Approximation of reliability for a large system with non-Markovian repair-times. (English) Zbl 0944.62090

The authors consider a system of many components with constant failure rate and general repair rate. Let \(q\) be the probability of failure of the system before complete restoration – this probability can be used to evaluate the reliability of the system when all the components are reliable and easily repairable. The authors derive some bounds on \(q\), and apply it when the repair times have the HNBUE (harmonic new better than used in expectation) ageing property.
Reviewer: M.Shaked (Tucson)


62N05 Reliability and life testing
60K10 Applications of renewal theory (reliability, demand theory, etc.)
90B25 Reliability, availability, maintenance, inspection in operations research
Full Text: DOI EuDML


[1] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing. Holt, Rhineart and Winston, New York ( 1975). Zbl0379.62080 MR438625 · Zbl 0379.62080
[2] J.-L. Bon, Méthodes Mathématiques de Fiabilité, Éditions Masson, Paris ( 1995).
[3] J.-L. Bon and E. Paltanea, Encadrement de la fiabilité d’un système markovien à partir des caractéristiques de ses composants, Actes des XXIXes Journées de Statistique, ASU ( 1997).
[4] K. Chen and Z. He, Reliability bounds for NBUE and NWUE distributions. Acta Mat. Appli. Sinica 4 ( 1989). Zbl0682.62078 MR1003622 · Zbl 0682.62078 · doi:10.1007/BF02006189
[5] D.R. Cox, Renewal Theory, J. Wiley ( 1967). Zbl0168.16106 · Zbl 0168.16106
[6] I.B. Gertsbakh, Asymptotic methods in reliability theory: A review. Adv. in Appl. Prob. 16 ( 1984) 47-175. Zbl0528.60085 MR732135 · Zbl 0528.60085 · doi:10.2307/1427229
[7] D.B. Gnedenko and A.D. Solovyev, Estimation de la fiabilité des systèmes réparables complexes. Teknicheskaia Kibernetika 3 ( 1975) 121-128 (en russe). Zbl0312.90022 MR423732 · Zbl 0312.90022
[8] V.V. Kalashnikov, Geometric sums: Bounds for rare events with applications, Kluwer academic Publishers ( 1997). Zbl0881.60043 MR1471479 · Zbl 0881.60043
[9] G.P. Klimov, Stokastiskie systemi obslujivanie, Nauka (in Russian) ( 1966). MR207064
[10] J. Keilson, Stochastic models in reliability theory, in Teoria dell affidabilita, Proc. Int. School Enrico Fermi, North-Holland ( 1984). Zbl0704.60086 MR891218 · Zbl 0704.60086
[11] I.N. Kovalenko, N.Yu. Kuznetsov and P.A. Pegg, Mathematical Theory of Reliability of Time dependent Systems with Practical Applications, J. Wiley ( 1997). Zbl0899.60074 · Zbl 0899.60074
[12] P. Pamphile, Calcul de fiabilité de grands systèmes hautement fiables, Thèse université Paris-Sud (Orsay), Paris ( 1994).
[13] A.D. Solovyev, Voprosi Matematicheskoi Teorii Nadejnosti, Gnedenko B.V., Ed., Radio i Sviaz, Moscow ( 1983) (in Russian).
[14] A.D. Solovyev and D.G. Konstant, Reliability estimation of a complex renewable system with an unbounded number of repair units. J. Appl. Probab. 28 ( 1991) 833-842. Zbl0746.60087 MR1133791 · Zbl 0746.60087 · doi:10.2307/3214686
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.