Serve the shortest queue and Walsh Brownian motion

TL;DR

This paper analyzes a queueing system with multiple classes under heavy traffic, revealing that the workload converges to a Walsh Brownian motion, which captures the dominance of one queue at a time and exhibits unique asymptotic properties.

Contribution

It introduces a novel diffusion limit for a multi-class queue with shortest queue priority, characterized by Walsh Brownian motion, highlighting unconventional heavy traffic behavior.

Findings

Workload converges to Walsh Brownian motion in heavy traffic

One queue becomes dominant during excursions

The angular distribution indicates class dominance probabilities

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 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 (1995) it is unconventional, in that the limit is not an 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).