How reaction time, update time, and adaptation time influence the stability of traffic flow

When modeling the acceleration and deceleration of drivers, there are three characteristic time constants that influence the dynamics and stability of traffic flow: The reaction time of the drivers, the velocity adaptation time needed to accelerate to a new desired velocity, and the numerical update time. By means of numerical simulations with a time-continuous car-following model, we investigate how these times interplay with each other and effectively influence the longitudinal instability mechanisms for a platoon of vehicles. The long-wavelength string instability is mainly driven by the velocity adaptation time while short-wavelength local instabilities arise for sufficiently high reaction and update times. Furthermore, we investigate the relation between large update time steps and finite reaction times as they both introduce delays in the reaction to the traffic situation. Remarkably, the numerical update time is dynamically equivalent to about half the reaction time, which clarifies the meaning of the time step in models formulated as iterated maps such as the Newell and the Gipps model. Furthermore, with respect to stability, there is an optimal adaptation time as a function of the reaction time.