Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-13T23:42:43.903Z Has data issue: false hasContentIssue false
  • English
  • Français

A comparison of convergence rates for three models in the theory of dams

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Robert Lund  and
Walter Smith
Robert Lund*
Affiliation:
The University of Georgia
Walter Smith*
Affiliation:
The University of North Carolina
*
∗Postal address: Department of Statistics, The University of Georgia, Athens, GA 30602–1952, USA.
∗∗Postal address: Department of Statistics, The University of North Carolina, Chapel Hill, NC 275993260, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.

Keywords

STORAGE MODELCONVERGENCE RATESMARKOV PROCESSPOISSON PROCESSEXPONENTIAL ERGODICITYSTORAGE EQUATION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 74 - 83
Copyright
Copyright © Applied Probability Trust 1997 

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] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Brockwell, P. J., Resnick, S. I. and Tweedie, R. L. (1982) Storage processes with general release rule and additive inputs. Adv. Appl. Prob. 14, 392–433.Google Scholar
[3] ÇInlar, E. and Pinsky, M. (1972) On dams with additive inputs and a general release rule. J. Appl. Prob. 9, 422–429.Google Scholar
[4] Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, London.Google Scholar
[5] Harrison, J. M. and Resnick, S. I. (1976) The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 4, 347–358.Google Scholar
[6] Lindvall, T. (1992) Lectures on the Coupling Method. Wiley, New York.Google Scholar
[7] Lund, R. B. (1993) Some limiting and convergence rate results in the theory of dams. PhD dissertation. The University of North Carolina.Google Scholar
[8] Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996) Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6, 218–237.Google Scholar
[9] Meyn, S. P. and Tweedie, R. L. (1993) Stability of Markovian processes III. Adv. Appl. Prob. 15, 518–548.Google Scholar
[10] Moran, P. A. P. (1967) Dams in series with continuous release. J. Appl. Prob. 4, 380–388.CrossRefGoogle Scholar
[11] Phatarfod, R. M. (1979) The bottomless dam. J. Hydrology 40, 337–363.Google Scholar
[12] Rubinovitch, M. and Cohen, J. W. (1980) Level crossings and stationary distributions for dams. J. Appl. Prob. 17, 218–226.Google Scholar
[13] Thorisson, H. (1983) The coupling of regenerative processes. Adv. Appl. Prob. 15, 531–561.CrossRefGoogle Scholar
[14] Tuominen, P. and Tweedie, R. L. (1979) Exponential ergodicity in Markovian queueing and dam models. J. Appl. Prob. 16, 867–880.CrossRefGoogle Scholar