A Diffusion Model of Dynamic Participant Inflow Management

This paper studies a diffusion control problem motivated by challenges faced by public health agencies who run clinics to serve the public. A key challenge for these agencies is to motivate individuals to participate in the services provided. They must manage the flow of (voluntary) participants so that the clinic capacity is highly utilized, but not overwhelmed. The organization can deploy costly promotion activities to increase the inflow of participants. Ideally, the system manager would like to have enough participants waiting in a queue to serve as many individuals as possible and efficiently use clinic capacity. However, if too many participants sign up, resulting in a long wait, participants may become irritated and hesitate to participate again in the future. We develop a diffusion model of managing participant inflow mechanisms. Each mechanism corresponds to choosing a particular drift rate parameter for the diffusion model. The system manager seeks to balance three different costs optimally: i) a linear holding cost that captures the congestion concerns; ii) an idleness penalty corresponding to wasted clinic capacity and negative impact on public health, and iii) costs of promotion activities. We show that a nested-threshold policy for deployment of participant inflow mechanisms is optimal under the long-run average cost criterion. In this policy, the system manager progressively deploys mechanisms in increasing order of cost, as the number of participants in the queue decreases. We derive explicit formulas for the queue length thresholds that trigger each promotion activity, providing the system manager with guidance on when to use each mechanism.