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Consider a k-out-of-nsystem which is subject to shocks that arrive at random times. This study develops δ-shock model, among the variants of well-known shock models, for such system which consists of independent components. In a δ-shock model, the system fails when the inter-arrival between two consecutive shocks is less than a critical threshold value of δ. Depending on the number of components that fail due to the occurrence of the shocks, we introduce two δ-shock models. The reliability function and mean time to failure of the system are discussed when the time between shocks has an arbitrary distribution. Furthermore, the problem of finding the optimal preventive time concerned with minimizing a mean cost per unit time is investigated. Numerical examples based on a simulation study are also provided to illustrate theoretical achievement.

AbstractThis paper is concerned with the optimal number of redundant allocation to n-component coherent systems consisting of heterogeneous dependent components. We assume that the system is built up of Lgroups of different components, $L\geq 1$ , where there are $n_i$ components in group i, and $\sum_{i=1}^{L}n_i=n$ . The problem of interest is to allocate $v_i$ active redundant components to each component of type i, $i=1,\dots, L$ . To get the optimal values of $v_i$ we propose two cost-based criteria. One of them is introduced based on the costs of renewing the failed components and the costs of refreshing the alive ones at the system failure time. The other criterion is proposed based on the costs of replacing the system at its failure time or at a predetermined time $\tau$ , whichever occurs first. The expressions for the proposed functions are derived using the mixture representation of the system reliability function based on the notion of survival signature. We assume that a given copula function models the dependency structure between the components. In the particular case that the system is a series-parallel structure, we provide the formulas for the proposed cost-based functions. The results are discussed numerically for some specific coherent systems.

Consider a k-out-of-nsystem consisting of Ltypes of components, L=1,2,…,nsuch that the number of components of the lth type is nl, l=1,2,…,L. Let this system is exposed to external shocks, and each entered shock may fail some components of the system. The system fails when at least n-k+1components fail. In this paper, we study the reliability of this system under a shock model. In addition to reliability analysis, we investigate a preventive maintenance policy for this system to reduce abrupt failures. Numerical examples are also provided to illustrate theoretical achievements.