Reliability, signature, and relative quality functions of systems under time‐homogeneous load‐sharing models

Abstract

For a coherent binary system made of binary components, we consider the assumption that the components' lifetimes are distributed according to a time‐homogeneous, load‐sharing model. Such models are characterized in terms of the so‐called multivariate conditional hazard rate functions. We aim to point out some related properties of the notions of signature, relative quality functions, and reliability functions. On this purpose, we preliminarily collect all the necessary background and review some related literature. This paper concludes with a discussion, also containing some hints for future work.

Stochastic Precedence and Minima Among Dependent Variables

“…Whence the thesis is obtained by recalling equations (32), (33) and by integrating the functions in (34) and (35) with respect to the variable ξ.…”

“…See e.g. [27] and also the reviews within the more recent papers [29,32,33]. This tool is different, but equivalent to the one based on the joint density function.…”

“…Several theoretical and applied aspects of such survival models have been studied in different fields and, in particular, in the reliability literature. See e.g [33]. and references cited therein.…”

The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay

“…Stochastic models with density f d in (5) are considered among others in [24], and are known as timehomogeneous load sharing models with reference to the reliability context. See [25] for a recent account on load sharing systems. We point out that, under the condition that i → r i is non-decreasing, they form a subclass of systems weakened by failure, as shown in [19].…”

On component failure in coherent systems with applications to maintenance strategies

“…The representation (1) is valid under the weaker condition that the component lifetimes are exchangeable (see Navarro and Rychlik [23]). For references on the signature-based properties of system lifetime, we refer the reader to [5], [11], [17], [20], [24], [25], [32], [36], [38], and [41]. For a recent work on different methods and algorithms for computing system signature, see Reed [29] and references therein.…”