Some Stochastic Bounds for Dams and Queues

Let X = {X(t), t > 0} be a process of the form X(t) = Z(t) − ct, where c is a positive constant and Z is an infinitely divisible, nondecreasing pure jump process. Assuming E[X(t)] < 0. let U be the d.f. of M = sup X(t). As is well known, U is the contents distribution of a dam with input Z and release rate c. If Z is compound Poisson, one can alternately view U as the waiting time distribution for an M/G/1 queue or 1 − U as the ruin function for a risk process.

Letting X and X₀ be two processes of the indicated form, it is shown that U ≤ U₀ if the two jump measures are ordered in a sense weaker than stochastic dominance. In the case where E(M) = E(M₀), a different condition on the jump measures yields E[f(M)] ≤ E[f(M₀)] for all concave f, this resulting from second-order stochastic dominance of the supremum distributions. By way of application, processes with deterministic jumps are shown to be extremal in certain ways, and those with exponential jumps are shown to be extremal among the class having IFR jump distribution. Finally, an extremal property of Brownian Motion (which is not among the class of processes considered) is demonstrated, this yielding simple bounds for E[f(M)] with f concave or convex. It is shown how all the bounds obtained for U or E[f(M)] can be further sharpened with additional computation.