## 4 Performance Evaluation
Further, both the proposed and ID models have a temporary decrease in speed due to the disturbance at 30.5 s. However, the proposed model recovers more quickly from this disturbance as the variations in speed are smaller. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{\(S/S^{\prime}\)} & **Maximum Traffic Flow** & **Critical** \\  & **(veh/s)** & **Density** \\ \hline
0.2 & 0.67 & 0.11 \\
0.5 & 0.83 & 0.08 \\
0.8 & 0.86 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum traffic flow and critical density for the proposed model. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{\(S/S^{\prime}\)} & **Maximum Traffic Flow** & **Critical** \\  & **(veh/s)** & **Density** \\ \hline
1 & 0.38 & 0.05 \\
4 & 0.45 & 0.03 \\
100 & 0.48 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum traffic flow and critical density for the ID model. Figure 7 presents the traffic acceleration on a 1 km circular road for \(S/S^{\prime}=0.2,\ 0.5\), and 0.8. This shows that when \(S/S^{\prime}=0.2\), CAV acceleration variations are small and range from 0.07 m/s\({}^{2}\) to \(-0.05\) m/s\({}^{2}\) at 1.0 s. This is because, in high-density traffic, CAVs operate at slower speeds and are required to stop frequently. Conversely, at moderate (\(S/S^{\prime}=0.5\)) and low (\(S/S^{\prime}=0.8\)) densities, CAV acceleration varies between 0.21 m/s\({}^{2}\) and \(-0.18\) m/s\({}^{2}\), and 0.30 m/s\({}^{2}\) and \(-0.30\) m/s\({}^{2}\), respectively, at 1.0 s. This is because a decrease in density reduces CAV constraints due to surrounding vehicles. This allows the CAVs to accelerate to higher speeds. However, for all values of \(S/S^{\prime}\), the variations in acceleration and deceleration decrease over time and are almost zero at 200 s. This is because CAVs can quickly stabilize their speeds and spacing due to their excellent reaction time. In addition, the adaptive behavior of CAVs based on sensor inputs such as radar and LiDAR allows them to adjust their speeds and distances relative to their surroundings. Conversely, the ID model results in larger acceleration and deceleration that increase over time as shown in Figure 8. With \(\delta=1\), the acceleration varies between 0.01 m/s\({}^{2}\) and \(-0.02\) m/s\({}^{2}\) at 200 s. These variations increase with \(\delta\) as with \(\delta=4\) and 100, it is between 0.06 m/s\({}^{2}\) and \(-0.08\) m/s\({}^{2}\) and 0.1 m/s\({}^{2}\) to \(-0.1\) m/s\({}^{2}\), respectively, at 200 s.  
Figure 9 gives the time-space trajectories of 40 CAVs with the proposed model and the results for the 1 st, 20 th, and 40 th (last) vehicles are summarized in Table 4. The black trajectory represents the 1 st vehicle, while the colored trajectories represent the other 39 vehicles. At high density (\(S/S^{\prime}=0.2\)), the first vehicle is in the queue for 0.5 s. It encounters minimal delay as it does not have to wait for other vehicles to move, whereas the 20 th and 40 th vehicles are in the queue for 31.5 s and 59.0 s, respectively. As the density decreases, the CAV queue times also decrease. With \(S/S^{\prime}=0.8\), the 1 st vehicle is in the queue for 0.5 s, whereas the 20 th and 40 th vehicles are in the queue for 30.0 s and 57.0 s,  
Figure 6: Traffic speed behavior with the ID model for (**a**) \(\delta=1\), (**b**) \(\delta=4\), and (**c**) \(\delta=100\), on a 1 km circular road. respectively. These results indicate that CAV queues are influenced by traffic density, and a higher density leads to longer queue times. Figure 8: Evolution of vehicle acceleration over time with the ID model for \(\delta=1,\,4,\) and \(100\) on a \(1\) km circular road. Figure 7: CAV acceleration evolution over time with the proposed model for \(S/S^{\prime}=0.2,\,0.5,\) and \(0.8\) on a \(1\) km circular road.