Rank Based Routing in Large Server Systems under Extreme Congestion
Sayan Banerjee, Amarjit Budhiraja, and Eva Loeser

TL;DR
This paper analyzes a novel rank-based routing policy in large server systems under extreme congestion, deriving diffusion limits, stationary distributions, and tradeoffs between communication cost and load balancing.
Contribution
It introduces a new routing policy with reduced communication costs, establishes its diffusion limit as an infinite-dimensional reflected Atlas process, and characterizes its stationary gap distributions.
Findings
Diffusive limit characterized as an infinite-dimensional reflected Atlas process.
Explicit stationary gap distributions parametrized by system parameters.
Quantified tradeoffs between communication cost and load balancing performance.
Abstract
We study parallel queues in an extreme heavy-traffic regime: each server works at rate , while jobs arrive to a dispatcher at rate , with fixed . Arrivals are routed by a marginal join-the-shortest-queue policy: a small stream of rate joins the current shortest queue, while the remaining stream of rate is routed uniformly at random. This policy greatly reduces communication cost relative to full JSQ, while improving load balancing and offering a natural mechanism for premium jobs to join shorter queues. Under diffusive scaling, we prove limit theorems for the ranked queue lengths and associated gap process. The limit is an infinite-dimensional reflected Atlas process, with reflection at the origin and rank-based drift acting on the lowest particle. Its dynamics depend only on , the shortest-queue arrival rate, while …
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