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February 2004 MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic
Alexander L. Stolyar
Ann. Appl. Probab. 14(1): 1-53 (February 2004). DOI: 10.1214/aoap/1075828046

Abstract

We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the input-queued cross-bar switch model and a discrete time version of a parallel server queueing system. Input flows n=1,,N are served in discrete time by a switch. The switch state follows a finite state, discrete time Markov chain. In each state m, the switch chooses a scheduling decision k from a finite set K(m), which has the associated service rate vector (μ1m(k),,μNm(k)).

We consider a heavy traffic regime, and assume a Resource Pooling (RP) condition. Associated with this condition is a notion of workload X=n\zenQn, where \ze=(\ze1,,\zeN) is some fixed nonzero vector with nonnegative components, and Q1,,QN are the queue lengths. We study the MaxWeight discipline which always chooses a decision k maximizing nγn[Qn]βμnm(k), that is, kargmaxinγn[Qn]βμnm(i), where β>0, γ1>0,,γN>0 are arbitrary parameters. We prove that under MaxWeight scheduling and the RP condition, in the heavy traffic limit, the queue length process has the following properties: (a) The vector (γ1Q1β,,γNQNβ) is always proportional to \ze (this is "State Space Collapse"), (b) the workload process converges to a Reflected Brownian Motion, (c) MaxWeight minimizes the workload among all disciplines. As a corollary of these properties, MaxWeight asymptotically minimizes the holding cost rate at all times, and cumulative cost (with this rate) over finite intervals.

Citation

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Alexander L. Stolyar. "MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic." Ann. Appl. Probab. 14 (1) 1 - 53, February 2004. https://doi.org/10.1214/aoap/1075828046

Information

Published: February 2004
First available in Project Euclid: 3 February 2004

zbMATH: 1057.60092
MathSciNet: MR2023015
Digital Object Identifier: 10.1214/aoap/1075828046

Subjects:
Primary: 60J70 , 60K25 , 90B15

Keywords: asymptotic optimality , equivalent workload formulation , Generalized switch , heavy traffic limit , MaxWeight scheduling , queuing networks , resource pooling , state space collapse

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.14 • No. 1 • February 2004
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