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On a Markov chain approach for the study of reliability structures

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

M. V. Koutras
M. V. Koutras*
Affiliation:
University of Athens
*
∗Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece.
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Abstract

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

Keywords

MARKOV IMBEDDINGMARKOV CHAINTRANSITION PROBABILITY MATRIXRELIABILITY GENERATING FUNCTIONCONSECUTIVE SYSTEMS

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 357 - 367
Copyright
Copyright © Applied Probability Trust 1996 

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References

Aki, S. and Hirano, K. (1989) Estimation of parameters in discrete distributions of order k. Ann. Inst. Statist. Math. 41, 47–61.CrossRefGoogle Scholar
Barlow, R. E. and Heidtmann, K. D. (1984) Computing k-out-of-n system reliability. IEEE Trans. Reliab. R33, 322–323.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Reinhart and Winston, New York.Google Scholar
Chao, M. T. and Fu, J. C. (1989) A limit theorem of certain repairable systems. Ann. Inst. Statist. Math. 41, 809–818.CrossRefGoogle Scholar
Chao, M. T. and Fu, J. C. (1991) The reliability of large series systems under Markov structure. Adv. Appl. Prob. 23, 894–908.CrossRefGoogle Scholar
Chao, M. T., Fu, J. C. and Koutras, M. V. (1995) Survey of reliability studies of consecutive-k-out-of-n: F and related systems. IEEE Trans. Reliab. R44, 120–127.CrossRefGoogle Scholar
Charalambides, Ch. A. (1986) On discrete distributions of order k. Ann. Inst. Statist. Math. 38, 557–568.CrossRefGoogle Scholar
Fu, J. C. (1986) Reliability of consecutive-k-out-of-n: F systems with (k-1)-step Markov dependence. IEEE Trans. Reliab. R35, 602–606.CrossRefGoogle Scholar
Fu, J. C. (1992) Poisson convergence in reliability of a large linearly connected system as related to coin tossing. Statist. Sinica. 3, 261–275.Google Scholar
Fu, J. C. and Koutras, M. V. (1994) Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058.CrossRefGoogle Scholar
Fu, J. C. and Lou, W. Y. (1991) On reliabilities of certain large linearly connected engineering systems. Statist. Prob. Lett. 12, 291–296.CrossRefGoogle Scholar
Godbole, A. P. (1990) Degenerate and Poisson convergence criteria for success runs. Statist. Prob. Lett. 10, 247–255.CrossRefGoogle Scholar
Godbole, A. P. (1991) Poisson approximations for runs and patterns of rare events. Adv. Appl. Prob. 23, 851–865.CrossRefGoogle Scholar
Godbole, A. P. (1993) Approximate reliabilities of m-consecutive-k-out-of-n: Failure systems. Statist. Sinica 3, 321–327.Google Scholar
Goulden, I. P. (1987) Generating functions and reliabilities for consecutive-k-out-of-n: F systems. Util. Math. 32, 141–147.Google Scholar
Greenberg, I. (1970) The first occurrence of n successes in N trials. Technometrics 12, 627–634.CrossRefGoogle Scholar
Griffith, W. S. (1986) On consecutive-k-out-of-n: failure systems and their generalizations. Reliability and Quality Control , pp. 157–165. ed. Basu, A. P. Elsevier, Amsterdam.Google Scholar
Iosifescu, M. (1980) Finite Markov Processes and their Applications. Wiley, New York.Google Scholar
Kontoleon, J. M. (1980) Reliability determination of a r-successive-out-of-n: F system. IEEE Trans. Reliab. R29, 437.CrossRefGoogle Scholar
Kounias, S. and Sfakianakis, M. (1991) The reliability of a linear system and its connection with the generalized birthday problem. Statist. Applic. 3, 531–543.Google Scholar
Papastavridis, S. G. (1990) m-consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R39, 386–388.CrossRefGoogle Scholar
Papastavridis, S. G. and Koutras, M. V. (1993) Consecutive-k-out-of-n systems. New Trends in System Reliability Evaluation , pp. 228–248. ed. Misra, K. B. Elsevier, Amsterdam.CrossRefGoogle Scholar
Philippou, A. N. and Makri, F. S. (1986) Success, runs and longest runs. Statist. Prob. Lett. 1, 171–175.CrossRefGoogle Scholar
Saperstein, B. (1973) On the occurrences of n successes within N Bernoulli trials. Technometrics 15, 809–818.Google Scholar
Saperstein, B. (1975) Note on a clustering problem. J. Appl. Prob. 12, 629–632.CrossRefGoogle Scholar
Seneta, E. (1981) Non-Negative Matrices and Markov Chains. 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Wu, J. S. and Chen, R. J. (1994a) An algorithm for computing the reliability of a weighted-k-out-of-n system. IEEE Trans. Reliab. R43, 327–328.Google Scholar
Wu, J. S. and Chen, R. J. (1994b) Reliability of consecutive-weighted-k-out-of-n: F systems. In Runs and Patterns in Probability: Selected Papers. pp. 205–211. ed. Godbole, A. and Papastavridis, S. Kluwer, Amsterdam.CrossRefGoogle Scholar