Binomial-Poisson entropic inequalities and the M/M/ queue
ESAIM: Probability and Statistics, Volume 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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     title = {Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue},
     journal = {ESAIM: Probability and Statistics},
     pages = {317--339},
     publisher = {EDP-Sciences},
     volume = {10},
     year = {2006},
     doi = {10.1051/ps:2006013},
     mrnumber = {2247924},
     language = {en},
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Chafaï, Djalil. Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 317-339. doi : 10.1051/ps:2006013.

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