R
Service center mean response time
Q
Average number of requests in the system
X
Service center throughput. If the system is ergodic,
we will always have X = lambda
p0
Steady-state probability that there are 0 requests in the system
pm
Steady-state probability that an arriving request has to wait in the queue
pk
Steady-state probability that there are k requests in the system (including the one being served).
If this function is called with less than four parameters, lambda, mu and m can be vectors of the same size. In this case, the results will be vectors as well.
REFERENCES
See also: erlangc,qsmm1,qsmminf,qsmmmk.
Next: The Erlang-C Formula, Previous: The M/M/m System, Up: Single Station Queueing Systems [Contents][Index]
Compute the steady-state blocking probability in the Erlang loss model.
The Erlang-B formula E_B(A, m) gives the probability that an open system with m identical servers, arrival rate \lambda, individual service rate \mu and offered load A = \lambda / \mu has all servers busy. This corresponds to the rejection probability of an M/M/m/0 system with m servers and no queue.
INPUTS
A
Offered load, defined as A = \lambda / \mu where \lambda is the mean arrival rate and \mu the mean service rate of each individual server (real, A > 0).
m
Number of identical servers (integer, m ≥ 1). Default m = 1
OUTPUTS
B
The value E_B(A, m)
A or m can be vectors, and in this case, the results will be vectors as well.
REFERENCES
See also: erlangc,engset,qsmmm.
Next: The Engset Formula, Previous: The Erlang-B Formula, Up: Single Station Queueing Systems [Contents][Index]
Compute the steady-state probability of delay in the Erlang delay model.
The Erlang-C formula E_C(A, m) gives the probability that an open queueing system with m identical servers, infinite wating space, arrival rate \lambda, individual service rate \mu and offered load A = \lambda / \mu has all the servers busy. This is the waiting probability in an M/M/m/\infty system with m servers and an infinite queue.
INPUTS
A
Offered load. A = \lambda / \mu where \lambda is the mean arrival rate and \mu the mean service rate of each individual server (real, 0 < A < m).
m
Number of identical servers (integer, m ≥ 1). Default m = 1
OUTPUTS
B
The value E_C(A, m)
A or m can be vectors, and in this case, the results will be vectors as well.
REFERENCES
See also: erlangb,engset,qsmmm.
Next: The M/M/inf System, Previous: The Erlang-C Formula, Up: Single Station Queueing Systems [Contents][Index]
Evaluate the Engset loss formula.
The Engset formula computes the blocking probability P_b(A,m,n) for a system with a finite population of n users, m identical servers, no queue, individual service rate \mu, individual arrival rate \lambda (i.e., the time until a user tries to request service is exponentially distributed with mean 1/\lambda), and offered load A=\lambda/\mu.
INPUTS
A
Offered load, defined as A = \lambda / \mu where \lambda is the mean arrival rate and \mu the mean service rate of each individual server (real, A > 0).
m
Number of identical servers (integer, m ≥ 1). Default m = 1
n
Number of requests (integer, n ≥ 1). Default n = 1
OUTPUTS
B
The value P_b(A, m, n)
A, m or n can be vectors, and in this case, the results will be vectors as well.
REFERENCES
See also: erlangb, erlangc.
Next: The M/M/1/K System, Previous: The Engset Formula, Up: Single Station Queueing Systems [Contents][Index]
The M/M/\infty system is a special case of M/M/m system with infinitely many identical servers (i.e., m = \infty). Each new request is always assigned to a new server, so that queueing never occurs. The M/M/\infty system is always stable.
Compute utilization, response time, average number of requests and throughput for an infinite-server queue.
The M/M/\infty system has an infinite number of identical servers. Such a system is always stable (i.e., the mean queue length is always finite) for any arrival and service rates.
INPUTS
lambda
Arrival rate (lambda>0
).
mu
Service rate (mu>0
).
k
Number of requests in the system (k ≥ 0
).
OUTPUTS
U
Traffic intensity (defined as \lambda/\mu). Note that this is different from the utilization, which in the case of M/M/\infty centers is always zero.
R
Service center response time.
Q
Average number of requests in the system (which is equal to the traffic intensity \lambda/\mu).
X
Throughput (which is always equal to X = lambda
).
p0
Steady-state probability that there are no requests in the system
pk
Steady-state probability that there are k requests in the system (including the one being served).
If this function is called with less than three arguments, lambda and mu can be vectors of the same size. In this case, the results will be vectors as well.
REFERENCES
See also: qsmm1,qsmmm,qsmmmk.
Next: The M/M/m/K System, Previous: The M/M/inf System, Up: Single Station Queueing Systems [Contents][Index]
In a M/M/1/K finite capacity system there is a single server, and there can be at most K jobs at any time (including the job currently in service), K > 1. If a new request tries to join the system when there are already K other requests, the 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.
Compute utilization, response time, average number of requests and throughput for a M/M/1/K finite capacity system.
In a M/M/1/K queue there is a single server and a queue with finite capacity: the maximum number of requests in the system (including the request being served) is K, and the maximum queue length is therefore K-1.
INPUTS
lambda
Arrival rate (lambda>0
).
mu
Service rate (mu>0
).
K
Maximum number of requests allowed in the system (K ≥ 1
).
n
Number of requests in the (0 ≤ n ≤ K
).
OUTPUTS
U
Service center utilization, which is defined as U = 1-p0
R
Service center response time
Q
Average number of requests in the system
X
Service center throughput
p0
Steady-state probability that there are no requests in the system
pK
Steady-state probability that there are K requests in the system (i.e., that the system is full)
pn
Steady-state probability that there are n requests in the system (including the one being served).
If this function is called with less than four arguments, lambda, mu and K can be vectors of the same size. In this case, the results will be vectors as well.
See also: qsmm1,qsmminf,qsmmm.
Next: The Asymmetric M/M/m System, Previous: The M/M/1/K System, Up: Single Station Queueing Systems [Contents][Index]
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 has K-m slots. The M/M/m/K system is always stable.
Compute utilization, response time, average number of requests and throughput for a M/M/m/K finite capacity system. In a M/M/m/K system there are m \geq 1 identical service centers sharing a fixed-capacity queue. At any time, at most K ≥ m requests can be in the system, including those being served. The maximum queue length is K-m. This function generates and solves the underlying CTMC.
INPUTS
lambda
Arrival rate (lambda>0
)
mu
Service rate (mu>0
)
m
Number of servers (m ≥ 1
)
K
Maximum number of requests allowed in the system,
including those being served (K ≥ m
)
n
Number of requests in the (0 ≤ n ≤ K
).
OUTPUTS
U
Service center utilization
R
Service center response time
Q
Average number of requests in the system
X
Service center throughput
p0
Steady-state probability that there are no requests in the system.
pK
Steady-state probability that there are K requests in the system (i.e., probability that the system is full).
pn
Steady-state probability that there are n requests in the system (including those being served).
If this function is called with less than five arguments, lambda, mu, m and K can be either scalars, or vectors of the same size. In this case, the results will be vectors as well.
REFERENCES
See also: qsmm1,qsmminf,qsmmm.
Next: The M/G/1 System, Previous: The M/M/m/K System, Up: Single Station Queueing Systems [Contents][Index]
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.
Compute approximate utilization, response time, average number of requests in service and throughput for an asymmetric M/M/m queue. In this type of system there are m different servers connected to a single queue. Each server has its own (possibly different) service rate. If there is more than one server available, requests are routed to a randomly-chosen one.
INPUTS
lambda
Arrival rate (lambda>0
)
mu
mu(i)
is the service rate of server
i, 1 ≤ i ≤ m.
The system must be ergodic (lambda < sum(mu)
).
OUTPUTS
U
Approximate service center utilization, U = \lambda / ( \sum_{i=1}^m \mu_i ).
R
Approximate service center response time
Q
Approximate number of requests in the system
X
Approximate system throughput. If the system is ergodic,
X = lambda
REFERENCES
See also: qsmmm.
Next: The M/Hm/1 System, Previous: The Asymmetric M/M/m System, Up: Single Station Queueing Systems [Contents][Index]
Compute utilization, response time, average number of requests and throughput for a M/G/1 system. The service time distribution is described by its mean xavg, and by its second moment x2nd. The computations are based on results from L. Kleinrock, Queuing Systems, Wiley, Vol 2, and Pollaczek-Khinchine formula.
INPUTS
lambda
Arrival rate
xavg
Average service time
x2nd
Second moment of service time distribution
OUTPUTS
U
Service center utilization
R
Service center response time
Q
Average number of requests in the system
X
Service center throughput
p0
Probability that there is not any request at system
lambda, xavg, t2nd can be vectors of the same size. In this case, the results will be vectors as well.
See also: qsmh1.
Previous: The M/G/1 System, Up: Single Station Queueing Systems [Contents][Index]
Compute utilization, response time, average number of requests and throughput for a M/H_m/1 system. In this system, the customer service times have hyper-exponential distribution:
___ m \ B(x) = > alpha(j) * (1-exp(-mu(j)*x)) x>0 /__ j=1
where \alpha_j is the probability that the request is served at phase j, in which case the average service rate is \mu_j. After completing service at phase j, for some j, the request exits the system.
