Mean field limits of large Jackson networks in heavy traffic
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914621436198912 |
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| author | Atar, Rami |
| author_facet | Atar, Rami |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01209 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mean field limits of large Jackson networks in heavy traffic Atar, Rami Probability 60K25, 60K35, 60J60, 60H10 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. |
| title | Mean field limits of large Jackson networks in heavy traffic |
| topic | Probability 60K25, 60K35, 60J60, 60H10 |
| url | https://arxiv.org/abs/2606.01209 |