Rank-Based Stochastic Differential Inclusions and Diffusion Limits for a Load-Balancing Model

Banerjee, Budhiraja and Estevez (2025) studied a randomized load balancing model in a heavy traffic asymptotic regime where the load balancing stream is thin compared to the total arrival stream. It was shown that the limit is given by a system of rank-based Brownian particles on the half-line. In this paper we extend this result from the case of exponential service time to an invariance principle, where service times have finite second moment. The main tool is a new notion of rank-based stochastic differential inclusion, which may be of interest in its own right. Funding: R. Atar is partially supported by the Israel Science Foundation [Grant 1035/20]. T. Ichiba was supported in part by the National Science Foundation [Grant NSF-DMS-2008427].