Dynamic Transfer Policies for Parallel Queues

We consider the problem of load balancing in parallel queues by transferring customers between them at discrete points in time. Holding costs accrue as customers wait in the queue, while transfer decisions incur both fixed (setup) costs and variable costs that increase with the number of transfers and travel distance, and vary by transfer direction. Our work is primarily motivated by inter-facility patient transfers to address imbalanced congestion and inequity in access to care during surges in hospital demand. Analyzing an associated fluid control problem, we show that under general assumptions, including time-varying arrivals and convex holding costs, the optimal policy partitions the state-space into a well-defined $\textit{no-transfer region}$ and its complement, implying that transferring is optimal if and only if the system is sufficiently imbalanced. In the absence of fixed transfer costs, an optimal policy moves the state to the no-transfer region's boundary; in contrast, with fixed costs, the state is moved to its relative interior. Leveraging our structural results, we propose a simulation-based approximate dynamic programming (ADP) algorithm to find effective transfer policies for the stochastic system. We investigate the performance and robustness of the fluid and ADP policies in a case study calibrated using data during the COVID-19 pandemic in the Greater Toronto Area, which demonstrates that transferring patients between hospitals could result in up to 27.7% reduction in total cost with relatively few transfers.