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

==== 8. Final answer: policy and assumptions ====

===== 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.

===== 8.2 Assumptions under which this policy is provably asymptotically optimal =====
# Heavy‑traffic regime. The ED operates near capacity in the standard heavy‑traffic sense: arrival rates are scaled up as r→∞r\to\inftyr→∞ while the number of physicians is fixed, with load 1+O(1/r)1 + O(1/r)1+O(1/r).
# Arrival processes. For each class: - Interarrival times are heavy‑tailed but have finite variance (e.g. Pareto with α>2\alpha>2α>2, lognormal, etc.). - Time‑variation of rates λj(t)\lambda_j(t)λj(t) is piecewise‑Lipschitz and does not oscillate on the diffusion time scale, so a time‑inhomogeneous diffusion approximation is valid.
# Service times. General i.i.d. per class with finite second moments.
# Work‑conserving, preemptive‑resume service. Physicians never idle when work is waiting and can preempt service (this is used for the theory; non‑preemptive versions approximate it closely in practice).
# Cost and constraint structure. - Objective is long‑run average congestion cost with linear per‑patient holding costs, so by Little’s law this coincides with minimising mean time‑to‑physician. - The 99th‑percentile constraint on the sickest class is encoded as an asymptotic throughput‑time (deadline) constraint, with deadlines scaling as in the heavy‑traffic framework.
# Diffusion approximation and state‑space collapse. The diffusion limit exists and satisfies the usual technical conditions (existence and uniqueness of the reflecting Brownian motion, etc.), and the Brownian control problem is well‑posed.

Under these assumptions, the threshold‑plus‑“relative age over deadline” triage rule combined with a Gcµ/h‑type rule for IP patients is asymptotically optimal in the sense of Huang–Carmeli–Mandelbaum and Plambeck–Kumar–Harrison:
* It is asymptotically compliant with the class‑1 (and, if desired, other classes’) throughput‑time (deadline) constraints.
* Among all asymptotically compliant policies, it minimises the limiting average congestion cost, hence minimises mean time‑to‑physician.

So, in short:

: Asymptotically optimal policy:
: In heavy traffic, with finite‑variance heavy‑tailed arrivals and finite physicians, the optimal triage/dispatch policy that minimises mean time‑to‑physician while imposing a hard bound on the 99th‑percentile wait for the sickest patients is a threshold‑plus‑index policy:
: 
: Use a queue‑length threshold on sickest‑class backlog to decide whether to work on triage versus IP patients.
: 
: Within triage, always serve next the patient with the largest ratio of waiting time to their class‑specific deadline (smallest slack / earliest‑deadline‑first).
: 
: Within IP, use a Gcµ/h‑type index on queue lengths and effective service requirements.
: 
: This policy is rigorously proven asymptotically optimal (and asymptotically enforces the SLA) under the heavy‑traffic and regularity assumptions listed above.