# A Macroscopic Traffic Model based on Driver Reaction and Traffic Stimuli
## 5 Performance Results
Figure 1: Normalized traffic density with the proposed model on a 300 m circular road with \(h=20\) m at 1 s, 20 s, 40 s and 60 s.  
Figure 2: Velocity with the proposed model on a 300 m circular road with \(h=20\) m at 1 s, 20 s, 40 s and 60 s.  
The density with the PW model at 1 s, 2 s, 4 s and 6 s on the circular road is shown in Figure 3 and given in Table 4. At 1 s, the density is 0.20 at 0 m, 0.01 between 5 m and 105 m, and 0.2 between 110 m and 300 m. At 2 s, the density is 0.17 at 0 m and decreases to 0.01 at 70 m. The density is 0.13 at 100 m, 0.12 between 165 m and 280 m, and then increases to 0.14 at 300 m. At 4 s, the density decreases from 0.17 at 0 m to 0.03 at 75 m. It is 0.05 at 115, 0.20 between 200 m and 280 m, and 0.19 at 300 m. At 6 s, the density is 0.17 at 0 m and decreases to 0.06 at 80 m. At 145 m it is 0.09, increases to 0.19 at 230 m and 0.20 at 278 m, and then deceases to 0.18 at 300 m.  
The velocity with the PW model at 1 s, 2 s, 4 s and 6 s is shown in Figure 4 and given in Table 4. At 1 s it is 8.0 m/s at 0 m, increases to 9.9 m/s at 5 m and stays constant to 105 m. The velocity decreases to 8.0 m/s at 110 m and remains at this value until 300 m. At 2 s, the velocity increases from 9.0 m/s at 0 m to 18.8 m/s at 45 m. It decreases to 10.4 m/s at 70 m and then to \(-\)1.2 m/s at 100 m, which is impossible. The velocity then increases to 7.9 m/s at 165 m and is 8.3 m/s at 300 m. At 4 s, it is 8.9 m/s at 0 m and increases to 16.6 m/s at 75 m, which is beyond the maximum of 10 m/s. At 115 m,  
Figure 3: Normalized density with the Payne–Whitham (PW) model on a 300 m circular road with \(C_{0}=10\) at 1 s, 2 s, 4 s and 6 s.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time (s)** & **Distance (m)** & **Normalized Density** & **Velocity (m/s)** \\ \hline \(1\) & \(1\) & 0.19 & 8.0 \\ \(1\) & \(10\)–\(104\) & 0.01 & 9.9 \\ \(1\) & \(111\)–\(300\) & 0.20 & 8.0 \\ \hline \(20\) & \(1\) & 0.20 & 8.0 \\ \(20\) &  
it decreases to 2.0 m/s and then increases to 7.9 m/s at 200 m and 8.4 m/s at 300 m. At 6 s, the velocity is 8.8 m/s at 0 m, increases to 14.4 m/s at 80 m and then decreases to 4.4 m/s at 145 m. At 230 m, it is 7.8 m/s and this increases to 8.5 m/s at 300 m.  
The proposed model traffic velocity over the 300 m road is given in Figure 5. This shows that the velocity becomes smooth over time. Further, the variations are small compared to the PW model, as shown in Figure 6. The velocity with the proposed model stays within the maximum of 10 m/s and minimum of 0 m/s. With the PW model, the velocity goes as high as 19.6 m/s and below 0 m/s due to a fixed speed constant, as shown in Figure 6. In general, the velocity with the proposed model evolves over time as expected, while the velocity with the PW model is unrealistic. \begin{table}
\begin{tabular}{c c c c} \hline
**Time (s)** & **Distance (m)** & **Normalized Density** & **Velocity (m/s)** \\ \hline
1 & 0 & 0.20 & 8.0 \\
1 & 5–105 & 0.01 & 9.9 \\
1 & 110–300 & 0.20 & 8.0 \\ \hline
2 & 0 & 0.17 & 9.0 \\
2 & 45 & 0.03 & 18.85 \\
2 & 70 & 0.01 & 10.4 \\
2 & 100 & 0.13 & \(-\)1.2 \\
2 & 165–280 & 0.12 & 7.9 \\
2 & 300 & 0.14 & 8.3 \\ \hline
4 & 0 & 0.17 & 8.9 \\
4 & 75 & 0.03 & 16.6 \\
4 & 115 & 0.05 & 2.0 \\
4 & 200–280 & 0.20 & 7.9 \\
4 & 300 & 0.19 & 8.4 \\ \hline
6 & 0 & 0.17 & 8.8 \\
6 & 80 & 0.06 & 14.4 \\
6 & 145 & 0.09 & 4.4 \\
6 & 230 & 0.19 & 7.8 \\
6 & 278 & 0.20 & 8.0 \\
6 & 300 & 0.18 & 8.5 \\ \hline \end{tabular}
\end{table} 
Table 4: Velocity and density with the PW model at 1 s, 2 s, 4 s and 6 s.  
Figure 4: Velocity with the Payne–Whitham (PW) model on a 300 m circular road with \(C_{0}=10\) at 1 s, 2 s, 4 s and 6 s.  
The spatial and temporal density evolution with the proposed model during congestion given by Equation (74) (the density is above the critical density 0.5 between 130 m and 180 m), for 60 s over the 300 m road is shown in Figure 7. These results show that the density still evolves smoothly over time. The normalized density with the proposed model stays within the minimum 0 and maximum 1, as required.