Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-15T17:49:33.767Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial methods in the theory of dams

Published online by Cambridge University Press:  14 July 2016

Lajos Takács
Lajos Takács*
Affiliation:
Columbia University, New York
Get accessRights & Permissions [Opens in a new window]

Extract

In this paper we shall be concerned with two mathematical models of infinite dams. In the first model independent random inputs occur at regular time intervals and in the second model independent random inputs occur in accordance with a Poisson process. The first model has already been studied by Gani, Yeo and others, and the second model by Gani and Prabhu, Gani and Pyke, Kendall, and others. For both models we shall find explicit formulas for the distribution of the content of the dam and that of the lengths of the wet periods and dry periods. The proofs are elementary and based on two generalizations of the classical ballot theorem.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 1 , Issue 1 , June 1964, pp. 69 - 76
Copyright
Copyright © Applied Probability Trust 1964 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Aeppli, A. (1924) Zur Theorie verketetter Wahrscheinlichkeiten. Thèse, Zürich.Google Scholar
[2] Andre, D. (1887) Solution directe du problème résolu par M. Bertrand. C.R. Acad. Sci. Paris 105, 436–437.Google Scholar
[3] Barbier, E. (1887) Généralisation du problème résolu par M. J. Bertrand. C.R. Acad. Sci. Paris 105, 407.Google Scholar
[4] Bertrand, J. (1887) Solution d'un problème. C.R. Acad. Sci. Paris 105, 369.Google Scholar
[5] Dwass, M. (1962) A fluctuation theorem for cyclic random variables. Ann. Math. Statist. 33, 1450–1454.CrossRefGoogle Scholar
[6] Gani, J. (1958) Elementary methods for an occupancy problem of storage. Math. Ann. 136, 454–465.Google Scholar
[7] Gani, J. and Prabhu, N.U. (1959) The time dependent solution for a storage model with Poisson input. J. Math. and Mech. 8, 653–663.Google Scholar
[8] Gani, J. and Pyke, R. (1960) The content of a dam as the supremum of an infinitely divisible process. J. Math. and Mech. 9, 639–651.Google Scholar
[9] Kendall, D. G. (1957) Some problems in the theory of dams. J.R. Statist. Soc. B, 19, 207–212.Google Scholar
[10] Takács, L. (1961) The probability law of the busy period for two types of queuing processes. Operat. Res. 9, 402–407.Google Scholar
[11] Takács, L. (1962) A generalization of the ballot problem and its application in the theory of queues. J. Amer. Statist. Ass. 57, 327–337.Google Scholar
[12] Takács, L. (1962) The time dependence of a single-server queue with Poisson input and general service times. Ann. Math. Statist. 33, 1340–1348.Google Scholar
[13] Takács, L. (1963) The distribution of majority times in a ballot. Z. Wahrscheinlichkeitstheorie 2, 118–121.Google Scholar
[14] Takács, L. (1964) The distribution of the content of a dam when the input process has stationary independent increments. Z. Wahrscheinlichkeitstheorie (to appear).Google Scholar
[15] Tanner, J. C. (1961) A derivation of the Borel distribution. Biometrika 48, 222–223.Google Scholar
[16] 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