INPUTS
lambda
Arrival rate
mu
mu(j)
is the phase j service rate. The total
number of phases m is length(mu)
.
alpha
alpha(j)
is the probability that a request
is served at phase j. alpha must have the same size
as mu.
OUTPUTS
U
Service center utilization
R
Service center response time
Q
Average number of requests in the system
X
Service center throughput
Next: References, Previous: Single Station Queueing Systems, Up: Top [Contents][Index]
• Introduction to QNs: | A brief introduction to Queueing Networks | |
• Single Class Models: | Queueing models with a single job class | |
• Multiple Class Models: | Queueing models with multiple job classes | |
• Generic Algorithms: | High-level functions for QN analysis | |
• Bounds Analysis: | Computation of asymptotic performance bounds | |
• QN Analysis Examples: | Queueing Networks analysis examples |
Next: Single Class Models, Up: Queueing Networks [Contents][Index]
Queueing Networks (QN) are a simple modeling notation that can be used to analyze many kinds of systems. In its simplest form, a QN is made of K service centers; center k has a queue connected to m_k (usually identical) servers. Arriving customers (requests) join the queue if there is at least one slot available. Requests are served according to a (de)queueing policy (e.g., FIFO). After service completes, requests leave the server and can join another queue or exit from the system.
Service centers where m_k = \infty are called delay centers or infinite servers. In this kind of centers, there is always one available server, so that queueing never occurs.
Requests join the queue according to a queueing policy, such as:
First-Come-First-Served
Last-Come-First-Served, Preemptive Resume
Processor Sharing
Infinite Server (m_k = \infty).
Queueing networks can be open or closed. In open networks there is an infinite population of requests; new customers are generated outside the system, and eventually leave the network. In closed networks there is a fixed population of request that never leave the system.
Queueing models can have a single request class (single class models), meaning that all requests behave in the same way (e.g., they spend the same average time on each particular server). In multiple class models there are multiple request classes, each with its own parameters (e.g., with different service times or different routing probabilities). Furthermore, in multiclass models there can be open and closed chains of requests at the same time.
A particular class of QN models, product-form networks, is of particular interest. Product-form networks fulfill the following assumptions:
Product-form networks are attractive because steady-state performance measures can be efficiently computed.
Next: Multiple Class Models, Previous: Introduction to QNs, Up: Queueing Networks [Contents][Index]
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
(Open models only) External arrival rate to service center k.
(Open models only) Overall external arrival rate to the system as a whole: \lambda = \sum_k \lambda_k.
(Closed models only) Total number of requests in the system.
Mean service time at center k. S_k is the average time elapsed from service start to service completion at center k.
Routing probability matrix. {\bf P} = [P_{i, j}] is a K \times K matrix where P_{i, j} is the probability that a request completing service at center i is routed to center j. The probability that a request leaves the system after being served at center i is \left(1-\sum_{j=1}^K P_{i, j}\right).
Mean number of visits to center k (also called visit ratio or relative arrival rate).
Model Outputs
Utilization of service center k. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). If center k is a single-server or multiserver node, then 0 ≤ U_k ≤ 1. If center k is an infinite server node (delay center), then U_k denotes the traffic intensity and is defined as U_k = X_k S_k; in this case the utilization may be greater than one.
Average response time of service center k, defined as the mean time between the arrival of a request in the queue and service completion of the same request.
Average number of requests in center k; this includes both the requests in the queue and those being served.
Throughput of service center k. The throughput is the rate of job completions, i.e., the average number of jobs completed over a given time interval.
Given the output parameters above, additional performance measures can be computed:
System throughput, X = X_k / V_k for any k for which V_k \neq 0
System response time, R = \sum_{k=1}^K R_k V_k
Average number of requests in the system, Q = \sum_{k=1}^K Q_k; for closed systems, this can be written as 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 V_j satisfy the following equation:
K ___ \ V_j = P_(0, j) + > V_i P_(i, j) j=1,...,K /___ i=1
where P_{0, j} is the probability that an external request goes to center j. If we denote with \lambda_j the external arrival rate to center j, and \lambda = \sum_j \lambda_j the overall external arrival rate, then P_{0, j} = \lambda_j / \lambda.
For closed models, the visit ratios satisfy the following equation:
/ | K | ___ | \ | V_j = > V_i P_(i, j) j=1,...,K | /___ | i=1 | | V_r = 1 for a selected reference station r \
Note that the set of traffic equations V_j = \sum_{i=1}^K V_i P_{i, j} alone can only be solved up to a multiplicative constant; to get a unique solution we impose an additional constraint V_r = 1 for some 1 ≤ r ≤ K. This constraint is equivalent to defining station r as the reference station; the default is r=1, see doc-qncsvisits. A job that returns to the reference station is assumed to have completed its activity cycle. The network throughput is set to the throughput of the reference station.
Compute the mean number of visits to the service centers of a single class, closed network with K service centers.
INPUTS
P(i,j)
probability that a request which completed service at center
i is routed to center j (K \times K matrix).
For closed networks it must hold that sum(P,2)==1
. The
routing graph must be strongly connected, meaning that each node
must be reachable from every other node.
r
Index of the reference station, r \in {1, …, K};
Default r=1
. The traffic equations are solved by
imposing the condition V(r) = 1
. A request returning to
the reference station completes its activity cycle.
OUTPUTS
V(k)
average number of visits to service center k, assuming r as the reference station.
Compute the average number of visits to the service centers of a single class open Queueing Network with K service centers.
INPUTS
P(i,j)
is the probability that a request which completed service at center i is routed to center j (K \times K matrix).
lambda(k)
external arrival rate to center k.
OUTPUTS
V(k)
average number of visits to server k.
EXAMPLE
Figure 5.1 shows a closed queueing network with a single class of requests. The network has three service centers, labeled CPU, Disk1 and Disk2, and is known as a central server model of a computer system. Requests spend some time at the CPU, which is represented by a PS (Processor Sharing) node. After that, requests are routed to Disk1 with probability 0.3, and to Disk2 with probability 0.7. Both Disk1 and Disk2 are FCFS nodes.
If we label the servers as CPU=1, Disk1=2, Disk2=3, we can define the routing matrix as follows:
/ 0 0.3 0.7 \ P = | 1 0 0 | \ 1 0 0 /
The visit ratios V, using station 1 as the reference station, can be computed with:
P = [0 0.3 0.7; ... 1 0 0 ; ... 1 0 0 ]; V = qncsvisits(P)
⇒ V = 1.00000 0.30000 0.70000
EXAMPLE
Figure 5.2 shows a open QN with a single class of requests. The network has the same structure as the one in Figure 5.1, with the difference that here we have a stream of jobs arriving from outside the system, at a rate \lambda. After service completion at the CPU, a job can leave the system with probability 0.2, or be transferred to other nodes with the probabilities shown in the figure.
The routing matrix is
/ 0 0.3 0.5 \ P = | 1 0 0 | \ 1 0 0 /
If we let \lambda = 1.2, we can compute the visit ratios V as follows:
p = 0.3; lambda = 1.2 P = [0 0.3 0.5; ... 1 0 0 ; ... 1 0 0 ]; V = qnosvisits(P,[1.2 0 0])
⇒ V = 5.0000 1.5000 2.5000
Function qnosvisits
expects a vector with K elements
as a second parameter, for open networks only. The vector contains the
arrival rates at each individual node; since in our example external
arrivals exist only for node S_1 with rate \lambda =
1.2, the second parameter is [1.2, 0, 0]
.
Jackson networks satisfy the following conditions:
We define the joint probability vector \pi(n_1, …, n_K) as the steady-state probability that there are n_k requests at service center k, for all k=1, …, N. Jackson networks have the property that the joint probability is the product of the marginal probabilities \pi_k:
joint_prob = prod( pi )
where \pi_k(n_k) is the steady-state probability that there are n_k requests at service center k.
Analyze open, single class BCMP queueing networks with K service centers.
This function works for a subset of BCMP single-class open networks satisfying the following properties:
m(k) ≥ 1
identical servers.
INPUTS
lambda
Overall external arrival rate (lambda>0
).
S(k)
average service time at center k (S(k)>0
).
V(k)
average number of visits to center k (V(k) ≥ 0
).
m(k)
number of servers at center i. If m(k) < 1
,
enter k is a delay center (IS); otherwise it is a regular
queueing center with m(k)
servers. Default is
m(k) = 1
for all k.
OUTPUTS
U(k)
If k is a queueing center,
U(k)
is the utilization of center k.
If k is an IS node, then U(k)
is the
traffic intensity defined as X(k)*S(k)
.
R(k)
center k average response time.
Q(k)
average number of requests at center k.
X(k)
center k throughput.
REFERENCES
See also: qnopen,qnclosed,qnosvisits.
From the results computed by this function, it is possible to derive other quantities of interest as follows:
R_s = dot(V,R);
Q_avg = sum(Q)
EXAMPLE
lambda = 3; V = [16 7 8]; S = [0.01 0.02 0.03]; [U R Q X] = qnos( 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
Analyze closed, single class queueing networks using the exact Mean Value Analysis (MVA) algorithm.
