# Macroscopic Traffic-Flow Modelling Based on Gap-Filling Behavior of Heterogeneous Traffic
## 5 Simulation Results and Discussion
### Straight Road with Ingress Traffic  
For straight road with ingress traffic, the ingress traffic follows a cosine function with densities ranging between 0.25 to 0.5 and velocities following the Greenshields distribution. The 3D plot for the PW model is depicted in Figure 1. The PW model does not exhibit any unrealistic behavior for densities, that is the densities at all the time instants and at all road steps are well within the defined range. However, it can be observed that, despite the very high initial densities and ingress traffic, the density curve smoothens very quickly, and no significant effect of the bottleneck is translated at any road segment over the entire simulation period.  
The velocity behavior of the PW model is illustrated in Figures 2 and 3. As discussed earlier, because of the constant anticipation term, the PW model results in negative velocities, which is not possible. Another unrealistic behavior is that the PW model exhibits velocities higher than the maximum velocity. Out of bound velocities have been highlighted in red in Figure 2. It can be further observed in Figure 3 that the problem of negative velocities persists till 0.35 s, while velocities exceeding the maximum velocity can be observed till 1.66 s, where the velocity is just above 25 m/s (90 km/h). At very early time instants, the velocities reach as high as 35.4 m/s (127 km/h). Beyond that, the velocities are well within the defined range.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Values** \\ \hline Equilibrium velocity distribution & Greenshields \\ \hline Road length & 100 m \\ Distance step & 1 m \\ \hline Simulation time for: & \\ Straight road, no ingress traffic & 2 s \\ Straight road, ingress traffic & 30 s \\ Circular road & 30 s \\ \hline Time step & 0.01 s \\ Relaxation time & 0.5 s \\ \hline Time headway for proposed model & 0.5 s \\ PW model anticipation constant & 25 m/s (90 km/h) \\ \hline Maximum velocity & 25 m/s (90 km/h) \\ Maximum density & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Input parameters for the numerical simulation of the proposed and PW models.  
Figure 1: Density behavior of PW model on straight road with ingress traffic.  
When simulated for similar traffic conditions, the proposed model exhibits robust and realistic results. In the case of density, it can be observed that the effect of the bottleneck has been projected backwards and an increasing density can be observed beyond the bottleneck (see Figures 4 and 5). At time \(t=15\) s, the density even reaches a value of 0.67. After that, the effect of the bottleneck starts diminishing steadily, as can be seen at time \(t=17\) s.  
The velocity behavior of the proposed model is even better. There is no unrealistic behavior at any road segment throughout the simulation period and the velocities are well within the defined range. As can be observed in Figures 6 and 7, the velocity starts increasing as the density decreases, which indicates that more gaps are produced ahead, and, hence, the drivers are encouraged to fill those gaps, resulting in the said increase of velocity.  
Figure 3: Velocity behavior of PW model on straight road with ingress traffic.  
Figure 2: Velocity behavior of PW model on straight road with ingress traffic.  
Figure 4: Density behavior of proposed model on straight road with ingress traffic.  
Figure 5: Density behavior of proposed model on straight road with ingress traffic.  
Figure 6: Velocity behavior of proposed model on straight road with ingress traffic.