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

==== 1. Model ====

===== 1.1 System =====
* One ED with a fixed, finite number SSS of identical physicians.
* Patients are triaged into JJJ acuity classes j∈{1,…,J}j\in\{1,\dots,J\}j∈{1,…,J} at arrival; class 111 is the sickest.
* We distinguish: - Triage-waiting patients (not yet seen by a physician). - Optionally, in‑process (IP) patients who have already been seen and may return for further checks (Huang–Carmeli–Mandelbaum use this explicitly). 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>

You only asked for time‑to‑first‑physician, so the key control is over triage‑waiting patients.

===== 1.2 Arrival and service processes =====
* For each class jjj, arrivals form a renewal process Aj(t)A_j(t)Aj(t) with interarrival times i.i.d. from a heavy‑tailed distribution FjF_jFj (e.g. lognormal or Pareto with tail index αj>2\alpha_j>2αj>2), so: - E[Xj]<∞E[X_j]<\inftyE[Xj]<∞, E[Xj2]<∞E[X_j^2]<\inftyE[Xj2]<∞, - but tails are subexponential / heavy relative to exponential.
* Arrival rates λj(t)\lambda_j(t)λj(t) are time‑varying, piecewise‑Lipschitz, reflecting ED arrival patterns (morning lull, evening peak, etc.).
* Service requirements for a physician’s first contact are i.i.d. within class: Vj∼Gj,E[Vj]=mj,Var⁡(Vj)<∞.V_j \sim G_j, \quad E[V_j]=m_j,\quad \operatorname{Var}(V_j)<\infty.Vj∼Gj,E[Vj]=mj,Var(Vj)<∞.
* Independence between all arrival and service sequences.

The “heavy‑tailed but finite variance” assumption is deliberate: it preserves functional central limit theorems (so Brownian/diffusion approximations still apply) while allowing high variability and burstiness consistent with ED data. See, e.g., generic diffusion approximations for G(t)/GI/1G(t)/GI/1G(t)/GI/1 queues and heavy‑traffic models of time‑varying arrivals. arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/pdf/2407.08765|publisher=arxiv.org|access-date=2025-12-04}}</ref>

===== 1.3 Control =====

A policy chooses, whenever a physician becomes free, which waiting patient to start or resume service (preemptive‑resume is allowed for the theory).

We restrict to work‑conserving policies: no idling while a patient is waiting.