One of the successes of the Brownian approximation approach to dynamic control of queueing networks is the design of a control policy for closed networks with two servers by Harrison and Wein. Adopting a Brownian approximation with only heuristic justification, theyinterpret the optimal control policy for the Brownian model as a static priority rule and conjecture that this priority rule is asymptotically optimal as the closed networks’s population becomes large. This paper studies closed queueing networks with two servers that are balanced, that is, networks that have the same relative load factor at each server. The validity of the Brownian approximation used by Harrison and Wein is established by showing that, under the policy they propose, the diffusion-scaled workload imbalance process converges weakly in the infinite population limit to the diffusion predicted by the Brownian approximation. This is accomplished by proving that the fluid limits of the queue length processes undergo state space collapse in finite time under the proposed policy, thereby enabling the application of a powerful new technique developed by Williams and Bramson that allows one to establish convergence of processes under diffusion scaling by studying the behavior of limits under fluid scaling. A natural notion of asymptotic optimality for closed queueing networks is defined in this paper.The proposed policy is shown to satisfy this definition of asymptotic optimality by showing that the performance under the proposed policy approximates bounds on the performance under every other policy arbitrarily well as the population increases without bound.
"Two-server closed networks in heavy traffic: diffusion limits and asymptotic optimality." Ann. Appl. Probab. 10 (3) 930 - 961, August 2000. https://doi.org/10.1214/aoap/1019487514