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

===== 6.2 Within triage: “relative age over deadline” (EDF‑type) rule =====

Conditioned on “serve triage now”, the policy chooses the head‑of‑the‑line patient from the triage class j∈{1,…,J}j\in\{1,\dots,J\}j∈{1,…,J} that maximises

τj(r)(t)dj(r),\frac{\tau_j^{(r)}(t)}{d_j^{(r)}},dj(r)τj(r)(t),
where τj(r)(t)\tau_j^{(r)}(t)τj(r)(t) is the age (waiting time so far) of the head‑of‑line class‑jjj patient, and dj(r)d_j^{(r)}dj(r) is the deadline (target time‑to‑physician) for that class. 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>

Equivalent interpretations:
* Earliest deadline / smallest slack first: choosing arg⁡max⁡τj/dj\arg\max \tau_j/d_jargmaxτj/dj is equivalent to choosing the class with smallest slack dj−τjd_j - \tau_jdj−τj if deadlines differ by fixed multiplicative constants.
* A “relative lateness” rule: the class that is proportionally closest to missing its target gets priority.

For the ED objective “minimize mean time‑to‑physician subject to sickest 99th‑percentile constraint”, a natural choice is:
* Class 1: set a firm deadline d1d_1d1 corresponding to the clinical SLA.
* Other classes j>1j>1j>1: choose larger deadlines djd_jdj (e.g., 15, 30, 60 minutes) that encode their looser service goals; or take dj→∞d_j\to\inftydj→∞ if they are unconstrained, in which case they effectively only get priority once high‑acuity classes are comfortably within their targets.

In particular, when class 1 has a much smaller d1d_1d1 than other classes, it has significantly higher priority whenever its patients have waited non‑trivially.