Some comparability results for waiting times in single- and many-server queues

Abstract

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S^′−x)₊ ≦ E(S^″−x)₊ (all x >0), are stochastically ordered as W^′≦_dW^″. The weaker conclusion, that E(W^′−x)₊ ≦ E(W^″−x)₊ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T^′)₊ ≦ E(x−T^″)₊ (all x). A sufficient condition for w_k≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.