Storage Models: Control of Dams Using P$_{\lambda ,\tau }^{M}$ Policies

Abstract

We discuss the problem of control of a dam using P$_{\lambda ,\tau }^{M}$ control policies when the input process is a subordinator, a spectrally positive Lévy process, and a spectrally positive Lévy process reflected at its infimum. We describe the content process by hitching Lévy processes and spectrally positive Lévy processes reflected at the full capacity of the dam, killed at the times of first up crossing and down crossing of levels $\lambda$ and $\tau$, respectively. Using the theory and methods of scale functions of spectrally Lévy processes, we give expressions on the first passage problem of the dam content. The potential measures for the content process up to the time of first passage through levels $\lambda$ and $\tau$ are obtained. Using these results we find the total discounted as well as the long-run average costs.