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

Djalil Chafai (1, 2)

1 LSP - Laboratoire de Statistique et Probabilités
2 UPTE - Physiopathologie et Toxicologie Expérimentales

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''.

Mots clés: Markov Processes; Jump Processes; Poisson processes; Diffusion Processes; Functional Inequalities; Logarithmic Sobolev Inequalities; Poincare Inequalities

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Djalil Chafai. Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue. ESAIM: Probability and Statistics, 2006, 10, pp.317-339. ⟨10.1051/ps:2006013⟩. ⟨hal-00012429v2⟩