Relative aging of (n − k + 1)‐out‐of‐n systems based on cumulative hazard and cumulative reversed hazard functions

Abstract

The notion of relative aging describes the rate at which one system ages relative to the other. Di Crescenzo, and Sengupta and Deshpande defined two notions of relative aging based on monotonicity of ratios of cumulative reversed hazard functions and cumulative hazard functions, respectively. In this paper, we discuss the relative aging of (n − k + 1)‐out‐of‐n systems in terms of these notions.

On Relative Aging Comparisons of Coherent Systems With Identically Distributed Components

“…Here, we provide some sufficient conditions under which one coherent system performs better than the other with respect to aging faster orders based on the cumulative hazard and the cumulative reversed hazard rates. Further, we show that the proposed results hold for the well known k -out-of-n systems thereby generalizing the results of Misra and Francis [26] to situations where we have general coherent systems comprising of components with d.i.d. (dependent and identically distributed) lifetimes.…”

“…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). It will be interesting to extend these studies to coherent systems with dependent component lifetimes.…”

On relative ageing of coherent systems with dependent identically distributed components

“…The second set of stochastic orders, called ageing faster orders, are defined based on monotonicity of ratios of some reliability measures, namely, hazard rate function, reversed hazard rate function, mean residual lifetime function, etc. For motivation and usefulness of these orders, we refer the reader to [13], [17], [24], [26], [28], [37], [39], [52], and [54]. Below we give the definitions of the ageing faster orders that are used in our paper.…”

Stochastic Comparisons of Weighted Distributions and Their Mixtures

“…Some relative stochastic orders including the relative (reversed) hazard rate and relative mean residual life orders have attracted the attention of researchers in the last decade (cf. Di-Crescenzo and Longobardi [20], Kayid et al [21], Misra and Francis [22], Misra et al [23], Ding et al [24], Ding and Zhang [25], Misra and Francis [26] and Misra and Francis [27]). We reminisce about the definition of these orders from Rezaei et al [28] and Kayid et al [21] [see, for example, Definition 1(v) and (vi)].…”