Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T09:18:34.040Z Has data issue: false hasContentIssue false
  • English
  • Français

An Infinite Dam with a Time-Dependent Release Rule

Published online by Cambridge University Press:  27 July 2009

M.S. Ali Khan
M.S. Ali Khan
Affiliation:
College of Science King Saud University, Riyadh, Saudi Arabia
Get access Rights & Permissions [Opens in a new window]

Abstract

An infinite dam with discrete inputs arriving at times 0 = T0 < T1 <…< Tn <… is considered, where {Tn;n = 0,1,2,…} forms a renewal process. There is a random release from the dam during the interrenewal times which depends on the content as well as on the length of the time after the last input. A relation is derived that connects the content distribution at any time t to that when t is a renewal point. This relation is used to obtain the content distribution in the limiting case and in the transient case. The probability of emptiness is obtained and an integral equation is derived for the probability of first emptiness in a special case.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 2 , Issue 4 , October 1988 , pp. 485 - 500
Copyright
Copyright © Cambridge University Press 1988

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

1.Ali, Khan M.S. (1986). An infinite dam with random withdrawal policy, Advances in Applied Probability 18: 933–951.Google Scholar
2.Birkhoff, G. & Rota, G.C. (1969). Ordinary differential equations. London: Blaisdell.Google Scholar
3.Çinlar, E. (1971). On dams with continuous semi-Markovian inputs. Journal of Mathematical Analysis and Applications 35: 434–448.CrossRefGoogle Scholar
4.Çinlar, E. (1973). Theory of continuous storage with Markov additive inputs and a general release rule. Journal of Mathematical Analysis and Applications 43: 207–231.CrossRefGoogle Scholar
5.Çinlar, E. (1975). A local time for a storage process, Annals of Probability 3: 930–950.CrossRefGoogle Scholar
6.çinlar, E. (1975). Introduction to stochastic processes. New Jersey: Prentice-Hall.Google Scholar
7.Çinlar, E. & Pinsky, M. (1971). A stochastic integral in storage theory. Zoilschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 17: 227–240.CrossRefGoogle Scholar
8.Çinlar, E. & Pinsky, M. (1972). On dams with additive inputs and a general release rule. Journal of Applied Probability 9: 422–429.CrossRefGoogle Scholar
9.Erugin, N.P. (1966). Linear systems of ordinary differential equations with periodic coefficients. New York: Academic Press.Google Scholar
10.Fedorov, G.F. (1953). Certain new cases of solutions of a system of two linear equations in closed form. Bestinik, Leningrad State University, No. 11.Google Scholar
11.Moran, P.A.P. (1969). A theory of dams with continuous input and a general release rule. Journal of Applied Probability 6: 88–98.CrossRefGoogle Scholar
12.Salakhova, I.M. & Chebotarev, X. (1960). Closed-form solvability of certain systems of linear differential equations. Izv. Vysshikh Uchebnykh Zavedeniy, Matem. 3.Google Scholar
13.Tricomi, F.C. (1957). Integral equations. New York: interscience.Google Scholar
14.Yeo, G.F. (1974). A finite dam with exponential release rate. Journal of Applied Probability 11: 122–133.CrossRefGoogle Scholar
15.Yeo, G.F. (1975). A finite dam with variable release rate. Journal of Applied Probability 12: 205–211.CrossRefGoogle Scholar