Editing Openai/6931fd85-fcec-8011-8057-6c5f7152feee (section)

===== 7.1 Step 1: Diffusion limit and state space collapse =====

Under the heavy‑traffic scaling, arrival and service processes satisfy functional central limit theorems (even with heavy‑tailed but finite‑variance inputs). After appropriate centering and scaling, the total workload and the vector of queue lengths converge in distribution to a SRBM in a convex polyhedron.

Heavy‑traffic “resource pooling” and state‑space collapse show that:
* all triage classes lie close to a one‑dimensional manifold parameterised by total triage workload;
* all IP classes lie close to another manifold parameterised by their total workload.

This is standard in multiclass queue control under heavy traffic.