A load balancing system in the many-server heavy-traffic asymptotics

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