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

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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 ′≦d W ″. 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 wk ≡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.

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