Optimal control of a finite dam using \(P^ M_{\lambda,\tau}\) policies and penalty cost: Total discounted and long run average cases. (English) Zbl 0726.60099

Abdel-Hameed, M.; Nakhi, Y.

J. Appl. Probab. 27, No. 4, 888-898 (1990).

A dam with capacity V receives water input consisting of a diffusion process with positive drift on \({\mathbb{R}}_+\), and appropriate behavior at the boundary. The release rate is either M or zero; water is released when the level reaches \(\lambda\), and release ceases when the level drops to \(\tau\). The penalty cost rate depends on the water level and the release rate, with additional starting and stopping costs. The authors describe the process by hitching killed Markov processes together, and describing the dam contents by the \(\alpha\)-potential of the killed process. The \(\alpha\)-potential appears in terms of the resolvent oprator, for which the differential equation is derived. From this, the authors obtain expressions for the expected total discounted and long term average costs. These results generalize the work of F. A. Attia [Stochastic Processes Appl. 25, 289-299 (1987; Zbl 0649.60086) and J. Appl. Probab. 26, No.2, 314-324 (1989; Zbl 0679.60076)], L. Yeh and L. J. Hua [ibid. 24, 186-189 (1987; Zbl 0617.93078)], and D. Zuckerman [ibid. 14, 421-425 (1977; Zbl 0359.93022)].

Reviewer: F.J.Beutler (Ann Arbor)

MSC:

60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60J35	Transition functions, generators and resolvents
90B22	Queues and service in operations research

Keywords:

optimal control of diffusion process; dam; Markov processes; long term average costs

Citations:

Zbl 0649.60086; Zbl 0679.60076; Zbl 0617.93078; Zbl 0359.93022