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On heavy traffic approximations for transient characteristics of M/M/∞ queues

Part of: Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Fabrice M. Guillemin ,
Ravi R. Mazumdar  and
Alain D. Simonian
Fabrice M. Guillemin*
Affiliation:
France Telecom
Ravi R. Mazumdar*
Affiliation:
INRS Télécom
Alain D. Simonian*
Affiliation:
France Telecom
*
∗Postal address: France Telecom, CNET Lannion-A, Route de Trégastel, 22300 Lannion, France.
∗∗Postal address: INRS Télécom, 16 P1. du Commerce, Ile des Soeurs, Verdun, Québec, Canada H3E 1H6.
∗∗∗Postal address: France Telecom, CNET Paris-A, 38–40 Rue du Général Leclerc, 92131 Issy-les-Mlx, France.
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Abstract

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

Keywords

M/M/∞QUEUESTRANSIENT BEHAVIOURHEAVY TRAFFIC APPROXIMATIONSORNSTEIN–UHLENBECK PROCESSCONVERGENCE OF DISTRIBUTION

MSC classification

Primary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60F05: Central limit and other weak theorems 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 490 - 506
Copyright
Copyright © Applied Probability Trust 1996 

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