@c -*- texinfo -*-

@c Copyright (C) 2008, 2009, 2010, 2011, 2012 Moreno Marzolla
@c
@c This file is part of the queueing toolbox, a Queueing Networks
@c analysis package for GNU Octave.
@c
@c The queueing toolbox is free software; you can redistribute it
@c and/or modify it under the terms of the GNU General Public License
@c as published by the Free Software Foundation; either version 3 of
@c the License, or (at your option) any later version.
@c
@c The queueing toolbox is distributed in the hope that it will be
@c useful, but WITHOUT ANY WARRANTY; without even the implied warranty
@c of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
@c GNU General Public License for more details.
@c
@c You should have received a copy of the GNU General Public License
@c along with the queueing toolbox; see the file COPYING.  If not, see
@c <http://www.gnu.org/licenses/>.

@node Single Station Queueing Systems
@chapter Single Station Queueing Systems

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

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

@item @math{M/M/1} single-server queueing station;

@item @math{M/M/m} multiple-server queueing station;

@item Asymmetric @math{M/M/m};

@item @math{M/M/\infty} infinite-server station (delay center);

@item @math{M/M/1/K} single-server, finite-capacity queueing station;

@item @math{M/M/m/K} multiple-server, finite-capacity queueing station;

@item @math{M/G/1} single-server with general service time distribution;

@item @math{M/H_m/1} single-server with hyperexponential service time distribution.

@end itemize

@end iftex

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

@c
@c M/M/1
@c
@node The M/M/1 System
@section The @math{M/M/1} System

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

@include help/qnmm1.texi

@noindent @strong{REFERENCES}

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

@auindex Bolch, G.
@auindex Greiner, S.
@auindex de Meer, H.
@auindex Trivedi, K.

@c
@c M/M/m
@c
@node The M/M/m System
@section The @math{M/M/m} System

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

@example
@code{mu(n) = min(m,n)*mu}
@end example

@include help/qnmmm.texi

@noindent @strong{REFERENCES}

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

@auindex Bolch, G.
@auindex Greiner, S.
@auindex de Meer, H.
@auindex Trivedi, K.

@c
@c M/M/inf
@c
@node The M/M/inf System
@section The @math{M/M/}inf System

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

@include help/qnmminf.texi

@noindent @strong{REFERENCES}

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

@auindex Bolch, G.
@auindex Greiner, S.
@auindex de Meer, H.
@auindex Trivedi, K.

@c
@c M/M/1/k
@c
@node The M/M/1/K System
@section The @math{M/M/1/K} System 

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

@include help/qnmm1k.texi

@c
@c M/M/m/k
@c
@node The M/M/m/K System
@section The @math{M/M/m/K} System 

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

@include help/qnmmmk.texi

@noindent @strong{REFERENCES}

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

@auindex Bolch, G.
@auindex Greiner, S.
@auindex de Meer, H.
@auindex Trivedi, K.

@c
@c Approximate M/M/m
@c
@node The Asymmetric M/M/m System
@section The Asymmetric @math{M/M/m} System 

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

@include help/qnammm.texi

@noindent @strong{REFERENCES}

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

@auindex Bolch, G.
@auindex Greiner, S.
@auindex de Meer, H.
@auindex Trivedi, K.

@c
@c
@c
@node The M/G/1 System
@section The @math{M/G/1} System 

@include help/qnmg1.texi

@c
@c
@c
@node The M/Hm/1 System
@section The @math{M/H_m/1} System
@include help/qnmh1.texi