Optimal Control of Auxiliary Service Queueing System

We consider an M/M/1 queueing system with auxiliary service in addition to primary service. When the auxiliary service is requested, the server allocates part of its capacity to process the additional service. We show that the possible optimal auxiliary capacity is actually on the convex envelope of the auxiliary cost function. Also, the optimal control policy is that the allocated capacity for auxiliary service is monotonically non-increasing in the number of people in the system.