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Ageing first-passage times of Markov processes: a matrix approach

Published online by Cambridge University Press:  14 July 2016

Haijun Li
Affiliation:
Washington State University
Moshe Shaked
Affiliation:
University of Arizona
Corresponding

Abstract

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Supported by NSF Grant DMS 9303891.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Brown, M. and Chaganty, N. R. (1983) On the first passage times distribution for a class of Markov chains. Ann. Prob. 11, 10001008.CrossRefGoogle Scholar
Cao, J. and Wang, Y. (1991) The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28, 473–179.CrossRefGoogle Scholar
Cheng, S. (1991) Pure jump shock models in the discrete time case. Syst. Sci. Math. Sci. 4, 309320.Google Scholar
Drosen, J. W. (1986) Pure jump shock models in reliability. Adv. Appl. Prob. 18, 423440.CrossRefGoogle Scholar
Durham, S., Lynch, J. and Padgett, W. J. (1990) TP2-orderings and the IFR property with applications. Prob. Eng. Inf. Sci. 4, 7388.CrossRefGoogle Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627649.CrossRefGoogle Scholar
Karasu, I. and Özekici, S. (1989) NBUE and NWUE properties of increasing Markov processes. J. Appl. Prob. 26, 827834.CrossRefGoogle Scholar
Karlin, S. (1968) Total Positivity 1. Stanford University Press, Palo Alto, CA.Google Scholar
Kijima, M. (1989) Uniform monotonicity of Markov processes and its related properties. J. Operat. Res. Soc. Japan 32, 475490.Google Scholar
Kijima, M. and Nakagawa, T. (1991) A cumulative damage shock model with imperfect preventive maintenance. Naval Res. Logist. 38, 145156.3.0.CO;2-D>CrossRefGoogle Scholar
Li, H. and Shaked, M. (1995) On the first passage times for Markov processes with monotone convex transition kernels. Stoch. Proc. Appl. 58, 205216.CrossRefGoogle Scholar
Pérez-Ocón, R. and Gámiz-Pérez, M. L. (1996) On first-passage times in increasing Markov processes. Statist. Prob. Lett. 26, 199203.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1987) IFRA properties of some Markov processes with general state space. Math. Operat. Res. 12, 562568.CrossRefGoogle Scholar
Shaked, M. and Shanthirumar, J. G. (1988) On the first passage times of pure jump processes. J. Appl. Prob. 25, 501509.CrossRefGoogle Scholar
Shared, M. and Shanthirumar, J. G. (1994) Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Shanthirumar, J. G. (1988) DFR property of first-passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–106.Google Scholar
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