Control of Dams Using Policies When the Input Process Is a Nonnegative L\primoslcurlybercurlybvy Process

We consider policy of a dam in which the water input is an increasing Lévy process. The release rate of the water is changed from 0 to and from to 0 at the moments when the water level upcrosses level and downcrosses level , respectively. We determine the potential of the dam content and compute the total discounted as well as the long-run average cost. We also find the stationary distribution of the dam content. Our results extend the results in the literature when the water input is assumed to be a Poisson process. 1. Introduction and Summary Lam and Lou [1] consider the control of a finite dam where the water input is a Wiener process, using policies. In these policies, the water release rate is assumed to be zero until the water reaches level as soon as this happens the water is released at rate until the water content reaches level Abdel-Hameed and Nakhi [2] discuss the optimal control of a finite dam using policies, using the total discounted as well as the long-run average costs. They consider the cases where the water input is a Wiener process and a geometric Brownian motion process. Lee and Ahn [3] consider the long-run average cost case when the water input is a compound Poisson process. Abdel-Hameed [4] treats the case where the water input is a compound Poisson process with a positive drift. He obtains the total discounted cost as well as the long-run average cost. Bae et al. [5] consider the policy in assessing the workload of an M/G/1 queuing system. Bae et al. [6] consider the log-run average cost for policy in a finite dam, when the input process is a compound Poisson process. In this paper, we consider the policy for the more general case where the water input is assumed to be an increasing Lévy process. At any time, the release rate can be increased from 0 to with a starting cost or decreased from to zero with a closing cost . Moreover, for each unit of output, a reward is received. Furthermore, there is a penalty cost which accrues at a rate , where is a bounded measurable function on the state space of the content process. We will use the term “increasing” to mean “nondecreasing” throughout this paper. In Section 2, we discuss the potentials of the processes of interest as well as the other results that are needed to compute the total discounted and long-run average costs. In Section 3, we obtain formulas for the cost functionals using the total discounted as well as the long-run average cost cases. In Section 4, we discuss the special cases where the water input is an increasing compound Poisson process as well as inverse Gaussian