Some aspects of the mathematical theory of storage

In this thesis, a storage model with infinite capacity, additive stochastic input and unit release per unit time is investigated. The content of the store in the deterministic case is defined as the unique solution of an integral equation. Properties of non-negative additive stochastic processes are obtained. These properties are used to study the distribution of the time of first emptiness when the input is stationary, and the distribution of the content. Applications to dams and queues with specific input laws are given. In particular, the waiting time for the queues M/M/1 and M/G/1, and the content of the dam with Gamma input are studied in detail. The dam with Inverse Gaussian input is introduced and its transient solution obtained explicitly. Finally, in the case of a Compound Poisson input, the continuity and differentiability of the distribution of the content are investigated. A non-stationary Compound Poisson input is considered, and it is shown that the probability of the store being empty and the Laplace transform of the content can be expanded in a power series. When the parameter of the input is periodic, it is shown that all terms of the series expansion are asymptotically periodic, and explicit expressions for the leading terms are obtained.