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

==== 3. Asymptotic (heavy‑traffic) framework ====

The structure result comes from a heavy‑traffic scaling:
* Consider a sequence of EDs indexed by r→∞r\to\inftyr→∞.
* In the rrr-th system: - Arrival rates are multiplied: λj(r)(t)=r λj(t)\lambda_j^{(r)}(t) = r\,\lambda_j(t)λj(r)(t)=rλj(t). - Number of physicians SSS is fixed and finite (this is “conventional heavy traffic”: fixed capacity, exploding demand). - Traffic intensity satisfies ρ(r)(t):=1S∑jλj(r)(t)mj≈1+β(t)r,\rho^{(r)}(t) := \frac{1}{S}\sum_j \lambda_j^{(r)}(t)m_j \approx 1 + \frac{\beta(t)}{r},ρ(r)(t):=S1j∑λj(r)(t)mj≈1+rβ(t), for bounded β(t)\beta(t)β(t). This is exactly the condition used in the ED control work of Huang–Carmeli–Mandelbaum. aviman.technion.ac.il<ref>{{cite web|title=aviman.technion.ac.il|url=https://aviman.technion.ac.il/files/References/INFORMS_2012_control_of_patient_flow.pdf|publisher=aviman.technion.ac.il|access-date=2025-12-04}}</ref>
* We time‑scale and space‑scale the queue length and “age” processes in the standard diffusion way: Q~j(r)(t)=1rQj(r)(r2t),τ~j(r)(t)=1rτj(r)(r2t),\tilde Q^{(r)}_j(t) = \frac{1}{r} Q^{(r)}_j(r^2 t),\quad \tilde \tau^{(r)}_j(t) = \frac{1}{r} \tau^{(r)}_j(r^2 t),Q~j(r)(t)=r1Qj(r)(r2t),τ~j(r)(t)=r1τj(r)(r2t), where τj(r)(t)\tau^{(r)}_j(t)τj(r)(t) is the age (waiting time so far) of the head‑of‑line triage patient of class jjj. aviman.technion.ac.il<ref>{{cite web|title=aviman.technion.ac.il|url=https://aviman.technion.ac.il/files/References/INFORMS_2012_control_of_patient_flow.pdf|publisher=aviman.technion.ac.il|access-date=2025-12-04}}</ref>

Under the finite‑second‑moment assumptions (even with heavy tails), standard heavy‑traffic arguments give:
* Convergence of the vector of scaled queue lengths and ages to a semimartingale reflecting Brownian motion (SRBM) in a convex polyhedron (the orthant), possibly with time‑inhomogeneous parameters because of λj(t)\lambda_j(t)λj(t). Georgia Tech ISyE<ref>{{cite web|title=Georgia Tech ISyE|url=https://link.springer.com/article/10.1023/A%3A1017983532376|publisher=Georgia Tech ISyE|access-date=2025-12-04}}</ref>
* State space collapse: in the heavy‑traffic limit the dynamics can be represented in a low‑dimensional subspace parameterised essentially by total workload and a small number of aggregate “backlog ratios”. aviman.technion.ac.il<ref>{{cite web|title=aviman.technion.ac.il|url=https://aviman.technion.ac.il/files/References/INFORMS_2012_control_of_patient_flow.pdf|publisher=aviman.technion.ac.il|access-date=2025-12-04}}</ref>

This gives a Brownian control problem (BCP): control a reflected diffusion representing workload allocation across classes to minimise a convex cost, subject to constraints on “throughput times” (waiting plus service), as in Plambeck–Kumar–Harrison. SpringerLink<ref>{{cite web|title=SpringerLink|url=https://link.springer.com/article/10.1023/A%3A1017983532376|publisher=link.springer.com|access-date=2025-12-04}}</ref>