A Note on Load Balancing in Many-Server Heavy-Traffic Regime

TL;DR

This paper uses Stein's method to analyze load balancing performance in many-server heavy-traffic regimes, deriving asymptotic queue length distributions under different scaling conditions as the number of servers grows large.

Contribution

It applies Stein's method to establish new asymptotic results for queue length distributions in large-scale load balancing systems under various moment growth conditions.

Findings

Queue length scaled by N^{- ext{alpha}} converges to an exponential distribution for alpha > 4.

Queue length scaled by N^{- ext{alpha}-1} converges to an exponential distribution for alpha > 3.

Results extend previous heavy-traffic analysis frameworks using Stein's method.

Abstract

In this note, we apply Stein's method to analyze the performance of general load balancing schemes in the many-server heavy-traffic regime. In particular, consider a load balancing system of $N$ servers and the distance of arrival rate to the capacity region is given by $N^{1-\alpha}$ with $\alpha > 1$. We are interested in the performance as $N$ goes to infinity under a large class of policies. We establish different asymptotics under different scalings and conditions. Specifically, (i) If the second moments linearly increase with $N$ with coefficients $\sigma_a^2$ and $\nu_s^2$, then for any $\alpha > 4$, the distribution of the sum queue length scaled by $N^{-\alpha}$ converges to an exponential random variable with mean $\frac{\sigma_a^2 + \nu_s^2}{2}$. (3) If the second moments quadratically increase with $N$ with coefficients $\tilde{\sigma}_a^2$ and $\tilde{\nu}_s^2$, then for any $\alpha > 3$, the distribution of the sum queue length scaled by $N^{-\alpha-1}$ converges to an exponential random variable with mean $\frac{\tilde{\sigma}_a^2 + \tilde{\nu}_s^2}{2}$. Both results are simple applications of our previously developed framework of Stein's method for heavy-traffic analysis in \cite{zhou2020note}.