An optimal PλM-service policy for an M/G/1 queueing system

We consider an M/G  /1 queueing system under PλM-service policy. As soon as the workload exceeds threshold λ > 0, the service rate is increased from 1 to M ⩾ 1 and is kept until the system becomes empty. After assigning several costs, we show that there exists a unique M minimizing the long-run average cost per unit time.