Optimal control of arrivals to a feedback queueing system

The performance analysis of ring type LANS has received much attention. The advantage of the ring type architecture is due to the fact that it is particularly suitable for the transport of real time traffic with guaranteed time delay performance. However, it can be shown that uncontrolled rings are unstable in the sense that the Markov chain that describes the behavior has adsorbing states which amount to deadlock. Thus, a natural question that arises is the control of ring type architectures. Motivated by this question, the authors consider a system of two coupled queues where a packet after being served in one queue can be fed into the other queue or leave the system. In addition there are external Poisson arrivals at each queue. These can in general be optimally controlled by applying a probabilistic rule minimizing an average discounted cost which is a linear function of the total amount of blocking as well as the number of packets in the system. It is shown that the optimal blocking mechanism is deterministic (bang-bang) and is characterized by two monotone switching curves in the state space associated with the system. The approach used relies on Markov decision theory and convexity arguments