On the First Failure Time of Dependent Multicomponent Reliability Systems

Abstract

In this paper multicomponent reliability systems are considered, where component failure and repair completion rates depend on the state, ages and current repair durations of the other components. This is a generalization of a model of Ross (Ross, S. M. 1984. A model in which component failure rates depend on working set. Naval Res. Logist. Quart. 31 297–300.). Sufficient conditions on the sets of rates which imply stochastic ordering between first failure times of two such systems are found. Sufficient conditions on the rates which imply that the first failure time of such a system is new better than used (NBU) are given. Some results of Barlow and Proschan (Barlow, R. E., Proschan, F. 1976. Theory of maintained systems: Distribution of time to first system failure. Math. Oper. Res. 1 32–42.), Chiang and Niu (Chiang, D. T., Niu, S. C. 1980. On the distribution of time to first system failure. J. Appl. Probab. 17 481–489.), and Ross (Ross, S. M. 1976. On the time to first failure in multicomponent exponential reliability systems. Stochastic Process. Appl. 4 167–173.) are obtained as special cases. A counterexample to an apparently stronger result of Miller (Miller, D. R. 1979. Almost sure comparison of renewal processes and poisson processes, with application to reliability theory. Math. Oper. Res. 4 406–413.) is also given. Further results and a discussion are included.