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 points space. They describe in particular the exponential dissipation of $\Phi$-entropies along the M/M/$\infty$ queue. 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. These inequalities are exactly the local inequalities of the M/M/$\infty$ process. Some of them are recovered by semigroup 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. 2005. ⟨hal-00012429v1⟩