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

We give in this paper two characterizations of the minimal exponential rate of survival $\theta$ * of the M/M/$\mathrm{\infty }$ 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{í}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.