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Monotone and associated Markov chains, with applications to reliability theory

Published online by Cambridge University Press:  14 July 2016

Bo Henry Lindqvist
Bo Henry Lindqvist*
Affiliation:
The Norwegian Institute of Technology
*
∗Postal address: Division of Mathematical Statistics, The Norwegian Institute of Technology, N-7034 Trondheim-NTH, Norway.
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Abstract

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

Keywords

NON-HOMOGENEOUS MARKOV CHAINSPARTIALLY ORDERED STATE SPACESSTOCHASTIC MONOTONICITYASSOCIATIONMARKOV MODELINGMULTISTATE SYSTEMSREPAIRABLE SYSTEMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 3 , September 1987 , pp. 679 - 695
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

This research is supported by the Royal Norwegian Council for Scientific and Industrial Research.

References

[1] Arjas, E. (1981) A stochastic process approach to multivariate reliability systems: Notions based on conditional stochastic order. Math. Operat. Res. 6, 263–276.CrossRefGoogle Scholar
[2] Arjas, E. and Norros, I. (1984) Life lengths and association: A dynamic approach. Math. Operat. Res. 9, 151–158.CrossRefGoogle Scholar
[3] Bergman, B. (1985) On reliability theory and its applications. Scand. J. Statist. 12, 1–41.Google Scholar
[4] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[5] Brown, M. and Chaganty, N. R. (1983) On the first passage time distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.CrossRefGoogle Scholar
[6] Cox, J. T. (1984) An alternate proof of a correlation inequality of Harris. Ann. Prob. 12, 272–273.CrossRefGoogle Scholar
[7] Daley, D. J. (1968) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.CrossRefGoogle Scholar
[8] Esary, J. D. and Proschan, F. (1970) A reliability bound for systems of maintained, interdependent components. J. Amer. Statist. Assoc. 65, 329–338.CrossRefGoogle Scholar
[9] Esary, J. D. and Proschan, F. (1972) Relationships among some notions of bivariate dependence. Ann. Math. Statist. 43, 651–655.CrossRefGoogle Scholar
[10] Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
[11] Funnemark, E. and Natvig, B. (1985) Bounds for the availabilities in a fixed time interval for multistate monotone systems. Adv. Appl. Prob. 17, 638–665.CrossRefGoogle Scholar
[12] Harris, T. E. (1974) Contact processes on a lattice. Ann. Prob. 2, 969–988.CrossRefGoogle Scholar
[13] Harris, T. E. (1977) A correlation inequality for Markov processes in partially ordered state spaces. Ann. Prob. 5, 451–454.CrossRefGoogle Scholar
[14] Hatoyama, Y. (1984) On Markov maintenance problems. IEEE Trans. Reliability 33, 280–283.CrossRefGoogle Scholar
[15] Hjort, N. L., Natvig, B. and Funnemark, E. (1985) The association in time of a Markov process with application to multistate reliability theory. J. Appl. Prob. 22, 473–479.CrossRefGoogle Scholar
[16] Iosifescu, M. (1980) Finite Markov Processes and Their Applications. Wiley, Bucuresti.Google Scholar
[17] Kamae, T., Krengel, U. and O'Brien, G. C. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
[18] Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.CrossRefGoogle Scholar
[19] Lindqvist, B. H. (1985) Association of probability measures on partially ordered spaces. Statistics No 2/85, Dept, of Math., Univ. of Trondheim.Google Scholar
[20] Lindqvist, B. H. (1986) Restricted maximum likelihood estimation of failure- and repair rates for certain systems of associated components. Statistics No 2/86, Dept, of Math., Univ. of Trondheim.Google Scholar
[21] Natvig, B. (1980) Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components. Adv. Appl. Prob. 12, 200–221.CrossRefGoogle Scholar
[22] Norros, I. (1985) Systems weakened by failures. Stoch. Proc. Appl. 20, 181–196.CrossRefGoogle Scholar
[23] Norros, I. (1986) A compensator representation of multivariate life length distributions, with applications, Scand. J. Statist. 13, 99–112.Google Scholar
[24] Ross, S. M. (1984) A model in which component failure rates depend on the working set. Naval Res. Log. Quart. 31, 301–308.CrossRefGoogle Scholar
[25] Sagsveen, A. (1982) Interaction in Markov chains. , Dept, of Math., University of Oslo.Google Scholar
[26] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Akademie-Verlag/Wiley, Berlin.Google Scholar