Optimal procedures for stochastically failing equipment

When a complex system such as the Apollo vehicle arrives on the Moon, certain parts may no longer function. To restore the system to a working state, these parts must be replaced by new ones; but the vehicle can carry only a limited number of spare parts and must also be able to leave at a specified time after arrival. This paper investigates an abstract (and simplified) version of the problem of the optimal maintenance procedure for such a system. As a bonus of the abstract formulation, the results obtained are applicable to a variety of specific systems.