Appointment-driven queueing systems with non-punctual customers

We consider a single-server queueing system where a finite number of customers arrive over time to receive service. Arrivals are driven by appointments, with a scheduled appointment time associated with each customer. However, customers are not necessarily punctual and may arrive either earlier or later than their scheduled appointment times or may not show up at all. Arrival times relative to scheduled appointments are random. Customers are not homogeneous in their punctuality and show-up behavior. The time between consecutive appointments is allowed to vary from customer to customer. Moreover, service times are assumed to be random with a \( \gamma \)-Cox distribution, a class of phase-type distributions known to be dense in the field of positive distributions. We develop both exact and approximate approaches for characterizing the distribution of the number of customers seen by each arrival. We show how this can be used to obtain the distribution of waiting time for each customer. We prove that the approximation provides an upper bound for the expected customer waiting time when non-punctuality is uniformly distributed. We also examine the impact of non-punctuality on system performance. In particular, we prove that non-punctuality deteriorates waiting time performance regardless of the distribution of non-punctuality. In addition, we illustrate how our approach can be used to support individualized appointment scheduling.