On the Distribution of Time to First System Failure

For a maintained reliability system, i.e., components may be repaired upon failures, we prove that the distribution of time to first system failure starting with all components new is NBU under the following conditions: (a) nonrepairable components have NBU uptime distributions, (b) repairable components have exponential uptime and DFR downtime distributions. This is a slight generalization of a theorem of Barlow and Proschan (1976). However, our method of proof is different from theirs. We also show that the initial up-period of the system is stochastically larger than any other system up-period. Finally, a method of finding a lower bound for the expected time until first system failure is outlined using our main result.