Jointly optimal allocation of a repairman and optimal control of service rate for machine repairman problem

Consider a production system with m unreliable machines, which are maintained by a single repairman. The time until failure of machine i and its repair time are exponentially distributed random variables with rates λ i and μ, respectively. Machine i earns at rate r i while it is working. The service rate can be controlled, and a cost c( μ) is charged when the service rate is μ. We assume the following compatibility condition: λ i < λ j implies that r i ⩾ r j . We consider both the optimal assignment of the repairman to the failed machines, and the optimal service rate. We demonstrate some monotone properties of the optimal policy.