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.

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Conference Proceedings of the 27th IEEE Conference on Decision and Control
Christidou, Ioanna (Ioanna), Lambadaris, I, & Mazumdar, Ravi (Ravi). (1988). Optimal control of arrivals to a feedback queueing system. In Proceedings of the IEEE Conference on Decision and Control (pp. 663–667).