This paper provides evidence that in an emergency department (ED), the arrival of an additional low-acuity patient substantially increases the wait time to start of treatment for high-acuity patients. This contradicts a long-standing prior conclusion in the medical literature that this effect is “negligible”. The prior methodology underestimates the effect by neglecting how delays are propagated in queueing systems. In contrast, this paper develops and validates a new estimation method based on queueing theory, machine learning, and causal inference. Wait time information displayed to low-acuity patients provide a quasi-randomized instrumental variable and is used to correct for omitted variable bias. Through a combination of empirical and queueing theoretic analyses, this paper identifies the two primary mechanisms by which a low-acuity patient increases the wait time for high-acuity patients: pre-triage delay and transition-delay. Thus the paper identifies ways to reduce high-acuity patients’ wait time, including reducing the standard deviation or mean of the transition delay in preemption, preventing transition delays by providing vertical or “fast track” treatment to more low-acuity patients; and designing wait time information systems to divert (especially when the ED is highly congested) low-acuity patients that do not need ED treatment.