Stochastic Models for the Amount of Overflow in a Finite Dam with Random Inputs,Random Outputs,and Exponential Release Policy

In this article, we discuss finite dam models to study the expected amount of overflow in a given time. The inputs into the dam are taken as random and there are two types of outputs—one is random and the other is deterministic which is proportional to the content of the dam. The master equation for the expected amount of overflow is an one dimensional equation with separable kernel. For this class of master equation, the integral equation for the expected amount of overflow has been transformed exactly into ordinary differential equation with variable coefficients. The imbedding method is used to study the expected amount of overflow in a given time without emptiness in this period. We also consider the model for the expected amount of overflow in a given time with any number of emptiness of the dam in this period. The results are derived in the form of a third order differential Equation for the Laplace transformation function for the expected overflow. The closed form analytical solutions are obtained in terms of beta functions and degenerate hyper-geometric functions of two variables.