An Infinite Dam with a Time-Dependent Release Rule

Abstract

An infinite dam with discrete inputs arriving at times 0 = T0 < T1 <…< Tn <… is considered, where {Tn;n = 0,1,2,…} forms a renewal process. There is a random release from the dam during the interrenewal times which depends on the content as well as on the length of the time after the last input. A relation is derived that connects the content distribution at any time t to that when t is a renewal point. This relation is used to obtain the content distribution in the limiting case and in the transient case. The probability of emptiness is obtained and an integral equation is derived for the probability of first emptiness in a special case.