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

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 $N^{1-α}$ with $α>0$. We show that for $α>4$ 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 $N^{1-α}$ 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.

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