Stability and Controllability of Revenue Systems via the Bode Approach

In online revenue systems, e.g. an advertising system, budget pacing plays a critical role in ensuring that the spend aligns with desired financial objectives. Pacing systems dynamically control the velocity of spending to balance auction intensity, traffic fluctuations, and other stochastic variables. Current industry practices rely heavily on trial-and-error approaches, often leading to inefficiencies and instability. This paper introduces a principled methodology rooted in Classical Control Theory to address these challenges. By modeling the pacing system as a linear time-invariant (LTI) proxy and leveraging compensator design techniques using Bode methodology, we derive a robust controller to minimize pacing errors and enhance stability. The proposed methodology is validated through simulation and tested by our in-house auction system, demonstrating superior performance in achieving precise budget allocation while maintaining resilience to traffic and auction dynamics. Our findings bridge the gap between traditional control theory and modern advertising systems in modeling, simulation, and validation, offering a scalable and systematic approach to budget pacing optimization.