Optimal control of a flexible server

We consider the dynamic scheduling of a multiclass queueing system with two servers, one dedicated (server 1) and one flexible (server 2), with no arrivals. Server 1 is dedicated lo processing type-1 jobs while server 2 is primarily responsible for processing type 2 jobs but can also aid server I with its work. We address when it is optimal for server 2 to aid server 1 with type-1 jobs rather than process type-2 jobs. The objective is to minimize the total holding costs incurred until all jobs in the system are processed and leave the System. We show that the optimal policy can exhibit one of three possible structures: ( an exhaustive policy for type-2 jobs, (lil) a nonincreasing switching curve in the number of type-1 jobs and (in) a nondecreasing switching curve in the number of type-1 jobs. We characterize the necessary and sufficient conditions under which each policy will be optimal. We also explore the use of the optimal policy for the problem with no arrivals as a heuristic for the problem with dynamic arrivals.