A complete solution for the optimal stochastic scheduling of a two-stage tandem queue with two flexible servers

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