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. (1992)

• Record Type: Journal Article
• Authors: Mehrez, A.; Brimberg, J.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 5:Number 4(1992)
• Page Start: 325
• Page End: 329
• Publication Date: 1992
• Subjects: M/M/s queue -- arrival rate -- expected queue length -- convexity -- Erlang delay formula; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953392000273
• Languages: English
• ISSNs: 1048-9533

Abstract

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.