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An infinite dam with random withdrawal policy

Published online by Cambridge University Press:  01 July 2016

M. S. Ali Khan
M. S. Ali Khan*
Affiliation:
King Saud University
*
∗Postal address: Department of Statistics, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia.
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Abstract

Most work in the theory of dams has dealt with deterministic withdrawals. Some more recent work, begun by Moran (1969), has been devoted to conditionally deterministic withdrawals which depend on the content of the dam, or in some cases on the last input. We consider in this paper an infinite dam fed by discrete additive inputs, under a totally random withdrawal policy. Explicit expressions are obtained for the time-dependent content distributions. Probabilities of emptiness are computed and stationary results are derived.

Keywords

INPUT PROCESSGENERATING FUNCTION: INVARIANT DISTRIBUTIONEMPTINESSDIFFERENTIAL-DIFFERENCE EQUATION
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 4 , December 1986 , pp. 933 - 951
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

This work was done during the author&s stay at Northwestern University as a Research Scholar.

References

Ali Khan, M. S. (1977) Infinite dams with discrete additive inputs. J. Appl. Prob. 14, 170–180.Google Scholar
Brockwell, P. J., Gani, J. M. and Resnick, S. I. (1982) Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.Google Scholar
Çinlar, E. (1971) On dams with continuous semi-Markovian inputs. J. Math. Anal. Appl. 35, 434–448.Google Scholar
Çinlar, E. (1973) Theory of continuous storage with Markov additive inputs and a general release rule. J. Math. Anal. Appl. 43, 207–231.Google Scholar
Çinlar, E. (1975) A local time for a storage process. Ann. Prob. 3, 930–950.Google Scholar
Çinlar, E. and Pinsky, M. (1971) A stochastic integral in storage theory. Z. Wahrscheinlichkeitsth. 17, 227–240.Google Scholar
Çinlar, E. and Pinsky, M. (1972) On dams with additive inputs and a general release rule. J. Appl. Prob. 9, 422–429.CrossRefGoogle Scholar
Collings, P. S. (1974) Dams with random outputs. J. Appl. Prob. 11, 838–861.CrossRefGoogle Scholar
Gani, J. (1955) Some problems in the theory of provisioning and of dams. Biometrika 42, 179–200.Google Scholar
Lehoczky, J. P. (1973) First emptiness problems in applied probability under first-order dependent inputs and general output conditions. J. Appl. Prob. 10, 330–342.Google Scholar
Moran, P. A. P. (1954) A probability theory of dams and storage systems. Austral. J. Appl. Sci. 5, 116–124.Google Scholar
Moran, P. A. P. (1969) A theory of dams with continuous input and a general release rule. J. Appl. Prob. 6, 88–98.Google Scholar
Pegram, G. G. S. (1975) Recurrence times of draft pattern from reservoirs. 7. Appl. Prob. 12, 647–652.Google Scholar
Phatarfod, R. M. (1979) The bottomless dam. J. Hydrology 40, 337–363.Google Scholar
Seneta, E. (1981) Non-negative Matrices and Markov Chains. Springer-Verlag, Berlin.Google Scholar
Van Harn, K. (1978) Classifying Infinitely Divisible Distributions by Functional Equations. Mathematisch Centrum, Amsterdam.Google Scholar
Yeo, G. F. (1961) The time dependent solution for an infinite dam with discrete additive inputs. J. R. Statist. Soc. B 23, 173–179.Google Scholar
Yeo, G. F. (1974) A finite dam with exponential release. J. Appl. Prob. 11, 122–133.Google Scholar
Yeo, G. F. (1975) A finite dam with variable release rate. J. Appl. Prob. 12, 205–211.Google Scholar