Load balancing system under Join the Shortest Queue: Many-Server-Heavy-Traffic Asymptotics

TL;DR

This paper analyzes the behavior of load balancing under Join the Shortest Queue in many-server heavy-traffic conditions, revealing asymptotic queue length distributions and convergence rates for different traffic regimes.

Contribution

It provides new asymptotic results for JSQ in many-server heavy-traffic regimes, including distribution convergence and state space collapse, with two proof techniques and convergence rate analysis.

Findings

Queue length distribution converges to exponential for er > 4

Average queue length scales with N^{1-\u03b1} in heavy traffic

Rate of convergence in Wasserstein distance is established

Abstract

We study the load balancing system operating under Join the Shortest Queue (JSQ) in the many-server heavy-traffic regime. If $N$ is the number of servers, we let the difference between the total service rate and the total arrival rate be $N^{1-\alpha}$ with $\alpha>0$. We show that for $\alpha>4$ the average queue length behaves similarly to the classical heavy-traffic regime. Specifically, we prove that the distribution of the average queue length multiplied by $N^{1-\alpha}$ converges to an exponential random variable. Moreover, we show a result analogous to state space collapse. We provide two proofs for our result: one using the one-sided Laplace transform, and one using Stein's method. We additionally obtain the rate of convergence in the Wasserstein's distance.