# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 3 Traffic Flow and Its Characteristics  
Traffic flow is a multifaceted concept that can be understood in both broad and narrow senses, each of which provides valuable insights into transportation systems. From a broad perspective, traffic flow encompasses the interaction between various modes of transport and the road infrastructure. This interaction involves not only vehicles but also pedestrians, cyclists, and unmanned systems, all of which navigate through a network of streets, highways, intersections, and traffic control devices. The goal of studying traffic flow at this level is to develop an optimal traffic network that balances the needs of all users while minimizing congestion and improving overall efficiency. Within this broad context, traffic flow can be viewed in terms of system optimum and user equilibrium. The system optimum seeks to minimize the total travel time for all users across the network, whereas user equilibrium refers to each driver choosing the shortest possible route from their current location to their destination. However, achieving both system optimum and user equilibrium simultaneously in real-world traffic networks can be challenging, if not impossible. Therefore, optimizing transport infrastructures is a key objective in transportation engineering, aimed at enhancing both efficiency and fairness within the system. In a narrower sense, traffic flow refers to the movement of vehicles along a one-dimensional pathway, such as a road or highway. This concept is often characterized by key parameters: speed (\(u\)), density (\(k\)), and flow (\(q\)). Understanding these parameters and their interrelationships is fundamental to traffic flow theory. A time-space diagram (Figure 1) is a useful tool for visualizing these relationships. It graphically represents the movement of vehicles over time along a specific travel path, with each vehicle's trajectory depicted as a curve with a non-negative slope. The position of each vehicle at any given time is shown along these trajectories. The average and instantaneous speeds can be defined for every vehicle. The instantaneous speed of the individual vehicle (\(u_{i}\)) is the slope of the trajectory at the appropriate point on the trajectory (slope of the red line in Figure 1, \(u_{i}=dx/dt\)). The average speed of the vehicle is measured in a selected section of the route (slope of the blue line in Figure 1, \(u_{ai}=\Delta x/\Delta t\)). Two average speeds for a group of vehicles are defined for traffic flow in a given period of time and road segment: space and time mean speeds. The time mean speed (\(u_{t}\)) is an average value of the instantaneous speeds (\(u_{i}\)) of vehicles measured at a reference point. If instantaneous vehicle speeds are measured at the reference point (\(x_{r}\)) (in Figure 1, the speed (\(u_{i}\)) at the point \(x_{r}\) is a slope at the point \(x_{r}\) that is similar to the red line), then the time mean speed is calculated as follows:The space mean speed (\(u_{s}\)) is an average value of the average vehicle speeds (\(u_{ai}\)) of the vehicles measured on the reference road segment. If the reference road segment is the whole space presented in Figure 1, then the average vehicle speeds (\(u_{ai}\)) is a slope similar to the blue line and is calculated as follows:  
\[u_{s}=\frac{1}{n}\sum_{i=1}^{n}u_{ai} \tag{2}\]  
The vertical distance between the trajectories is the vehicle spacing (s) between the centers of two vehicles. The horizontal distance represents the vehicle headway (\(h\)). The density (\(k\)) is the number of vehicles per unit length of the reference road segment. The density is presented in Figure 1 as a vertical cross-section of the space-time diagram at time \(t_{r}\). It is inverse to the average spacing:  
\[k=\frac{1}{\frac{1}{n}\sum_{i=1}^{n}s_{i}} \tag{3}\]  
Flow (\(q\)) is the number of vehicles that pass through the reference point per unit of time. The flow is presented in Figure 1 as a horizontal cross-section of the space-time diagram at point \(x_{r}\).