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

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}.

arXiv admin note: text overlap with arXiv:2003.06454

References in corpus (1)

• On Universal Scaling of Distributed Queues under Load Balancing

Cited by in corpus (1)

• Optimal Load Balancing in Bipartite Graphs