Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-𝑟+ 1)-out-of-𝑛 systems

Abstract

Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

Ordering and ageing properties of developed sequential order statistics governed by the Archimedean copula

“…Now, we have that W (j) and D j X j−1:n are independent, and T (j) and B j Z j−1:n are independent. Again, by using (5) and the condition that G(nu)/R(u) − G(u)/R(u) is positive and increasing in u > 0, we get…”

“…This implies that and are independent. Again, by using (5) and the condition that is positive and increasing in , we get which implies that is ILR. Furthermore, from the induction hypothesis, we have that is ILR.…”

“…Similarly, and are also independent. Again, by using (5) and the condition that is increasing in , we get that This implies that which implies that is IFR. By using these two facts and the condition , from Lemma 2.4(a) we get that .…”

“…Furthermore, Burkschat and Navarro [13] studied closure properties of different ageing classes under the formation of sequential k-out-of-n systems. Later, Barmalzan et al [5] studied various ordering and ageing properties of residual lifetimes of live components in sequential k-out-of-n systems. One may note that all of the aforementioned studies were carried out for sequential k-out-of-n systems with independent components (or equivalently, sequential order statistics with independent random variables).…”

Ordering and aging properties of systems with dependent components governed by the Archimedean copula

“…Among all, Pledger and Proschan [43], to the best of our knowledge, are the first who studied stochastic comparisons of two -out-of- systems with heterogeneous components. Different variations of this problem were further studied by numerous researchers (see [2,6,10,12,22,23,25,45,46,50,51], to name a few). Stochastic comparisons of coherent systems with independent and nonidentically distributed components were considered in [8,9,34].…”