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August 2000 Two-server closed networks in heavy traffic: diffusion limits and asymptotic optimality
Sunil Kumar
Ann. Appl. Probab. 10(3): 930-961 (August 2000). DOI: 10.1214/aoap/1019487514


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.


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Sunil Kumar. "Two-server closed networks in heavy traffic: diffusion limits and asymptotic optimality." Ann. Appl. Probab. 10 (3) 930 - 961, August 2000.


Published: August 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1073.60538
MathSciNet: MR1789984
Digital Object Identifier: 10.1214/aoap/1019487514

Primary: 60K25
Secondary: 60F17 , 90F35

Keywords: diffusion limits , fluid limits , Functional limit theorems , Queueing networks , scheduling policies

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 3 • August 2000
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