The following queueing disciplines are supported: FCFS, LCFS-PR, PS
and IS (Infinite Server). This function supports fixed-rate service
centers or multiple server nodes. For general load-dependent service
centers, use the function qncsmvald
instead.
Additionally, the normalization constant G(n), n=0, …, N is computed; G(n) can be used in conjunction with the BCMP theorem to compute steady-state probabilities.
INPUTS
N
Population size (number of requests in the system, N ≥ 0
).
If N == 0
, this function returns
U = R = Q = X = 0
S(k)
mean service time at center k (S(k) ≥ 0
).
V(k)
average number of visits to service center k (V(k) ≥ 0
).
Z
External delay for customers (Z ≥ 0
). Default is 0.
m(k)
number of servers at center k (if m is a scalar, all
centers have that number of servers). If m(k) < 1
,
center k is a delay center (IS); otherwise it is a regular
queueing center (FCFS, LCFS-PR or PS) with m(k)
servers. Default is m(k) = 1
for all k (each
service center has a single server).
OUTPUTS
U(k)
If k is a FCFS, LCFS-PR or PS node (m(k) ≥
1
), then U(k)
is the utilization of center k,
0 ≤ U(k) ≤ 1. If k is an IS node
(m(k) < 1
), then U(k)
is the traffic
intensity defined as X(k)*S(k)
. In this case the
value of U(k)
may be greater than one.
R(k)
center k response time. The Residence Time at center
k is R(k) * V(k)
. The system response
time Rsys can be computed either as Rsys =
N/Xsys - Z
or as Rsys =
dot(R,V)
Q(k)
average number of requests at center k. The number of
requests in the system can be computed either as
sum(Q)
, or using the formula
N-Xsys*Z
.
X(k)
center K throughput. The system throughput Xsys can be
computed as Xsys = X(1) / V(1)
G(n)
Normalization constants. G(n+1)
contains the value of
the normalization constant G(n), n=0, …, N as
array indexes in Octave start from 1. G(n) can be used in
conjunction with the BCMP theorem to compute steady-state
probabilities.
NOTES
In presence of load-dependent servers (i.e., if m(k)>1
for some k), the MVA algorithm is known to be numerically
unstable. Generally, this issue manifests itself as negative values
for the response times or utilizations. This is not a problem of
the queueing
toolbox, but of the MVA algorithm, and has
currently no known solution. This function prints a warning if
numerical problems are detected; the warning can be disabled with
the command warning("off", "qn:numerical-instability")
.
REFERENCES
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".
See also: qncsmvald,qncscmva.
EXAMPLE
S = [ 0.125 0.3 0.2 ]; V = [ 16 10 5 ]; N = 20; m = ones(1,3); Z = 4; [U R Q X] = qncsmva(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 );
Mean Value Analysis algorithm for closed, single class queueing
networks with K service centers and load-dependent service
times. This function supports FCFS, LCFS-PR, PS and IS nodes. For
networks with only fixed-rate centers and multiple-server
nodes, the function qncsmva
is more efficient.
INPUTS
N
Population size (number of requests in the system, N ≥ 0
).
If N == 0
, this function returns U = R = Q = X = 0
S(k,n)
mean service time at center k
where there are n requests, 1 ≤ n
≤ N. S(k,n)
= 1 / \mu_{k}(n),
where \mu_{k}(n) is the service rate of center k
when there are n requests.
V(k)
average number of visits to service center k (V(k) ≥ 0
).
Z
external delay ("think time", Z ≥ 0
); default 0.
OUTPUTS
U(k)
utilization of service center k. The utilization is defined as the probability that service center k is not empty, that is, U_k = 1-\pi_k(0) where \pi_k(0) is the steady-state probability that there are 0 jobs at service center k.
R(k)
response time on service center k.
Q(k)
average number of requests in service center k.
X(k)
throughput of service center k.
NOTES
In presence of load-dependent servers, the MVA algorithm is known to be numerically unstable. Generally this problem manifests itself as negative response times or utilization.
REFERENCES
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-Dependent Service: Closed Networks”.
See also: qncsmva.
Conditional MVA (CMVA) algorithm, a numerically stable variant of MVA. This function supports a network of M ≥ 1 service centers and a single delay center. Servers 1, …, (M-1) are load-independent; server M is load-dependent.
INPUTS
N
Number of requests in the system, N ≥ 0
. If
N == 0
, this function returns U = R
= Q = X = 0
S(k)
mean service time on server k = 1, …, (M-1)
(S(k) > 0
). If there are no fixed-rate servers, then
S = []
Sld(n)
inverse service rate at server M (the load-dependent server)
when there are n requests, n=1, …, N.
Sld(n) =
1 / \mu(n).
V(k)
average number of visits to service center k=1, …, M,
where V(k) ≥ 0
. V(1:M-1)
are the
visit rates to the fixed rate servers; V(M)
is the
visit rate to the load dependent server.
Z
External delay for customers (Z ≥ 0
). Default is 0.
OUTPUTS
U(k)
center k utilization (k=1, …, M)
R(k)
response time of center k (k=1, …, M). The
system response time Rsys can be computed as Rsys
= N/Xsys - Z
Q(k)
average number of requests at center k (k=1, …, M).
X(k)
center k throughput (k=1, …, M).
REFERENCES
Analyze closed, single class queueing networks using the Approximate
Mean Value Analysis (MVA) algorithm. This function is based on
approximating the number of customers seen at center k when a
new request arrives as Q_k(N) \times (N-1)/N. This function
only handles single-server and delay centers; if your network
contains general load-dependent service centers, use the function
qncsmvald
instead.
INPUTS
N
Population size (number of requests in the system, N > 0
).
S(k)
mean service time on server k
(S(k)>0
).
V(k)
average number of visits to service center
k (V(k) ≥ 0
).
m(k)
number of servers at center k
(if m is a scalar, all centers have that number of servers). If
m(k) < 1
, center k is a delay center (IS); if
m(k) == 1
, center k is a regular queueing
center (FCFS, LCFS-PR or PS) with one server (default). This function
does not support multiple server nodes (m(k) > 1
).
Z
External delay for customers (Z ≥ 0
). Default is 0.
tol
Stopping tolerance. The algorithm stops when the maximum relative difference between the new and old value of the queue lengths Q becomes less than the tolerance. Default is 10^{-5}.
iter_max
Maximum number of iterations (iter_max>0
.
The function aborts if convergenge is not reached within the maximum
number of iterations. Default is 100.
OUTPUTS
U(k)
If k is a FCFS, LCFS-PR or PS node (m(k) == 1
),
then U(k)
is the utilization of center k. If
k is an IS node (m(k) < 1
), then
U(k)
is the traffic intensity defined as
X(k)*S(k)
.
R(k)
response time at center k.
The system response time Rsys
can be computed as Rsys = N/Xsys - Z
Q(k)
average number of requests at center k. The number of
requests in the system can be computed either as
sum(Q)
, or using the formula
N-Xsys*Z
.
X(k)
center k throughput. The system throughput Xsys can be
computed as Xsys = X(1) / V(1)
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").
See also: qncsmva,qncsmvald.
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:
n = [n1, … nK]; population vector
p = 1/G(N+1) \prod F(k,k);
Here \pi(n_1, …, n_K) is the joint probability of having n_k requests at node k, for all k=1, …, K; we have that \sum_{k=1}^K n_k = N
The convolution algorithms computes the normalization constants
{\bf G} = \left[G(0), …, G(N)\right] for single-class,
closed networks with N requests. The normalization constants
are returned as vector G=[G(1), …
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
qncsconv
and qncsconvld
. The first one
supports single-station nodes, multiple-station nodes and IS nodes.
The second one supports networks with general load-dependent service
centers.
Analyze product-form, single class closed networks with K service centers using the convolution algorithm.
Load-independent service centers, multiple servers (M/M/m
queues) and IS nodes are supported. For general load-dependent
service centers, use qncsconvld
instead.
INPUTS
N
Number of requests in the system (N>0
).
S(k)
average service time on center k (S(k) ≥ 0
).
V(k)
visit count of service center k (V(k) ≥ 0
).
m(k)
number of servers at center k. If m(k) < 1
,
center k is a delay center (IS); if m(k) ≥
1
, center k it is a regular M/M/m queueing center
with m(k)
identical servers. Default is
m(k) = 1
for all k.
OUTPUT
U(k)
center k utilization.
For IS nodes, U(k)
is the traffic intensity
X(k) * S(k)
.
R(k)
average response time of center k.
Q(k)
average number of customers at center k.
X(k)
throughput of center k.
G(n)
Vector of normalization constants. G(n+1)
contains the value of
the normalization constant with n requests
G(n), n=0, …, N.
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
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.
See also: qncsconvld.
