Coordinated Care: Capacity Allocation to Improve Itinerary Completion in Queueing Networks

Coordinated care is a burgeoning paradigm where patients receive diagnosis and treatment planning involving collaboration between two or more medical specialties to facilitate rapid and effective solutions to complex conditions. A key service metric for coordinated care organizations is how quickly they can move patients through their sequence of appointments at multiple clinical services in the network. Each patient's care path is uncertain when the appointment capacities are being planned, because information about the patient's condition evolves over the course of the patient's care process. Hence, the planning of root (first) appointments is the primary operational lever, as the other appointments evolve stochastically. In this work, we develop a discrete-time queueing network to optimize this root appointment allocation over a cyclic time horizon to maximize the proportion of patients that can complete their care by a class-dependent deadline. The model accounts for several salient features of coordinated care networks, including parallel appointment requests, stochastic paths, and time-varying features. We provide an {exact} characterization of the sojourn time in the network with a doubly-stochastic phase-type distribution and leverage a mean-field model with convergence guarantees to address intractability. We then develop a policy improvement framework that approximates the original stochastic optimization by a sequence of linear programs (LP), where the sojourn time model is parameterized in the policy evaluation step. The LPs are computationally efficient, and our algorithm can solve large-scale stochastic optimization for networks of realistic sizes (e.g., 26 service stations). In a case study of the Mayo Clinic, our solution improves on-time completion to more than 93\%, from 60\% under the current plan. We demonstrate that this is a multifaceted problem, and that ignoring any those facets can lead to poor performance. Simultaneously accounting for all these complexities makes manual template design challenging and highlights the practical significance of our optimization algorithm.