We consider a two-stage tandem queue with two parallel servers and two queues. We assume that initially all jobs are present and that no further arrivals take place at any time. The two servers are identical and can serve both types of job. The processing times are exponentially distributed. After being served, a job of queue 1 joins queue 2, whereas a job of queue 2 leaves the system. Holding costs per job and per unit time are incurred if there are jobs holding in the system. Our goal is to find the optimal strategy that minimizes the expected total holding costs until the system is cleared. We give a complete solution for the optimal control of all possible parameters (costs and service times), especially for those parameter regions in which the optimal control depends on how many jobs are present in the two queues.

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 to processing type-1 jobs while server 2 is primarily responsible for processing type-2 jobs but can also aid server 1 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: (i) an exhaustive policy for type-2 jobs, (ii) a nonincreasing switching curve in the number of type-1 jobs and (iii) 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.