Appointments for care pathways the Geo^x/D/1 queue with slot reservations

Motivated by the increasing popularity of care pathways in outpatient clinics, where some patients complete a significant part of the path in one day, hospitals aim to optimize the flow of these patients by prioritizing them in the appointment planning process. This paper considers the Geo$^x /D/1$ queue with slot reservations that serves regular patients and priority patients. Priority patients reserve a time slot in a reservation window and are blocked when all slots in the reservation window are occupied by other priority patients. The reservation window models the advance reservation of service slots by patients on a care pathway. We model the Geo$^x /D/1$ queue as a $M/G/1$-type queue and apply a matrix-analytic approach, which simplifies to a matrix-geometric solution. We use the vector generating functions to derive the patients’ waiting times. Numerical experiments illustrate the influence of the reservation window on the number of regular and priority patients present and the blocking probabilities for priority patients.