First Emptiness of Two Dams in Parallel

Abstract

This paper considers the probabilities of first emptiness of two dams in parallel, both subject to a steady release at constant unit rate, and fed by a discrete additive input process such that unit inputs are always directed to the dam with lesser content. The problem is equivalent to that of the single dam fed by two ordered inputs, and a recurrence relation for the probabilities of first emptiness in this process is obtained. Equations for the generating functions of the probabilities are derived, and a formal solution to these is given. A more convenient method of evaluating probabilities of first emptiness is found by reducing the process to an associated occupancy problem; it is shown how the probabilities of first emptiness for Poisson inputs are then obtained by a rapid computational procedure. The paper concludes with a general formulation of the problem when the times of arrival for two ordered non-negative inputs of random size form a Poisson process.