On the residual lifetimes of dependent components upon system failure

In this paper, we consider coherent systems composed of identical yet possibly dependent components. The dependence structure among components is modeled via copulas. We investigate residual lifetimes of components that survived the failure of the system. The main result of the paper is a new general formula describing the joint distribution of these residual lifetimes. Using the new formula we look more closely at this distribution in the cases of Clayton, Gumbel–Hougaard and Frank copulas and standard exponential and Weibull margins. In particular, we present some comparative numerical results that enable us to examine the influence of the dependence among components on the residual lifetimes of the surviving ones. We observe that the results obtained under the assumption of dependence and independence may differ significantly, especially when the system breaks down just after it started working.