Showing papers by "J. Michael Harrison published in 1976"

The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule

Abstract: Consider a storage process X = {Xt, t ≥ 0} with compound Poisson input and a state-dependent release rule r· which is arbitrary except for the requirement that state zero be reachable in finite time from any positive starting state. We show that there exists a stationary distribution for X if and only if there is a limiting distribution independent of the initial state, in which case the stationary distribution is unique and coincides with the limiting distribution. A necessary and sufficient condition for the existence of a stationary distribution, as well as a general solution for the distribution when it exists, is given. We also give a general formula for Ux, the probability that level b is exceeded before level a is reached, starting from state x ∈ a, b]. Both the stationary distribution and Ux are expressed in terms of a certain positive kernel.

Dynamic Scheduling of a Two-Class Queue: Small Interest Rates

Abstract: We consider a single server queuing system with two classes of customers who arrive according to independent Poisson processes. The two service time distributions are arbitrary, and we assume a linear holding cost and fixed service reward for each class. The problem is to decide, at the completion of each service and given the state of the system, which class (if any) to admit next into service. We seek a policy, called Blackwell optimal, which will maximize for all sufficiently small interest rates the expected net present value of service rewards received minus holding costs incurred over an infinite planning horizon. For a variety of different cases, it is shown that there exists a Blackwell optimal policy which simply enforces a static priority ranking of the classes, choosing idleness only when the system is empty. Criteria for ranking the classes are derived, extending classical results on optimal priority rules. For other cases, involving one zero holding cost and/or an unstable system, it is shown...