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

==== 6. Structure of the asymptotically optimal policy ====

The asymptotically optimal policy for this ED model is essentially the ED policy proved optimal by Huang–Carmeli–Mandelbaum, specialised so that:
* triage waiting‑time deadlines are chosen so that the sickest class has deadline d1d_1d1 corresponding to the desired 99th percentile,
* congestion costs are linear (Ck(x)=ckxC_k(x)=c_k xCk(x)=ckx), so minimising congestion cost coincides with minimising mean time‑to‑physician.

Their main result: a simple threshold‑plus‑index policy is asymptotically optimal (in heavy traffic) while being asymptotically compliant with all deadlines. aviman.technion.ac.il<ref>{{cite web|title=aviman.technion.ac.il|url=https://pubsonline.informs.org/doi/10.1287/opre.2015.1389|publisher=pubsonline.informs.org|access-date=2025-12-04}}</ref>

===== 6.1 Triage vs. IP: threshold on a reference triage class =====

Choose a single triage class as a reference; with a tight SLA for the sickest class, it is natural to choose class 111.

In the rrr-th system, let Q1(r)(t)Q_1^{(r)}(t)Q1(r)(t) be the number of class‑1 triage patients in queue at time ttt, and d1(r)d_1^{(r)}d1(r) the corresponding deadline.

The policy uses a queue‑length threshold:
* 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), then every idle physician attends triage patients.
* Otherwise, physicians may prioritize IP patients according to a certain index rule (below).

Intuition: the threshold λ1(r)d1(r) \lambda_1^{(r)} d_1^{(r)}λ1(r)d1(r) is exactly the number of class‑1 patients that would be waiting if everyone were just on time relative to the deadline. Keeping Q1Q_1Q1 near this threshold in diffusion scale enforces the deadline in the limit. 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 is how the 99th‑percentile constraint is enforced: in heavy traffic, whenever the backlog of class‑1 patients threatens to grow beyond what is compatible with hitting deadline d1d_1d1, all physicians switch their effort to triage.

===== 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.

===== 6.3 Within in‑process (IP) patients: a Gcµ/h‑type rule =====

Conditioned on “serve IP now”, Huang–Carmeli–Mandelbaum use a generalised cμc\mucμ rule with feedback, equivalent to the Gcµ/h rule later analysed by Long et al. ResearchGate<ref>{{cite web|title=ResearchGate|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>

Specifically, with queue length Qk(t)Q_k(t)Qk(t) of IP class kkk, cost functions Ck(⋅)C_k(\cdot)Ck(⋅) (here taken linear), and “effective mean service time” mkem_k^emke that accounts for expected downstream work from feedback, an IP patient is chosen from class

k∈arg⁡max⁡kCk′(Qk(t))mke.k \in \arg\max_{k} \frac{C_k'(Q_k(t))}{m_k^e}.k∈argkmaxmkeCk′(Qk(t)).
With linear costs Ck(x)=ckxC_k(x)=c_k xCk(x)=ckx, this reduces to a weighted cμc\mucμ‑type rule: always serve an IP class with the largest ratio ck/mkec_k/m_k^eck/mke, effectively identical to the generalised cμc\mucμ-rule of van Mieghem. AIMS Press<ref>{{cite web|title=AIMS Press|url=https://www.aimspress.com/article/doi/10.3934/math.2025196?viewType=HTML|publisher=AIMS Press|access-date=2025-12-04}}</ref>

If you only care about time‑to‑first‑physician, you can ignore the IP piece; the triage part and threshold are the core of the answer.