A note on non-existence of diffusion limits for serve-the-longest-queue when the buffers are equal in size

TL;DR

This paper investigates the diffusion limits of serve-the-longest-queue discipline in multiclass queues with equal buffer sizes, revealing that limits depend on finer properties under heavy traffic regimes, unlike the conventional regime.

Contribution

It demonstrates the non-existence of diffusion limits in certain heavy traffic regimes for serve-the-longest-queue with equal buffers, highlighting the importance of finer system properties.

Findings

Diffusion limits are determined by first and second order data in conventional regimes.

Finer properties influence limits under Halfin-Whitt heavy traffic regimes.

A deterministic arrival pattern is used to establish the results.

Abstract

We consider the serve-the-longest-queue discipline for a multiclass queue with buffers of equal size, operating under (i) the conventional and (ii) the Halfin-Whitt heavy traffic regimes, and show that while the queue length process' scaling limits are fully determined by the first and second order data in case (i), they depend on finer properties in case (ii). The proof of the latter relies on the construction of a {\it deterministic} arrival pattern.