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

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: 10 November 2005

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”.