## 4 Model Hyperbolicity  
The proposed traffic system (10) should satisfy the hyperbolicity condition, which says that the rate of change during transition for a congestion buildup should be smaller than the rate of change in free flow [55]. In other words, the free flow traffic evolution is at a higher rate than during congestion. The traffic dissemination is quick during free flow and is slow during congestion. This hyperbolicity property of the proposed traffic system should ensure that velocity fluctuates in a finite range of 0 and the minimum limit. To check the hyperbolicity, the conserved form of (10) for \(\rho\) and \(\rho v\) is  
\[\begin{split}\rho_{t}&+(\rho v)_{x}=0\\ (\rho v)_{t}+\left(\frac{(\rho v)^{2}}{\rho}+\left(\frac{(d_{t}- d_{s})\tau}{t^{2}}\right)^{2}\rho\right)_{x}&=\frac{v(\rho)-v}{\tau}. \end{split} \tag{11}\]  
Equation (11) is in a quasi-linear form and is appropriate for the Jacobian matrix to acquire eigenvalues. The two variables in (11) are \(\rho\) and \(\rho v\). The spatial components are partially differentiated with \(\rho\) and \(\rho v\) such that the spatial components are linearized. Then, the Jacobian matrix is  
\[\left(\begin{array}{cc}j_{11}&j_{12}\\ j_{21}&j_{22}\\ \end{array}\right), \tag{12}\]  
where \(j_{11}=\frac{\partial}{\partial\rho}\rho\), \(j_{12}=\frac{\partial}{\partial\rho}\rho\), \(j_{21}=\frac{\partial}{\partial\rho}\left(\frac{(\rho v)^{2}}{\rho}+\left( \frac{(d_{t}-d_{s})\tau}{t^{2}}\right)^{2}\rho\right)\), and \(j_{22}=\frac{\partial}{\partial\rho v}\left(\frac{(\rho v)^{2}}{\rho}+\left( \frac{(d_{t}-d_{s})\tau}{t^{2}}\right)^{2}\rho\right)\). The Jacobian matrix of the proposed system (10) from (12) is  
\[\left(\begin{array}{cc}0&1\\ -v^{2}+\left(\frac{(d_{t}-d_{s})\tau}{t^{2}}\right)^{2}&2v\\ \end{array}\right), \tag{13}\]  
then the eigenvalues are  
\[\begin{split}\lambda_{1}&=v+\frac{(d_{t}-d_{s})\tau}{t^{2}},\\ \lambda_{2}&=v-\frac{(d_{t}-d_{s})\tau}{t^{2}}.\end{split} \tag{14}\]  
Eigenvalues show that the proposed traffic system (10) is hyperbolic, as \(\lambda_{1,2}\) are real and distinct. Further, the velocity fluctuations are dependent on \(\frac{(d_{t}-d_{s})\tau}{t^{2}}\), that is, for a larger traversed time, the expected traffic evolution is slower, whereas for a smaller traversed time, the expected traffic evolution is faster. The changes in the forward direction of traffic are disseminated at the rate \(\lambda_{2}\). That is, the traffic changes in acceleration and deceleration in a smooth flow travel with the speed \(\lambda_{2}\). The acceleration and deceleration traveling speed in the forward direction reduces with a higher driver sensitivity than with a lower driver sensitivity. In the rearward direction, the changes are disseminated at the rate \(\lambda_{1}\). In other words, during congestion, the changes in acceleration and deceleration travel down the traffic with the speed \(\lambda_{1}\). That is, the acceleration and deceleration traveling speed in the rearward direction is always greater than the average speed with changes in driver sensitivity. At \(d_{t}=d_{s}\), the changes in both the forward and rearward directions travel with average speed \(v\).  
The PW system (3) Jacobian matrix from (12) is  
\[\left(\begin{array}{cc}0&1\\ -v^{2}+A^{2}&2v\\ \end{array}\right), \tag{15}\]and the eigenvalues are  
\[\lambda_{1} =v+A,\] \[\lambda_{2} =v-A. \tag{16}\]  
Eigenvalues show that the PW traffic system (3) is hyperbolic, as \(\lambda_{1,2}\) are real and distinct. Further, the velocity fluctuations are dependent on \(A\), that is, traffic evolution is at a uniform constant rate, which is not appropriate and contradictory to the traffic physics. The traffic evolution with (10) changes as the transition distance, relaxation time, and traversed time changes. In other words, the proposed traffic system (10) considers the traffic system, and the traffic evolution is impacted by the conditions.