On the application and extension of system signatures in engineering reliability

Abstract

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 in Samaniego, IEEE Trans Reliabil R-34 (1985) 69-72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes.

Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

“…A mixed system of order n is a stochastic mixture of coherent systems of order n and may be realized by selecting a system at random according to a fixed probability distribution over the class of coherent systems of order n (see [4]). Navarro et al [20] proved that the coherent systems with k components are equal in law to mixed systems with n (n > k) components. For example, the system with one component, X 1,1 , is equal in distribution (see [20]) to the mixed system with signature p = ( ) and, hence, under the condition that it is not working at time t, (2) holds with the vector…”

“…Navarro et al [20] proved that the coherent systems with k components are equal in law to mixed systems with n (n > k) components. For example, the system with one component, X 1,1 , is equal in distribution (see [20]) to the mixed system with signature p = ( ) and, hence, under the condition that it is not working at time t, (2) holds with the vector…”

“…component lifetimes can be written as a mixture of distribution functions of order statistics (see [11] and [23]). Navarro et al [18] (see also [16] and [20]) observed that it holds when the components of the system are absolutely continuous and exchangeable (i.e. the joint survival function R(x 1 , .…”

Signatures of Coherent Systems Built with Separate Modules

“…. , T n are exchangeable; in this respect see, in particular, [11], [12], [14], [16], [17], and [21].…”

“…. ,S Nm−1 can then be obtained by comparing (16) with the expression of the reliability of the recurrent system written in terms of its basic components (recall (10)). Namely, we can writē…”