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Improved availability bounds for binary and multistate monotone systems with independent component processes

Part of: Survival analysis and censored data Operations research and management science

Published online by Cambridge University Press:  15 September 2017

Jørund Gåsemyr  and
Bent Natvig
Jørund Gåsemyr*
Affiliation:
University of Oslo
Bent Natvig*
Affiliation:
University of Oslo
*
* Postal address: Department of Mathematics, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway.
* Postal address: Department of Mathematics, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway.
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Abstract

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities of the system in an interval, i.e. the probabilities that the system performs above the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities exactly, but if the component performance processes are independent, it is possible to construct lower bounds based on the component availabilities to the different levels over the interval. In this paper we show that by treating the component availabilities over the interval as if they were availabilities at a single time point, we obtain an improved lower bound. Unlike previously given bounds, the new bound does not require the identification of all minimal path or cut vectors.

Keywords

Availability boundbinary monotone systemflow network systemminimal cut vectorminimal path vectormultistate monotone system

MSC classification

Primary: 62N05: Reliability and life testing 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 750 - 762
Copyright
Copyright © Applied Probability Trust 2017 

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References

Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Google Scholar
Block, H. W. and Savits, T. H. (1982). A decomposition for multistate monotone systems. J. Appl. Prob. 19, 391–402. Google Scholar
Bodin, L. D. (1970). Approximations to system reliability using a modular decomposition. Technometrics 12, 335–344. Google Scholar
Butler, D. A. (1982). Bounding the reliability of multistate systems. Operat. Res. 30, 530–544. Google Scholar
Esary, J. D. and Proschan, F. (1970). A reliability bound for systems of maintained, interdependent components. J. Amer. Statist. Assoc. 65, 329–338. Google Scholar
Ford, L. R., Jr. and Fulkerson, D. R. (1956). Maximal flow through a network. Canadian J. Math. 8, 399–404. Google Scholar
Funnemark, E. and Natvig, B. (1985). Bounds for the availabilities in a fixed time interval for multistate monotone systems. Adv. Appl. Prob. 17, 638–665. Google Scholar
Gåsemyr, J. (2012). Bounds for the availabilities of multistate monotone systems based on decomposition into stochastically independent modules. Adv. Appl. Prob. 44, 292–308. CrossRefGoogle Scholar
Gåsemyr, J. and Natvig, B. (2005). Probabilistic modelling of monitoring and maintenance of multistate monotone systems with dependent components. Methodol. Comput. Appl. Prob. 7, 63–78. Google Scholar
Huseby, A. B. and Natvig, B. (2013). Discrete event simulation methods applied to advanced importance measures of repairable components in multistate network flow systems. Reliab. Eng. System Safety 119, 186–198. CrossRefGoogle Scholar
Huseby, A. et al. (2010). Advanced discrete event simulation methods with application to importance measure estimation in reliability. In Discrete Event Simulations, ed. A. Goti, InTech, pp. 205–222. Google Scholar
Natvig, B. (1980). Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components. Adv. Appl. Prob. 12, 200–221. Google Scholar
Natvig, B. (1982). Two suggestions of how to define a multistate coherent system. Adv. Appl. Prob. 14, 434–455. Google Scholar
Natvig, B. (1986). Improved upper bounds for the availabilities in a fixed time interval for multistate monotone systems. Adv. Appl. Prob. 18, 577–579. Google Scholar
Natvig, B. (1993). Strict and exact bounds for the availabilities in a fixed time interval for multistate monotone systems. Scand. J. Statist. 20, 171–175. Google Scholar
Natvig, B. (2011). Multistate Systems Reliability Theory with Applications. John Wiley, Chichester. Google Scholar