# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 5 Traffic Flow Models
### Microscopic Models  
#### 5.2.1 Car-Following Models  
[PERSON] and [PERSON] [121] provided a comprehensive genealogical tree of vehicle headway modeling, particularly focusing on car-following models within microscopic traffic modeling. Their work delineates three primary branches--safe distance, stimulus response, and psychophysical-psychological models--highlighting their development and evolution from the late 1950s to the present. These branches have largely evolved independently, illustrating the diverse theoretical foundations within car-following models. Earlier, [PERSON] et al. [120] had also explored these modeling branches, identifying similar patterns in safe-distance and stimulus-response models, while also introducing a cellular automata branch. This addition reflects the increasing complexity and diversity of approaches in traffic modeling. [PERSON] et al. [153; 154] further enriched this discourse by providing a historical review of car-following models, particularly in the context of vehicle-to-everything (V2X) environments. They mapped key developments in the car-following theory, including the introduction of artificial intelligence models and intelligent driving models, onto a timeline, thus offering a broader context for understanding the evolution of these models. In car-following models, the dynamics between a leader and a follower vehicle are central, with the follower's behavior modeled in terms of position, velocity, acceleration, or combinations thereof. The foundational work by [PERSON] [155] in 1953 introduced the first microscopic car-following model, establishing a rule based on maintaining a safe following distance. The law specifies the distance from the leader for the follower. This distance (\(d\)) is the sum of the length of the leading vehicle (\(l\)), a minimum distance at rest (\(d_{0}\)), and a distance proportional to the velocity of the follower (\(u\)):  
\[d=l+d_{0}+ku \tag{20}\]  
Building on [PERSON] model, researchers such as [PERSON] et al. [156], [PERSON] et al. [157], [PERSON] [158], [PERSON] and [PERSON] [159], and others further refined the car-following theory, each contributing to its theoretical and practical evolution. Notably, [PERSON] [160] and [PERSON] et al. [161] developed models that introduced new dynamics and feedback mechanisms, which have been foundational in the continued development of microscopic traffic models. [PERSON] [162] and [PERSON] [163] also made significant contributions, particularly in the context of traffic simulations and the modeling of driver behaviors. These foundational works are discussed in greater detail in subsequent sections. Recent advancements in car-following models have introduced new methodologies and technologies. For example, [PERSON] and [PERSON] [164] discretized a continuous car-following model into a difference equation and proposed a feedback control system, enhancing the model's applicability in various traffic scenarios. [PERSON] [165] incorporated headway memory and evolution trends into existing models, further refining the predictive accuracy of car-following behaviors. [PERSON] et al. [166] extended these models for intelligent transportation systems, integrating vehicle-to-vehicle communication and considering driver characteristics to improve the realism of simulations. Emerging technologies have also influenced car-following models. [PERSON] et al. [167] proposed a model combining Markov theory with a gated recurrent unit neural network, enabling trajectory predictions in micro-traffic simulations. This approach exemplifies the integration of deep learning techniques into traditional traffic modeling. Similarly, [PERSON] et al. [168] analyzed a mixed traffic flow based on discrete following intervals, proposing a discrete car-following interval model tailored for autonomous vehicles. Other researchers, such as [PERSON] et al. [169; 170], have focused on nonlinear dynamics within car-following models.