Two methods to approximate the superposition of imperfect failure processes

Suppose a series system is composed of a number of repairable components. If a component fails, it is repaired immediately and the effectiveness of the repair may be imperfect. Then the failure process of the component can be modelled by an imperfect failure process and the failure process of the system is the superposition of the failure processes of all components. In the literature, there is a bulk of research on the superimposed renewal process (SRP) for the case where the repair on each component is assumed perfect. For the case that the component causing the system to fail is unknown and that repair on a failed component is imperfect, however, there is little research on modelling the failure process of the system. Typically, the likelihood functions for the superposition of imperfect failure processes cannot be given explicitly. Approximation-based models have to be sought. This paper proposes two methods to model the failure process of a series system in which the failure process of each component is assumed an arithmetic reduction of intensity and an arithmetic reduction of age model, respectively. The likelihood method of parameter estimation is given. Numerical examples and real-world data are used to illustrate the applicability of the proposed models.