Improving series and parallel systems through mixtures of duplicated dependent components

Abstract

We discuss suitable conditions such that the lifetime of a series or of a parallel system formed by two components having nonindependent lifetimes may be stochastically improved by replacing the lifetimes of each of the components by an independent mixture of the individual components’ lifetimes. We also characterize the classes of bivariate distributions where this phenomenon arises through a new weak dependence notion

Stochastic comparisons of series and parallel systems with randomized independent components

“…It should be pointed out that the results given in the present paper have a narrow intersection with the contribution of Di Crescenzo and Pellerey [7]. First of all, papers [6] and [7] deal with 2-unit series and parallel systems, whereas the systems we examine here have arbitrary fixed sizes. Moreover, the single common case involves random variables K 1 = 1 a.s. and K 2 having binomial distribution with parameters n = 2 and p = 1/2.…”

“…. , n − 1 (by (7) and Theorem 3.A.34 in Shaked and Shanthikumar [17]), so that K j ≤ icx K j+1 . Hence, if X ≥ st Y then from Theorem 1 we obtain S K j ≤ st S K j+1 for j = 0, 1, .…”

“…It is shown in Di Crescenzo [6] that, if the first system can be modified by a random choice of its components, allowing each of the two units to be randomly chosen from a set of components identical to those in the series, then its reliability can be improved, obtaining a system more reliable the the second one, although the original one was worse. An extension to the case of dependent components has been treated recently in Di Crescenzo and Pellerey [7]. See also Navarro and Spizzichino [14], where stochastic comparisons of series and parallel systems with vectors of component lifetimes sharing the same copula are studied.…”

Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

“…Results involving the fact that increasing the randomness in the structure of reliability systems causes a stochastical improvement of the system have been given recently in [11], where it was shown that the lifetime of a series or of a parallel system may be stochastically improved by means of suitable mixtures. See also [12], where the reliability of a system improves by introducing randomness in the number of system components extracted from different batches.…”

Analysis of reliability systems via Gini-type index

“…They provided sufficient conditions on the components lifetimes and on the random number of components chosen from the two stocks in order to improve the reliability of the whole system. See, also, Navarro and Spizzichino [16] and Di Crescenzo and Pellerey [5] for the analysis and the comparison of parallel and series systems with heterogeneous components sharing the same copula, or with components linked via suitable mixtures.…”