Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-19T19:07:39.760Z Has data issue: false hasContentIssue false
  • English
  • Français

A new look at the Moran dam

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

W. Stadje
W. Stadje*
Affiliation:
University of Osnabrück
*
∗∗ Postal address: Fachbereich Mathematik/Informatik, Universität Osnabrück, Postfach 4469, D-4500 Osnabrück, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

For the original Moran dam with independent and identically distributed inputs a representation of the stationary distribution is given which readily provides a geometric rate of convergence to this distribution. For the integer-valued case the stationary distribution can be expressed in terms of simple boundary crossing probabilities for the underlying random walk.

Keywords

STATIONARY DISTRIBUTIONGEOMETRIC ERGODICITYBOUNDARY CROSSING PROBABILITY

MSC classification

Primary: 90B05: Inventory, storage, reservoirs
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 489 - 495
Copyright
Copyright © Applied Probability Trust 1993 

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

Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, New York.Google Scholar
Cohen, J. W. (1982) The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
Eger, K.-H. (1985) Sequential Tests. Teubner, Leipzig.Google Scholar
Enkawa, T. and Mori, M. (1985) Exact expressions for OC and ASN functions of Poisson sequential probability ratio test. Rep. Stat. Appl. Res. JUSE 32, 1–16.Google Scholar
Gani, J. (1957) Problems in the probability theory of storage systems. J. R. Statist. Soc. B 19, 181–206.Google Scholar
Girshick, M. A. (1946) Contributions to the theory of sequential analysis II, III. Ann. Math. Statist. 17, 282–298.Google Scholar
Gnedenko, B. W. and König, D. (1984) Handbuch der Bedienungstheorie. Akademie-Verlag, Berlin.Google Scholar
Kohlruss, D. (1991) Berechnungsmethoden für die Gütefunktion und die ASN-Funktion von SPRT's für ganzzahlige Zufallsgrößen–ein Vergleich. Preprint 4/91, Department of Applied Mathematics and Computer Science, University of Münster.Google Scholar
Lloyd, E. H. and Saleem, S. D. (1979) A note on seasonal Markov chains with gamma or gamma-like distributions. J. Appl. Prob. 16, 117–128.Google Scholar
Lloyd, E. H. and Warren, D. (1982) The linear reservoir with seasonal gamma-distributed Markovian inflows. In Time Series Methods in Hydrosciences, ed. El-Shaarawi, A. H. and Esterby, S. R. Elsevier, Amsterdam.Google Scholar
Moran, P. A. P. (1955) A probability theory of dams and storage systems: modifications of the release rules. Austral. J. Appl. Sci. 6, 117–130.Google Scholar
Moran, P. A. P. (1959) The Theory of Storage. Methuen, London.Google Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-Negative Operators. p.Google Scholar
Nummelin, E. and Tweedie, R. L. (1978) Geometric ergodicity and R-positivity for general Markov chains. Ann. Prob. 6, 404–420.Google Scholar
Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Proc. Appl. 12, 187–202.Google Scholar
Pegg, P. A. and Phatarfod, R. M. (1977) Dams with additive inputs revisited. J. Appl. Prob. 14, 367–374.CrossRefGoogle Scholar
Phatarfod, R. M. (1982) On some applications of Wald's identity to dams. Stoch. Proc. Appl. 13, 279–292.Google Scholar
Phatarfod, R. M. (1988) Sums and weighted sums of a gamma Markov sequence. J. Appl. Prob. 25, 204–209.Google Scholar
Prabhu, N. U. (1958a) Some exact results for the finite dam. Ann. Math. Statist. 29, 1234–1243.CrossRefGoogle Scholar
Prabhu, N. U. (1958b) On the integral equation for the finite dam. Quart. J. Math. 9, 183–188.CrossRefGoogle Scholar
Prabhu, N. U. (1964) Time-dependent results in storage theory. J. Appl. Prob. 1, 1–46.CrossRefGoogle Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Siegmund, D. (1985) Sequential Analysis. Springer-Verlag, New York.Google Scholar
Tweedie, R. L. (1983) Criteria for rates of convergence of Markov chains, with an application to queueing and storage theory. Probability, Statistics and Analysis, London Math. Soc. Lecture Notes Series 79, 260–276.Google Scholar
Wald, A. (1947) Sequential Analysis. Wiley, New York.Google Scholar