On the application and extension of system signatures in engineering reliability.

*(English)*Zbl 1153.90386Summary: Following a review of the basic ideas in structural reliability, including signature-based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes \(m < n\) with exchangeable component lifetimes, it is shown that the distribution of an \(m\)-component system’s lifetime can be written as a mixture of the distributions of \(k\)-out-of-\(n\) systems. When the system has \(n\) components, the vector of coefficients in this mixture representation is precisely the signature of the system defined by the second author [IEEE Trans. Reliabil. R-34, 69–72 (1985)]. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes.

##### MSC:

90B25 | Reliability, availability, maintenance, inspection in operations research |

##### Keywords:

coherent system \(k\)-out-of-\(n\) system; exchangeability; order statistics; stochastic ordering; hazard rate ordering; likelihood ratio ordering
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\textit{J. Navarro} et al., Nav. Res. Logist. 55, No. 4, 313--327 (2008; Zbl 1153.90386)

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