On relative ageing of coherent systems with dependent identically distributed components

Abstract

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

On Relative Aging Comparisons of Coherent Systems With Identically Distributed Components

“…It is worth mentioning here that the results stated in Proposition 3.1 and Theorem 3.1 also follow from Theorem 3.1 of Hazra and Misra [15] under a similar set of conditions, as in Theorem 3.1, except that there condition (iv) of Theorem 3.1 is replaced by the condition X ≤ c Y and Y ≤ rh X. Note that this assumption is much stronger as compared to (iv) because ≤ c =⇒ ≤ c * and ≤ rh =⇒ ≤ st , and the reverse implications are not necessarily true.…”

“…We derive sufficient conditions under which the use of components having lifetime vector X makes the system age faster than if the components with lifetime vector Y are used. The following lemma, obtained in Hazra and Misra [15,Proposition 3.2], will be used in proving the next result.…”

“…Before stating the next result, we give the following lemma obtained in Hazra and Misra [15,Proposition 3.1].…”

“…Note that this assumption is much stronger as compared to (iv) because ≤ c =⇒ ≤ c * and ≤ rh =⇒ ≤ st , and the reverse implications are not necessarily true. The methodology used in the proofs of this paper differs from that in Hazra and Misra [15]. Thus, it is meaningful to study the results given in Proposition 3.1 and Theorem 3.1.…”

“…There are a few articles where aging faster orders were considered to compare two coherent systems. The study of k -out-of-n systems using aging faster orders (in the hazard and the reversed hazard rates) was done by Misra and Francis [25], Li and Li [24], and Ding and Zhang [11], whereas a similar study for coherent systems was done by Ding et al [10] and Hazra and Misra [15]. Recently, Misra and Francis [26] studied k -out-of-n systems with independent components in terms of aging faster orders (in the cumulative hazard and the cumulative reversed hazard rates).…”