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Perturbation analysis of a variable M/M/1 queue: a probabilistic approach

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Nelson Antunes ,
Christine Fricker ,
Fabrice Guillemin  and
Philippe Robert
Nelson Antunes*
Affiliation:
INRIA
Christine Fricker*
Affiliation:
INRIA
Fabrice Guillemin*
Affiliation:
France Télécom
Philippe Robert*
Affiliation:
INRIA
*
∗ Postal address: INRIA-Rocquencourt, RAP project, Domaine de Voluceau, 78153 Le Chesnay, France.
∗ Postal address: INRIA-Rocquencourt, RAP project, Domaine de Voluceau, 78153 Le Chesnay, France.
∗∗∗∗ Postal address: France Télécom R&D, CORE/CPN, 22300 Lannion, France. Email address: fabrice.guillemin@francetelecom.com
∗ Postal address: INRIA-Rocquencourt, RAP project, Domaine de Voluceau, 78153 Le Chesnay, France.
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Abstract

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In this paper, motivated by the problem of the coexistence on transmission links of telecommunications networks of elastic and unresponsive traffic, we study the impact on the busy period of an M/M/1 queue of a small perturbation in the service rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter ε ≪ 1. We specifically compute the two first terms of the power series expansion in ε of the mean value of the busy period duration. This allows us to study the validity of the reduced service rate approximation, which consists in comparing the perturbed M/M/1 queue with the M/M/1 queue whose service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of the environment process (X(t)) play a key role.

Keywords

Perturbation analysisexpansion of cycle formulaM/M/1 queue

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 263 - 283
Copyright
Copyright © Applied Probability Trust 2006 

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