Discrete dams with Markovian inputs

It is known that, subject to some technical conditions, the deficit process of an infinitely deep dam with a Markov chain input process and unit withdrawals has a limiting zero-modified geometric distribution. In this paper it is shown that, provided the state space of the input process is finite, the limiting distribution of the deficit process of a finite dam tends to the zero-modified distribution above as the capacity tends to infinity. A convergence rate is established when the input process is an independent sequence. When the input process is a semi-Markov chain we find a simple condition ensuring that the limiting deficit distribution is a zero-modified geometric distribution. Some results are obtained for infinitely deep, and high, dams when the input process is a first order discrete autoregressive process. Nearly all examples of Markovian input processes have linear conditional expectations. The final section is a brief expository essay on such processes and mentions some open problems.