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SYSTEM RELIABILITY AND WEIGHTED LATTICE POLYNOMIALS

Published online by Cambridge University Press:  27 May 2008

Alexander Dukhovny  and
Jean-Luc Marichal
Alexander Dukhovny
Affiliation:
Mathematics DepartmentSan Francisco State UniversitySan Francisco, CA 94132 E-mail: dukhovny@math.sfsu.edu
Jean-Luc Marichal
Affiliation:
Mathematics Research UnitUniversity of LuxembourgL-1511 Luxembourg, Luxembourg E-mail: jean-luc.marichal@uni.lu
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Abstract

The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in the case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of “indicator” variables. A connection is studied between Y and order statistics of the set of arguments.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 3 , July 2008 , pp. 373 - 388
Copyright
Copyright © Cambridge University Press 2008

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References

1David, H. & Nagaraja, H. (2003). Order statistics, 3rd ed.Chichester, UK: Wiley.CrossRefGoogle Scholar
2Dukhovny, A. (2007). Lattice polynomials of random variables. Statistics and Probability Letters, 77(10): 989–994.CrossRefGoogle Scholar
3Grabisch, M., Marichal, J.-L. & Roubens, M. (2000). Equivalent representations of set functions. Mathematics of Operations Research 25(2): 157–178.CrossRefGoogle Scholar
4Marichal, J.-L. (2006). Cumulative distribution functions and moments of lattice polynomials. Statistics and Probability Letters, 76(12): 1273–1279.CrossRefGoogle Scholar
5Marichal, J.-L. (2008). Weighted lattice polynomials. http://arxiv.org/abs/0706.0570://arxiv.org/abs/0706.0570.Google Scholar
6Marichal, J.-L. (2008). Weighted lattice polynomials of independent random variables. Discrete Applied Mathematics 156(5): 685–694.CrossRefGoogle Scholar
7Ovchinnikov, S. (1996). Means on ordered sets. Mathematical Social Sciences, 32(1): 39–56.CrossRefGoogle Scholar
8Rausand, M. & Høyland, A. (2004). System reliability theory, 2nd ed.Hoboken, NJ: Wiley–Interscience.Google Scholar
9Sugeno, M. (1974). Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology, Tokyo.Google Scholar
10Sugeno, M. (1977). Fuzzy measures and fuzzy integrals—a survey. In Gupta, M.M., Saridis, G.N. & Gaines, B.R. (eds.), Fuzzy automata and decision processes New York: North-Holland, pp. 89–102.Google Scholar