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 and N requests. Let
n = [n_1, …, n_K]
be a valid
population vector, \sum_{k=1}^K n_k = N. Then, the steady-state
probability p(k)
to have n(k)
requests at
service center k can be computed as:
n = [1 2 0]; N = sum(n); # 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] = qncsconv( N, S, V, m ); p = [0 0 0]; # initialize p # Compute the probability to have n(k) jobs at service center k for k=1:3 p(k) = (V(k)*S(k))^n(k) / G(N+1) * ... (G(N-n(k)+1) - V(k)*S(k)*G(N-n(k)) ); printf("Prob( n(%d) = %d )=%f\n", k, n(k), p(k) ); endfor
-| Prob( n(1) = 1 ) = 0.17975 -| Prob( n(2) = 2 ) = 0.48404 -| Prob( n(3) = 0 ) = 0.52779
(recall that G(N+1)
represents G(N), since
in Octave array indices start at one).
Convolution algorithm for product-form, single-class closed queueing networks with K general load-dependent service centers.
This function computes steady-state performance measures for
single-class, closed networks with load-dependent service centers
using the convolution algorithm; the normalization constants are also
computed. The normalization constants are returned as vector
G=[G(1), …, G(N+1)]
where
G(i+1)
is the value of G(i).
INPUTS
N
Number of requests in the system (N>0
).
S(k,n)
mean service time at center k where there are n
requests, 1 ≤ n ≤ N. S(k,n)
= 1 / \mu_{k,n}, where \mu_{k,n} is the service rate of center
k when there are n requests.
V(k)
visit count of service center k
(V(k) ≥ 0
). The length of V is the number of
servers K in the network.
OUTPUT
U(k)
center k utilization.
R(k)
average response time at center k.
Q(k)
average number of requests in center k.
X(k)
center k throughput.
G(n)
Normalization constants (vector). G(n+1)
corresponds to G(n), as array indexes in Octave start
from 1.
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. 313–317. Function qncsconvld
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 qncsconvld
which has been made similar to
function f_i defined in Schwetman, Some Computational
Aspects of Queueing Network Models.
See also: qncsconv.
Approximate MVA algorithm for closed queueing networks with blocking.
INPUTS
N
number of requests in the system. N must be strictly greater
than zero, and less than the overall network capacity: 0 <
N < sum(M)
.
S(k)
average service time on server k (S(k) > 0
).
M(k)
capacity of center k. The capacity is the maximum number of requests in a service
center, including the request in service (M(k) ≥ 1
).
P(i,j)
probability that a request which completes service at server i will be transferred to server j.
OUTPUTS
U(k)
center k utilization.
R(k)
average response time of service center k.
Q(k)
average number of requests in service center k (including the request in service).
X(k)
center k throughput.
REFERENCES
See also: qnopen, qnclosed.
Compute utilization, response time, average queue length and throughput for open or closed queueing networks with finite capacity and a single class of requests. Blocking type is Repetitive-Service (RS). This function explicitly generates and solve the underlying Markov chain, and thus might require a large amount of memory.
More specifically, networks which can me analyzed by this function have the following properties:
INPUTS
lambda(k)
N
If the first argument is a vector lambda, it is considered to be
the external arrival rate lambda(k) ≥ 0
to service center
k of an open network. If the first argument is a scalar, it is
considered as the population size N of a closed network; in this case
N must be strictly
less than the network capacity: N < sum(C)
.
S(k)
average service time at service center k
C(k)
capacity of service center k. The capacity includes both
the buffer and server space m(k)
. Thus the buffer space is
C(k)-m(k)
.
P(i,j)
transition probability from service center i to service center j.
m(k)
number of servers at service center
k. Note that m(k) ≥ C(k)
for each k.
If m is omitted, all service centers are assumed to have a
single server (m(k) = 1
for all k).
OUTPUTS
U(k)
center k utilization.
R(k)
response time on service center k.
Q(k)
average number of customers in the service center k, including the request in service.
X(k)
throughput of service center k.
NOTES
The space complexity of this implementation is O(\prod_{k=1}^K (C_k + 1)^2). The time complexity is dominated by the time needed to solve a linear system with \prod_{k=1}^K (C_k + 1) unknowns.
Next: Generic Algorithms, Previous: Single Class Models, Up: Queueing Networks [Contents][Index]
In multiple class queueing models, we assume that there exist C different classes of requests. Each request from class c spends on average time S_{c, k} in service at center k. For open models, we denote with {\bf \lambda} = \lambda_{c, k} the arrival rates, where \lambda_{c, k} is the external arrival rate of class c requests at center k. For closed models, we denote with {\bf N} = \left[N_1, …, N_C\right] the population vector, where N_c is the number of class c requests in the system.
The transition probability matrix for multiple class networks is a C \times K \times C \times K matrix {\bf P} = [P_{r, i, s, j}] where P_{r, i, s, j} is the probability that a class r request which completes service at center i will join server j as a class s request.
Model input and outputs can be adjusted by adding additional indexes for the customer classes.
Model Inputs
(open networks) External arrival rate of class-c requests to service center k
(open networks) Overall external arrival rate to the whole system: \lambda = \sum_{c=1}^C \sum_{k=1}^K \lambda_{c, k}
(closed networks) Number of class c requests in the system.
Average service time. S_{c, k} is the average service time on service center k for class c requests.
Routing probability matrix. {\bf P} = [P_{r, i, s, j}] is a C \times K \times C \times K matrix such that P_{r, i, s, j} is the probability that a class r request which completes service at server i will move to server j as a class s request.
Mean number of visits of class c requests to center k.
Model Outputs
Utilization of service center k by class c requests. The utilization is defined as the fraction of time in which the resource is busy (i.e., the server is processing requests). If center k is a single-server or multiserver node, then 0 ≤ U_{c, k} ≤ 1. If center k is an infinite server node (delay center), then U_{c, k} denotes the traffic intensity and is defined as U_{c, k} = X_{c, k} S_{c, k}; in this case the utilization may be greater than one.
Average response time experienced by class c requests on service center k. The average response time is defined as the average time between the arrival of a customer in the queue, and the completion of service.
Average number of class c requests on service center k. This includes both the requests in the queue, and the request being served.
Throughput of service center k for class c requests. The throughput is defined as the rate of completion of class c requests.
It is possible to define aggregate performance measures as follows:
Utilization of service center k:
Uk = sum(U,k);
System response time for class c requests:
Rc = sum( V.*R, 1 );
Average number of class c requests in the system:
Qc = sum( Q, 2 );
Class c throughput:
X(c) = X(c,k) ./ V(c,k);
for any k for which V(c,k) != 0
For closed networks, we can define the visit ratios V_{s, j} for class s customers at service center j as follows:
V_sj = sum_r sum_i V_ri P_risj s=1,...,C, j=1,...,K V_s r_s = 1 s=1,...,C
where r_s is the class s reference station. Similarly to single class models, the traffic equation for closed multiclass networks can be solved up to multiplicative constants unless we choose one reference station for each closed class and set its visit ratio to 1.
For open networks the traffic equations are as follows:
V_sj = P_0sj + sum_r sum_i V_ri P_risj s=1,...,C, j=1,...,K
where P_{0, s, j} is the probability that an external arrival goes to service center j as a class-s request. If \lambda_{s, j} is the external arrival rate of class s requests to service center j, and \lambda = \sum_s \sum_j \lambda_{s, j} is the overall external arrival rate, then P_{0, s, j} = \lambda_{s, j} / \lambda.
Compute the average number of visits for the nodes of a closed multiclass network with K service centers and C customer classes.
INPUTS
P(r,i,s,j)
probability that a class r request which completed service at center i is routed to center j as a class s request. Class switching is allowed.
r(c)
index of class c reference station,
r(c) \in {1, …, K}, 1 ≤ c ≤ C.
The class c visit count to server r(c)
(V(c,r(c))
) is conventionally set to 1. The reference
station serves two purposes: (i) its throughput is assumed to be the
system throughput, and (ii) a job returning to the reference station
is assumed to have completed one cycle. Default is to consider
station 1 as the reference station for all classes.
OUTPUTS
V(c,i)
number of visits of class c requests at center i.
ch(c)
chain number that class c belongs
to. Different classes can belong to the same chain. Chains are
numbered sequentially starting from 1 (1, 2, …). The
total number of chains is max(ch)
.
Compute the visit ratios to the service centers of an open multiclass network with K service centers and C customer classes.
INPUTS
P(r,i,s,j)
probability that a class r request which completed service at center i is routed to center j as a class s request. Class switching is supported.
lambda(r,i)
external arrival rate of class r requests to center i.
OUTPUTS
V(r,i)
visit ratio of class r requests at center i.
Exact analysis of open, multiple-class BCMP networks. The network can be made of single-server queueing centers (FCFS, LCFS-PR or PS) or delay centers (IS). This function assumes a network with K service centers and C customer classes.
INPUTS
lambda(c)
If this function is invoked as qnom(lambda, S, V, …)
,
then lambda(c)
is the external arrival rate of class
c customers (lambda(c) ≥ 0
). If this
function is invoked as qnom(lambda, S, P, …)
, then
lambda(c,k)
is the external arrival rate of class
c customers at center k (lambda(c,k)
≥ 0
).
S(c,k)
mean service time of class c customers on the service center
k (S(c,k)>0
). For FCFS nodes, mean service
times must be class-independent.
