A Continuous Time Storage Model with Markov Net Inputs

摘要

A model for a dam is considered wherein the net input rate (input minus output rate) follows a finite Markov chain in continuous time, Xt, and the dam contents process, Ct, is the integral of the Markov chain. The dam is then modelled with the bivariate Markov process (Xt,Ct), of which three variations are considered. These are the doubly-infinite dam with no top or bottom, the semi-infinite dam with only one boundary, and the finite dam with both a top and a bottom. Some of the analysis is performed under the most general situation in which Xt is defined on m states and has an arbitrary generator, while other analysis is performed under the restricted case when m = 2. For the doubly-infinite dam, the first and second moment functions and the maximum and minimum variables are studied. The expected range function is explicitly derived in a special two-state case. Also in the two-state case, weak convergence to the Wiener process is established in D(0 to infinity), from which the asymptotic distribution of the range is obtained. For the semi-infinite and finite dams, techniques of invariance used in the physical sciences are introduced to study first passage times. These techniques are used to derive the distribution and moments of the wet period of the dam in special cases, and the limiting probabilities of emptiness and overflow. (Author)