Customer Scheduling Under Queueing Constraints

We consider a scheduling problem of an exponential single server with a finite queueing capacity that serves customers from n heterogeneous classes. Arrivals are Poissonian and every class has its own rate and its own finite waiting room. The waiting rooms could be of arbitrary sizes. Arriving customers that find a full queue are lost. We are interested in finding a scheduling policy that allows service preemption and has a weighted throughput which is close enough to the optimal one. As an optimal scheduling is extremely hard to find, we apply a different methodology to tackle the problem. First, we bound the optimal weighted throughput from above and find the asymptotically optimal policy. Then, based on our bounding technique and the asymptotically optimal policy, we propose a new policy, the overflow scheduling policy, that provides a weighted throughput which is very close to the upper bound. The quality of the policy is demonstrated by various examples.