## 3 Performance Results  
The proposed and ID models are evaluated using the Euler scheme [40] implemented in MATLAB. A circular road of length 2000 m is considered with periodic boundary conditions. The simulation time is 200 s and the time step is 0.5 s. The desired velocity is 30 m/s and the maximum and the minimum deceleration (negative deceleration) are 0.5 m/s\({}^{2}\) and 3 m/s\({}^{2}\), respectively [29]. The jam spacing is 2 m [37], and the forward and rearward distance headways are both 25 m [2]. The time headway varies depending on the traffic conditions and is usually between 0.5 s and 2.6 s [42]. Here, the forward time headway is 1.5 s, the rearward time headway is 1.6 s, and \(\tau_{s}\) is 1.4 s. The ID model is evaluated for \(\tau=2\) s [6], while the proposed model is evaluated for \(\tau=0.6,1.15,2\), and 2.2 s. The acceleration exponent varies from 1 to \(\infty\) and is commonly set to 4 [22], so \(\delta=1,4\) and 100 is considered. There are 52 vehicles on the road, each with a length of 4.5 m [43]. The simulation parameters are given in Table 1.  
Figure 3 presents the ID model velocity on a 2000 m circular road for \(\delta=1,4\), and 100. When \(\delta=1\), the initial velocity is 6.7 m/s and increases to 22.5 m/s at 97 s. It then decreases to 0.86 m/s at 127 s and then increases to 8.5 m/s at 200 s. When \(\delta=4\), the velocity is 3.1 m/s at 138.5 s, increases to 15.9 m/s at 200 s, and then decreases to 0.07 m/s at 107.5 s. The highest velocity is 27.1 m/s at 78 s. When \(\delta=100\), the velocity increases \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Desired velocity, \(v_{d}\) & 30 m/s \\ Time headway for ID model, \(\tau\) & 2 s \\ Forward and rearward distance headways, \(h_{f}\) and \(h_{r}\) & 25 m \\ Forward time headway, \(\tau_{f}\) & 1.5 s \\ Rearward time headway, \(\tau_{r}\) & 1.6 s \\ Safe time headway, \(\tau_{s}\) & 1.4 s \\ Jam spacing, \(J\) & 2 m \\ Maximum deceleration, \(a_{m}\) & 0.5 m/s\({}^{2}\) \\ Minimum deceleration, \(b\) & 3 m/s\({}^{2}\) \\ Time headway for the proposed model, \(\tau\) & 0.6, 1, 1.5, 2, and 2.2 s \\ Vehicle length, \(L\) & 4.5 m \\ Acceleration exponent, \(\delta\) & 1.4, and 100 \\ Time step, \(\Delta t\) & 0.5 s \\ Maximum normalized density, \(\rho_{m}=1/J\) & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameters. from 0.74 m/s at 1.5 s to 29.8 m/s from 62 s to 73.5 s. It decreases to 0.12 m/s at 101.5 s, increases to 1.5 m/s at 127 s, and is 16.7 m/s at 200 s.  
The velocity on a 2000 m circular road with the proposed model for \(\tau=0.6\), 1, 1.5, 2, and 2.2 s is shown in Figure 4. When \(\tau=0.6\) s, the initial velocity is 8.1 m/s and increases to 19.5 m/s at 127.5 s. It decreases to 11.0 m/s at 163 s and then increases to 15.0 m/s at 200 s. When \(\tau=1\) s, the initial velocity is 9.74 m/s and increases to 21.5 m/s at 108 s. It is 9.5 m/s at 142 s and 15.0 m/s at 200 s. When \(\tau=1.5\) s, the velocity increases from 10.7 m/s at 95 s to 23.0 m/s at 25.5 s. It decreases to 3.8 m/s at 125.5 s and then increases to 13.2 m/s at 200 s. When \(\tau=2\) s, the initial velocity is 10.1 m/s and increases to 24.1 m/s at 87.5 s. It decreases to 0.44 m/s at 119 s and then increases to 11.1 m/s at 200 s. When \(\tau=2.2\) s, the initial velocity is 10.4 m/s and increases to 24.3 m/s at 86 s. It is 0.12 m/s at 86 s and increases to 9.9 m/s at 200 s.  
Figure 5 presents the flow with the ID model on a 2000 m circular road for \(\delta=1,4\), and 100.