A. Mehrez, J. Brimberg, "A note on the convexity of the expected queue length of the queue with respect to the arrival rate: a third proof", International Journal of Stochastic Analysis, vol. 5, Article ID 843287, 5 pages, 1992. https://doi.org/10.1155/S1048953392000273
A note on the convexity of the expected queue length of the queue with respect to the arrival rate: a third proof
The convexity of the expected number in an 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, 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.
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