## 6 Simulation Results  
The performance of the proposed and [PERSON]'s models is presented in this section for periods of 75 s. In these simulations, the temporal and spatial traffic evolution on a 300 m circular road was considered to assess the worst traffic scenario [16]. The circular road was chosen as the traffic dynamics within the circle are easy to observe. In addition, the worst traffic scenario can be observed on a circular rather than a straight road. The time period was 0.5 s, while the distance length was 10 m to guarantee Courant-Friedrichs-Levy conditions [28]. The relaxation time was 1 s [16]. The safe distance headway was 15 m. The traffic for both the proposed and [PERSON]'s models achieved Greenshield's equilibrium velocity (speed, Equation (11)) distribution during alignment [25]. The traffic density on the road was normalized between zero and one. The maximum velocity and density were 9 m/s and 1, respectively. The initial density condition was  
\[\rho_{0}= \begin{cases}0.1\ for\ x<100\\ 0.6\ for\ 100\leq\ x\leq 200.\\ 0.1\ for\ x>200\end{cases}  \tag{48}\]  
A drastic change in the initial density was considered to assess the traffic behavior of [PERSON]'s and the proposed models. The simulation parameters for both models are summarized in Table 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Values** \\ \hline Road length, \(x\) & 300 m \\ \hline Type of road & Circular Road \\ \hline Total time of simulation & 50 s \\ \hline Time step, \(\Delta\)t & 0.5 s \\ \hline Distance step, \(\Delta\)x & 10 m \\ \hline Relaxation time, \(\tau\) & 1 s \\ \hline Safe distance headway, \(h_{s}\) & 15 m \\ \hline Equilibrium velocity distribution, \(v(\rho)\) & Greenshield’s \\ \hline Maximum velocity, \(v_{m}\) & 9m/s \\ \hline Maximum density (normalized), \(\rho_{m}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Simulation parameters for the proposed and [PERSON]’s models. The traffic behavior of [PERSON]'s density model on a circular 300 m road at 5 s, 15 s, 37.5 s, and 50 s is shown in Figure 14. At 5 s, the range of traffic density was from 0.05 to 0.45. The density increased from 0.093 at 0 m to 0.43 at 170 m, then smoothly decreased to 0.154 from 170 m up to 260 m. The density was more oscillatory and abruptly changed over short distances between 240 m and 300 m, as shown in Figure 13. At 15 s, the range of density was from 0.16 to 0.32. The density reduced from 0.21 at 0 m to 0.16 at 60 m. It oscillated over a short distance of 10 m and increased between 60 m and 160 m. It rose to 0.32 at 200 m and fell to 0.24 at 300 m. At 37.5 s, the range of density was from 0.24 at 0 m to 0.25 at 70 m. The oscillatory behavior was observed with [PERSON]'s model at 5 s and 15 s, as shown in Figure 14.  
The corresponding velocity of [PERSON]'s model on a circular 300 m road at 5 s, 15 s, 37.5 s, and 50 s is shown in Figure 15. At 5 s, the velocity range was from 12.6 m/s to -1.5 m/s,  
Figure 14: The [PERSON] model’s density behavior on a 300 m circular road at 1 s at 5 s,15 s, 37.5 s, and 50 s.  
Figure 13: Velocity behavior of the proposed model on a 300 m circular road at 5 s, 15 s, 37.5 s, and 50 s.  
oscillating from 8.32 m/s at 0 m to 3.89 m/s at 100 m. The velocity decreased from 4.05 m/s at 110 m to 10.61 m/s at 270 m. At 280 m, the velocity was negative, \(-1.50\) m/s, and at 290 m, it was 12.60 m/s. At 15 s, the velocity ranged between 8.10 m/s at 0 m and 5.5 m/s at 120 m. It oscillated over a short distance of 10 m from 60 m to 160 m and smoothly aligned from 6.0 m/s at 160 m to 7.8 m/s at 300 m. At 37.5 s and 50 s, the velocity was nearly at 6.8 m/s. The velocity behavior of [PERSON]'s model was unusual as a velocity cannot be negative, and it did not stay within the prescribed range. Furthermore, changes over a shorter distance of 10 m did not accurately reflect traffic behavior. Figure 16 shows the proposed model density behavior on a 300 m circular road for 75 s. The density varied from 0.07 to 0.62 within the minimum of zero and maximum of one range.