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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 $n=1,\ldots,N$ 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 $(\mu_1^m(k),\ldots,\mu_N^m(k))$.

We consider a heavy traffic regime, and assume a Resource Pooling (RP) condition. Associated with this condition is a notion of workload $X=\sum_n \zen Q_n$, where $\ze=(\ze_1,\ldots,\ze_N)$ is some fixed nonzero vector with nonnegative components, and $Q_1,\ldots,Q_N$ are the queue lengths. We study the MaxWeight discipline which always chooses a decision $k$ maximizing $\sum_n \gamma_n [Q_n]^{\beta} \mu_n^m(k)$, that is, \[ k \in \mathop{\arg\max}_{i} \sum_n \gamma_n [Q_n]^{\beta} \mu_n^m(i), \] where $\beta>0$, $\gamma_1>0,\ldots,\gamma_N>0$ 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 $(\gamma_1 Q_1^{\beta},\ldots,\gamma_N Q_N^{\beta})$ 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 \[ \sum_n \gamma_n Q_{n}^{\beta+1} \] 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

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Vol.14 • No. 1 • February 2004
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