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A Recurrent Solution of PH/M/c/N-Like and PH/M/c-Like Queues

Part of: Computer system organization Operations research and management science Special processes

Published online by Cambridge University Press:  04 February 2016

Alexandre Brandwajn  and
Thomas Begin
Alexandre Brandwajn*
Affiliation:
University of California Santa Cruz
Thomas Begin*
Affiliation:
Université Lyon 1
*
∗ Postal address: Baskin School of Engineering, University of California Santa Cruz, 1156 High Street, Mail Stop SOE3, Santa Cruz, CA 95064, USA. Email address: alexb@soe.ucsc.edu
∗∗ Postal address: LIP UMR CNRS - ENS Lyon - UCB Lyon 1 - INRIA, Université Lyon 1, 46 allée d'Italie, 69364 Lyon Cedex 7, France. Email address: thomas.begin@ens-lyon.fr
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Abstract

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We propose an efficient semi-numerical approach to compute the steady-state probability distribution for the number of requests at arbitrary and at arrival time instants in PH/M/c-like systems with homogeneous servers in which the inter-arrival time distribution is represented by an acyclic set of memoryless phases. Our method is based on conditional probabilities and results in a simple computationally stable recurrence. It avoids the explicit manipulation of potentially large matrices and involves no iteration. Owing to the use of conditional probabilities, it delays the onset of numerical issues related to floating-point underflow as the number of servers and/or phases increases. For generalized Coxian distributions, the computational complexity of the proposed approach grows linearly with the number of phases in the distribution.

Keywords

G/M/c-like queuephase-type distributionconditional probabilityqueue length distributionrecurrent solutionnumerical stabilitycomputational efficiencyasymptotic geometric distribution

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service 68M20: Performance evaluation; queueing; scheduling
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 84 - 99
Copyright
© Applied Probability Trust 

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