Relationships Between Importance Measures and Redundancy in Systems With Dependent Components

Abstract

The paper shows the connections between some importance indices for the components in an engineering coherent system and the performance of the system obtained when a redundancy mechanism is applied to a specific component. A copula approach is used to model the dependency among the components. This approach includes the popular case of independent components. Under some assumptions, it is proved that if component i is more important than component j, then the system obtained by applying a redundancy procedure to the ith component is better, under different stochastic criteria, than that obtained with the jth component. These results can be applied to several redundancy mechanisms. A new importance index is defined to study active redundancies. Some illustrative examples are provided.

On relative ageing of coherent systems with dependent identically distributed components

“…a coherent system made from a set of components which have already been used for time t > 0). It is a fact that a coherent system of new components does not always have a longer lifetime than a coherent system made out of used components (see [46]). Similarly, a used coherent system may or may not perform better than a coherent system of used components.…”

Redundancy mechanisms to systems with statistically dependent component and redundancy lifetimes

“…As in Navarro and Fernández-Martínez (2021), a survival copula is employed to model the statistical dependence of redundancy and component lifetimes, and a distortion function is utilized to represent reliability functions of redundant system and the redundancy mechanism at component level. Through two numerical examples we find that the framework of Navarro et al (2019) and Navarro and Fernández-Martínez (2021) are not suitable for coherent systems when redundancy and other component lifetimes are statistically dependent. Then, we derive the new distortion function of coherent system with one redundancy mechanism at component level.…”

“…through a distortion transform of the component reliability in the context of mutually dependent component lifetimes. Navarro et al (2019) introduced a distortion function to represent a redundancy mechanism and then investigated the relation between the importance measure and redundancy of coherent systems with heterogeneous and dependent component lifetimes. Subsequently, Navarro and Fernández-Martínez (2021) and Torrado et al (2021) employed the distortion function to unify different redundancy mechanisms and thus examined how the dependence structure of component lifetimes plays a part in better allocating the redundancy for coherent systems of homogeneous and dependent component lifetimes.…”

“…The rest of this paper rolls out as follows: Section 2 reviews some important notions that are to be used, including stochastic orders, copula functions and distortion functions of coherent systems. In Section 3, we illustrate by two numerical examples that the typical representations of the reliability function of coherent system with a redundancy mechanism at one component in Navarro et al (2019) and Navarro and Fernández-Martínez (2021) are not valid when the active redundancy lifetime and other component lifetimes are dependent. In Section 4, we derive the distortion function for coherent systems with one redundancy mechanism at component level, and then as applications we present in Section 5 the usual stochastic order and the hazard rate order on the redundant symmetric coherent system lifetime.…”

Orderings of component level versus system level at active redundancies for coherent systems with dependent components

“…The research of this topic is mainly divided into two levels: active redundancy at the component level () and at the system level (). In the former case, the main problem is where to allocate active spares in a system to improve the reliability; see, for example, Boland et al [6], Li and Ding [25], Li et al [27], Zhao et al [52], Zhuang and Li [54], You et al [48], Fang and Li [12,13], Chen et al [8], Yan and Luo [43], You and Li [47], Zhang [49], Ling et al [28], Yan et al [45], Kim [21], Navarro et al [36] and Zhang et al [51]. In the later case, Barlow and Proschan [1] first proposed a well-known BP principle that is more reliable in the sense of usual stochastic ordering for the coherent system with independent components.…”