## 3 Performance Results
When \(\delta=1\), the initial flow is 0.04 veh/s and increases to 0.54 veh/s at 122.5 s. It is 0.21 veh/s at 128 s and increases to 0.34 veh/s at 199.5 s. When \(\delta=4\), the initial flow is 0.05 veh/s and increases to 0.56 veh/s at 104.5 s. It decreases to 0.01 veh/s at 108.5 s and then increases to 0.31 veh/s at 140 s and 0.37 veh/s at 200 s. When \(\delta=100\), the initial flow is 0.04 veh/s, increases to 0.57 veh/s at 99 s, and then decreases to 0.002 veh/s at 104 s.  
Figure 4: Velocity with the proposed model for \(\tau=0.6\), 1, 1.5, 2, and 2.2 s on a 2000 m circular road. Figure 3: Velocity with the ID model for \(\delta=1,4\), and 100 on a 2000 m circular road. The flow on a 2000 m circular road with the proposed model for \(\tau=0.6,1,1.5,2\), and 2.2 s is given in Figure 6. When \(\tau=0.6\) s, the flow is 0.07 veh/s at 130.5 s, increases to 0.83 veh/s at 161.5 s, and then decreases to 0.44 veh/s at 199.5 s. When \(\tau=1\) s, the flow is 0.05 veh/s at 106 s, increases to 0.72 veh/s at 140 s, and then decreases to 0.49 veh/s at 200 s. When \(\tau=1.5\) s, the flow is 0.06 veh/s at 99 s and increases to 0.66 veh/s at 122.5 s. It decreases to 0.39 veh/s at 134.5 s and then increases to 0.42 veh/s at 200 s. When \(\tau=2\) s, the flow is 0.07 veh/s at 98 s and then increases to 0.54 veh/s at 115.5 s. It decreases to 0.15 veh/s at 120 s and then increases to 0.36 veh/s at 200 s. When \(\tau=2.2\) s, the flow is 0.04 veh/s at 90.5 s and increases to 0.51 veh/s at 114 s. It is 0.05 veh/s at 119 s and increases to 0.33 veh/s at 200 s.  
Figure 7 presents the trajectories of a platoon of 52 vehicles with the ID model on a 2000 m circular road, and the results at 90 s are given in Table 2. The thick black line is the trajectory of the 1 st vehicle while the pink lines show the trajectories of the following 51 vehicles. When \(\delta=1\), Figure 7a shows that the position of the 1 st and 15 th vehicles is 1286 m and 266.9 m, respectively, while the position of the 30 th and 50 th vehicles is \(-23.1\) m and \(  
Figure 9 presents the density with the ID model over time and space on a 2000 m circular road and Table 4 gives the congestion results. When \(\delta=1\), Figure 9a shows that there is congestion from 0 s to 101 s as the density is 0.50. At \(-93.0\) m, it decreases to 0.29 at 125.5 s and 0.15 at 198.5 s. At 199 s, the density is 0.15 at \(-9.0\) m and decreases to 0.012 at 1791 m. When \(\delta=4\), Figure 9b shows that there is congestion from 0 s to 103 s as the density is 0.50. It is between 0.26 and 0.50 from 110.5 s to 200 s. At 198 s, the density is 0.50 at \(-132.0\) m and decreases to 0.10 at \(-54.0\) m and 0.02 at 1785 m. When \(\delta=100\), Figure 9c shows that there is congestion between \(-99.0\) m and \(-3.0\) m as the density is 0.50, and at \(-144.0\) m this continues until 200 s. At 198 s, the density is 0.5 at \(-144.0\) and decreases to 0.07 at \(-21.0\) m and 0.01 at 1791 m.  
The density with the proposed model over time and space on a 2000 m circular road is given in Figure 10 and Table 5 gives the congestion results. When \(\tau=0.6\) s, Figure 10a shows that there is congestion from 0 s to 102 s as the density is 0.50. It decreases to 0.01 at \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Acceleration Exponent**} & \multirow{2}{*}{\begin{tabular}{c} **Time during Which** \\ **Congestion Occurs** \\ **(s)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Density during** \\ **Congestion** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Density after** \\ **Congestion Dissipates** \\ **(s)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Time after Congestion** \\ **Dissipates** \\ **(s)** \\ \end{tabular} } \\ \hline
1 & 0–101 & 0.50 & 0.29 & 125.5 \\
4 & 0–103 & 0.50 & 0.26–0.50 & 110.5–200 \\
100 & 0–200 & 0.50 &
\begin{tabular}{c} Congestion does not \\ dissipate \\ \end{tabular} & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ID model density and time during and after congestion.