Research on Mathematical Modeling and Control Strategy of Urban Traffic Flow Congestion Evolution with Time Delay Stochastic Differential Equations

Abstract

Urban road congestion is not simply caused by traffic demand exceeding capacity, but is a dynamic evolutionary result of the coupling of vehicle arrival randomness, driving response lag, signal control delay, path guidance feedback, accident disturbance, and road network bottlenecks. The traditional deterministic traffic flow model can describe the average relationship between density, speed, and flow, but it lacks sufficient explanation for phenomena such as peak period fluctuations amplification, sudden congestion transfer, and control lag failure. This article uses stochastic differential equations with time delay as a tool to construct an urban traffic congestion evolution model based on the absorption of macroscopic traffic flow theory, stochastic dynamical systems, and feedback control ideas. The model takes road density, average speed, queue length, and congestion intensity as state variables, introduces demand disturbance, capacity disturbance, and perceived noise, and characterizes the delay of driving response, detection and acquisition, signal execution, and information release through discrete and distributed time delays. Further research provides methods for congestion equilibrium state, mean square stability, threshold discrimination, and control strategy design, and proposes a governance framework that combines signal coordination, speed induction, demand regulation, path diversion, and model predictive control. The results show that the combined effect of time delay and noise will significantly reduce the system stability margin. Only by incorporating the control advance, feedback period and disturbance intensity into the unified model can the reliability of urban traffic congestion governance be improved.