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

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 $\mathrm{M}\mathrm{/}\mathrm{M}\mathrm{/}\mathrm{\infty }$ queuing processes on the set of non-negative integers. Our approach is global in nature and yields the sharp constant.