Volume 10, September 2006
|Page(s)||317 - 339|
|Published online||08 September 2006|
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.
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”.
Mathematics Subject Classification: 26D15 / 46E99 / 47D07 / 60J27 / 60J60 / 60J75 / 94A17
Key words: Functional inequalities / Markov processes / entropy / birth and death processes / queues.
© EDP Sciences, SMAI, 2006
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