An Optimal Maintenance Policy for a Queueing System Server Under Periodic Observation

This paper describes an optimal maintenance policy for an M/M/1 queueing system server. Customers arrive at the system in a Poisson stream and are served by the exponential server. After a random time, the server is interrupted by a failure and this failure is detected through regularly timed observations. We begin corrective maintenance when we detect the failure. Through the failure of the server, we lose the customers in the system at the time of failure, as well as the customers that arrive between the failure and the completion of corrective maintenance. However, it is possible to avoid the failure and subsequent corrective maintenance by performing preventive maintenance at observation time. It is true that customers in the system at the start of preventive maintenance and those that arrive prior to its completion are lost. Since the queueing system should serve as many customers as possible, our objective is to minimize the number of lost customers. We then formulate this problem as a semi-Markov decision process and prove the switch curve structure of the optimal policy.