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Generalized reliability bounds for coherent structures

Part of: Special processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Michael V. Boutsikas  and
Markos V. Koutras
Michael V. Boutsikas*
Affiliation:
University of Athens
Markos V. Koutras*
Affiliation:
University of Athens
*
∗Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece
∗Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece
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Abstract

In this article we introduce generalizations of several well-known reliability bounds. These bounds are based on arbitrary partitions of the family of minimal path or cut sets of the system and can be used for approximating the reliability of any coherent structure with i.i.d. components. An illustration is also given of how the general results can be applied for a specific reliability structure (two-dimensional consecutive-k1 x k2-out-of-n1x n2 system) along with extensive numerical calculations revealing that, in most cases, the generalized bounds perform better than other available bounds in the literature for this system.

Keywords

Reliability boundscut setspath setsassociationbinary coherent structurestwo-dimensional consecutive-k1 × k2-out-of-n1 × n2 system

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 778 - 794
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

∗∗∗

Research supported by the National Scholarship Foundation of Greece.

References

Barbour, A. D., Chryssaphinou, O., and Roos, M. (1996). Compound Poisson approximation in systems reliability. Naval Res. Logist. 43, 251–264.3.0.CO;2-9>CrossRefGoogle Scholar
Barlow, R. E., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Boutsikas, M. V., and Koutras, M. V. (2000). A bound for the distribution of the sum of discrete associated or NA random variables. To appear in Ann. Appl. Prob.Google 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. Reliability 44, 120–127.CrossRefGoogle Scholar
Esary, J. D., and Proschan, F. (1963). Coherent structures of non-identical components. Technometrics 5, 191–209.CrossRefGoogle Scholar
Esary, J. D., Proschan, F., and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Fu, J. C., and Koutras, M. V. (1994). Poisson approximations for two-dimensional patterns. Ann. Inst. Statist. Math. 46, 179–192.CrossRefGoogle Scholar
Fu, J. C., and Koutras, M. V. (1995). Reliability bounds for coherent structures with independent components. Statist. Prob. Lett. 22, 137–148.CrossRefGoogle Scholar
Glaz, J., and Naus, J. I. (1991). Tight bounds and approximations for scan statistic probabilities for discrete data. Ann. Appl. Prob. 1, 306–318.CrossRefGoogle Scholar
Godbole, A. P., Potter, L. K., and Sklar, J. K. (1998). Improved upper bounds for the reliability of d-dimensional consecutive-k-out-of-n:F systems. Naval Res. Logist. 45, 219–230.3.0.CO;2-B>CrossRefGoogle Scholar
Koutras, M. V., and Papastavridis, S. G. (1993). Application of the Chen–Stein method for bounds and limit theorems in the reliability of coherent structures. Naval Res. Logist. 40, 617–631.3.0.CO;2-4>CrossRefGoogle Scholar
Koutras, M. V., Papastavridis, S. G., and Papadopoulos, G. K. (1993). Reliability of 2-dimensional consecutive-k-out-of-n:F systems. IEEE Trans. Reliability 42, 658–661.CrossRefGoogle Scholar
Koutras, M. V., Papastavridis, S. G., and Petakos, K. I. (1996). Bounds for coherent reliability structures. Statist. Prob. Lett. 26, 431–448.CrossRefGoogle Scholar
Makri, F. S., and Psillakis, Z. M. (1997). Bounds for reliability of k-within 2-dimensional consecutive-r-out-of-n failure systems. Microelectronics and Reliability 37, 1217–1224.CrossRefGoogle Scholar
Papastavridis, S. G., and Koutras, M. V. (1993). Consecutive-k-out-of-n systems. In New Trends in System Reliability Evaluation, ed. Misra, K. B. Elsevier, Amsterdam, pp. 228–248.CrossRefGoogle Scholar
Salvia, A., and Lasher, W. (1990). 2-dimensional consecutive-k-out-of-n:F models. IEEE Trans. Reliability 39, 382–385.CrossRefGoogle Scholar
Yamamoto, H., and Miyakawa, M. (1995). Reliability of a linear connected-(r, s)-out-of-(m, n):F lattice system. IEEE Trans. Reliability 44, 333–336.CrossRefGoogle Scholar