Binomial-Poisson entropic inequalities and the M/M/∞ queue
ESAIM: Probability and Statistics, Tome 10 (2006), pp. 317-339

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”.

DOI : 10.1051/ps:2006013
Classification : 26D15, 46E99, 47D07, 60J27, 60J60, 60J75, 94A17
Keywords: functional inequalities, Markov processes, entropy, birth and death processes, queues 
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Chafaï, Djalil. Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 317-339. doi: 10.1051/ps:2006013

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