A note on the convexity of the expected queue length of the \(M/M/s\) queue with respect to the arrival rate: A third proof (Q2365561)

Summary: The convexity of the expected number in an \(M/M/s\) queue with respect to the arrival rate (or traffic intensity) is well-known. \textit{W. K. Grassmann} [J. Appl. Probab. 20, 916-919 (1983; Zbl 0526.60087)] proved this result directly by making use of a bound on the probability that all servers are busy. Independently, \textit{H. L. Lee} and \textit{M. A. Cohen} [ibid. 20, 920-923 (1983; Zbl 0526.60084)] derived this result by showing that the Erlang delay formula is a convex function. In this note, we provide a third method of proof, which exploits the relationship between the Erlang delay formula and the Poisson probability distribution. Several interesting intermediate results are also obtained.