# A New Driver Model Based on Driver Response
## 5 Performance Results
The density between 0 s and 34 s was 0.14 at \(-\)96.57 m, which indicates a traffic queue. After the queue dissipated, the density at 117 s gradually decreased to 0.03 at 296.3 m from 0.1 at \(-\)93.2 m at 41.5 s. With \(\delta=4\), the density between 0 m and 66.6 m at 120 s was 0.01. It increased to 0.03 at 306.3 m and then decreased to 0.015 at 602.7 m, as shown in Figure 7b. The density between 0 s and 33.5 s at \(-\)96.5 m was 0.14, which indicates a queue. After the queue dissipated, the density at 115 s gradually decreased to 0.07 to 0.08 at 592.7 m as shown in Figure 7c. The density between 0 s and 33.5 s at \(-\)96.6 m was 0.14, which indicates a queue. After the queue dissipated, the density at 68.5 s gradually decreased to 0.07 at \(-\)23.3 m from 0.1 at \(-\)93.2 at 40.5 s. It then increased to 0.14 at 77.5 s at 20.0 m and was constant between 63.3 m and 86.5 m to 120 s. These results show that as \(\delta\) increased, the change in density also increased with time as shown in Figure 7a-c, so the ID model was not stable. For the proposed model with \(T=0.1\) s, the density at 0 m was 0.03 at 119.5 s. It decreased to 0.01 at 466.2 m and then to 0.009 at 559.4 m as shown in Figure 7d. The density at 0 s between \(-\)96.5 m and \(-\)3.3 m was 0.14 and was constant up to 120 s at \(-\)109 m, which indicates a queue. With \(T=0.5\) s, the density between 0 m and 452.8 m at 119.5 s was 0.01 and increased to 0.03 at 606 m, as shown in Figure 7e. The density between 0 s and 37.5 s at \(-\)96.5 m was 0.14, which indicates a queue. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **1 st Vehicle** & **4 th Vehicle** & **10 th Vehicle** \\  & **Position (m)** & **Position (m)** & **Position (m)** \\ \hline \(\delta=1\) & 261.9 & 80.8 & \(-\)58.9 \\ \(\delta=4\) & 314.5 & 101.1 & \(-\)58.3 \\ \(\delta=20\) & 317.7 & 101.3 & \(-\)58.3 \\ \(T=0.1\) s & 88.7 & \(-\)17.5 & \(-\)63.0 \\ \(T=0.5\) s & 207.4 & 64.7 & \(-\)60.9 \\ \(T=1\) s & 262.2 & 93.2 & \(-\)42.2 \\ \(T=1.6\) s & 289.7 & 94.1 & \(-\)58.3 \\ \(T=2.7\) s & 307.0 & 75.8 & \(-\)61.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Position of the 1 st, 4 th, and 10 th vehicles at 30 s with the ID and proposed models. Figure 7: _Cont._  
Figure 7: Traffic density with the ID and proposed models over a circular road of length 800 m: (**a**) ID model, \(\delta=1\); (**b**) ID model, \(\delta=4\); (**c**) ID model, \(\delta=20\). Traffic density with the ID and proposed models over a circular road of length 800 m: (**d**) proposed model, \(T=0.1\) s; (**e**) proposed model, \(T=0.5\) s. Traffic density with the ID and proposed models over a circular road of length 800 m: (**f**) proposed model, \(T=1\) s; (**g**) proposed model, \(T=1.6\) s; (**h**) proposed model, \(T=2.7\) s.  
With \(T=1.6\) s the density between 0 m and 156.5 m at 119 s was 0.01. It increased to 0.29 at 356.3 m and then decreased to 0.01 at 606.0 m, as shown in Figure 7g. The density between 0 s and 33.5 s at \(-\)96.5 m was 0.14, which indicates a queue. After the queue dissipated, the density at 114.5 s gradually decreased to 0.01 at \(-\)179.8 m from 0.08 at \(-\)93.2 m at 42.5 s. With \(T=2.7\) s, the density at 0 m was 0.04 at 119.5 s and then decreased to 0.01 at 362.9 m, as shown in Figure 7h. The density between 0 s and 36.5 s at \(-\)96.5 m was 0.14, which indicates a queue. After the queue dissipated, the density at 56.5 s gradually decreased to 0.06 at \(-\)89.9 m from 0.1 at \(-\)93.2 m at 45.5 s. It then decreased to 0.03 at 117.5 s at \(-\)46.6 m.  
These results indicate that with the proposed model, changes in density were small and the velocity and density evolve realistically over time based on the time headway. Conversely, the velocity and density with the ID model are based on a constant \(\delta\), which produced unrealistic traffic behaviour, as shown in the results with \(\delta=20\). In this case, the changes in density increased over time whereas they should have decreased.