*(English)*Zbl 0708.60084

Summary: We consider a single-server queueing system with Poisson arrivals and general service times. While the server is up, it is subject to breakdowns according to a Poisson process. When the server breaks down, we need to repair the server immediately by initiating one of two available repair operations. The operating costs of the system include customer holding costs, repair costs and running costs. The objective is to find a corrective maintenance policy that minimizes the long-run average operating costs of the system.

The problem is formulated as a semi-Markov decision process. Under some mild conditions on the repair time and service time distributions and the customer holding cost rate function, we prove that there exists an optimal stationary policy which is monotone, i.e., which is characterized by a single threshold parameter: The stochastically faster repair is initiated if and only if the number of customers in the system exceeds this threshold.

We also present an efficient algorithm for the determination of an optimal monotone policy and its average cost. We then extend the problem to allow the system to postpone the repair until some future point in time. We provide a partial characterization of an optimal policy and show that monotone policies are, in general, not optimal. The latter problem also extends the authorsâ€™ previous work.

##### MSC:

60K10 | Applications of renewal theory |

60K25 | Queueing theory |

90C40 | Markov and semi-Markov decision processes |

90B25 | Reliability, availability, maintenance, inspection, etc. (optimization) |

90B22 | Queues and service (optimization) |