Reservoirs with Mixed Markovian-Independent Inflows

Abstract

The paper discusses a stochastic reservoir with a mixed inflows system, consisting of a Markovian component and an independent-sequence component, generalizing results earlier established for a single Markovian inflow stream. These results are: an explicit form for the generating function of the stationary probabilities of water levels in a semi-infinite reservoir, and the ratio theorem relating these probabilities to those in a finite reservoir subject to the same inflows; and a functional equation for the generating function of the times to first emptiness of a semi-infinite reservoir. The functional equation involves a latent root $\lambda ( \theta )$ of the matrix generating function of the transition probabilities of the Markovian component $\{ X_t \}$ of the inflow. A minor result, which might not be devoid of more general interest, is that $\dot \lambda ( 1 ) = E_\infty ( X_t )$.