The Annals of Applied Probability

Serve the shortest queue and Walsh Brownian motion

Rami Atar and Asaf Cohen

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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).

Article information

Source
Ann. Appl. Probab., Volume 29, Number 1 (2019), 613-651.

Dates
Received: February 2018
Revised: July 2018
First available in Project Euclid: 5 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1544000438

Digital Object Identifier
doi:10.1214/18-AAP1432

Mathematical Reviews number (MathSciNet)
MR3910013

Zentralblatt MATH identifier
07039134

Subjects
Primary: 60F05: Central limit and other weak theorems 93E03: Stochastic systems, general 60K25: Queueing theory [See also 68M20, 90B22] 60J65: Brownian motion [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Serve the shortest queue heavy traffic diffusion limits Walsh Brownian motion

Citation

Atar, Rami; Cohen, Asaf. Serve the shortest queue and Walsh Brownian motion. Ann. Appl. Probab. 29 (2019), no. 1, 613--651. doi:10.1214/18-AAP1432. https://projecteuclid.org/euclid.aoap/1544000438


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