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Reliability analysis of complex repairable systems by means of marked point processes

Published online by Cambridge University Press:  14 July 2016

Peter Franken*
Affiliation:
Humboldt-Universität, Berlin
Arnfried Streller*
Affiliation:
Humboldt-Universität, Berlin
*
Postal address: Sektion Mathematik, Humboldt-Universität, 1086 Berlin PSF 1297, German Democratic Republic.
Postal address: Sektion Mathematik, Humboldt-Universität, 1086 Berlin PSF 1297, German Democratic Republic.

Abstract

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1980 

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References

Arndt, K. (1977) Calculation of availability of two-unit systems with repair and preventive maintenance by the method of embedded semi-Markov processes (in Russian). Izv. Akad. Nauk SSSR Techn. Kibernet. 7079.Google Scholar
Arndt, K. and Franken, P. (1977) Random point processes applied to availability analysis of redundant systems with repair. IEEE Trans. Reliability R–26, 266269.Google Scholar
Arndt, K. and Franken, P. (1978) Construction of a class of stationary processes with applications in reliability. Zast. Mat. 16, 379393.Google Scholar
Arndt, U. and Franken, P. (1979) A continuity theorem for stationary generalized regenerative processes (in Russian). Izv. Akad. Nauk SSSR Techn. Kibernet., 9497.Google Scholar
Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Belyayev, Yu. K. (1962) Linear-like Markov processes and their applications in reliability theory (in Russian). In Proc. All-Union Conf. Probability and Statistics, Vilnius, 309323.Google Scholar
Brown, M. and Ross, S. (1972) Asymptotic properties of cumulative processes. SIAM J. Appl. Math. 25, 93106.Google Scholar
Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems, 121. Springer-Verlag, Berlin.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, Wiley, New York, 299383.Google Scholar
Franken, P. (1976) A generalization of regenerative processes with application to stable single server queues. Preprint No. 5/76, Humboldt-Universität Berlin, Sektion Mathematik.Google Scholar
Franken, P. (1978) A remark on the stationary availability. Math. Operations forsch. Statist. Ser. Optimization 9, 143144.Google Scholar
Franken, P. and Streller, A. (1978) A general method for calculation of stationary interval reliability of complex systems with repair. Elektron. Informationsverarbeit. Kybernetik 14, 283290.Google Scholar
Franken, P. and Streller, A. (1979) Stationary generalized regenerative processes (in Russian). Teor. Veroyat. Primerien. 24, 7890.Google Scholar
Gaver, D. P. (1963) Time to failure and availability of parallel redundant systems with repair. IEEE Trans. Reliability R–12, 3038.Google Scholar
Gaver, D. P. (1972) Point processes in reliability. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P.A.W., Wiley, New York, 774800.Google Scholar
Gnedenko, B. W., Beljajew, J. K. and Solowjow, A. D. (1968) Mathematische Methoden in der Zuverlässigkeitstheorie, I. Akademie–Verlag, Berlin.Google Scholar
Gnedenko, B. W. and Kowalenko, I. N. (1971) Einführung in die Bedienungstheorie. Akademie–Verlag, Berlin.Google Scholar
Keilson, J. (1974) Monotonicity and convexity in systems survival functions and metabolic disappearance curves. In Reliability and Biometry, SIAM, Philadelphia.Google Scholar
König, D. (1977) Stochastic processes with basic stationary marked point processes (abstract) Adv. Appl. Prob. 9, 440442.Google Scholar
König, ?., Matthes, K. and Nawrotzki, K. (1971) Unempfindlichkeitseigenschaften von Bedienungsprozessen. Supplement to Gnedenko and Kowalenko (1971).Google Scholar
König, D., Rolski, T., Schmidt, V. and Stoyan, D. (1978) Stochastic processes with embedded marked point processes and their application in queueing. Math. Operations forsch. Statist., Ser. Optimization 9, 125141.Google Scholar
Nawrotzki, K. (1975) Markovian random marked sequences and their application in queueing theory (in Russian). Math. Operations forsch. Statist. 6, 445477.Google Scholar
Nawrotzki, K. (1978) Einige Bemerkungen zur Verwendung der Palmschen Verteilung in der Bedienungstheorie. Math. Operations forsch. Statist. Ser. Optimization 9, 241253.CrossRefGoogle Scholar
Pyke, R. and Schaufele, R. (1966) The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Statist. 37, 14391462.Google Scholar
Ross, S. (1975) On the calculation of asymptotic system reliability characteristics. In Reliability and Fault Tree Analysis, SIAM, Philadelphia, 331350.Google Scholar
Streller, A. (1979) On the stationary interval reliability of two-unit systems with repair and preventive maintenance (in Russian). Izv. Akad. Nauk SSSR Techn. Kibernet. 98103.Google Scholar