# A Macroscopic Traffic Model based on Driver Reaction and Traffic Stimuli
## 5 Performance Results
The maximum density with the proposed model at 0.1 s is 0.8 at 131 m. At 60 s, the density is very smooth. The corresponding velocity with the proposed model is given in Figure 8. These results show that the velocity evolves smoothly over time and stays within the maximum of 10 m/s and minimum of 0 m/s. At 0.1 s, the velocity is 0.5 m/s at 131 m when the density is 0.8. With the PW model, the velocity is as high as 19.6 m/s and below 0 m/s, as shown in Figure 6. Thus, the proposed model provides more realistic behavior than the PW model. Figure 5: Velocity with the proposed model on a 300 m circular road for 60 s with \(\tau=2.5\) s and distance headway \(h=20\) m.  
Figure 6: Velocity with the PW model on a 300 m circular road with \(\tau=2.5\) s and \(C_{0}=10\). The density behavior with the proposed model for time step 0.01 s and road step 2 m over the 300 m road at 1 s, 20 s, 40 s and 60 s is shown in Figure 9 and given in Table 5. Comparing the results from 1 s to 60 s, the density becomes smoother over time. At 1 s, the density is 0.20 at 1 m, and from 28 m to 102 m it is 0.01. It increases to 0.2 at 118 m and stays at this level to 300 m. At 20 s, the density is 0.20 at 1 m, and decreases to 0.01 at 234 m, and then increases to 0.20 at 300 m. At 40 s, the density is 0.12 at 1 m, decreases to 0.02 at 96 m, and is 0.2 at 142 m. It is 0.12 at 300 m. At 60 s, the density is 0.20 at 1 m and decreases to 0.05 at 250 m. From 260 m and 288 m, the density varies between 0.06 and 0.19 and is 0.20 at 300 m. The density is smoother at density discontinuities than the results in Figure 1 for time step 0.1 s and road step 1 m, however there are no significant differences. Thus, the numerical scheme is stable. Figure 8: Velocity with the proposed model during congestion (\(\rho>0.5\)) over a 300 m circular road with \(\tau=2.5\) s and \(h=20\) m.  
Figure 7: Density behavior with the proposed model during congestion (\(\rho>0.5\)) over a 300 m circular road for 60 s with \(\tau=2.5\) s and distance headway \(h=20\) m.  
The velocity behavior with the proposed model for time step 0.01 s and road step 2 m over the 300 m road at 1 s, 20 s, 40 s and 60 s is shown in Figure 10 and given in Figure 4:0 At 1 s, the velocity is 8.0 m/s at 1 m and increases to 9.9 m/s at 28 m. The velocity is a constant 8.0 m/s between 118 m and 300 m. At 20 s, the velocity is 8.0 m/s at 1 m and increases to 9.9 m/s at 234 m. It is 8.0 m/s at 300 m. At 40 s, the velocity increases from 8.8 m/s at 1 m to 9.7 m/s at 96 m. The velocity is 8.0 m/s at 142 m, then increases to 8.8 m/s at 300 m. At 60 s, the velocity is 8.0 m/s at 1 m and smoothly increases to 9.5 m/s at 250 m. The velocity varies between 9.4 m/s and 8.06 m/s from 260 m to 288 m. It is 8.0 m/s at 300 m. The velocity behavior of the proposed model with time step 0.01 s and road step 2 m is smoother at abrupt changes than the results in Figure 2 for time step 0.1 s and road step 1 m. However, there are no significant differences, which confirms that the numerical scheme is stable. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time (s)** & **Distance (m)** & **Normalized Density** & **Velocity (m/s)** \\ \hline \(1\) & \(1\) & 0.20 & 8.0 \\ \(1\) & 28–102 & 0.10 & 9.9 \\ \(1\) & 118–300 & 0.20 & 8.0 \\ \hline \(20\) & \(1\) & 0.20 & 8.0 \\ \(20\) & 234 & 0.01 & 9.9 \\ \(20\) & 300 & 0.20 & 8.0 \\ \hline \(40\) & \(1\) & 0.12 & 8.8 \\ \(40\) & 96 & 0.02 & 9.7 \\ \(40\) & 142 & 0.20 & 8.0 \\ \(40\) & 300 & 0.12 & 8.8 \\ \hline \(60\) & \(1\) & 0.20 & 8.0 \\ \(60\) & 250 & 0.05 & 9.5 \\ \(60\) & 260 & 0.06 & 9.4 \\ \(60\) & 288 & 0.19 & 8.1 \\ \(60\) & 300 & 0.20 & 8.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Velocity and density of the proposed model for the resolution test at 1 s, 20 s, 40 s and 60 s.  
Figure 9: Density behavior with the proposed model for time step 0.01 s and road step 2 m on a 300 m circular road with \(h=20\) m at 1 s, 20 s, 40 s and 60 s.