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February 2019 Serve the shortest queue and Walsh Brownian motion
Rami Atar, Asaf Cohen
Ann. Appl. Probab. 29(1): 613-651 (February 2019). DOI: 10.1214/18-AAP1432

Abstract

We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the nominal workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^{N}$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the system, and it remains so until the system becomes (nearly) empty again. The radial part of the WBM, given as a reflected Brownian motion (RBM) on the half-line, captures the total workload asymptotics, whereas its angular distribution expresses how likely it is for each class to become dominant on excursions.

As a heavy traffic result, it is nonstandard in three ways: (i) In the terminology of Harrison (In Stochastic Networks (1995) 1–20 Springer), it is unconventional, in that the limit is not a RBM. (ii) It does not constitute an invariance principle, in that the limit law (specifically, the angular distribution) is not determined solely by the first two moments of the data, and is sensitive even to tie breaking rules. (iii) The proof method does not fully characterize the limit law (specifically, it gives no information on the angular distribution).

Citation

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Rami Atar. Asaf Cohen. "Serve the shortest queue and Walsh Brownian motion." Ann. Appl. Probab. 29 (1) 613 - 651, February 2019. https://doi.org/10.1214/18-AAP1432

Information

Received: 1 February 2018; Revised: 1 July 2018; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039134
MathSciNet: MR3910013
Digital Object Identifier: 10.1214/18-AAP1432

Subjects:
Primary: 60F05, 60J65, 60J70, 60K25, 93E03

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 1 • February 2019
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