## 3 Traffic Flow Modeling  
[PERSON] and [PERSON] (PW) devised a system that characterizes the driver response, acceleration/deceleration, and the traffic changes conservation for a large ideal road. This system includes two equations. The first equation considers the changes on an ideally long road with conserved (negligible) traffic changes. The second equation considers the driver response and the traffic acceleration. The driver response contemplates a constant, which dictates a uniform alignment of vehicles to different forward traffic conditions. The PW system is [37; 38; 39; 40]  
\[\rho_{t}+(\rho v)_{x}=0 \tag{3}\] \[(\rho v)_{t}+\left(\rho v^{2}+A^{2}\rho\right)_{x}=\frac{v(\rho )-v}{\tau},\]  
where \(t\) and \(x\) presents the partial spatio-temporal changes; \(\rho\) is density; \(v\) is velocity (speed); and \(v(\rho)\) is the desirable equilibrium velocity. Relaxation time \(\tau\) is required by the following vehicles to achieve \(v(\rho)\) and align to the forward conditions; \(A\) is a uniform (constant) velocity at which drivers align to different traffic conditions. Traffic density is measured in vehicles per meter (veh/m). The average speed is measured in m/s, and the unit of \(\tau\) is s. The unit of \(A\) is m/s. The expression (\(\rho v\)) is the traffic flow and is measured in vehicles per second (veh/s). This is not an appropriate approach, as it ignores the transition distance and the distance headway between vehicles during the spatial and temporal traffic evolution. The following vehicles preserve the safe distance headway \(d_{s}\) with the forward vehicles in order to avoid from an unfortunate event. This behavior is analogous to a spring with a mass \(m\) attached. As a force is applied to the mass, the spring is extended, which stores potential energy to return to its stable position. This is an analogous scenario \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Description** & **Equation** & \begin{tabular}{c} **Coefficient of** \\ **Determination** \\ \end{tabular} & \begin{tabular}{c} **Mean Squared** \\ **Error (MSE)** \\ \end{tabular} & \begin{tabular}{c} **Mean Absolute** \\ **Error (MAE)** \\ \end{tabular} \\ \hline \begin{tabular}{c} Logarithmic \\ regression, \\ traversed time \\ \end{tabular} & \(-10.77\ln t\) + 42.76 & 0.963 & 1.218 & 0.904 \\ \
\begin{tabular}{c} Exponential \\ regression, \\ traversed time \\ \end{tabular} & \(36.397 exp^{-0.06t}\) & 0.960 & 1.283 & 1.022 \\
\begin{tabular}{c} Linear regression, \\ traversed time \\ \end{tabular} & \(-0.455t+19.571\) & 0.855 & 3.798 & 1.836 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Determination coefficient, mean squared errors, and mean absolute errors for best fitness of the data. with vehicle traffic. At a faster speed, vehicles have larger distances between vehicles. As the speed reduces, the distance between vehicles reduces, until it reaches safe distance headway \(d_{s}\). That is, the vehicle transition distance \(d_{t}\) at different speeds fluctuates about \(d_{s}\). As \(d_{t}\) increases relative to \(d_{s}\), the vehicles accelerate, whereas the vehicles decelerate as the transition distance is smaller than \(d_{s}\). Considering the analogous mass-spring and vehicle flow behavior, [PERSON]'s law [54] for vehicle acceleration/deceleration \(\alpha_{t}\) during the transition can be devised as  
\[\alpha_{t}=\frac{k}{m}(d_{t}-d_{s}) \tag{4}\]  
where \(m\) and \(k\) represent the spring mass and constant, respectively; \(\eta=\frac{k}{m}\) is the sensitivity. \[\alpha_{t}=\eta(d_{t}-d_{s}), \tag{5}\]  
where \(\eta\) is analogous to the driver sensitivity during the traffic transition to align to the desirable velocity \(v(\rho)\) of the forward vehicles. The traffic transition is based on the driver sensitivity \(\eta\). For a small traversed time over a road section, the change to achieve the desirable velocity is large, and driver sensitivity is high.