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 [1] proves this result directly by making use of a bound on
the probability that all servers are busy. Independently, Lee and Cohen
[2] derive 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.