Journal of Applied Mathematics and Stochastic Analysis
Volume 5 (1992), Issue 4, Pages 325-329

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

A. Mehrez1 and J. Brimberg2

1Kent State University, Graduate School of Management, Kent 44242, Ohio, USA
2Royal Military College of Canada, Department of Engineering Management, Ontario, Kingston K7K 5L0, Canada

Received 1 December 1992

Copyright © 1992 A. Mehrez and J. Brimberg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.