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

Rami Atar,Subhamay Saha

doi:10.1214/16-ecp4370

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 deterministic arrival pattern.

Highlights

We analyze the multi-class queue in two different diffusion regimes, namely the conventional and the Halfin-Whitt (HW) heavy traffic regimes, operating under the serve-the-longest-queue (SLQ) scheduling policy
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)
In both regimes the traffic intensity is asymptotic to unity, where in conventional heavy traffic, the model is based on a single server and the arrival rate and service time distributions are scaled up, while in the HW regime, the arrival rate and number of servers are scaled up and the service time distributions are kept fixed; see [2] and references therein for more on these regimes

Summary

Introduction

We analyze the multi-class queue in two different diffusion regimes, namely the conventional and the Halfin-Whitt (HW) heavy traffic regimes, operating under the serve-the-longest-queue (SLQ) scheduling policy In both regimes the traffic intensity is asymptotic to unity, where in conventional heavy traffic, the model is based on a single server and the arrival rate and service time distributions are scaled up, while in the HW regime, the arrival rate and number of servers are scaled up and the service time distributions are kept fixed; see [2] and references therein for more on these regimes. A diffusion limit does not always exist under the ‘usual’ set of assumptions. This stands in contrast to the conventional regime where, as we show, the limit is fully determined by the first and second order data. BM starting from zero, having drift m and infinitesimal covariance matrix A

A counterexample to existence of limits in the Halfin-Whitt regime

A limit result in conventional heavy traffic