Analysis of patient waiting time governed by a generic maximum waiting time policy with general phase-type approximations

Waiting time can affect patient satisfaction and quality of care in the emergency department (ED). Studies have shown that waiting time accounted for more than 50 % of total patient turnaround time at ED. The objective of this study is to examine a maximum waiting time policy such that patients who would experience a long wait are assumed to be processed in a threshold period. In particular, we are interested to investigate the associated factors of the policy such as new mean waiting time and the threshold period and their interaction. Under the policy, original patient waiting distribution is transformed to a piecewise distribution where one piecewise discontinuous and one piecewise continuous distributions are further investigated. Under the phase-type (PH) distribution assumption on the original waiting time, we establish closed-form expressions concerning new mean waiting time and time points of the threshold period. By fitting PH distributions to patient waiting data of an emergency department in Singapore, the factors are then estimated under various scenarios using the obtained analytical expressions. Specifically, for a given target mean waiting time, the threshold period needed in the policy is estimated. New mean waiting time is assessed with different choices of the threshold period. Analytical expressions in terms of the variance of the transformed waiting time and the threshold period are also presented.