On Stochastic Bounds for the Delay Distribution in the GI/G/s Queue

A counterexample is constructed to show that the steady-state delay distribution in the GI/G/s queue with the FIFO discipline need not be stochastically less (in the sense of first-order stochastic dominance) than the steady-state delay distribution in the same system with the cyclic queue discipline. Such an ordering was conjectured by R. W. Wolff. However, it is shown that the stochastic dominance does hold for many GI/M/s queues. When it does, this bound on the delay distribution is often better than the bound for GI/M/s systems obtained by S. L. Brumelle, but neither bound appears to be very good, as is illustrated here for the M/M/s case.