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A finite dam with exponential release

Published online by Cambridge University Press:  14 July 2016

G. F. Yeo
G. F. Yeo*
Affiliation:
University of Melbourne
*
* The author is currently at Linköping University, Sweden.
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Abstract

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam.

Keywords

FINITE DAMSPOISSON INPUTSEXPONENTIAL RELEASEFIRST PASSAGE TIMESRECURRENCE RELATIONSNUMERICAL INTEGRATIONSERIES SOLUTIONSIMULATION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 122 - 133
Copyright
Copyright © Applied Probability Trust 1974 

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References

Chover, J. and Yeo, G. F. (1965) Solutions of some two-sided boundary problems for sums of variables with alternating distributions. J. Appl. Prob. 2, 377–395.CrossRefGoogle Scholar
Cinlar, E. and Pinsky, M. (1972) On dams with additive inputs and a general release rule. J. Appl. Prob. 9, 422–429.Google Scholar
Gaver, D. P. and Miller, R. G. (1962) Limiting distributions for some storage problems. Studies in Applied Probability and Management Science. (Ed. Arrow, et al.) Stanford University Press.Google Scholar
Hasofer, A. M. (1963) On the integrability, continuity, and differentiability of a family of functions introduced by L. Takacs. Ann. Math. Statist. 34, 1045–1049.CrossRefGoogle Scholar
Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Keilson, J. and Mermin, N. D. (1959) The second-order distribution of integrated shot noise. IRE Trans. on Information Theory IT-5, 75–77.CrossRefGoogle Scholar
Phatarfod, R. M. (1969) A note on the first emptiness problem of a finite dam with Poisson type inputs. J. Appl. Prob. 6, 227–230.CrossRefGoogle Scholar
Prabhu, N. U. (1965) Queues and Inventories, Wiley, New York.Google Scholar
Saaty, T. M. (1961) Elements of Queueing Theory. McGraw-Hill, New York.Google Scholar