Source: Ann. Appl. Probab.
Volume 10, Number 3
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
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