Mean field limits of large Jackson networks in heavy traffic

Abstract

We consider an open Jackson network with $n$ exchangeable single-server stations and weak all-to-all interaction through routing: upon service completion at station $i$, a job is routed to station $j$ with probability $p/n$, where $p\in (0,1)$, or leaves the system with probability $q=1-p$. We study a joint asymptotic regime in which the number of stations tends to infinity while the system approaches heavy traffic. Under the critical-load condition and diffusive scaling of time and queue length, we prove propagation of chaos for the queue-length and cumulative-idleness processes. The limiting McKean--Vlasov dynamics are described by the nonlinear reflected Brownian motion \[ \bar X(t)=\bar X_0+\bar W(t)+\hatγt+\bar L(t)-p\,\mathbb{E} \bar L(t), \] where $\bar{W}$ is a Brownian motion with variance parameter $2$ and $\bar{L}$ is the reflection term at the origin. The proof proceeds by coupling the rescaled queueing network to a Brownian particle system interacting through boundary local times. A key step is a decoupling construction that replaces the correlated routing noise in the prelimit by asymptotically equivalent independent driving noises.