## 3 Results
### Discussion of Results
#### 3.2.4 Parametric Study of the ANN-PSO Hybrid Model for Delay in the Queue and Delay in the System  
The ANN-PSO model was also evaluated for its ability to predict delay in the queue, where the best results were obtained with six neurons, a swarm population size of 200, and acceleration factors C1 = 1 and C2 = 2.75. This setup produced a training R2 of 0.89247 and an MSE of 0.0031, with a testing R2 of 0.8654. These results suggest that the model performs well in estimating delays, although slight variations in swarm population sizes resulted in different levels of accuracy (Results highlighted in bold and italic in Table 7). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Neurons**} & \multirow{2}{*}{**Swarm Population**} & \multicolumn{2}{c}{**Acceleration Factor**} & \multirow{2}{*}{**Training R2**} & \multirow{2}{*}{**MSE**} & \multirow{2}{*}{**Testing R2**} \\ \cline{3-3} \cline{5-6}  & & \(\mathbf{c_{1}}\) & & & & \\ \hline
8 & 100 & 1 & 2.50 & 0.8927 & 47.9088 & 0.6287 \\ \hline
8 & 200 & 1 & 2.75 & 0.9097 & 40.7382 & 0.5268 \\ \hline
8 & 400 & 1 & 2.25 & 0.9281 & 32.6434 & 0.4674 \\ \hline
9 & 10 & 1 & 2.75 & 0.8539 & 63.9547 & 0.391 \\ \hline
9 & 20 & 1 & 3.00 & 0.9357 & 29.5123 & 0.5934 \\ \hline
9 & 50 & 1.5 & 2.25 & 0.8848 & 51.1429 & 0.6248 \\ \hline
9 & 100 & 2 & 2.00 & 0.9298 & 32.0031 & 0.4377 \\ \hline
9 & 200 & 1.5 & 2.25 & 0.8924 & 47.9906 & 0.4536 \\ \hline
9 & 400 & 1 & 2.50 & 0.9087 & 41.0479 & 0.6865 \\ \hline
10 & 10 & 1 & 2.75 & 0.8210 & 81.3186 & 0.6978 \\ \hline
10 & 20 & 1.5 & 2.50 & 0.7698 & 100.8901 & 0.2325 \\ \hline
10 & 50 & 1.5 & 2.50 & 0.8986 & 45.3610 & 0.3447 \\ \hline
10 & 100 & 1 & 2.75 & 0.9197 & 36.3546 & 0.5877 \\ \hline
10 & 200 & 1 & 2.75 & 0.9383 & 28.1631 & 0.7121 \\ \hline
10 & 400 & 1.5 & 2.50 & 0.9260 & 33.6574 & 0.7940 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Neurons**} & \multirow{2}{*}{**Swarm Population Size**} & \multicolumn{2}{c}{**Acceleration Factor**} & \multirow{2}{*}{**Training R2**} & \multirow{2}{*}{**MSE**} & \multirow{2}{*}{**Testing R2**} \\ \cline{3-3} \cline{5-6}  & & \(\mathbf{c_{1}}\) & & & & \\ \hline
5 & 10 & 2.25 & 2.00 & 0.8721 & 0.0037 & 0.78 \\ \hline
5 & 20 & 2.25 & 2.00 & 0.8861 & 0.0032 & 0.7847 \\ \hline
5 & 50 & 1.5 & 2.25 & 0.7869 & 0.0058 & 0.3988 \\ \hline
5 & 100 & 1 & 2.75 & 0.8490 & 0.0042 & 0.7261 \\ \hline
5 & 200 & 1.5 & 2.00 & 0.8835 & 0.0033 & 0.8386 \\ \hline
5 & 400 & 1.5 & 2.00 & 0.8837 & 0.0033 & 0.8139 \\ \hline
6 & 10 & 1 & 3.00 & 0.8118 & 0.0052 & 0.4746 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: ANN-PSO sensitive analysis for delay in queue. For delay in the system, the optimal configuration was found with six neurons, a swarm population size of 400, and acceleration factors of C\({}_{1}\) = 1 and C\({}_{2}\) = 2.25. The model achieved a training R\({}^{2}\) of 0.9114, an MSE of 0.0028, and a testing R\({}^{2}\) of 0.9024. Compared to queue length predictions, the delay-based outputs exhibited lower error rates, highlighting the model's ability to track variations in waiting times efficiently (Results highlighted in bold and italic in Table 8). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \ \begin{tabular}{c} **Number of** \\ **Neurons** \\ \end{tabular}  & \begin{tabular}{c} **Swarm Population** \\ **Size** \\ \end{tabular}  & \multicolumn{2}{c}{**Acceleration Factor**} & \multirow{2}{*}{**Training \(R^{2}\)**} & \multirow{2}{*}{**MSE**} & \multirow{2}{*}{**Testing \(R^{2}\)**} \\ \cline{1-1} \cline{5-7} \(6\) & & & & & & \\ \cline \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number of** & **Swarm Population** & \multicolumn{2}{c}{**Acceleration Factor**} & \multirow{2}{*}{**Training \(R^{2}\)**} & \multirow{2}{*}{**MSE**} & \multirow{2}{*}{**Testing \(R^{2}\)**} \\
**Neurons** & **Size** & & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \\ \hline  
5 & 2.50 & 0.8954 & 0.0033 & 0.6539 \\ \hline \hline \end{tabular}
\end{table}