ANALYSIS OF A REPAIRABLE 2-UNIT PARALLEL REDUNDANT SYSTEM WITH FAILURE AND REPAIR TIMES ARBITRARILY DISTRIBUTED

This paper considers systems composed of two elements, or subsystems arranged in parallel, each of which may be at any moment in one of two states: a functioning state or the failed state. The case where the outage time and the repair time are exponentially distributed is treated by Feller [An Introduction to Probability Theory and Its Applications. Wiley, New York, 2nd Edn (1957)]. The generalization of the case where only the repair time is arbitrarily distributed is given by Gaver [IEEE Trans. Reliab. R-11, 30-39 (1963)]; unfortunately, his method is not operational because one of his equations, which describes the evolution of the system, is not correct. We give details in the introduction, We can also find a variant of this method in [IEEE Trans, Reliab. R-35 (1986)], without the preceding difficulty arising. However, it is possible to solve this problem by a totally different method, which utilizes a result from the semi-Markovian theory. Our first purpose is to give details of this approach. We calculate the catastrophic probabilities of the system. The second purpose of our paper is to treat the same problem in a more general framework, where the outage time and the repair time are both distributed in any way. In this case, the process representing the evolution of the system is no longer Markovian or semi-Markovian.