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

The convexity of the expected number in an M/M/s queue with respect to the arrival rate (or traffic intensity) is well known. Grassmann proves this result directly by making use of a bound on the probability that all servers are busy. Independently, Lee and Cohen derive this result by showing that the Erlang delay formula is a convex function. In this note, the authors 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.