Comparisons of series and parallel systems with components sharing the same copula

Abstract

SUMMARYThe paper is devoted to study stochastic comparisons of series and parallel systems with vectors of component lifetimes sharing the same copula. We show that, under some conditions on the common copula, the series system with heterogeneous components is worse than the series system with homogeneous components having a common reliability function, which is equal to the average of the reliability functions of the heterogeneous components. However, we show that this property is not necessarily true for arbitrary copulas. We obtain similar properties for parallel systems and for general coherent systems. For these purposes, we introduce in our analysis the notion of the mean function of a copula.

Coherent systems based on sequential order statistics

“…These representations were used to compare stochastically coherent system lifetimes and to obtain approximations and bounds for their characteristics (see Refs. [20,22,24,25,[27][28][29]36]). Some specific results for consecutive k-out-of-n systems with dependent components were obtained in Ref.…”

Improving series and parallel systems through mixtures of duplicated dependent components

“…Because of this reason, conditions under which the above mentioned Parrondo's paradox in reliability still holds for dependent components sharing the same copula has been studied in Navarro and Spizzichino [18]. The aim of this note is to further investigate the conditions leading to the stochastic improvements of systems proposed in Di Crescenzo [5] for dependent units; it is shown here that this is actually the case under the assumption of a new weak dependence condition, defined in Section 2.…”

Signature-Based Representations for the Reliability of Systems with Heterogeneous Components

“…Also, in this case, some comparison results were obtained in [14] and [21]. In the case of dependent heterogeneous components, representations and bounds for k-out-of-n systems (order statistics) were obtained in [24,Chapter 5], and comparisons between parallel and series systems were given in [17] and [18].…”

“…This reliability function is related to the well-known concept of mean function associated with a real-valued function (see [6], [8, p. 65], and [18]), which is defined as follows.…”