## 3 Results
### Discussion of Results
#### 3.2.5 Parametric Study of the ANN-PSO Hybrid Model for Probability of Idle Servers  
For the probability of idle servers, the best configuration included seven neurons, a swarm population size of 100, and acceleration factors C\({}_{1}\) = 1 and C\({}_{2}\) = 2.5. This resulted in a training R\({}^{2}\) of 0.77514, with a very low MSE of 0.0009785, and a testing R\({}^{2}\) of 0.7092. Unlike other outputs, the probability of idle servers had a slightly lower R\({}^{2}\) value, suggesting that the model struggled slightly with generalization. This is likely due to the complexity of predicting idle server behavior, which depends on multiple interacting factors (Results highlighted in bold and italic in Table 9). \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 \(R^{2}\)**} & \multirow{2}{*}{**MSE**} & \multirow{2}{*}{**Testing \(R^{2}\)**} \\ \cline{3-3} \cline{5-6}  & & \(\mathbf{c_{1}}\) & & & & \\ \hline
5 & 10 & 2.25 & 2.00 & 0.8340 & 0.0008 & 0.41 \\ \hline
5 & 20 &  
A critical takeaway from this analysis is that optimal configurations varied across different outputs, emphasizing the need to tailor the ANN-PSO model to specific performance metrics. This ensures both accuracy and efficiency in addressing telemedicine demands. The results confirm the robustness of the model, though no single configuration fits all outputs, making adaptive optimization strategies essential. The influence of the ANN-PSO parameters (C\({}_{1}\), C\({}_{2}\), and swarm size) on model performance was significant, highlighting the necessity of fine-tuning these values during the optimization process. For instance, complex outputs like queue length required larger swarm sizes (e.g., 200) to improve prediction accuracy, whereas simpler metrics such as system utilization performed optimally with smaller swarm sizes (e.g., 20), ensuring computational efficiency. A key and somewhat counterintuitive finding of our parametric analysis was that smaller swarm populations (e.g., 10-50 particles) often yielded superior performance in predicting queue intensity compared to larger swarms. This can be explained by the nature of our optimization problem: the search space defined by tuning a single hidden layer. An ANN with 5-10 neurons and a handful of acceleration coefficients is relatively low-dimensional. In such contexts, smaller swarms concentrate on the search around promising regions, accelerating convergence and avoiding the excessive exploration that larger populations can introduce. By contrast, very large swarms may continually scatter particles into less relevant areas of the solution space, destabilizing convergence without yielding proportional gains in model accuracy. These dynamics align with established PSO tuning principles, which recommend reducing the swarm size when the problem dimensionality is modest to favor exploitation over exploration [28; 29]. The ANN-PSO model effectively predicts multiple performance metrics, allowing telemedicine systems to adapt to fluctuating demand and improve overall operational efficiency. Each optimized configuration represents a balance between high accuracy and computational efficiency. Its adaptability to different outputs makes it a reliable tool for real-time decision-making, enhancing efficiency and responsiveness in telemedicine operations. Overfitting was monitored through the evaluation of the model's test set performance across 36 configurations. The best-performing models exhibited close R\({}^{2}\) values between training and testing sets, e.g., 0.9403 (train) and 0.9164 (test) for queue intensity, and 0.9450 (train) and 0.9406 (test) for system utilization (see Tables 3 and 4). Such consistent results indicate effective generalization and confirm the absence of overfitting. These findings reinforce the importance of parameter optimization in achieving robust and generalizable predictions.