Infinite dams with discrete additive inputs

Abstract

This paper considers an infinite dam fed by a discrete input Xt during the time interval [t, t + 1), t = 0, 1, 2, ···. At time t – 0 there is an output Yt = min(Zt–1, + Xt–1, r) from the dam leaving behind the amount Zt = max(0, Zt–1, + Xt–1, r). The probability Pr(Zt = i), i = 0, 1, 2, ··· is discussed under the strict assumption that r > 1 and the given initial condition that Z0 = u, u = 1, 2, ···. The generating function technique has been used throughout the paper.