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

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\mathrm{\infty }$ queue. They describe in particular the exponential dissipation of $\mathrm{\Phi }$-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 $\mathrm{\Phi }$-calculus’'.