The minimal quasi-stationary distribution of the absorbed M/M/$\infty$ queue

TL;DR

This paper characterizes the minimal exponential survival rate of the absorbed M/M/∞ queue using two methods, one based on excursion durations and the other on complex analysis involving the incomplete gamma function.

Contribution

It provides a new proof of a known result and confirms a conjecture by Martínez and Ycart through complex analysis techniques.

Findings

Derived a known result on excursion durations.

Proved a conjecture relating to the minimal exponential survival rate.

Connected the problem to the incomplete gamma function.

Abstract

We give in this paper two characterizations of the minimal exponential rate of survival $\theta$ * of the M/M/$\infty$ queue. The first one is a derivation of a known result on the duration of excursions of this process. The second one was conjectured by Mart{\'i}nez and Ycart [8] and is proved using complex analysis by establishing a connection with the first characterization in terms of the incomplete gamma function.