First exit times and diffusion approximations for storage models with Poisson inputs, Poisson outputs and deterministic release rule

Abstract

In this paper we discuss a number of finite storage problems with random inputs and random outputs together with linear release policy. This class of problems forms one dimensional master equations with separable kernel. For this class of problems the first passage time for overflow or emptiness can be transformed into ordinary differential equations. Closed form analytical solutions are obtained for first passage times for emptiness and overflow, treating the barriers as absorbing or reflecting. The imbedding technique is used to obtain the closed form solution for all the models. The results are derived in the form of third order differential equations for Laplace Transform functions for the first passage times. Diffusion approximation for these models is also obtained using suitable statistical conditions.