  We study a control problem for queueing systems where customers may return for additional episodes of service after their initial service completion. At each service completion epoch, the decision maker can choose to reduce the probability of return for the departing customer but at a cost that is convex increasing in the amount of reduction in the return probability. Other costs are incurred as customers wait in the queue and every time they return for service. Our primary motivation comes from post-discharge Quality Improvement (QI) interventions (e.g., follow up phone-calls, appointments) frequently used in a variety of healthcare settings to reduce unplanned hospital readmissions. Our objective is to understand how the cost of interventions should be balanced with the reductions in congestion and service costs. To this end, we consider a fluid approximation of the queueing system and characterize the structure of optimal long-run average and bias-optimal transient control policies for the fluid model. Our structural results motivate the design of intuitive surge protocols whereby different intensities of interventions (corresponding to different levels of reduction in the return probability) are provided based on the congestion in the system. Through extensive simulation experiments, we study the performance of the fluid policy for the stochastic system and identify parameter regimes where it leads to significant cost savings compared to a fixed long-run average optimal policy that ignores holding costs and a simple policy that uses the highest level of intervention whenever the queue is non-empty. In particular, we find that in a parameter regime relevant to our motivating application, dynamically adjusting the intensity of interventions could result in up to 25.4% reduction in long-run average cost and 33.7% in finite-horizon costs compared to the simple aggressive policy. 
Dynamic Control of Service Systems with Returns: Application to Design   of Post-Discharge Hospital Readmission Prevention Programs