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

==== 7. Why this policy is asymptotically optimal (proof sketch) ====

The full proofs are long (they use reflected Brownian motions, martingale problems, and variational inequalities), but the structure can be described succinctly:

===== 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.

===== 7.2 Step 2: Brownian control problem =====

The limiting control problem becomes:
* Control the allocation of instantaneous service rate among triage classes jjj and IP classes kkk as a function of the current state (Qj,τj,Qk)(Q_j,\tau_j,Q_k)(Qj,τj,Qk).
* Objective: minimise E∫0T∑jcjQj(t)+∑kCk(Qk(t)) dt\mathbb{E}\int_0^T \sum_j c_j Q_j(t) + \sum_k C_k(Q_k(t)) \, dtE∫0Tj∑cjQj(t)+k∑Ck(Qk(t))dt subject to the reflected Brownian dynamics and the hard constraint that the class‑1 age process τ1(t)\tau_1(t)τ1(t) stays below d^1\hat d_1d^1 (asymptotically).

Using Lagrange multipliers for the constraint and standard arguments for convex stochastic control, the problem decomposes into:
* A workload allocation problem across triage vs IP.
* A triage sequencing problem across classes jjj.
* An IP sequencing problem across IP classes kkk.

The KKT conditions of the associated static optimisation (the “inner loop” of the diffusion control) yield index structures: at any state, the marginal value of putting one more unit of service capacity into class jjj is a monotone function of τj/dj\tau_j/d_jτj/dj; similarly, for IP classes, the marginal value is proportional to Ck′(Qk)/mkeC_k'(Q_k)/m_k^eCk′(Qk)/mke.

===== 7.3 Step 3: Threshold plus relative‑age rule for triage =====

Plambeck–Kumar–Harrison (single‑server model) and Huang–Carmeli–Mandelbaum (multi‑server ED) solve exactly this kind of BCP with throughput‑time (deadline) constraints and show:
* There exists a reference triage class whose backlog relative to its deadline effectively controls all other triage classes in the limit.
* An asymptotically optimal policy is: - Route service to triage as soon as the reference class backlog exceeds the “just‑in‑time” level λ1d1\lambda_1 d_1λ1d1. - Within triage, sequence classes so as to equalise their scaled “relative backlogs” (ratios of queue length and/or age to λjdj\lambda_j d_jλjdj), which is equivalent to prioritising the class with the largest τj/dj\tau_j/d_jτj/dj.

They show two key properties:
# Asymptotic compliance: under this policy, the diffusion limit of the head‑of‑line age processes satisfies sup⁡0≤t≤T(τ~j(t)−d^j)+=0\sup_{0\le t\le T}(\tilde \tau_j(t)-\hat d_j)^+ = 00≤t≤Tsup(τ~j(t)−d^j)+=0 for all jjj almost surely, so all deadlines are respected in the limit.
# Asymptotic optimality: among all asymptotically compliant policies, this policy minimises the limiting cost functional J(π)J(\pi)J(π); equivalently, it minimises the heavy‑traffic limit of the congestion cost, and thus (under linear costs) the average time‑to‑physician.

===== 7.4 Step 4: Lifting back to the original queue =====

Finally, standard interchange‑of‑limits and stability arguments (as in Harrison’s and Atar–Mandelbaum–Reiman’s work on multiclass queues) show that if a policy is asymptotically optimal for the diffusion model and the diffusion model is a valid heavy‑traffic approximation, then the corresponding sequence of prelimit policies is asymptotically optimal for the original queues.

Under the finite‑variance heavy‑tailed assumption and time‑varying but “slow” arrival rates (piecewise‑Lipschitz), the diffusion approximation is valid on any fixed time horizon, and the asymptotic compliance w.r.t. deadlines translates into achieving arbitrarily tight service‑level targets for percentiles such as the 99th.