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

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 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.