Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T01:32:00.973Z Has data issue: false hasContentIssue false

Optimal Control of a Large Dam

Published online by Cambridge University Press:  14 July 2016

Vyacheslav M. Abramov*
Affiliation:
Monash University
*
Postal address: School of Mathematical Sciences, Monash University, Building 28M, Clayton Campus, Clayton, VIC 3800, Australia. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A large dam model is the object of study of this paper. The parameters Llower and Lupper define its lower and upper levels, L = Lupper - Llower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let J1 and J2 denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let Lt denote the dam level at time t, and let p1 = limt→∞P{Lt = Llower} and p2 = limt→∞P{Lt > Lupper} exist. The long-run average cost, J = p1J1 + p2J2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize J.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2007 

References

Abdel-Hameed, M. (2000). Optimal control of a dam using policies and penalty cost when the input process is a compound Poisson process with positive drift. J. Appl. Prob. 37, 408416.Google Scholar
Abdel-Hameed, M. and Nakhi, Y. (1990). Optimal control of a finite dam using policies and penalty cost: total discounted and long-run average cases. J. Appl. Prob. 27, 888898.Google Scholar
Abramov, V. M. (1991). Investigation of a Queueing System with Service Depending on a Queue-Length. Donish, Dushanbe, Tadzhikistan (in Russian).Google Scholar
Abramov, V. M. (1997). On a property of a refusals stream. J. Appl. Prob. 37, 800805.CrossRefGoogle Scholar
Abramov, V. M. (2002). Asymptotic analysis of the GI/M/1/n queueing system as n increases to infinity. Ann. Operat. Res. 112, 3541.Google Scholar
Abramov, V. M. (2004). Asymptotic behavior of the number of lost messages. SIAM J. Appl. Math. 64, 746761.CrossRefGoogle Scholar
Bae, J., Kim, S. and Lee, E. Y. (2002). A policy for an M/G/1 queueing system. Appl. Math. Modelling 26, 929939.CrossRefGoogle Scholar
Bae, J., Kim, S. and Lee, E. Y. (2003). Average cost under the policy in a finite dam with compound Poisson inputs. J. Appl. Prob. 40, 519526.Google Scholar
Boxma, O., Kaspi, H., Kella, O. and Perry, D. (2005). On/off storage systems with state-dependent input, output, and switching rates. Prob. Eng. Inf. Sci. 19, 114.Google Scholar
Faddy, M. J. (1974). Optimal control of finite dams: discrete (2-stage) output procedure. J. Appl. Prob. 11, 111121.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York.Google Scholar
Kaspi, H., Kella, O. and Perry, D. (1996). Dam processes with state-dependent batch sizes, and intermittent production processes with state-dependent rates. Queueing Systems 24, 3757.Google Scholar
Lam, Y. and Lou, J. H. (1987). Optimal control for a finite dam. J. Appl. Prob. 24, 196199.Google Scholar
Lee, E. Y. and Ahn, S. K. (1998). policy for a dam with input formed by a compound Poisson process. J. Appl. Prob. 35, 482488.Google Scholar
Phatarfod, R. M. (1989). Riverflow and reservoir storage models. Math. Comput. Modelling 12, 10571077.CrossRefGoogle Scholar
Postnikov, A. G. (1979). Tauberian theory and its applications. Trudy Mat. Inst. Steklov 144, 147 pp. (in Russian). English translation: Proc. Steklov Inst. Math. 1980, 138 pp.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Subhankulov, M. A. (1976). Tauberian Theorems with Remainder. Nauka, Moscow (in Russian).Google Scholar
Sznajder, R. and Filar, J. A. (1992). Some comments on a theorem of Hardy and Littlewood. J. Optimization Theory Appl. 75, 201208.Google Scholar
Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar
Zukerman, D. (1977). Two-stage output procedure of a finite dam. J. Appl. Prob. 14, 421425.Google Scholar