A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals

Abstract

A well-known theorem (Moran, 1959) relating to the equilibrium condition of a semi-infinite “Moran reservoir” with stationary independent inputs and unit output, gives (a) the probability of emptiness, and (b) the generating function of the distribution of levels, in terms of the input distribution. A further theorem (Prabhu, 1958) points out that, in a finite reservoir, the ratio of the probabilities of any two comparable levels is independent of the size of the reservoir, and and is in fact the same as the corresponding ratio for the semi-infinite reservoir. In the present note the theorems are extended to deal with the case of Markovian inputs.