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 \(N^{1-\alpha}\) with \(\alpha>0\). We show that for \(\a...

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Bibliographic Details
Published in:arXiv.org 2021-05
Main Authors: Hurtado-Lange, Daniela, Maguluri, Siva Theja
Format: Article
Language:English
Subjects:
Convergence
Load balancing
Markov analysis
Queues
Queuing theory
Random variables
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2004.04826