The Stationary Distribution of a Stochastic Clearing Process

This research grew out of an investigation of utilization in capacity expansion. The utilization at any time is the demand divided by the capacity. When there is uncertainty about the evolution of demand, it is appropriate to model the demand as a stochastic process, and thus the utilization also becomes a stochastic process. It was found that a utilization stochastic process associated with exponentially growing stochastic demand is closely related to the stochastic clearing processes introduced and investigated by Stidham. Interest in the impact of uncertainty on utilization led to this study of the impact of uncertainty on the stationary distribution of a stochastic clearing process. Stidham showed for a large class of clearing processes that the stationary distribution is never the uniform distribution, which is characteristic of deterministic models with continuous linear input. Here it is shown for a larger class of clearing processes that the stationary distribution is always stochastically less than or equal to the uniform distribution in the sense of second-order stochastic dominance (characterized by the expected value of all nondecreasing concave functions). For various special cases, stronger stochastic order relations are established. For a related capacity expansion model, it is shown that greater uncertainty lowers the expected utilization.