Delay Distributions in Appointment Systems with Generally and Non-Identically Distributed Service Times and No-Shows

Abstract

We derive the probability distribution of each patient's wait time and doctor's idle time and overtime in single-server systems where healthcare providers respond to appointment-driven arrivals with generally and non-identically distributed service times. Patient no-show probabilities and appointment intervals are also non-identical. The formulation is due to a convolution-based busy period analysis that can be implemented via numerical integration methods. A special case yields the distribution of customer wait times and server idle times in transient-phase D/M/1 queues for appointments at fixed-intervals, where analytical distribution functions facilitate a mathematical proof that delays are unfair to patients. We demonstrate more generally that appointment system assessment of delay tail probabilities (rather than expectations) considers the non-linear disutility for delays, and indeed reverses conclusions made in prior analysis.