Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T03:01:30.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic solution of a finite dam governed by a general input

Published online by Cambridge University Press:  14 July 2016

S. K. Srinivasan
S. K. Srinivasan*
Affiliation:
Indian Institute of Technology, Madras
Get access Rights & Permissions [Opens in a new window]

Abstract

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.

Keywords

FINITE DAMPOINT PROCESSREGENERATION POINTMARKOV CHAINPRODUCT DENSITYRENEWAL DENSITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 134 - 144
Copyright
Copyright © Applied Probability Trust 1974 

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.)

References

Bellman, R. E. and Harris, T. E. (1948) On the theory of age dependent stochastic branching processes Proc. Nat. Acad. Sci. U.S.A. 34, 601–604.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North Holland, Amsterdam.Google Scholar
Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Gani, J. (1957) Problems in the probability theory of storage systems J. R. Statist. Soc. B 19, 181–206.Google Scholar
Moran, P. A. P. (1954) A probability theory of dams and storage systems Austral. J. Appl. Sci. 5, 116–124.Google Scholar
Pollaczek, P. (1957) Problèmes stochastiques posés par le phénomène de formation d'une queue d'attente à un guichet et par des phénomènes apparentés. Mémorial des Sciences Mathématiques 136. Ganthier-Villars, Paris.Google Scholar
Prabhu, N. U. (1964) Time dependent results in storage theory J. Appl. Prob. 1, 1–46.Google Scholar
Ramakrishnan, A. (1950) Stochastic processes relating to particles distributed in a continuous infinity of states Proc. Camb. Phil. Soc. 46, 595–602.Google Scholar
Ramakrishnan, A. (1953) Stochastic processes associated with random divisions of a line Proc. Camb. Phil. Soc. 49, 473–485.Google Scholar
Rice, S. O. (1954) Mathematical analysis of random noise Bell Syst. Tech. J. 25, 46–156.Google Scholar
Roes, P. B. M. (1970) The finite dam J. Appl. Prob. 7, 316–325, 599–616.Google Scholar
Roes, P. B. ?. (1970) On the expected number of crossings of a level in certain stochastic processes J. Appl. Prob. 7, 766–770.CrossRefGoogle Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar