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”.

Keywords: functional inequalities, Markov processes, entropy, birth and death processes, queues

@article{PS_2006__10__317_0, author = {Chafa{\"\i}, Djalil}, 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}, url = {http://www.numdam.org/articles/10.1051/ps:2006013/} }

TY - JOUR AU - Chafaï, Djalil TI - Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue JO - ESAIM: Probability and Statistics PY - 2006 SP - 317 EP - 339 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006013/ DO - 10.1051/ps:2006013 LA - en ID - PS_2006__10__317_0 ER -

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. http://www.numdam.org/articles/10.1051/ps:2006013/

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