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

===== 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.