V(c,k)
visit ratio of class c customers to service center k
(V(c,k) ≥ 0
). If you pass this argument,
class switching is not allowed
P(r,i,s,j)
probability that a class r job completing service at center
i is routed to center j as a class s job.
If you pass argument P, class switching is allowed;
however, all servers must be fixed-rate or infinite-server nodes
(m(k) ≤ 1
for all k).
m(k)
number of servers at center k. If m(k) < 1
,
enter k is a delay center (IS); otherwise it is a regular
queueing center with m(k)
servers. Default is
m(k) = 1
for all k.
OUTPUTS
U(c,k)
If k is a queueing center, then U(c,k)
is the
class c utilization of center k. If k is an IS
node, then U(c,k)
is the class c traffic
intensity defined as X(c,k)*S(c,k)
.
R(c,k)
class c response time at center k. The system
response time for class c requests can be computed as
dot(R, V, 2)
.
Q(c,k)
average number of class c requests at center k. The
average number of class c requests in the system Qc
can be computed as Qc = sum(Q, 2)
X(c,k)
class c throughput at center k.
NOTES
If the function call specifies the visit ratios V, class switching is not allowed. If the function call specifies the routing probability matrix P, then class switching is allowed; however, all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported. Note that the meaning of parameter lambda is different from one case to the other (see below).
REFERENCES
See also: qnopen,qnos,qnomvisits.
Return the set of population mixes for a closed multiclass queueing
network with exactly k customers. Specifically, given a
closed multiclass QN with C customer classes, where there
are N(c)
class c requests, c = 1, …, C
a k-mix M is a vector of length C with the following
properties:
In other words, a k-mix is an allocation of k
requests to C classes such that the number of requests
assigned to class c does not exceed the maximum value
N(c)
.
pop_mix is a matrix with C columns, such that each row represents a valid mix.
INPUTS
k
Size of the requested mix (scalar, k ≥ 0
).
N(c)
number of class c requests (k ≤ sum(N)
).
OUTPUTS
pop_mix(i,c)
number of class c requests in the i-th population
mix. The number of mixes is rows(pop_mix)
.
If you are interested in the number of k-mixes only, you can
use the funcion qnmvapop
.
REFERENCES
The slightly different problem of enumerating all tuples k_1, …, k_N such that \sum_i k_i = k and k_i ≥ 0, for a given 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
See also: qncmnpop.
EXAMPLE
Let us consider a multiclass network with C=2 customer classes; the maximum number of class 1 requests is 2, and the maximum number of class 2 requests is 3. How is it possible to allocate 3 requests to the two classes so that the maximum number of requests per class is not exceeded?
N = [2 3]; mix = qncmpopmix(3, N)
-| mix = [ [2 1] [1 2] [0 3] ]
Given a network with C customer classes, this function
computes the number of k-mixes H(r,k)
that can
be constructed by the multiclass MVA algorithm by allocating
k customers to the first r classes.
See doc-qncmpopmix for the definition of k-mix.
INPUTS
N(c)
number of class-c requests in the system. The total number
of requests in the network is sum(N)
.
OUTPUTS
H(r,k)
is the number of k mixes that can be constructed allocating k customers to the first r classes.
REFERENCES
See also: qncmmva,qncmpopmix.
Compute steady-state performance measures for closed, multiclass queueing networks using the Mean Value Analysys (MVA) algorithm.
Queueing policies at service centers can be any of the following:
(First-Come-First-Served) customers are served in order of arrival; multiple servers are allowed. For this kind of queueing discipline, average service times must be class-independent.
(Processor Sharing) customers are served in parallel by a single server, each customer receiving an equal share of the service rate.
(Last-Come-First-Served, Preemptive Resume) customers are served in reverse order of arrival by a single server and the last arrival preempts the customer in service who will later resume service at the point of interruption.
(Infinite Server) customers are delayed independently of other customers at the service center (there is effectively an infinite number of servers).
INPUTS
N(c)
number of class c requests; N(c) ≥ 0
. If
class c has no requests (N(c) == 0
), then for
all k, this function returns
U(c,k) = R(c,k) = Q(c,k) = X(c,k) = 0
S(c,k)
mean service time for class c requests at center k
(S(c,k) ≥ 0
). If the service time at center
k is class-dependent, then center k is assumed
to be of type -/G/1–PS (Processor Sharing). If center
k is a FCFS node (m(k)>1
), then the service
times must be class-independent, i.e., all classes
must have the same service time.
V(c,k)
average number of visits of class c requests at
center k; V(c,k) ≥ 0
, default is 1.
If you pass this argument, class switching is not allowed
P(r,i,s,j)
probability that a class r request completing service at center
i is routed to center j as a class s request; the
reference stations for each class are specified with the paramter
r. If you pass argument P, class switching is
allowed; however, you can not specify any external delay (i.e.,
Z must be zero) and all servers must be fixed-rate or
infinite-server nodes (m(k) ≤ 1
for all
k).
r(c)
reference station for class c. If omitted, station 1 is the
reference station for all classes. See qncmvisits
.
m(k)
If m(k)<1
, then center k is assumed to be a delay
center (IS node -/G/\infty). If m(k)==1
, then
service center k is a regular queueing center
(M/M/1–FCFS, -/G/1–LCFS-PR or -/G/1–PS).
Finally, if m(k)>1
, center k is a
M/M/m–FCFS center with m(k)
identical servers.
Default is m(k)=1
for each k.
Z(c)
class c external delay (think time); Z(c) ≥
0
. Default is 0. This parameter can not be used if you pass a
routing matrix as the second parameter of qncmmva
.
OUTPUTS
U(c,k)
If k is a FCFS, LCFS-PR or PS node (m(k) ≥
1
), then U(c,k)
is the class c utilization at
center k, 0 ≤ U(c,k) ≤ 1. If k is an
IS node, then U(c,k)
is the class c traffic
intensity at center k, defined as U(c,k) =
X(c,k)*S(c,k)
. In this case the value of
U(c,k)
may be greater than one.
R(c,k)
class c response time at center k. The class c
residence time at center k is R(c,k) *
C(c,k)
. The total class c system response time is
dot(R, V, 2)
.
Q(c,k)
average number of class c requests at center k. The
total number of requests at center k is
sum(Q(:,k))
. The total number of class c
requests in the system is sum(Q(c,:))
.
X(c,k)
class c throughput at center k. The class c
throughput can be computed as X(c,1) / V(c,1)
.
NOTES
If the function call specifies the visit ratios V, then class switching is not allowed. If the function call specifies the routing probability matrix P, then class switching is allowed; however, in this case all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported.
In presence of load-dependent servers (e.g., if m(i)>1
for some i), the MVA algorithm is known to be numerically
unstable. Generally this problem shows up as negative values for the
computed response times or utilizations. This is not a problem with the
queueing
package, but with the MVA algorithm;
as such, there is no known workaround at the moment (aoart from using a
different solution technique, if available). This function prints a
warning if it detects numerical problems; you can disable the warning
with the command warning("off", "qn:numerical-instability")
.
Given a network with K service centers, C job classes
and population vector {\bf N}=\left[N_1, …, N_C\right], 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 qncmmvaap
function, which implements
the approximate MVA algorithm. Note however that qncmmvaap
will only provide approximate results.
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 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").
See also: qnclosed, qncmmvaapprox, qncmvisits.
Approximate Mean Value Analysis (MVA) for closed, multiclass queueing networks with K service centers and C customer classes.
This implementation uses Bard and Schweitzer approximation. It is based on the assumption that the queue length at service center k with population set {\bf N}-{\bf 1}_c is approximated with
Q_k(N-1c) ~ (n-1)/n Q_k(N)
where \bf N is a valid population mix, {\bf N}-{\bf 1}_c is the population mix \bf N with one class c customer removed, and n = \sum_c N_c is the total number of requests.
This implementation works for networks with infinite server (IS) and single-server nodes only.
INPUTS
N(c)
number of class c requests in the system (N(c) ≥ 0
).
S(c,k)
mean service time for class c customers at center k
(S(c,k) ≥ 0
).
V(c,k)
average number of visits of class c requests to center
k (V(c,k) ≥ 0
).
m(k)
number of servers at center k. If m(k) < 1
,
then the service center k is assumed to be a delay center
(IS). If m(k) == 1
, service center k is a
regular queueing center (FCFS, LCFS-PR or PS) with a single server
node. If omitted, each service center has a single server. Note
that multiple server nodes are not supported.
Z(c)
class c external delay (Z ≥ 0
). Default is 0.
tol
Stopping tolerance (tol>0
). The algorithm stops if
the queue length computed on two subsequent iterations are less than
tol. Default is 10^{-5}.
iter_max
Maximum number of iterations (iter_max>0
.
The function aborts if convergenge is not reached within the maximum
number of iterations. Default is 100.
OUTPUTS
U(c,k)
If k is a FCFS, LCFS-PR or PS node, then U(c,k)
is the utilization of class c requests on service center
k. If k is an IS node, then U(c,k)
is the
class c traffic intensity at device k,
defined as U(c,k) = X(c)*S(c,k)
R(c,k)
response time of class c requests at service center k.
