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ESAIM: P&S, September 2006, Vol. 10, pp. 317-339
DOI: 10.1051/ps:2006013

Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue

Djalil Chafaï

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


(Received November 10, 2005. Published online: 8 September 2006.)

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/$\infty$ queue. They describe in particular the exponential dissipation of $\Phi$-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/$\infty$ queues. Proofs are elementary and rely essentially on the development of a "$\Phi$-calculus".


Mathematics Subject Classification. 26D15, 46E99, 47D07, 60J27, 60J60, 60J75, 94A17

Key words: Functional inequalities, Markov processes, entropy, birth and death processes, queues.


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