Showing papers by "J. Michael Harrison published in 1977"

Limit Theorems for Periodic Queues

Abstract: Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE ( S ) W = { W 1 , W 2 , · ··} and the server load (or virtual waiting-time process) Z = { Z ( t ), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takacs) relating the limiting virtual and actual waiting-time distributions of a GI / G /1 queue.

Independence and Calibration in Decision Analysis

Abstract: An individual is said to be potentially miscalibrated if he is not sure whether his future subjective probability assessments will agree with observed frequency Alternately, the individual is said to be uncertain about his own calibration It is argued that such a person will never perceive any two events as probabilistically independent, in the same sense that an ignorant person does not perceive events as certain Uncertainty about one's own calibration does not prevent rational behavior in the decision theoretic sense, but it may make much more difficult the process of translating decision theoretic principles into practical procedures for analysis of real decision problems

The supremum distribution of a Lévy process with no negative jumps

Abstract: Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ(s) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W(0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W, this yields an elementary proof that the Laplace transform of M is µs/ψ(s). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.

Some Stochastic Bounds for Dams and Queues

Abstract: Let X = {Xt, t > 0} be a process of the form Xt = Zt-ct, where c is a positive constant and Z is an infinitely divisible, nondecreasing pure jump process. Assuming E[Xt] < 0. let U be the d.f. of M = sup Xt. 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 EM = EM0, a different condition on the jump measures yields E[fM] ≤ E[fM0] 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[fM] with f concave or convex. It is shown how all the bounds obtained for U or E[fM] can be further sharpened with additional computation.