Stock Levels Generated by a Controlled-Variability Arrival Process

Controlled-variability arrivals are of widespread occurrence, and a model for discrete controlled-variability arrivals, called the (a, b) process, has been discussed in the literature. This paper considers the marginal distribution of stock levels generated by (a, b) arrivals of equally sized cargoes, assuming that discharge into stock is instantaneous and that offtake from stock is at a continuous uniform rate equal to the mean rate of planned arrivals. It is shown that, provided the arrival pattern is not too close to regularity, the resulting distribution of stock levels is effectively normal and has a standard deviation that can be simply expressed in terms of the parameters of the (a, b) process. The paper also discusses some aspects of the use of this result for the justification of storage facilities in some real systems.