# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 3 Traffic Flow and Its Characteristics
### Shape of the Fundamental Relation  
The concept of the fundamental relation in traffic flow, which describes the relationship between key variables such as spacing, speed, density, and flow, was first proposed by [PERSON] [27]. This relation is crucial for understanding how traffic behaves under different conditions and forms the basis of many traffic flow models. In 1955, [PERSON] and [PERSON], followed by [PERSON] in 1956, developed the LWR model, which uses hydrodynamic analogies and kinematic wave theory to describe the relationship between traffic density (\(k\)), flow (\(q\)), and speed (\(u\)) [29,30]. The static hydrodynamic equation for this model is as follows:  
\[q=ku \tag{5}\]  
Vehicle preservation involves the equation of dynamic continuity, for example, for the homogeneous case:  
\[\frac{\partial k}{\partial t}+\frac{\partial q}{\partial x}=\frac{\partial k} {\partial t}+\frac{\partial(ku)}{\partial x}=0 \tag{6}\]  
To solve a macroscopic flow model, a third equation, often referred to as the fundamental relation or diagram, is necessary. [PERSON] [27] proposed a linear relation that leads to a parabolic relationship between flow and density, where flow is zero when density is either zero or at its maximum (\(k_{\text{max}}\)) and reaches a peak at critical density (\(k_{\text{cr}}\)):  
\[u=u_{\text{max}}\bigg{(}1-\frac{k}{k_{\text{max}}}\bigg{)} \tag{7}\]  
The maximum density (\(k_{\text{max}}\)), when the flow is stopped, is named the jam density: \(k_{\text{jam}}=k_{\text{max}}\). Then, the fundamental relation is of the following form:  
\[q=4q_{\text{cap}}k\bigg{(}1-\frac{k}{k_{\text{jam}}}\bigg{)} \tag{8}\]  
The density-speed and density-flow according to this model are shown in Figure 2. The critical density (\(k_{\text{cr}}\)) divides traffic flow into two areas (or phases): free flow and congestion. In contrast to Greenshields, [PERSON] [31] proposed a logarithmic relation between flow and density that better aligns with experimental data in certain scenarios. This model introduces an adjusting parameter to account for variations in traffic behavior:  
\[q=q_{\text{cap}}\frac{ek}{k_{\text{jam}}}\text{ln}\left(\frac{k_{\text{jam}} }{k}\right) \tag{9}\]  
where \(e\) is the adjusting parameter, \(q_{\text{cap}}\) is the maximum flow, and \(k_{\text{jam}}\) is the jam (maximum) density. Because this equation is not defined at \(k=0\), some initial zone with instant speed is introduced. [PERSON] et al. extended the car-following theory to derive a relationship similar to [PERSON]'s model, emphasizing that the sensitivity of a following vehicle is inversely proportional to the spacing between vehicles [32]. [PERSON] further developed the fundamental relations by proposing an exponential model for speed-flow functions [33]:  
\[u_{s}=u_{s0}\text{exp}(-bk) \tag{10}\]  
which was generalized in the following form [34]:  
\[u_{s}=u_{s0}\text{exp}\left[-\left(\frac{k}{b}\right)^{a}\right] \tag{11}\]  
where \(u_{s}\) is the space mean speed, \(u_{s0}\) is the space mean speed at zero flow (maximum speed), and \(a\) and \(b\) are the parameters. Then, the critical density and capacity are as follows:  
\[k_{cr}=b\left(\frac{1}{a}\right)^{\frac{1}{2}} \tag{12}\]  
\[q_{\text{max}}=u_{s0}b(ae)^{-\frac{1}{a}} \tag{13}\]  
This model can be extended to represent capacity drop:  
\[u_{s}=\left\{ \begin{matrix}(u_{s0}-\Delta u)\text{exp}\left[-\left(\frac{k}{b} \right)^{a}\right]+\Delta u&for\;k<k_{cr}\\ (u_{s0}-\Delta u)\text{exp}\left[-\left(\frac{k}{b}\right)^{a}\right]&for\;k \geq k_{cr}\end{matrix} \right. \tag{14}\]  
where \(a\), \(b\), and \(\Delta u\) are the parameters. [PERSON] addressed the shortcomings of [PERSON]'s model [31] in low-density areas by proposing a discontinuous exponential form that combines elements of both [PERSON] and [PERSON]'s relations [35]:  
\[u_{s}=\left\{ \begin{matrix}u_{\text{max}}\text{exp}\left(-\frac{k}{k_{m}} \right)&for\;k<k_{cr}\\ c\ln\frac{k_{\text{jam}}}{k}&for\;k\geq k_{cr}\end{matrix} \right. \tag{15}\]  
where c and \(k_{m}\) are the parameters.