Strategic Idling in Appointment Systems with Sequential Servers

Problem definition: This paper addresses an appointment scheduling problem involving multiple sequential servers using a distributionally robust optimization (DRO) approach. Two decisions are optimized: the schedule for patient visits and an adjusting policy to rebalance customers’ waiting time across servers. Methodology/results: We formulate the distributionally robust appointment scheduling problem in sequential-servers systems using conic optimization, incorporating service time correlations. We find that the traditional cost-minimization approach results in imbalanced waiting times, concentrated at downstream servers. To address this, we propose strategically idling upstream servers, inspired by queueing literature, and develop a DRO model to jointly optimize the schedule and strategic idling (SI) policy. Through extensive numerical studies, we thoroughly examine the role of SI in a system’s performance and the effect of correlation information on the optimal schedule and SI policy. Finally, using data from a clinic, we conduct a case study to demonstrate the performance advantage of our SI policy, over existing SI policies in the literature. Managerial implications: First, our SI model provides a jointly optimal schedule and SI policy that is effective in balancing waiting times across servers, in some scenarios, and also reduces total waiting time, with the cost borne by the service provider’s overtime. Second, the typical dome-shaped schedule provides an advantage in systems with multiple congested servers, as it evenly distributes congestion across servers. Finally, incorporating correlations lessens the expected cost and improves the patients’ waiting experience with balanced waiting time across servers. Funding: Z. Yan received financial support from the Ministry of Education, Singapore [AcRF Tier 1 Grant RG20/23] and the School of Physical and Mathematical Sciences Collaborative Research Award (2024). Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0278 .