## 6 Discussion of Results  
Figures 2-5 show that the density and velocity with the proposed system are within the employed range for the worst case scenario of maximum density 1. The traffic adjustment during the density and velocity changes are according to the employed conditions. That is, as the density grows, velocity drops, and the opposite behavior of velocity is observed with a decay in the density on the road. In this illustration, the driver sensitivity is 0.0031 s\({}^{-2}\) during transitions at multiple locations on a 2000 m circular road. Conversely, with the PW traffic system, it is observed in Figures 8 and 9 that the velocity does not evolve according to the density behavior as depicted in Figures 6 and 7, although the simulations conditions for both the PW and proposed traffic systems are the same. The velocity with the PW traffic system goes negative and above the maximum velocity limit. The maximum velocity observed with PW traffic system is 25.5 m/s, which is inaccurate, as the maximum allowable velocity is 0.964 m/s. The minimum velocity observed is \(-\)20 m/s, although the minimum allowable velocity is 0 m/s. As expected, the traffic velocity is unrealistic as the driver behavior adjusts to the following conditions uniformly. In addition, traffic adjustment during the density and velocity transition is not based on traffic physics. Figures 10 and 11 show the proposed traffic system evolution with the driver sensitivities of 0.0025 s\({}^{-2}\) and 1 s\({}^{-2}\) for 200 s. The results show that the traffic evolves differently at 0.0025 s\({}^{-2}\) than with 1 s\({}^{-2}\). Figure 10 show that the velocity evolves more quickly with 1 s\({}^{-2}\) than with 0.0025 s\({}^{-2}\). As expected, the smaller traversed time during transition aligns more quickly than with a larger traversed time. A sensitivity of 1 s\({}^{-2}\) represents that the flow is smooth and has smaller tendencies for larger velocity changes. Whereas with 0.0025 s\({}^{-2}\), the driver takes longer during transitions to align. This is a presentation of congested flow and has tendencies for larger changes in velocity. Figure 11 shows that the density with 1 s\({}^{-2}\) aligns quickly. The changes in density are smaller than with 0.0025 s\({}^{-2}\). The density with 0.0025 s\({}^{-2}\) has larger changes as expected than with 1 s\({}^{-2}\). The density for both of the driver sensitivities is within the employed range of the maximum 1 and 0. The velocity also stays within the bounds. The velocity changes with the proposed traffic system follow  
Figure 11: Traffic velocity behavior with the proposed system on a 2000 m circular road. The solid lines represent the proposed system behavior with the driver sensitivity of 0.0025 s\({}^{-2}\) at 0.2 s, 12 s, 100 s, and 200 s. The dotted lines represent the proposed system behavior with the driver sensitivity of 1 s\({}^{-2}\) at 0.2 s, 12 s, 100 s, and 200 s.  
the changes in density and have the tendency to acquire the Greenshields velocity-density behavior. The realistic and appropriate behavior of the proposed traffic system is due to the appropriate characterization of driver behavior based on traffic physics. That is, the proposed system considers the traffic density, transition distance, and distance headway during transitions. The inaccurate and inappropriate traffic evolution with the PW traffic system is due to the traffic characterization as a constant at traffic transitions. This shows that, unlike the PW traffic system, the proposed traffic system is more robust. The proposed model is applicable to the traffic flow where lane discipline is not followed. This can produce better results in different road conditions, as the performance results were evaluated for the worst traffic conditions in a circular road, where the traffic changes remain in the circle. These changes are more prone to further larger changes and cause excessive acceleration and deceleration. In addition, the traffic conditions are more vulnerable to traffic accidents.