Q(c,k)
average number of class c requests at service center k.
X(c,k)
class c throughput at service center k.
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 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.
See also: qncmmva.
Mean Value Analysis for mixed queueing networks. The network consists of K service centers (single-server or delay centers) and C independent customer chains. Both open and closed chains are possible. lambda is the vector of per-chain arrival rates (open classes); N is the vector of populations for closed chains.
Class switching is not allowed. Each customer class must correspond to an independent chain.
If the network is made of open or closed classes only, then this
function calls qnom
or qncmmva
respectively, and
prints a warning message.
INPUTS
lambda(c)
N(c)
For each customer chain c:
N(c)>0
is the
number of class c requests and lambda(c)
must be
zero;
lambda(c)>0
is the arrival rate of class c
requests and N(c)
must be zero;
In other words, for each class c the following must hold:
(lambda(c)>0 && N(c)==0) || (lambda(c)==0 && N(c)>0)
S(c,k)
mean class c service time at center k,
S(c,k) ≥ 0
. For FCFS nodes, service times must be
class-independent.
V(c,k)
average number of visits of class c customers to center
k (V(c,k) ≥ 0
).
m(k)
number of servers at center k. Only single-server
(m(k)==1
) or IS (Infinite Server) nodes
(m(k)<1
) are supported. If omitted, each center is
assumed to be of type M/M/1-FCFS. Queueing discipline for
single-server nodes can be FCFS, PS or LCFS-PR.
OUTPUTS
U(c,k)
class c utilization at center k.
R(c,k)
class c response time at center k.
Q(c,k)
average number of class c requests at center k.
X(c,k)
class c throughput at center k.
REFERENCES
See also: qncmmva, qncm.
Next: Bounds Analysis, Previous: Multiple Class Models, Up: Queueing Networks [Contents][Index]
The queueing
package provides a high-level function
qnsolve
for analyzing QN models. qnsolve
takes as
input a high-level description of the queueing model, and delegates
the actual solution of the model to one of the lower-level
function. qnsolve
supports single or multiclass models, but at
the moment only product-form networks can be analyzed. For non
product-form networks See Non Product-Form QNs.
qnsolve
accepts two input parameters. The first one is the list
of nodes, encoded as an Octave cell array. The second parameter
is the vector of visit ratios V, which can be either a vector
(for single-class models) or a two-dimensional matrix (for
multiple-class models).
Individual nodes in the network are structures build using the
qnmknode
function.
Creates a node; this function can be used together with
qnsolve
. It is possible to create either single-class nodes
(where there is only one customer class), or multiple-class nodes
(where the service time is given per-class). Furthermore, it is
possible to specify load-dependent service times. String literals
are case-insensitive, so for example "-/g/inf", "-/G/inf"
and "-/g/INF" are all equivalent.
INPUTS
S
Mean service time.
S(c)
is assumed to
the the load-independent service time for class c customers.
S(n)
is assumed to be
the class-independent service time at the node, when there are n
requests.
S(c,n)
is assumed to be the class c service time
when there are n requests at the node.
m
Number of identical servers at the node. Default is m=1
.
s2
Squared coefficient of variation for the service time. Default is 1.0.
The returned struct Q should be considered opaque to the client.
See also: qnsolve.
After the network has been defined, it is possible to solve it using
qnsolve
.
High-level function for analyzing QN models.
INPUTS
N
N(c)
Number of requests in the system for closed networks. For
single-class networks, N must be a scalar. For multiclass
networks, N(c)
is the population size of closed class
c.
lambda
lambda(c)
External arrival rate (scalar) for open networks. For single-class
networks, lambda must be a scalar. For multiclass networks,
lambda(c)
is the class c overall arrival rate.
QQ{i}
List of queues in the network. This must be a cell array
with N elements, such that QQ{i}
is
a struct produced by the qnmknode
function.
Z
External delay ("think time") for closed networks. Default 0.
OUTPUTS
U(k)
If k is a FCFS node, then U(k)
is the utilization
of service center k. If k is an IS node, then
U(k)
is the traffic intensity defined as
X(k)*S(k)
.
R(k)
average response time of service center k.
Q(k)
average number of customers in service center k.
X(k)
throughput of service center k.
Note that for multiclass networks, the computed results are per-class
utilization, response time, number of customers and throughput:
U(c,k)
, R(c,k)
, Q(c,k)
,
X(c,k)
.
String literals are case-insensitive, so "closed", "Closed" and "CLoSEd" are all equivalent.
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:
[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;
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 );
This function computes steady-state performance measures of closed queueing networks using the Mean Value Analysis (MVA) algorithm. The qneneing network is allowed to contain fixed-capacity centers, delay centers or general load-dependent centers. Multiple request classes are supported.
This function dispatches the computation to one of
qncsemva
, qncsmvald
or qncmmva
.
S(k)
is the average service time of center k, and this function
calls qncsmva
which supports load-independent
service centers. If S is a matrix, S(k,i)
is the
average service time at center k when i=1, …, N
jobs are present; in this case, the network is analyzed with the
qncmmvald
function.
qncmmva
function.
See also: qncsmva, qncsmvald, qncmmva.
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(size(S)); # All centers are single-server Z = 2; # External delay N = 15; # Maximum population to consider V = qncsvisits(P); # Compute number of visits X_bsb_lower = X_bsb_upper = X_ab_lower = X_ab_upper = X_mva = zeros(1,N); for n=1:N [X_bsb_lower(n) X_bsb_upper(n)] = qncsbsb(n, S, V, m, Z); [X_ab_lower(n) X_ab_upper(n)] = qncsaba(n, S, V, m, 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]); legend("location","southeast"); legend("boxoff"); xlabel("Number of Requests n"); ylabel("System Throughput X(n)");
Compute utilization, response time, average number of requests in the
system, and throughput for open queueing networks. If lambda is
a scalar, the network is considered a single-class QN and is solved
using qnopensingle
. If lambda is a vector, the network
is considered as a multiclass QN and solved using qnopenmulti
.
See also: qnos, qnom.
Next: QN Analysis Examples, Previous: Generic Algorithms, Up: Queueing Networks [Contents][Index]
Compute Asymptotic Bounds for open, single-class networks with K service centers.
INPUTS
lambda
Arrival rate of requests (scalar, lambda ≥ 0
).
D(k)
service demand at center k.
(vector of length K, D(k) ≥ 0
).
S(k)
mean service time at center k.
(vector of length K, S(k) ≥ 0
).
V(k)
mean number of visits to center k.
(vector of length K, V(k) ≥ 0
).
m(k)
number of servers at center k.
This function only supports M/M/1 queues, therefore
m must be ones(size(S))
.
OUTPUTS
Xl
Xu
Lower and upper bounds on the system throughput. Xl is always set to 0 since there can be no lower bound on the throughput of open networks (scalar).
Rl
Ru
Lower and upper bounds on the system response time. Ru
is always set to +inf
since there can be no upper bound on the
throughput of open networks (scalar).
See also: qnomaba.
Compute Asymptotic Bounds for open, multiclass networks with K service centers and C customer classes.
INPUTS
lambda(c)
class c arrival rate to the system (vector of length
C, lambda(c) > 0
).
D(c, k)
class c service demand at center k (C \times K
matrix, D(c, k) ≥ 0
).
S(c, k)
mean service time of class c requests at center k
(C \times K matrix, S(c, k) ≥ 0
).
V(c, k)
mean number of visits of class c requests at center k
(C \times K matrix, V(c, k) ≥ 0
).
OUTPUTS
Xl(c)
Xu(c)
lower and upper bounds of class c throughput.
Xl(c)
is always 0 since there can be no lower
bound on the throughput of open networks (vector of length
C).
Rl(c)
Ru(c)
lower and upper bounds of class c response time.
Ru(c)
is always +inf
since there can be no
upper bound on the response time of open networks (vector of length
C).
Compute Asymptotic Bounds for the system throughput and response time of closed, single-class networks with K service centers.
Single-server and infinite-server nodes are supported. Multiple-server nodes and general load-dependent servers are not supported.
INPUTS
N
number of requests in the system (scalar, N>0
).
D(k)
service demand at center k
(D(k) ≥ 0
).
S(k)
mean service time at center k
(S(k) ≥ 0
).
V(k)
average number of visits to center
k (V(k) ≥ 0
).
m(k)
number of servers at center k
(if m is a scalar, all centers have that number of servers). If
m(k) < 1
, center k is a delay center (IS);
if m(k) = 1
, center k is a M/M/1-FCFS server.
This function does not support multiple-server nodes. Default
is 1.
Z
External delay (scalar, Z ≥ 0
). Default is 0.
OUTPUTS
Xl
Xu
Lower and upper bounds on the system throughput.
Rl
Ru
Lower and upper bounds on the system response time.
See also: qncmaba.
Compute Asymptotic Bounds for closed, multiclass networks with K service centers and C customer classes. Single-server and infinite-server nodes are supported. Multiple-server nodes and general load-dependent servers are not supported.
INPUTS
N(c)
number of class c requests in the system
(vector of length C, N(c) ≥ 0
).
