Semi-log-convexity of ${\rm M}/{\rm M}/\infty$ queues on $\mathbb{Z}_+$

TL;DR

This paper proves a sharp semi-log-convexity property for the ${\rm M}/{\rm M}/\infty$ queue semigroup on $\mathbb{Z}_+$, improving the previous bound by removing the constant $\tfrac{1}{12}$ and establishing $\Delta_{\rm d}[\log A_t f](n) \ge \log\left(1-\frac{p^{2}}{[p+\rho(1-p)^{2}]^{2}}\right)$ for all $t>0$, $n\in\mathbb{N}$ and nonzero $f$. The proof uses a Mehler-type decomposition $X_t=Y_t+Z_t$ with $Y_t\sim\mathcal B(k,p)$ and $Z_t\sim\pi_{\rho q}$ to obtain a key inequality $G_k(n)^2 \le K G_k(n+1)G_k(n-1)$, plus a generalization to binomial–Poisson convolutions, and then applies Cauchy–Schwarz together with reversibility to deduce the global bound. The results connect to discrete Li–Yau-type inequalities and to Talagrand convolution phenomena, with the sharp constant reflecting a global, time-asymptotic optimum.

Abstract

We solve the problem left in the recent paper by N. Gozlan et al [Potential Analysis 58, 2023, 123--158], establishing the semi-log-convexity of semigroups associated with ${\rm M}/{\rm M}/\infty$ queuing processes on the set of non-negative integers. Our approach is global in nature and yields the sharp constant.

Theorems & Definitions (6)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['keylemma']}
• proof : Proof of Theorem \ref{['main']}