Joint Reliability Function of Coherent Systems with Shared Heterogeneous Components

In this paper, we consider two coherent systems having shared components. We assume that in the two systems there are three different types of components; components of type one that just belong to the first system, components of type two that lie only in the second system and components of type three that are shared by the two systems. We use the concept of joint survival signature to assess the joint reliability function of the two systems. Using this concept, some representations for the joint reliability function of the system lifetimes are obtained under two different scenarios of component failures. In the first scenario, we assume that the components of the systems fail according to different counting processes such as non-homogeneous Poisson processes. In the second scenario, it is assumed that the component lifetimes of each type are exchangeable while the three types of component lifetimes can be independent or dependent. To illustrate the theoretical results, two systems with shared components are studied numerically and graphically.