A Network-Modeling View of Emergency-Department Patient Flow: Calibrated Admission-Risk Screening at Triage With Distribution-Free, Acuity-Conditional Coverage Guarantees

An emergency department (ED) is a network: patient streams are sorted by acuity, queued, and routed through a finite pool of physicians and beds toward admission or discharge. We study admission-risk screening on 558,029 adult ED visits from the Yale-New Haven Health System, using the information present at the triage node: the Emergency Severity Index (ESI) and six triage vital signs, with age and sex retained for stratified validity checks. A gradient-boosted classifier ranks admission with AUC = 0.809, accuracy = 0.760, and Brier score = 0.157 on a held-out set of 111,607 visits. The methodological contribution is a Mondrian (per-acuity) split-conformal layer with a finite-sample, class-conditional coverage guarantee (Theorem 1) and a matching upper bound. Empirical coverage is 0.910 against a nominal 0.90, with mean prediction-set size 1.43. Residual audits show that ESI-level coverage can conceal within-class gaps, most clearly for abnormal vital signs inside ESI-4 (coverage 0.549 versus 0.943 when vitals are not abnormal). A joint Mondrian on ESI x age restores those cells to the nominal band whenever the calibration count meets n_c >= 99, the size that keeps the additive slack 1/(n_c+1) at most 0.01 (Proposition 1). We map each prediction set to an explicit routing action and a three-term decision loss (false preparation, delayed request, unresolved hedge). At a five-to-one delay-to-false-prep cost ratio, the conformal policy has mean cost 0.544, against 0.899 for a hard 0.50 threshold. Theorem 4 places the predictor in the queueing dynamics: committed false bed requests occur at a rate at most alpha, and MaxWeight stability is preserved when the oracle bed load plus that extra load stays below one. Scheduling and surge rules (Theorems 2 and 3) remain framework theory pending timestamped data. Every number is computed from real records.