Diagnostic clinics are among healthcare facilities that suffer from long waiting times which can cause medical issues and lead to increases in patient no-shows. Reducing waiting times without significant capital investments is a challenging task. We tackle this challenge by proposing a new appointment scheduling policy for such clinics that does not require significant investments. The clinic in our study serves outpatients, inpatients, and emergency patients. Emergency patients must be seen on arrival, and inpatients must be given next day appointments. Outpatients, however, can be given later appointments. The proposed policy takes advantage of this by allowing the postponement of the acceptance of appointment requests from outpatients. The appointment scheduling process is modeled as a two-stage stochastic programming problem where a portion of the clinic capacity is allocated to inpatients and emergency patients in the first stage.In the second stage, outpatients are scheduled based on their priority classes. After a detailed analysis of the solutions obtained from the two-stage stochastic model, we develop a simple, non-anticipative policy for patient scheduling. We evaluate the performance of this proposed, easy-to-implement policy in a simulation study which shows significant improvements in outpatient indirect waiting times. ) classify waiting time of patients into two categories. They define direct waiting time as the time the patient waits in the healthcare facility on the day of appointment and indirect waiting time as the time between the day the patient requests an appointment and the appointment day. Unfortunately, long indirect waiting times are common in practice. For instance, Kesling and Nissenbaum (2014) reported that 84% of patients in Veterans Affairs (VA) hospitals wait more than 14 days to see a physician. In addition to the medical issues that long indirect waiting times cause, they can also lead to increases in patient no-shows (Green et al. 2006) which have significant effect on annual revenues (Moore et al. 2001). Thus, healthcare managers face the challenge of improving their appointment systems to decrease waiting times and no-shows without incurring major capital costs. Diagnostic clinics are among the healthcare facilities that generally suffer from long indirect waiting times (McCarthy et al. 2000). One such clinic is the Radiology Department at PrismaHealth, our collaborator on this study. The clinic provides service to outpatients, inpatients, and emergency patients. The requests for appointments are handled on a first-come-first-served (FCFS) basis. The emergency patients are the highest priority group, followed by inpatients and then outpatients. The outpatients are further categorized into a number of priority classes based on co-morbidities and chronic conditions. The emergency patients are seen as soon as they arrive if there is capacity or immediately referred to another clinic. The inpatients are either given a next day appointment during regular hours or seen during overt...