Early Reservation for Follow-up Appointments in a Slotted-Service Queue

We study an appointment-based slotted-service queue with the goal of maximizing service volume. Returning customers prefer to be served by the same service agent as in their previous visit. This model captures aspects of a whole host of settings, including medical clinics, law firms, and tutoring services. We consider a simple strategy that a service provider may use to reduce balking among returning customers—designate some returning customers as high-priority customers. These customers are placed at the head of the queue when they call for a follow-up appointment. In an appointment-based system, this policy can be implemented by booking a high-priority returning customer’s appointment right before he or she leaves the service facility. We focus on a need-based policy in which the decision to prioritize some customers depends on their return probability. We analyze three systems: an open-access system, a traditional appointment system, and a carve-out system. We show that in an open-access system, the service provider should never prioritize returning customers in order to maximize the throughput rate. However, it is always optimal to prioritize some customers in a traditional appointment system. In the carve-out system, which may be modeled as a system with two parallel queues, the optimal policy varies depending on which queue is more congested. In the traditional system, we prove that the throughput rate is a quasi-concave function of the threshold under the assumption that returning customers see time averages. This allows service systems to determine optimal operating policies that are both easy to implement and provably optimal.