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

DOI : https://doi.org/10.1051/ps:2006013
Classification : 26D15,  46E99,  47D07,  60J27,  60J60,  60J75,  94A17
Mots clés : 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. http://www.numdam.org/articles/10.1051/ps:2006013/

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