Copula-based representations for the reliability of the residual lifetimes of coherent systems with dependent components

In the context of coherent systems, we obtain representations for the reliability function of the residual lifetime at time t under different assumptions. Specifically, four cases are considered based on system information at time t: the system is working (case 1); all the components of the system are working (case 2); some components have failed at given times, t(1), ..., t(r) < t but the other components and the system are working (case 3); some components have failed at some unknown failure times (case 4). In each of these cases, we obtain a representation for the reliability function of the associated system residual lifetime, based on the copula representation for the component lifetimes and the structure of the system. These representations can be applied to systems with generally dependent components. The representations obtained are also used to compare the respective system residual lifetimes under different stochastic orders. Some illustrative examples are included. (C) 2017 Elsevier Inc. All rights reserved.