Optimal control of a queueing system with P^M_lambda-serivce policy

We consider an M/G/1 queue with P_M_lambda-service policy, which is a two-stage service policy. The server starts to serve with rate 1 if a job arrives to the sever in idle state. If the workload of the system upcrosses lambda, then the service rate is changed to M and this rate continues until the system is empty. It costs to change the service rate to M and maintaining the rate. When the expectation of the stationary workload is supposed to be less than a given value, we derive the optimal value of M.