D(c, k)
class c service demand
at center k (C \times K matrix, D(c,k) ≥ 0
).
S(c, k)
mean service time of class c
requests at center k (C \times K matrix, S(c,k) ≥ 0
).
V(c,k)
average number of visits of class c
requests to center k (C \times K matrix, V(c,k) ≥ 0
).
m(k)
number of servers at center k
(if m is a scalar, all centers have that number of servers). If
m(k) < 1
, center k is a delay center (IS);
if m(k) = 1
, center k is a M/M/1-FCFS server.
This function does not support multiple-server nodes. Default
is 1.
Z(c)
class c external delay
(vector of length C, Z(c) ≥ 0
). Default is 0.
OUTPUTS
Xl(c)
Xu(c)
Lower and upper bounds for class c throughput.
Rl(c)
Ru(c)
Lower and upper bounds for class c response time.
REFERENCES
See also: qncsaba.
Compute Balanced System Bounds for single-class, open networks with K service centers.
INPUTS
lambda
overall arrival rate to the system (scalar, lambda ≥ 0
).
D(k)
service demand at center k (D(k) ≥ 0
).
S(k)
service time at center k (S(k) ≥ 0
).
V(k)
mean number of visits at center k (V(k) ≥ 0
).
m(k)
number of servers at center k. This function only supports
M/M/1 queues, therefore m must be
ones(size(S))
.
OUTPUTS
Xl
Xu
Lower and upper bounds on the system throughput. Xl is always set to 0, since there can be no lower bound on open networks throughput.
Rl
Ru
Lower and upper bounds on the system response time.
See also: qnosaba.
Compute Balanced System Bounds on system throughput and response time for closed, single-class networks with K service centers.
INPUTS
N
number of requests in the system (scalar, N ≥ 0
).
D(k)
service demand at center k (D(k) ≥ 0
).
S(k)
mean service time at center k (S(k) ≥ 0
).
V(k)
average number of visits to center k (V(k)
≥ 0
). Default is 1.
m(k)
number of servers at center k. This function supports
m(k) = 1
only (single-eserver FCFS nodes); this
parameter is only for compatibility with qncsaba
. Default is
1.
Z
External delay (Z ≥ 0
). Default is 0.
OUTPUTS
Xl
Xu
Lower and upper bound on the system throughput.
Rl
Ru
Lower and upper bound on the system response time.
REFERENCES
See also: qncmbsb.
Compute Balanced System Bounds for closed, multiclass networks with K service centers and C customer classes. Only single-server nodes are supported.
INPUTS
N(c)
number of class c requests in the system (vector of length C).
D(c, k)
class c service demand at center k (C \times K
matrix, D(c,k) ≥ 0
).
S(c, k)
mean service time of class c
requests at center k (C \times K matrix, S(c,k) ≥ 0
).
V(c,k)
average number of visits of class c
requests to center k (C \times K matrix, V(c,k) ≥ 0
).
OUTPUTS
Xl(c)
Xu(c)
Lower and upper class c throughput bounds (vector of length C).
Rl(c)
Ru(c)
Lower and upper class c response time bounds (vector of length C).
See also: qncsbsb.
Compute Composite Bounds (CB) on system throughput and response time for closed multiclass networks.
INPUTS
N(c)
number of class c requests in the system.
D(c, k)
class c service demand
at center k (S(c,k) ≥ 0
).
S(c, k)
mean service time of class c
requests at center k (S(c,k) ≥ 0
).
V(c,k)
average number of visits of class c
requests to center k (V(c,k) ≥ 0
).
OUTPUTS
Xl(c)
Xu(c)
Lower and upper bounds on class c throughput.
Rl(c)
Ru(c)
Lower and upper bounds on class c response time.
REFERENCES
Compute PB Bounds (C. H. Hsieh and S. Lam, 1987) for single-class, closed networks with K service centers.
INPUTS
number of requests in the system (scalar, N > 0
).
D(k)
service demand of service center k (D(k) ≥ 0
).
S(k)
mean service time at center k (S(k) ≥ 0
).
V(k)
visit ratio to center k (V(k) ≥ 0
).
m(k)
number of servers at center k. This function only supports
M/M/1 queues, therefore m must be
ones(size(S))
.
Z
external delay (think time, Z ≥ 0
). Default 0.
OUTPUTS
Xl
Xu
Lower and upper bounds on the system throughput.
Rl
Ru
Lower and upper bounds on the system response time.
REFERENCES
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.
See also: qncsaba, qbcsbsb, qncsgb.
Compute Geometric Bounds (GB) on system throughput, system response time and server queue lenghts for closed, single-class networks with K service centers and N requests.
INPUTS
N
number of requests in the system (scalar, N > 0
).
D(k)
service demand of service center k (vector of length
K, D(k) ≥ 0
).
S(k)
mean service time at center k (vector of length K,
S(k) ≥ 0
).
V(k)
visit ratio to center k
(vector of length K, V(k) ≥ 0
).
m(k)
number of servers at center k. This function only supports
M/M/1 queues, therefore m must be
ones(size(S))
.
Z
external delay (think time, Z ≥ 0
, scalar). Default is 0.
OUTPUTS
Xl
Xu
Lower and upper bound on the system throughput. If Z>0
,
these bounds are computed using Geometric Square-root Bounds
(GSB). If Z==0
, these bounds are computed using Geometric Bounds (GB)
Rl
Ru
Lower and upper bound on the system response time. These bounds
are derived from Xl and Xu using Little’s Law:
Rl = N / Xu - Z
,
Ru = N / Xl - Z
Ql(k)
Qu(k)
lower and upper bounds of center K queue length.
REFERENCES
In this implementation we set X^+ and X^- as the upper
and lower Asymptotic Bounds as computed by the qncsab
function, respectively.
Previous: Bounds Analysis, Up: Queueing Networks [Contents][Index]
In this section we illustrate with a few examples how the
queueing
package can be used to analyze queueing network
models. Further examples can be found in the functions demo blocks,
and can be inspected with the demo function
Octave
command.
Let us consider again the network shown in Figure 5.1. We denote with S_k the average service time at center k, k=1, 2, 3. Let the service times be S_1 = 1.0, S_2 = 2.0 and S_3 = 0.8. The routing of jobs within the network is described with a routing probability matrix \bf P: a request completing service at center i is enqueued at center j with probability P_{i, j}. We use the following routing matrix:
/ 0 0.3 0.7 \ P = | 1 0 0 | \ 1 0 0 /
The network above can be analyzed with the qnclosed
function
see doc-qnclosed. qnclosed
requires the following
parameters:
Number of requests in the network (since we are considering a closed network, the number of requests is fixed)
Array of average service times at the centers: S(k)
is
the average service time at center k.
Array of visit ratios: V(k)
is the average number of
visits to center k.
We can compute V_k from the routing probability matrix
P_{i, j} using the qncsvisits
function
see doc-qncsvisits. Therefore, we can analyze the network for a
given population size N (e.g., N=10) as follows:
N = 10; S = [1 2 0.8]; P = [0 0.3 0.7; 1 0 0; 1 0 0]; V = qncsvisits(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 vectors 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 highest
among the service centers. Thus, center 1 is the bottleneck
device.
This network can also be analyzed with the qnsolve
function
see doc-qnsolve. qnsolve
can handle open, closed or
mixed networks, and allows the network to be described in a very
flexible way. First, let Q1, Q2 and Q3 be the
variables describing the service centers. Each variable is
instantiated with the qnmknode
function.
Q1 = qnmknode( "m/m/m-fcfs", 1 ); Q2 = qnmknode( "m/m/m-fcfs", 2 ); Q3 = qnmknode( "m/m/m-fcfs", 0.8 );
The first parameter of qnmknode
is a string describing the
type of the node; "m/m/m-fcfs"
denotes a M/M/m–FCFS
center (this parameter is case-insensitive). 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
Let us consider an open network with K=3 service centers and the following routing probabilities:
/ 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 compute the visit counts
V_k to center k, k = 1, 2, 3. We use the
qnosvisits
function as follows:
P = [0 0.3 0.5; 1 0 0; 1 0 0]; lambda = [0.15 0 0]; V = qnosvisits(P, lambda) ⇒ V = 5.00000 1.50000 2.50000
where lambda(k)
is the arrival rate at center k,
and \bf P is the routing matrix. Assuming the same service times as
in the previous example, the network can be analyzed with the
qnopen
function see doc-qnopen, as follows:
S = [1 2 0.8]; [U R Q X] = qnopen( sum(lambda), S, V ) ⇒ U = 0.75000 0.45000 0.30000 ⇒ R = 4.0000 3.6364 1.1429 ⇒ Q = 3.00000 0.81818 0.42857 ⇒ X = 0.75000 0.22500 0.37500
The first parameter of the qnopen
function is the (scalar)
aggregate arrival rate.
