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 queue with respect to the arrival rate: a third proof
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 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.