Preservation of reliability classes under the formation of coherent systems

Abstract

The preservation of reliability aging classes under the formation of coherent systems is a relevant topic in reliability theory. Thus, it is well known that the new better than used class is preserved under the formation of coherent systems with independent components. However, surprisingly, the increasing failure rate class is not preserved in the independent and identically distributed case, that is, the components may have the (negative) aging increasing failure rate property, but the system does not have this property. In this paper, we study conditions for the preservation of the main reliability classes under the formation of general coherent systems. These results can be applied both for systems with independent or dependent components. We consider both the case of systems with identically distributed components and the case of systems with components having different distributions. Copyright © 2013 John Wiley & Sons, Ltd.

Comparisons of mixed systems with decreasing failure rate component lifetimes using dispersive order

“…This property is not always true (see Navarro and Shaked ). Also, similar results can be derived for systems with dependent identically distributed components by using the conditions given in Navarro et al , (Theorem 2.4) for the hazard rate order and the conditions given in Navarro et al , for the preservation of the DFR class under the formation of the systems.…”

“…□ Remark In the proof of Theorem , a system with just one unit ( X ) was used as a ‘facilitator’ to obtain comparisons in the dispersive order when the component lifetimes are DFR. From Theorem and the results given in Navarro et al , , one can use as a ‘facilitator’ any mixed system which preserves the DFR class such as the series systems with i.i.d. components. Remark Under Condition (iii) in Theorem , that is, when the component lifetimes are i.i.d.…”

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

“…Remark Still for a binary system with binary components, an alternative representation of the reliability function can be expressed in terms of a distorted distribution, or aggregation, of the marginal survival functions of the different components. The form of such a function of course depends on both the system's structure function ϕ and on the survival copula K of the vector of components' lifetimes (see, in particular, other works). By comparing, for any fixed ϕ , such a copula‐based representation of reliability with the one based on a potential extension of formula , one may establish a relation between the probabilities and on one side and the copula K on the other side.…”

Sharp bounds for the reliability of systems and mixtures with ordered components

“…Hence, by applying the inclusion–exclusion formula to (1), we infer (see, e.g., ) that the system reliability function can be written as for all , where is a continuous increasing function such that and , where and are the n ‐dimensional vectors and , respectively. is called a generalized distortion (or a continuous aggregation function ).…”