# A Microscopic Traffic Model Considering Driver Reaction and Sensitivity
## 4 Simulation Results
With \(\delta=1\), the maximum velocity is 20.0 m/s, whereas with \(\delta=4\) and 20, it is 25.0 and 30.0 m/s, respectively. These results indicate that the traffic queue with the ID model dissipates based on the constant \(\delta\) rather than driver response, which is unrealistic. Figure 7 presents the temporal and spatial evolution of a queue with the proposed model, and the congestion results are summarized in Table 7. This shows that the velocity within the queue is zero. For \(\frac{\tau}{\tau_{i}}<1\) (aggressive driver), with \(h=0.3\), the queue dissipates at 91.5 s as shown in Figure 7a, and the corresponding velocity is 1.3 m/s. With \(h=0.5\) and 1.0, the queue dissipates at 68.0 and 54.5 s, respectively, as shown in Figure 7b,c. With \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Acceleration Exponent \(\delta\)**} & **Time during Which** & **Velocity after** & **Time after** \\  & **Congestion Occurs** & **Congestion** & **Congestion** \\  & **(s)** & **Dissipates** & **Dissipates** \\ \hline
1 & \(0-57.5\) & 1.3 & 57.5 \\
4 & \(0-56.0\) & 1.2 & 56.0 \\
20 & \(0-55.5\) and \(84.5-150.0\) & \(1.1-5.2\) & \(84.5-150.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Velocity and time for the ID model during and after congestion. Figure 6: Velocity versus time and space with the ID model over a 1200 m circular road for (**a**) \(\delta=1\), (**b**) \(\delta=4\), and (**c**) \(\delta=20\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Driver**} & \multirow{2}{*}{\(h=\frac{\delta}{D}\)} & **Maximum** & **Maximum** & **Critical Velocity** \\
**Sensitivity** & & **Flow (veh/s)** & **Density** & **(m/s)** \\ \hline \(\frac{\tau}{\tau_{i}}<1\) & 0.3 & 0.32 & 0.03 & 9.4 \\ (aggressive driver) & 0.5 & 0.40 & 0.04 & 10.2 \\  & 1.0 & 0.52 & 0.05 & 11.3 \\ \hline \(\frac{\tau}{\tau_{i}}>1\) & 0.3 & 0.30 & 0.03 & 10.3 \\ (sluggish driver) & 0.5 & 0.33 & 0.02 & 13.2 \\  & 1.0 & 0.35 & 0.02 & 17.2 \\ \hline \(\frac{\tau}{\tau_{i}}=1\) & 0.3 & 0.33 & 0.04 & 9.2 \\ (typical driver) & 0.5 & 0.37 & 0.03 & 11.3 \\  & 1.0 & 0.41 & 0.03 & 12.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Maximum flow, density, and critical velocity for the proposed model. \(h=0.5\), the velocity after the queue dissipates is 1.6 m/s, while with \(h=1.0\), it is 2.5 m/s. For \(\frac{\tau}{\tau_{s}}>1\) (sluggish driver), with \(h=0.3\) and 0.5, the queue dissipates at 62.5 and 61.0 s, respectively, as shown in Figure 7d,e. With \(h=0.3\), the velocity after the queue dissipates is 1.4 m/s, while with \(h=0.5\), it is 1.2 m/s. With \(h=1.0\), the initial queue dissipates at 60.0 s and the corresponding velocity is between 1.0 and 1.9 m/s as shown in Table 7. A queue again develops at 125.0 s and lasts until 150.0 s between \(-43.3\) and \(-56.6\) m as shown in Figure 7f. For \(\frac{\tau}{\tau_{s}}=1\) (typical driver), with \(h=0.3\), the queue dissipates at 60.0 s as shown in Figure 7g, and the corresponding velocity is 1.7 m/s. With \(h=0.5\) and 1.0, the queue dissipates at 58.5 and 59.0 s, respectively, as shown in Figure 7h,i. With \(h=0.5\), the velocity after the queue dissipates is 1.8 m/s, while with \(h=1.0\), the velocity is 2.1 m/s. The largest velocity after the queue dissipates is 2.5 m/s. These results indicate that queue dissipation is quick with an aggressive driver compared to a sluggish driver. Furthermore, the dissipation with a typical driver is between that of aggressive and sluggish drivers, as expected. Figure 8 presents the temporal and spatial trajectories of the vehicles with the ID model, and the corresponding congestion results are given in Table 8. The thick red line represents the 1 st vehicle, while the black lines correspond to the other 20 vehicles. At 55.0 s, with \(\delta=1\), the 1 st vehicle is at 742.0 m, while the 10 th and 20 th vehicles are at 112.0 and \(-\)89.9 m, respectively. With \(\delta=4\), the 1 st vehicle is at 919.3 m, while the 10 th and 20 th vehicles are at 166.4 and \(-\)88.6 m, respectively.