## 4 Performance Evaluation
Figure 10 gives the vehicle trajectories over space and time for the ID model and the results for the 1 st, 20 th, and 36 th (last) vehicles are summarized in Table 5. With \(\delta=1\), the queue dissipates after 113.0 s at \(-\)73.2 m, so all 36 vehicles are in the queue during this time. With \(\delta=4\) and 100, the queue does not dissipate and vehicles are in the queue for 200 s. Thus, with the ID model, queue dissipation is based on \(\delta\), i.e., the queue dissipates faster with a small \(\delta\), but it may not dissipate with a large \(\delta\). Figures 11 and 12 give the traffic flow evolution over time of a vehicle on a 1 km circular road with the proposed and ID models, respectively. This shows that the flow \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(S/S^{\prime}\)} & **1 st Vehicle Time** & **20 th Vehicle Time** & **40 th Vehicle Time** \\  & **in Queue (s)** & **in Queue (s)** & **in Queue (s)** \\ \hline
0.2 & 0.5 & 31.5 & 59.0 \\
0.5 & 0.5 & 30.5 & 58.0 \\
0.8 & 0.5 & 30.5 & 57.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Time for which the 1 st, 20 th, and 40 th (last) vehicles are in the queue with the proposed model. Figure 10: Trajectories of 36 vehicles with the ID model over a 1 km circular road for (**a**) \(\delta=1\), (**b**) \(\delta=4\), and (**c**) \(\delta=100\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\delta\)} & **1 st Vehicle Time** & **20 th Vehicle Time** & **36 th Vehicle Time** \\  & **in Queue (s)** & **in Queue (s)** & **in Queue (s)** \\ \hline
1 & 113 & 113 & 113 \\
4 & 200 (Queue does not dissipate) & \\
100 & 200 (Queue does not dissipate) & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Time for which the 1 st, 20 th, and 36 th (last) vehicles are in the queue with the ID model. Figure 9: Trajectories of 40 CAVs with the proposed model over a 1 km circular road for (**a**) \(S/S^{\prime}=0.2\), (**b**) \(S/S^{\prime}=0.5\), and (**c**) \(S/S^{\prime}=0.8\). with the proposed model increases as the density decreases. In dense traffic (\(S/S^{\prime}=0.2\)), the maximum flow is 0.67 veh/s at 79.0 s. At lower densities (\(S/S^{\prime}=0.5\) and 0.8), the maximum flow achieved is 0.74 veh/s at 73.5 s and 0.80 veh/s at 163.5 s, respectively. With the ID model, the maximum flow for \(\delta=1\) is 0.4 veh/s at 200 s. The flow increases with \(\delta\), as for \(\delta=4\) and 100, the flow is 6.9 veh/s at 159.0 s and 10.5 veh/s at 154.0 s, respectively. In addition, the variations in flow with the ID model are larger than with the proposed model. Figures 13 and 14 show the traffic flow behavior on a 1 km circular road with the proposed and ID models, respectively. Figure 13 indicates that the flow with the proposed model increases as the density decreases. In dense traffic (\(S/S^{\prime}=0.2\)), at \(-22\) m, the  
Figure 11: Temporal flow evolution of a vehicle on a 1 km circular road with the proposed model. Figure 12: Temporal flow evolution of a vehicle on a 1 km circular road with the ID model. maximum flow is 0.75 veh/s at 80.0 s. At a lower density (\(S/S^{\prime}=0.5\)), at \(-26\) m, the maximum flow is 0.83 veh/s at 73.5 s, whereas for \(S/S^{\prime}=0.8\), it is 0.91 veh/s at \(-20\) m and \(-79.0\) s. Figure 14 shows that the maximum flow with the ID model for \(\delta=1\) is 0.95 veh/s at \(-75.0\) m and 100 s. For \(\delta=4\), the maximum flow is 7.5 veh/s and there are variations from 58.0 s to 187.0 s between \(-84\) m and \(-138\) m, and from 167.5 s to 200 s between 837 m and 864 m. When \(\delta=100\), the maximum flow is 11.2 veh/s with variations from 56.0 s to 182.0 s between \(-27\) m and \(-138\) m, and from 163.0 s to 200 s between 849 m and 858 m. This shows that an increase in \(\delta\) increases the traffic flow as well as the variations. Figure 15 presents the CAV flow behavior under a cyberattack at \(c=100\) on a 1 km circular road with the proposed model.