Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-16T12:25:36.006Z Has data issue: false hasContentIssue false
  • English
  • Français

New improved bounds for reliability of consecutive-k-out-of-n:F systems

Part of: Operations research and management science Combinatorial probability Special processes

Published online by Cambridge University Press:  14 July 2016

Marco Muselli
Marco Muselli*
Affiliation:
Italian National Research Council
*
∗ Postal address: Institute for Electronic Circuits, Italian National Research Council, 16149 Genoa, Italy. Email address: muselli@ice.ge.cnr.it
Get access Rights & Permissions [Opens in a new window]

Abstract

New bounds are found for the reliability of consecutive-k-out-of-n:F systems with equal component failure probabilities. The expressions involved are simple, thus allowing a direct use in the derivation of theoretical properties.

These bounds can also be employed in numerical computations when the value of n or k is so large that the exact calculation of the reliability is not achievable. Comparisons show that the approximation errors exhibited by these new formulas are lower than those of other widely used bounds.

Keywords

Consecutive k-out-of-n:F systemreliabilitylower boundupper boundapproximation error

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1164 - 1170
Copyright
Copyright © by the Applied Probability Trust 2000 

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.)

References

Arrati, R., Goldstein, L., and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen-Stein Method. Ann. Prob. 17, 9–25.Google Scholar
Barbour, A. D., Chrysaphinou, O., and Roos, M. (1995). Compound Poisson approximation in reliability theory. IEEE Trans. Rel. 44, 398–402.CrossRefGoogle Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Clarendon Press, Oxford.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. Rel. 44, 120–127.CrossRefGoogle Scholar
Chen, L. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Földes, A. (1979). The limit distribution of the length of the longest head-run. Period. Math. Hungar. 10, 301–310.CrossRefGoogle Scholar
Fu, J. C. (1985). Reliability of a large consecutive-k-out-of-n:F system. IEEE Trans. Rel. 34, 127–130.CrossRefGoogle Scholar
Fu, J. C., and Hu, B. (1987). On reliability of large consecutive-k-out-of-n:F system with k – 1step Markov dependence. IEEE Trans. Rel. 36, 75–77.CrossRefGoogle Scholar
Godbole, A. P. (1991). Poisson approximations for runs and patterns of rare events. Adv. Appl. Prob. 23, 851–865.CrossRefGoogle Scholar
Hwang, F. K. (1986). Simplified reliabilities for consecutive-k-out-of-n:F systems. SIAM J. Algebraic Discrete Meth. 7, 258–264.CrossRefGoogle Scholar
Kopociński, B. (1991). On the distribution of the longest success-run in Bernoulli trials. Mat. Stos. 34, 3–13.Google Scholar
Lambiris, M., and Papastavridis, S. G. (1985). Exact reliability formulas for linear & circular consecutive-k-out-of-n:F systems. IEEE Trans. Rel. 34, 124–126.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.CrossRefGoogle Scholar
Muselli, M. (1997). On convergence properties of pocket algorithm. IEEE Trans. Neural Networks 8, 623–629.CrossRefGoogle ScholarPubMed
Muselli, M. (1999). New bounds and approximations for reliability of consecutive-k-out-of-n:F systems. Tech. Rept 4/99, Institute for Electronic Circuits, Italian National Research Council.Google Scholar
Muselli, M. (2000). Useful inequalities for the longest run distribution. Statist. Prob. Lett. 46, 239–249.CrossRefGoogle Scholar
Papastavridis, S. G. (1986). Upper and lower bounds for the reliability of a consecutive-k-out-of-n:F system. IEEE Trans. Rel. 35, 607–610.CrossRefGoogle Scholar
Papastavridis, S. G., and Chrysaphinou, O. (1988). An approximation for large consecutive-k-out-of-n:F systems. IEEE Trans. Rel. 37, 386–387.CrossRefGoogle Scholar