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

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