Summary
This paper considers a finite dam in continuous time fed by inputs, with a negative exponential distribution, whose arrival times form a Poisson process; there is a continuous release at unit rate, and overflow is allowed. Various results have been obtained by appropriate limiting methods from an analogous discrete time process, for which it is possible to find some solutions directly by determinantal methods.
First the stationary dam content distribution is found. The distribution of the probability of first emptiness is obtained both when overflow is, and is not allowed. This is followed by the probability the overflow before emptiness, which is then applied to determine the exact solution for an insurance risk problem with claims having a negative exponential distribution. The time-dependent content distribution is found, and the analogy with queueing theory is discussed.
Article PDF
Similar content being viewed by others
References
Barlett, M. S.: An introduction to Stochastic processes. Cambridge University Press 1955.
Cramér, H.: Mathematical methods of Statistics. Princeton University Press 1946.
- Collective risk theory. FörsÄkringsaktielolaget Skandia 1955.
Doob, J. L.: Stochastic processes. New York: John Wiley 1953.
Erdelyi, A.: Tables of integral transforms 1. McGraw Hill 1954.
Gani, J.: Problems in the probability theory of storage systems. J. Roy. statist. Soc., Ser. B 19, 181–207 (1957).
Ghosal, A.: Emptiness in the finite dam. Ann. math. Statistics 31, 863–868 (1960).
Kendall, D. G.: Some problems in the theory of dams. J. Roy. statist. Soc., Ser. B 19, 207–212 (1957).
Moran, P. A. P.: A probability theory of dams and storage systems. Austral. J. appl. Sci., 5, 116–124 (1954).
- The theory of storage. Methuen 1959.
Prabhu, N. U.: Some exact results for the finite dam. Ann. math. Statistics 29, 1234–1243 (1958).
Segerdahl, C. O.: On homogeneous random processes and collective risk theory. Uppsala: Almquist and Wiksells 1939.
Takács, L.: Investigation of waiting time problems by reduction to Markov processes. Acta math. Acad. Sci. Hungar. 6, 101–129 (1955).
Weesakul, B.: First emptiness in a finite dam. J. Roy. statist. Soc., Ser. B. 23, 343–351.
- The time-dependent solution for a finite dam with geometric inputs. Austral. Math. Soc. Summer Research Institute report 1961.
Widder, D. V.: The Laplace transform. Princeton University Press 1946.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Weesakul, B., Yeo, G.F. Some problems in finite dams with an application to insurance risk. Z. Wahrscheinlichkeitstheorie verw Gebiete 2, 135–146 (1963). https://doi.org/10.1007/BF00531967
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00531967