Again, it is possible to use the qnsolve
high-level function:
Q1 = qnmknode( "m/m/m-fcfs", 1 ); Q2 = qnmknode( "m/m/m-fcfs", 2 ); Q3 = qnmknode( "m/m/m-fcfs", 0.8 ); lambda = [0.15 0 0]; [U R Q X] = qnsolve( "open", sum(lambda), { Q1, Q2, Q3 }, V ) ⇒ U = 0.75000 0.45000 0.30000 ⇒ R = 4.0000 3.6364 1.1429 ⇒ Q = 3.00000 0.81818 0.42857 ⇒ X = 0.75000 0.22500 0.37500
The following example is taken from 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.
Let us consider the following multiclass QN with three servers and two classes
Servers 1 and 2 (labeled APL and IMS, respectively) are infinite server nodes; server 3 (labeled SYS) is Processor Sharing (PS). Mean service times are given in the following table:
APL | IMS | SYS | |
---|---|---|---|
Class 1 | 1 | - | 0.025 |
Class 2 | - | 15 | 0.500 |
There is no class switching. If we assume a population of 15 requests for class 1, and 5 requests for class 2, then the model can be analyzed as follows:
S = [1 0 .025; 0 15 .5]; P = zeros(2,3,2,3); P(1,1,1,3) = P(1,3,1,1) = 1; P(2,2,2,3) = P(2,3,2,2) = 1; V = qncmvisits(P,[3 3]); # reference station is station 3 N = [15 5]; m = [-1 -1 1]; [U R Q X] = qncmmva(N,S,V,m)
⇒ U = 14.32312 0.00000 0.35808 0.00000 4.70699 0.15690 R = 1.00000 0.00000 0.04726 0.00000 15.00000 0.93374 Q = 14.32312 0.00000 0.67688 0.00000 4.70699 0.29301 X = 14.32312 0.00000 14.32312 0.00000 0.31380 0.31380
The following example is from M. Marzolla, The qnetworks Toolbox: A Software Package for Queueing Networks Analysis, Technical Report UBLCS-2010-04, Department of Computer Science, University of Bologna, Italy, February 2010.
The model shown in Figure 5.4 shows a three-tier enterprise system with K=6 service centers. The first tier contains the Web server (node 1), which is responsible for generating Web pages and transmitting them to clients. The application logic is implemented by nodes 2 and 3, and the storage tier is made of nodes 4–6.The system is subject to two workload classes, both represented as closed populations of N_1 and N_2 requests, respectively. Let D_{c, k} denote the service demand of class c requests at center k. We use the parameter values:
Serv. no. | Name | Class 1 | Class 2 |
---|---|---|---|
1 | Web Server | 12 | 2 |
2 | App. Server 1 | 14 | 20 |
3 | App. Server 2 | 23 | 14 |
4 | DB Server 1 | 20 | 90 |
5 | DB Server 2 | 80 | 30 |
6 | DB Server 3 | 31 | 33 |
We set the total number of requests to 100, that is N_1 + N_2 = N = 100, and we study how different population mixes (N_1, N_2) affect the system throughput and response time. Let 0 < \beta_1 < 1 denote the fraction of class 1 requests: N_1 = \beta_1 N, N_2 = (1-\beta_1)N. The following Octave code defines the model for \beta_1 = 0.1:
N = 100; # total population size beta1 = 0.1; # fraction of class 1 reqs. S = [12 14 23 20 80 31; \ 2 20 14 90 30 33 ]; V = ones(size(S)); pop = [fix(beta1*N) N-fix(beta1*N)]; [U R Q X] = qncmmva(pop, S, V);
The qncmmva(pop, S, V)
function invocation uses the
multiclass MVA algorithm to compute per-class utilizations U_{c,
k}, response times R_{c,k}, mean queue lengths Q_{c,k}
and throughputs X_{c,k} at each service center k, given
a population vector pop, mean service times S and visit
ratios V. Since we are given the service demands D_{c, k}
= S_{c, k} V_{c,k}, but function qncmmva
requires separate
service times and visit ratios, we set the service times equal to the
demands, and all visit ratios equal to one. Overall class and system
throughputs and response times can also be computed:
X1 = X(1,1) / V(1,1) # class 1 throughput ⇒ X1 = 0.0044219 X2 = X(2,1) / V(2,1) # class 2 throughput ⇒ X2 = 0.010128 XX = X1 + X2 # system throughput ⇒ XX = 0.014550 R1 = dot(R(1,:), V(1,:)) # class 1 resp. time ⇒ R1 = 2261.5 R2 = dot(R(2,:), V(2,:)) # class 2 resp. time ⇒ R2 = 8885.9 RR = N / XX # system resp. time ⇒ RR = 6872.7
dot(X,Y)
computes the dot product of two vectors.
R(1,:)
is the first row of matrix R and V(1,:)
is
the first row of matrix V, so dot(R(1,:), V(1,:))
computes \sum_k R_{1,k} V_{1,k}.
We can also compute the system power \Phi = X / R, which defines how efficiently resources are being used: high values of \Phi denote the desirable situation of high throughput and low response time. Figure 5.6 shows \Phi as a function of \beta_1. We observe a “plateau” of the global system power, corresponding to values of \beta_1 which approximately lie between 0.3 and 0.7. The per-class power exhibits an interesting (although not completely surprising) pattern, where the class with higher population exhibits worst efficiency as it produces higher contention on the resources.
We now consider an example of multiclass network with class switching. The example is taken from Sch82, and is shown in Figure Figure 5.7.
The system consists of three devices and two job classes. The CPU node is a PS server, while the two nodes labeled I/O are FCFS. Class 1 mean service time at the CPU is 0.01; class 2 mean service time at the CPU is 0.05. The mean service time at node 2 is 0.1, and is class-independent. Similarly, the mean service time at node 3 is 0.07. Jobs in class 1 leave the CPU and join class 2 with probability 0.1; jobs of class 2 leave the CPU and join class 1 with probability 0.2. There are N=3 jobs, which are initially allocated to class 1. However, note that since class switching is allowed, the total number of jobs in each class does not remain constant; however the total number of jobs does.
C = 2; K = 3; S = [.01 .07 .10; ... .05 .07 .10 ]; P = zeros(C,K,C,K); P(1,1,1,2) = .7; P(1,1,1,3) = .2; P(1,1,2,1) = .1; P(2,1,2,2) = .3; P(2,1,2,3) = .5; P(2,1,1,1) = .2; P(1,2,1,1) = P(2,2,2,1) = 1; P(1,3,1,1) = P(2,3,2,1) = 1; N = [3 0]; [U R Q X] = qncmmva(N, S, P)
⇒ U = 0.12609 0.61784 0.25218 0.31522 0.13239 0.31522 R = 0.014653 0.133148 0.163256 0.073266 0.133148 0.163256 Q = 0.18476 1.17519 0.41170 0.46190 0.25183 0.51462 X = 12.6089 8.8262 2.5218 6.3044 1.8913 3.1522
Next: Copying, Previous: Queueing Networks, Up: Top [Contents][Index]
Ian F. Akyildiz, Mean Value Analysis for Blocking Queueing Networks, IEEE Transactions on Software Engineering, vol. 14, n. 2, april 1988, pp. 418–428. DOI 10.1109/32.4663
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.
F. Baskett, K. Mani Chandy, R. R. Muntz, and F. G. Palacios. 1975. Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. J. ACM 22, 2 (April 1975), 248Ã¢Â€Â”260, DOI 10.1145/321879.321887
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.
J. P. Buzen, Computational Algorithms for Closed Queueing Networks with Exponential Servers, Communications of the ACM, volume 16, number 9, september 1973, pp. 527–531. DOI 10.1145/362342.362345
G. Casale, A note on stable flow-equivalent aggregation in closed networks. Queueing Syst. Theory Appl., 60:193Ã¢Â€Â“-202, December 2008, DOI 10.1007/s11134-008-9093-6
G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008. DOI 10.1109/TC.2008.37
C. M. Grinstead, J. L. Snell, (July 1997). Introduction to Probability. American Mathematical Society. ISBN 978-0821807491; this excellent textbook is available in PDF format and can be used under the terms of the GNU Free Documentation License (FDL)
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C. H. Hsieh and S. Lam, Two classes of performance bounds for closed queueing networks, PEVA, vol. 7, n. 1, pp. 3–30, 1987
T. Kerola, The Composite Bound Method (CBM) for Computing Throughput Bounds in Multiple Class Environments, Performance Evaluation, Vol. 6 Isue 1, March 1986, DOI 10.1016/0166-5316(86)90002-7; also available as Technical Report CSD-TR-475, Department of Computer Sciences, Purdue University, mar 13, 1984 (Revised aug 27, 1984).
E. D. Lazowska, J. Zahorjan, G. Scott Graham, and K. C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. available online.
M. Reiser, H. Kobayashi, On The Convolution Algorithm for Separable Queueing Networks, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29–31, 1976). SIGMETRICS ’76. ACM, New York, NY, pp. 109–117. DOI 10.1145/800200.806187
M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. DOI 10.1145/322186.322195
P. Schweitzer, Approximate Analysis of Multiclass Closed Networks of Queues, Proc. Int. Conf. on Stochastic Control and Optimization, jun 1979, pp. 25Ã¢Â€Â”29
H. D. Schwetman, Testing Network-of-Queues Software, Technical Report CSD-TR 330, Department of computer Sciences, Purdue University, 1980
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