## 2 Materials and Methods
It has a minimal correlation with other outputs. The number of servers is negatively correlated with delay in the queue/system (\(-\)0.57/\(-\)0.55), confirming that more servers reduce wait time. It is negatively correlated with queuing utilization (\(-\)0.32), reflecting distributed load. The heatmap in Figure 1 reveals strong positive correlations between arrival rate and most performance indicators such as queue intensity (\(p\) = 0.93), queue length in the system (\(p\) = 0.91), and delay in the queue (\(\rho\) = 0.84), indicating that higher arrival rates significantly intensify system congestion. Conversely, the probability of idle servers shows a moderate to strong negative correlation with input variables, particularly with arrival rate (\(\rho\) = \(-\)0.73) and queue intensity (\(\rho\) = \(-\)0.70), suggesting reduced idleness as the system load increases. These non-linear, monotonic relationships confirm that the results are not trivially explained by linear regression alone and validate the complexity and variability inherent in the system dynamics. The presented analysis supports the robustness of the scenario design and highlights the need for advanced modeling approaches beyond simple regression. \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & \begin{tabular}{c} **Queue** \\ **Intensity** \\ \end{tabular}  & \begin{tabular}{c} **Queuing** \\ **Utilization** \\ \end{tabular}  & \begin{tabular}{c} **Queue Length** \\ **in Queue** \\ \end{tabular}  & \begin{tabular}{c} **Queue Length** \\ **in System** \\ \end{tabular}  & \begin{tabular}{c} **Delay in** \\ **Queue** \\ \end{tabular}  & \ \begin{tabular}{c} **Delay in** \\ **System** \\ \end{tabular}  & \begin{tabular}{c} **Probability of** \\ **Idle Server** \\ \end{tabular}  \\ \hline Arrival rate & 0.7274 & 0.12166 & 0.105613 & 0.335902 & \(-\)0.33008 & \(-\)0.40173 & \(-\)0.46655 \\ \hline Service rate & \(-\)0.12751 & \(-\)0.29166 & \(-\)0.16621  
2. **Model Creation**: The ANN model was defined in MATLAB, specifying the input and output layers. This step established the fundamental structure of the network, ensuring compatibility with the optimization process. The dataset was split into **training (70%), validation (15%)**, and **testing (15%)** subsets. For ANN-PSO optimization, a **90% training and 10% testing** split was adopted to allow the swarm to better explore the search space during fitness evaluation. 3. Parameter Configuration: * We then set the parameters for the network as follows: * The number of neurons in the hidden layer was set within the range of 5 to 10 to determine the optimal network complexity. This range was selected based on previous empirical studies and is consistent with the recommendations of [PERSON] et al. [25], which suggest limiting the neuron count to avoid overfitting while maintaining sufficient model capacity. * The swarm population size was defined based on exploratory testing, ranging between 10 and 400, to enhance optimization effectiveness. Smaller swarm sizes (e.g., 10-50) facilitate faster convergence and lower computation time, while larger swarms (e.g., 200-400) enable broader exploration of the solution space for more complex outputs. * Acceleration factors (C1 and C2) were adjusted to balance global and local search capabilities, improving convergence. These factors were tested within the range of 1.0 to 3.0 (C1 1-2,5 and C2 2-3), aligning with best practices to maintain a balance between cognitive (individual learning) and social (group learning) influences in PSO search dynamics, as suggested by [PERSON] [26]. * The maximum number of iterations is set to 1000, determined through convergence analysis and stabilization of results. This upper limit was chosen to allow a sufficient search depth without incurring excessive computational overhead. Hyperparameter tuning was conducted using a sensitivity analysis approach, as recommended by [PERSON] et al.