# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 5 Traffic Flow Models
### Microscopic Models
#### 5.2.2 Safe-Distance Models  
Pipes' model (Equation (19)), which is a foundational safe-distance model, introduces acceleration that is proportional to the relative velocity between vehicles. This model set the stage for subsequent developments in car-following theories by establishing a basic rule for maintaining safe distances between vehicles.  
Building on this, [PERSON] and [PERSON] [159] introduced an important refinement by incorporating a driver's tendency to avoid collisions in all situations. They added a time delay to account for the follower's reaction to the leader's changes, making the distance term (\(d_{0}\)) velocity-dependent. This adjustment provided a more realistic depiction of driver behaviors, acknowledging the delays inherent in human reaction times.  
[PERSON] [160] further advanced the field by proposing a nonlinear relationship between acceleration and both the distance to the leader and the difference between the follower's current and optimal velocities. This approach introduced a more complex dynamic into the modeling of car-following behaviors, allowing for a better understanding of how drivers adjust their speed based on the distance and speed differential relative to the vehicle ahead. Similarly, [PERSON] et al. [161] also explored nonlinear dependencies but focused on the leader's velocity, the relative velocity, and the distance between vehicles, adding another layer of sophistication to car-following models by integrating multiple factors into the decision-making process of drivers.  
[PERSON] [176] contributed a significant advancement by dividing vehicle movements into two regimes: one where the vehicle's velocity is limited by its own capabilities (such as the maximum velocity and acceleration), and another where the velocity is limited by the safe distance to the leader. This dual-regime approach offered a more comprehensive framework for understanding vehicle dynamics under varying traffic conditions.  
[PERSON] [177] later proposed a simplified car-following model, which became a pivotal development in the field. In this model, vehicles are assumed to follow the same trajectory, merely translated in space and time. This simplification was particularly important as it allowed researchers to bypass the complexities of the transition process when a leader changes speed, which had been the focus of earlier models. By focusing instead on the sources of speed changes, [PERSON]'s model facilitated the integration of car-following behaviors into multiscale models, thereby broadening the model's applicability in more complex traffic simulations.  
[PERSON] and [PERSON] [84] introduced further nuance by incorporating driver behaviors, such as timid and aggressive driving, into their models. This addition recognized the variability in driver behaviors, which can significantly impact traffic flow and safety and allowed for a more realistic simulation of traffic dynamics.  
Collectively, these studies illustrate the evolution of car-following models from simple safe-distance rules to more complex and nuanced approaches that account for various factors influencing driver behaviors and vehicle dynamics. The relationships between these models demonstrate a clear progression in the sophistication of car-following theories, each building on and refining the concepts introduced by its predecessors. By integratingthese models into broader traffic simulations, researchers have been able to develop more accurate and versatile tools for understanding and managing traffic flow.