Optimal restarting distribution after repair for a Markov deteriorating system

We consider a repairable system such that different completeness degrees are possible for the repair (or corrective maintenance) that go from a ‘minimal’ up to a ‘complete’ repair. Our question is: to what extent must the system be repaired in case of failure for the long-run availability to be optimal? The system evolves in time according to a Markov process as long as it is running, whereas the duration of repairs follows general distributions. After repair, the system starts again in the up-state i with probability d(i). We observe from numerical examples that the optimal restarting distribution dopt (such that the long-run availability is optimal) is generally random and does not correspond to a new start in a fixed up-state. Sufficient conditions under which the optimal restarting distribution is non-random are given. Also, the optimal restarting distribution is provided for two classical structures in reliability.