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

===== 8.1 Policy =====

Under the assumptions below, the asymptotically optimal triage/dispatch policy that minimises mean time‑to‑physician subject to a hard upper bound on the 99th‑percentile wait for the sickest patients has the following structure:
# Service‑level encoding. Fix a target d1d_1d1 for the 99th‑percentile time‑to‑physician for class‑1 patients. Choose class‑specific deadlines djd_jdj for other classes (possibly large / loose).
# Threshold for triage vs IP. Designate class 1 as the reference triage class. When a physician becomes idle in the rrr-th system: - If Q1(r)(t)≥λ1(r)d1(r)Q_1^{(r)}(t) \ge \lambda_1^{(r)} d_1^{(r)}Q1(r)(t)≥λ1(r)d1(r), serve triage patients. - Otherwise, serve IP patients. Intuitively: as soon as the high‑acuity backlog threatens the SLA d1d_1d1, all physicians turn to triage until that backlog comes down.
# Within triage (who gets the next physician). If triage has been chosen, assign the physician to the head‑of‑line triage patient in the class j∈arg⁡max⁡k∈{1,…,J}τk(r)(t)dk(r).j \in \arg\max_{k\in\{1,\dots,J\}} \frac{\tau_k^{(r)}(t)}{d_k^{(r)}}.j∈argk∈{1,…,J}maxdk(r)τk(r)(t). That is, give priority to the patient who is proportionally closest to missing their class‑specific target time djd_jdj. When all service rates are similar across classes, this is an earliest‑deadline / smallest‑slack‑first rule.
# Within IP (if you model IP states). If IP has been chosen, assign the physician according to a Gcµ/h‑type index: k∈arg⁡max⁡lCl′(Ql(t))mle,k \in \arg\max_{l} \frac{C_l'(Q_l(t))}{m_l^e},k∈arglmaxmleCl′(Ql(t)), which, under linear congestion costs, reduces to serving the IP class with the largest cl/mlec_l/m_l^ecl/mle.

In practice: for time‑to‑physician, steps (2)–(3) are the key parts; step (4) matters only if you explicitly model returning patients and want full ED‑wide optimality.