Analytical Formulation of Autonomous Vehicle Freeway Merging Control With State-Dependent Discharge Rates

The core of the freeway merging control problem lies in dynamic queue propagation and dissipation linked to merging vehicle behavior. Traditionally, queueing is modeled through demand-supply interactions with time-varying demand and fixed capacity. However, field observations indicate that flow rates drop during freeway congestion at merges due to the impact of intersecting traffic—a factor often overlooked in fundamental diagrams. This manuscript introduces an analytical approach to characterize and control the dynamic multi-stage merging of autonomous vehicles, prioritizing traffic efficiency and safety. For the first time, the effective discharge rate at the merging point is analytically derived in a closed-form, accounting for the reduction caused by the dynamic multi-stage merging process. Leveraging this expression, performance metrics such as queue length and traffic delay are derived as the first optimization objective. Additionally, a crash risk function is formulated to quantitatively assess potential collisions during the merging process, serving as the second objective. Finally, the problem is formulated as a dynamic programming model to jointly minimize delay and crash risk, with the merging location and speed as decision variables. Given the terminal state, the ramp vehicle merging task is formulated as a recursive optimization problem, employing backward induction to find the minimum-cost solution. Numerical experiments using the NGSIM dataset validate the derived effective discharge rate. The results indicate that the proposed model outperforms two benchmark algorithms, leading to a more efficient and safer merging process.