Yield and overflow from a finite stochastic reservoir

Abstract

The paper discusses the overflow (spillage) and yield rates and the total overflow and total yield over a specified time from a finite discrete stochastic reservoir, in which the yieldY _t during the working interval (t,t+1) is a function of the storageZ _t at timet, the inflow sequence {X _t } being IID.

The distribution vector of the spillage rate at timet is a telescoped version of the distribution of a certain Markovian variable whose transition matrix is derived. Formulae are given for the distribution of the total spillageW _h given suitable initial conditions, forh=1,2,3; and a simple expression derived forE(W _h ).

The distribution of the yield rateY _t is a trivial modification of the storage distribution. As for the total yieldR _t =Y ₁+...+Y _t , it is shown that the bivariate sequence {R _t ,Z _t } is first-order Markovian, whereZ _t is the storage at timet. The transition matrix of this process is obtained and the method of evaluating the marginal distribution of the total yieldR _t is exemplified.