A Study on Repairable Series Systems with Markov Repairable Units

Consider a repairable series system consisting of n units and each of them follows a Markov process with finite state and continuous time. Under independence assumption among units, the repairable series system has been widely studied by using the Markov process method and Lz-transform method. However, both methods have faced the problem of state exploration although some approximation methods have been used. Thus, it is still an interesting and significant problem to be explored. In this chapter, we investigate repairable series systems by using matrix method which has been widely used in aggregated stochastic processes especially in ion channel modeling and aggregated repairable systems. The formulas for reliability, instantaneous and interval availabilities are given in matrix form for four kinds of repairable series systems, general repairable series system, general repairable series system with neglected failures, phased-mission repairable series system and phased-mission repairable series system with neglected failures, respectively. Numerical examples are shown to illustrate the results for the four kinds of systems and present how matrix method is used to solve the problem of state exploration in the Lz-transform method. Finally, the conclusions and some future possible applications are given.