Weak aging properties for coherent systems with statistically dependent component lifetimes

Abstract

As a relevant topic in reliability theory, the preservation of aging properties under the formation of various coherent structures contributes to improving system performance through better structure design and more effective system maintenance. The classical research in this line usually focuses upon coherent systems with independent component lifetimes. Recently, some authors discussed the preservation of IFR, NBU, and DMRL in the setting of dependent component lifetimes. This paper further investigates sufficient conditions for coherent systems with dependent component lifetimes to preserve aging properties including NBUC, NBU (2), DMRL, and their dual versions. Some examples are presented to illustrate coherent structures and typical copula functions fulfilling the present sufficient conditions as well.

On relevation redundancy to coherent systems at component and system levels

“…In addition, such an assumption integrates both the system structure and the statistical dependence among component lifetimes. It is worth pointing out the following conditions due to [13], which are sufficient to this assumption:…”

“…In addition, such an assumption integrates both the system structure and the statistical dependence among component lifetimes. It is worth pointing out the following conditions due to [13], which are sufficient to this assumption: a series system with system component lifetimes linked by Archimedean survival copula such that is log-concave and is log-concave, e.g. with ; a parallel system with component lifetimes linked by Archimedean copula such that is log-convex and is log-convex, e.g.…”

On relevation transform involved with statistical dependence between two component lifetimes

“…Given that the active redundancy survives the primary component, the system lifetime is represented as the relevation $$$$ X\curlyvee S=X+{S}_X, $$$$ where $$$ {S}_X=\left[S-X\;|\;S>X\right] $$$ denotes the residual lifetime of $$$ S $$$ at the random time $$$ X $$$. For more on the residual lifetime at random time please refer to Gupta et al, 31 Li and Li, 32 Li and Fang, 33 Li and Lu, 34 Li and Zuo 35 . It should be remarked here that the operation “$$$ \curlyvee $$$” is clearly not commutative.…”

“…denotes the residual lifetime of S at the random time X. For more on the residual lifetime at random time please refer to Gupta et al, 31 Li and Li, 32 Li and Fang, 33 Li and Lu, 34 Li and Zuo. 35 It should be remarked here that the operation "⋎" is clearly not commutative.…”

Resolving an old problem on the preservation of the IFR property under the formation of -out-of- systems with discrete distributions

“…components. Preservation properties for NBUE/NWUE classes can be seen in [15] and [16]. Recent preservation results for systems under the exponential distribution were obtained in [24].…”