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Binomial-Poisson entropic inequalities and the M/M/∞ queue

Published online by Cambridge University Press:  08 September 2006

Djalil Chafaï
Djalil Chafaï*
Affiliation:
UMR 181 INRA/ENVT Physiopathologie et Toxicologie Experimentales, École Nationale Vétérinaire de Toulouse, 23 Chemin des Capelles, 31076, Toulouse Cedex 3, France, and UMR 5583 CNRS/UPS Laboratoire de Statistique et Probabilités, Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, Cedex 4, France. chafai@math.ups-tlse.fr.nospam
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Abstract

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues. Proofs are elementary and rely essentially on the development of a “Φ-calculus”.

Keywords

Functional inequalitiesMarkov processesentropybirth and death processesqueues.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 317 - 339
Copyright
© EDP Sciences, SMAI, 2006

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