Developing nonlinear queuing regressions to increase emergency department patient safety: Approximating reneging with balking

Administrators know when Emergency Department (ED) overcrowding is a problem in their hospital. Lead times to change ED capacity are long and require strategic tools. ED patients who Leave WithOut Treatment (LWOT) before seeing a physician are, in queuing nomenclature, ‘reneging’ from an overcrowded situation and are an important measure of ED patient safety. We propose to enable strategic decision making on future ED capacity on the basis of patient safety (rather than congestion measures). We hypothesize that the LWOT reneging percentage is captured by the balking probability (pK) relationship of an M/M/1/K queue. If true, this relationship is superior to the typical ad hoc regression relationships commonly found. Since it is based on a physical scientific mechanism, the sample size requirements and extrapolation power are improved. We derive the form of a binomial response nonlinear weighted regression model that best fits pK for predicting LWOT to long-term ED performance by means of Gauss–Newton linearization. Our results include asymptotic Wald confidence intervals on prediction, specific Pearson and Deviance model goodness-of-fit tests, and residual analysis that facilitate identification of outlying data points. None of these features exist for reneging (or balking) models previously presented in the literature.