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Bounds for Stopping Times with Application to the Approximation of Distribution Functions

Published online by Cambridge University Press:  27 July 2009

Rhonda Richter  and
J. George Shanthikumar
Rhonda Richter
Affiliation:
Department of Decision and Information Sciences, Santa Clara University, Santa Clara, California 95053
J. George Shanthikumar
Affiliation:
Walter A. Haas School of Business, University of California at Berkeley, Berkeley, California 94720
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Abstract

We consider stopping times associated with sequences of non-negative random variables and Poisson processes. With sufficient conditions on the dependence property between the sequences of non-negative random variables (or the Poisson processes) and the stopping times, we develop easily computable stochastic bounds for the stopping times. We use these bounds to develop approximations within the class of generalized phase-type distribution functions for arbitrary distribution functions. The computational tractability of generalized phase-type distribution functions facilitate the computational analysis of complex stochastic systems using these approximate distribution functions. Applications of these approximations to renewal processes are illustrated.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 8 , Issue 1 , January 1994 , pp. 21 - 31
Copyright
Copyright © Cambridge University Press 1994

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References

1.Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing: Probability models. Silver Spring, MD: To Begin With.Google Scholar
2.Cao, X.R. (1993). Decomposition of random variables with bounded hazard rates. Operations Research Letters 13: 113–120.CrossRefGoogle Scholar
3.Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables, with applications. Annals of Mathematical Statistics 38: 1466–1474.CrossRefGoogle Scholar
4.Ross, S.M. (1983). Stochastic processes. New York: Wiley.Google Scholar
5.Ross, S.M. (1987). Approximations for renewal processes. Probability in the Engineering and Informational Sciences 1: 163–174.CrossRefGoogle Scholar
6.Ross, S.M., Shanthikumar, J.G., & Zhang, X. (1993). Some pitfalls of black box queue inference: The case of state-dependent server queues. Probability in the Engineering and Informational Sciences 7: 149–157.CrossRefGoogle Scholar
7.Shanthikumar, J.G. (1985). Bilateral phase type distributions. Naval Research Logistics Quarterly 32: 119–136.CrossRefGoogle Scholar
8.Shanthikumar, J.G. (1986). Uniformization and hybrid simulation/analytic methods of renewal processes. Operations Research 34: 573–580.CrossRefGoogle Scholar
9.Wolff, R.W. (1988). Stochastic modelling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar