Hostname: page-component-5c6d5d7d68-ckgrl Total loading time: 0 Render date: 2024-08-21T13:19:11.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic methods in reliability theory: a review

Published online by Cambridge University Press:  01 July 2016

I. B. Gertsbakh
I. B. Gertsbakh*
Affiliation:
Ben Gurion University of the Negev
*
∗Postal address: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beersheva 84 105, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

Section 1 of this paper reviews some works related to reliability evaluation of systems without repair. The assumption that element failure rates are low enables one to obtain an expression for the main term in the asymptotic representation of system reliability function. Section 2 is devoted to repairable systems. The main index of interest in reliability is the time to the first system failure. A typical situation in reliability is that the repair time is much smaller than the element lifetime. This ‘fast repair' property leads to an interesting phenomenon, that for many renewable systems the time to system failure converges in distribution, under appropriate norming, to an exponential random variable. Some basic theorems explaining this fact are presented and a series of typical examples is considered. Special attention is paid to reviewing the works describing the exponentiality phenomenon in birth-and-death processes. Some important aspects of computing the normalizing constants are considered, among them the role played by the so-called ‘main event'. Section 2 also reviews various bounds on the deviation from exponentiality. Section 3 gives brief comments on some works and techniques related to asymptotic reliability analysis. In particular, attention is paid to the works presenting upper and lower bounds on the reliability function.

A considerable part of this review is based on sources originally published in Russian.

Keywords

RELIABILITYEXPONENTIALITYFAST REPAIRSMALL PARAMETERMAIN EVENTREPAIRABLE SYSTEMSASYMPTOTICS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 1 , March 1984 , pp. 147 - 175
Copyright
Copyright © Applied Probability Trust 1984 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was carried out during the author's visit to the University of Delaware in 1981–82 and was supported in part by the National Science Foundation under Grant No. ENG-7908351 and the Air Force Office of Scientific Research under Grant No. AFOSR-77–3236.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt Rinehart and Winston, New York.Google Scholar
Brown, M. (1975) The first passage time distribution for parallel exponential systems with repair. In Reliability and Fault Tree Analysis, SIAM, Philadelphia, 365–396.Google Scholar
Brown, M. (1983) Approximating IMRL distributions by exponential distributions, with applications to first passage time. Ann. Prob. 11, 419–427.Google Scholar
Burtin, Yu. D. and Pittel, B. G. (1972) Asymptotic estimates of the reliability of a complex system. Engineering Cybernetics 10(3), 445–451.Google Scholar
Dukhovny, I. M. and Koenigsberg, E. (1981) Invariance properties of queueing networks and their application to computer/communications systems. INFOR 19, 185–204.Google Scholar
Epstein, B. (1969) Mathematical Models for Systems Reliability. Publishing House of the Student Association, Technion, Haifa.Google Scholar
Epstein, B. (1971) Sojourn time distributions for two-unit repairable systems. In Developments in Operations Research, ed. Avi-Itzhak, B., Gordon and Breach, New York, 1, 183–190.Google Scholar
Epstein, B. and Hosford, J. (1960) Reliability of some two unit redundant systems. In. Proc. 6th National Symp. Reliability and Quality Control, 466–476.Google Scholar
Genis, Ya. G. (1978) The convergence of a certain class of distribution functions to an exponential distribution function in reliability and queueing problems. Automation and Remote Control 39(6), 94–101.Google Scholar
Gertsbakh, I. B. (1982) Confidence limits for highly reliable coherent systems with exponentially distributed component life. J. Amer. Statist. Assoc. 77, 673–679.Google Scholar
Gnedenko, B. V., Belyayev, Yu. K. and Solovyev, A. D. (1969) Mathematical Methods in Reliability Theory. Academic Press, New York.Google Scholar
Gnedenko, D. B. and Solovyev, A. D. (1974) A general model for standby with renewal. Engineering Cybernetics 12(6), 82–86.Google Scholar
Gnedenko, D. B. and Solovyev, A. D. (1975) Estimation of the reliability of complex renewable systems. Engineering Cybernetics 13(3), 89–96.Google Scholar
Heyde, C. C. and Leslie, J. R. (1976) On moment measures of departure from the normal and exponential laws. Stoch. Proc. Appl. 4, 317–328.Google Scholar
Keilson, J. (1966) A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 886–870.Google Scholar
Keilson, J. (1975) Systems of independent Markov components and their transient behavior. In Reliability and Fault Tree Analysis, SIAM, Philadelphia, 351–364.Google Scholar
Keilson, J. (1979) Markov Chain Models–Rarity and Exponentiality. Springer-Verlag, Berlin.Google Scholar
Kovalenko, I. N. (1975) Investigation and Analysis of Reliability of Complex Systems (in Russian). Naukova Dumka, Kiev.Google Scholar
Kovalenko, I. N. (1976) Analyticostatistical method for calculating the characteristics of highly reliable systems. Cybernetics 11(6), 895–907.Google Scholar
Kovalenko, I. N. (1977) Limit theorems for reliability theory. Cybernetics 12(6), 902–914.Google Scholar
Kovalenko, I. N. (1980) Asymptotic state enlargement for random processes. Cybernetics 15(6), 876–886.Google Scholar
Kozlov, B. A. and Ushakov, I. A. (1975) Handbook of Reliability Computation (in Russian). Sovietskoje Radio, Moscow.Google Scholar
Locks, M. O. (1980) Recursive disjoint products, inclusion-exclusion and min-cut approximations. IEEE Trans. Reliability R-29, 368–371.Google Scholar
Lomonosov, ?. V. and Polesskii, V. P. (1971) An upper bound for the reliability of information network. Problems of Information Transmission 7(4), 337–339.Google Scholar
Lomonosov, ?. V. and Polesskii, V. P. (1972) Lower bound for network reliability. Problems of Information Transmission 8(2), 118–123.Google Scholar
McGregor, M. A. (1963) Approximation formulas for reliability with repair. IEEE Trans. Reliability R-12, 64–92.Google Scholar
Ovchinnikov, V. N. (1976) Asymptotic behavior of the time to first failure in the model of nonhomogeneous redundancy with rapid repair. Engineering Cybernetics 14(2), 76–83.Google Scholar
Pavlov, I. V. and Ushakov, I. A. (1978) The asymptotic distribution of the time until a semi-Markov process gets out of a kernel. Engineering Cybernetics 16(5), 68–72.Google Scholar
Sakhobov, O. and Solvyev, A. D. (1977) Two-sided estimates of reliability in a general standby model with one renewal unit. Engineering Cybernetics 15(4), 58–63.Google Scholar
Sevastyanov, B. A. (1957) An ergodic theorem for Markov processes and its application to telephone lines with failure. Theory Prob. Appl. 2, 104–112.CrossRefGoogle Scholar
Shakhbazov, A. A. (1982) Limiting distribution of the time of the first entrance for semi-Markov processes and its application to reliability theory. Engineering Cybernetics 20(3).Google Scholar
Solovyev, A. D. (1970) Standby with rapid renewal. Engineering Cybernetics 8(1), 49–64.Google Scholar
Solovyev, A. D. (1971) Asymptotic behavior of the time of first occurence of a rare event. Engineering Cybernetics 9(6), 1038–1048.Google Scholar
Solovyev, A. D. (1972) Asymptotic distribution of the moment of first crossing of a high level by a birth and death process. Proc. 6th Berkeley Symp. Math Statist. Prob. 3, 71–86.Google Scholar
Solovyev, A. D. and Sakhobov, O. (1976) Two-sided reliability estimates of renewal systems (in Russian). Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, No. 5.Google Scholar
Solovyev, A. D. and Zaitsev, V. A. (1975) Standby with incomplete renewal. Engineering Cybernetics 13(1), 58–62.Google Scholar
Vinogradov, O. P. (1968) Limiting distributions for the moment of first loss of an order in a single-line queueing system with a limited number of positions. Math. Notes Acad. Sci. USSR 3(5), 345–348.Google Scholar
Vinogradov, O. P. (1974) Asymptotic distribution of the instant of first loss of an order in the case of fast servicing. Engineering Cybernetics 12(6), 86–92.Google Scholar
Waksman, Z. (1982) Flow-in-network technique for asymptotic reliability investigation of two-terminal network. Submitted for publication.Google Scholar
Zaitsev, V. A. and Solovyev, A. D. (1975) Redundancy of complex systems. Engineering Cybernetics 13(4), 66–76.Google Scholar