Queueing Systems

, Volume 71, Issue 1–2, pp 97–136 | Cite as

Log-weight scheduling in switched networks

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Abstract

We consider switched queueing networks in which there are constraints on which queues may be served simultaneously. The scheduling policy for such a network specifies which queues to serve at any point in time. We introduce and study a variant of the popular maximum weight or backpressure policy which chooses the collection of queues to serve that has maximum weight. Unlike the maximum weight policies studied in the literature, the weight of a queue depends on logarithm of its queue-size in this paper. For any multihop switched network operating under such maximum log-weighted policy, we establish that the network Markov process is positive recurrent as long as it is underloaded. As the main result of this paper, a meaningful fluid model is established as the formal functional law of large numbers approximation. The fluid model is shown to be work-conserving. That is, work (or total queue-size) is nonincreasing as long as the network is underloaded or critically loaded. We identify invariant states or fixed points of the fluid model. When underloaded, null state is the unique invariant state. For a critically loaded fluid model, the space of invariant states is characterized as the solution space of an optimization problem whose objective is lexicographic ordering of total queue-size and the negative entropy of the queue state. An important contribution of this work is in overcoming the challenge presented by the log-weight function in establishing meaningful fluid model. Specifically, the known approaches in the literature primarily relied on the “scale invariance” property of the weight function that log-function does not possess.

Keywords

Scheduling Maximum weight Fluid models Log-weight Work-conservation 

Mathematics Subject Classification

60K25 60K30 90B36 
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Notes

Acknowledgements

DS is supported by NSF CAREER CNS-0546590 and NSF Collaborative project on Flow Level Models. DJW is supported by a Royal Society university research fellowship. We are grateful for further support from the British Council Researcher Exchange program, and the Newton Institute programme on Stochastic Processes in Communication Sciences.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Dept of EECSMITCambridgeUSA
  2. 2.Dept of Computer ScienceUCLLondonUK

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