Car-following models, crucial for understanding traffic flow dynamics, have been evolving since the 1950s, offering microscopic insights into vehicle behaviors. These models, based on ordinary, delayed and stochastic differential equations, depict how vehicles adjust speed and acceleration in response to spacing and speed of the preceding vehicle, with applications ranging from adaptive cruise control (ACC) to traffic engineering. The adaptive time gap (ATG) model introduces a safety parameter for time gap regulation, ensuring stable uniform dynamics. The ATG model is a good candidate for ACC systems. The collision-free optimal velocity (CFOV) model is a simplified version of the Optimal Velocity (OV) model by Bando et al. (1995) that mitigates unrealistic collisions and backward movements while describing stop-and-go traffic patterns (see the simulations #1, #2, #3). Derivation of the microscopic CFOV model results in a parabolic convection-diffusion partial differential macroscopic equation while derivation of the ATG models leads to the Lighthill-Whitham-Richards (LWR) macroscopic model.

Stop-and-go wave and traffic instability

Stop-and-go waves, inherent in congested traffic (see the experiments by Sugiyama et al. or more recently by Stern et al.), pose safety and environmental concerns. Stop-and-go dynamics in single-file motion can emerge due to various factors including delay and latency in nonlinear car-following models, and stochastic noise effects. They induce a phase transition from a laminar traffic to an oscillating traffic with stop-and-go waves, including capacity drop and hysteresis phenomena. Stability analysis of car-following models is vital for ensuring the effectiveness of adaptive cruise control systems. Several recent experiments (see e.g. the experiments by Gunter et al. or the OpenACC experiments) have shown that current ACC systems describe unstable collective behaviors. Local and global (collective) stability considerations essential for understanding oscillating dynamics on infinite or periodic roadways, and . Mathematical techniques range from Laplace transforms for platooning system, to linear algebra with circulant matrix and exponential ansatz for finite periodic and infinite systems, respectively. Inspired by pioneer works with cellular automata models by Nagel, Schreckenberg, and Schadschneider, continuous-time noisy model range from white noise models to colored noises such as Brownian noises given by the Ornstein-Uhlenbeck process (see this presentation).