Time to Equilibrium in a Car-Following Scenario under Local Stable Conditions

In traffic flow stability analysis, extensive research has been conducted on stability criteria, offering binary classifications of stability, that is, defining flow as stable or unstable. Despite being informative, this classification falls short of providing detailed characteristics of stability, such as the time required for a vehicle to regain equilibrium subject to a disturbance from a preceding vehicle. To address this problem, in this study, a quantitative metric, the time to equilibrium (TTE), is introduced under the condition of local stability. In a car-following scenario of two vehicles, considering that the preceding vehicle undergoes a short-term deceleration–acceleration change, an analytical formulation of the TTE is derived by employing linear stability analysis with the disturbance approximated using the Dirac delta function. The bisection method is then applied to approximate analytical solutions. Subsequent simulation experiments, utilizing various car-following parameters and disturbance settings, demonstrate the general validity of the proposed analytical TTE, barring some large errors in extreme scenarios (unlikely in real-world driving) and the intrinsic features of the Dirac delta function. We then provide applicable ranges for car-following parameters with different selection criteria. Lastly, by using real-world vehicle trajectory data, the proposed TTE is further validated.