The Epochs of Emptiness of a Semi-Infinite Discrete Reservoir

Abstract

This note discusses, on a discrete time scale, the waiting times to first emptiness and tonth emptiness of a semi-infinite Moran reservoir subject to mutually independent stationary integral-valued inputs, and unit output. Discrete analogues are obtained of Kendall's (1957) integral equation for the first-empty time, and intuitive arguments are given to exemplify the non-uniqueness of the solutions of integral and difference equations of this type. Kendall's solution to his integral equation is shown to give the distribution of first-empty time for the semi-infinite discrete reservoir with positive initial contents. Simple expressions are obtained for the distributions of the duration of an empty period and of the duration of a non-empty period.