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

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 $N^{1-α}$ with $α> 1$. 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 $σ_a^2$ and $ν_s^2$, then for any $α> 4$, the distribution of the sum queue length scaled by $N^{-α}$ converges to an exponential random variable with mean $\frac{σ_a^2 + ν_s^2}{2}$. (3) If the second moments quadratically increase with $N$ with coefficients $\tildeσ_a^2$ and $\tildeν_s^2$, then for any $α> 3$, the distribution of the sum queue length scaled by $N^{-α-1}$ converges to an exponential random variable with mean $\frac{\tildeσ_a^2 + \tildeν_s^2}{2}$. Both results are simple applications of our previously developed framework of Stein's method for heavy-traffic analysis in \cite{zhou2020note}.