# 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.