Throughput maximization for two station tandem systems: a proof of the Andradóttir-Ayhan conjecture

We study a tandem queueing network with two stations, M heterogeneous flexible servers, and a finite intermediate buffer. The objective is to dynamically assign the servers to the stations in order to maximize the throughput of the system. The form of the optimal policy for M ≤ 3 was derived in two previous papers. In one of those papers, Andradóttir and Ayhan (Operations Research, Vol. 53, pp. 516-531, 2005) provide a conjecture on the form of the optimal policy for M ≥ 4. We prove their conjecture in this paper, showing that the optimal policy is defined by monotone thresholds and the ratios of the service rates among the servers. For M > 1, we also prove that the optimal policy always uses the entire intermediate buffer.