Reliability of Systems with Dependent Components Based on Lattice Polynomial Description

Abstract

Reliability of a system is considered where the components' random lifetimes may be dependent. The structure of the system is described by an associated “lattice polynomial” function. Based on that descriptor, general framework formulas are developed and used to obtain direct results for the cases where a) the lifetimes are “Bayes-dependent,” that is, their interdependence is due to external factors (in particular, where the factor is the “preliminary phase” duration) and b) where the lifetimes' dependence is implied by upper or lower bounds on lifetimes of components in some subsets of the system. (The bounds may be imposed externally based, say, on the connections environment.) Several special cases are investigated in detail.

Keywords:

• Lattice polynomial function
• Reliability
• Semicoherent system

Mathematics Subject Classification:

• Primary 62N05, 90B25
• Secondary 28B15, 94C10

ACKNOWLEDGMENTS

The authors thank the referees for constructive suggestions. Jean-Luc Marichal acknowledges support by the internal research project F1 R-MTH-PUL-09MRDO of the University of Luxembourg.

Notes

A similar expression was obtained for the general case in Section 5 of Ref.^{[ 14 ]}

For the case of exchangeable component lifetimes, this corollary was actually obtained in Ref.^{[ 13 ]} by using the concept of signature.