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Chaoticity on path space for a queueing network with selection of the shortest queue among several

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Carl Graham
Carl Graham*
Affiliation:
École Polytechnique, Palaiseau
*
∗Postal address: CMAP, École Polytechnique, 91128 Palaiseau, France (UMR CNRS 7641). Email address: carl@cmapx.polytechnique.fr
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Abstract

We consider a network with N infinite-buffer queues with service rates λ, and global task arrival rate Nν. Each task is allocated L queues among N with uniform probability and joins the least loaded one, ties being resolved uniformly. We prove Q-chaoticity on path space for chaotic initial conditions and in equilibrium: any fixed finite subnetwork behaves in the limit N goes to infinity as an i.i.d. system of queues of law Q. The law Q is characterized as the unique solution for a non-linear martingale problem; if the initial conditions are q-chaotic, then Q0 = q, and in equilibrium Q0 = qρ is the globally attractive stable point of the Kolmogorov equation corresponding to the martingale problem. This result is equivalent to a law of large numbers on path space with limit Q, and implies a functional law of large numbers with limit (Qt)t≥0. The significant improvement in buffer utilization, due to the resource pooling coming from the choices, is precisely quantified at the limit.

Keywords

Mean-field interactionchaoticityequilibriumnon-linear martingale problemsresource pooling

MSC classification

Primary: 60K25: Queueing theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 198 - 211
Copyright
Copyright © 2000 by The Applied Probability Trust 

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