## 4 Numerical Experiments
### Model Comparisons  
This section compares the fitting effect of the traffic density model derived from the queue based on threshold control with other models. We take the flow-density-speed data under the benchmark, fitting the [PERSON]'s model [21] and NF model [14] in Table 1, and [PERSON] model [24] and Vandaele model [23] in Table 2 (\(c_{a}\)\({}^{2}\)\(\leq\) 1).  
The comparison of the model fitting effect includes fitting errors (including MSE, SSE, MAE and RMSE), fitness (R-square) and measurement results. For fitness, we calculate not only the global fitness but also the local fitness, including the low-density area (\(0\leq k<0.96\)), middle-density area (\(0.96\leq k<1.44\)) and high-density area (\(k\geq 1.44\)). The results are shown in Table 5.  
Then we plotted the comparison figure, as shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Errors, Fitness and Results** & **Our Model** & \begin{tabular}{c} **[PERSON]** \\ **Model** \\ \end{tabular} & \
\begin{tabular}{c} **[PERSON]** \\ **Model** \\ \end{tabular} &
\begin{tabular}{c} **[PERSON] Model** \\ \end{tabular} & **NF Model** \\ \hline \multirow{3}{*}{Fitting errors} & MSE & 0.81 & 1.33 & 1.17 & 1.47 & 1.23 \\  & SSE & 566.9 & 927.1 & 817.0 & 1025.5 & 855.0 \\  & MAE & 0.69 & 0.75 & 0.73 & 0.9 & 0.78 \\  & PMSE & 0.90 & 1.15 & 1.08 & 1.21 & 1.11 \\ \hline \multirow{3}{*}{Fitness * (R-square)} & G & 0.74 & 0.57 & 0.62 & 0.53 & 0.61 \\  & L & 0.82 & 0.56 & 0.75 & 0.71 & 0.72 \\ \cline{1-1}  & M & 0.77 & 0.68 & 0.74 & 0.73 & 0.73 \\ \cline{1-1}  & H & \(-\)1.97 & \(-\)9.02 & \(-\)7.71 & \(-\)7.65 & \(-\)7.43 \\ \hline \multirow{3}{*}{Measurement results} & C & 91.54 & 72.98 & 83.52 & 81.50 & 81.07 \\  & \(k_{C}\) & 0.888 & 0.912 & 0.768 & 0.660 & 0.780 \\ \cline{1-1}  & \(\varphi_{d}\) & 281.927 & 16.138 & 28.404 & 22.398 & 24.734 \\ \hline \hline \end{tabular}  
* G = global; L = low-density area; M = middle-density area; H = high-density area.  
\end{table}
Table 5: The fitting errors index of different models. Numbers in bold indicate the minimum error or maximum fitness. (Design parameters: \(\lambda_{a}=0.06,\ n=3,\ m=2,\ N=165\).)  
Figure 6: Comparison with other models. (**a**) is comparison with [PERSON] model and NF model; (**b**) is comparison with [PERSON] model and [PERSON] model.  
From Table 5 and Figure 6, it can be seen that the fitting errors (MSE, SSE, MAE and RMSE) of our model are the smallest of all models, which indicates that our model has better adaptability on the fitting data.  
According to the comparison of R-square, our model has the best fit with the highest global fitness and is greater than 0.7. In addition, the fitness is highest in the low-density and middle-density regions, 0.82 and 0.77, respectively, but the fitness is lower in the high-density region. This indicates that, compared with other models, our model has a better fit in the low-density and middle-density regions but a poor fit in the high-density region (the fitness of all models in the low-density region is less than 0, which shows that all models have this problem).  
In addition, the proposed model can capture the capacity point and the capacity drop phenomenon better than other models, which indicates that our model can better reflect the actual capacity of the system.