Abstract
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.
Highlights
Optimization of queueing systems is becoming an increasingly important topic in several applied areas, such as flexible manufacturing and telecommunications
We provide a third method of proof, which exploits the relationship between the Erlang delay formula and the Poisson probability distribution
Grassmann’s method uses a bound on the probability that all servers are busy to show that the second partial derivative is positive
Summary
Optimization of queueing systems is becoming an increasingly important topic in several applied areas, such as flexible manufacturing and telecommunications. Ohio 44342, U.S.A. Royal Military College of Canada Department of Engineering Management The convexity of the expected number in an M/M/s queue with respect to the arrival rate (or traffic intensity) is well known. Grassmaan [1] proves this result directly by making use of a bound on the probability that all servers are busy.
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