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

===== 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.