## 4 Results  
### Model Performance Analysis  
The deep learning model showed strong predictive accuracy on the test dataset, effectively estimating sanitary fixture needs under varying conditions. Table 3 presents the primary performance metrics:  
A **mean squared error (MSE)** of 0.342 and root mean squared error (RMSE) of 0.585 indicate that deviations between predictions and actual usage typically hover below one fixture unit. Meanwhile, the mean absolute percentage error (MAPE) of 7.21% suggests that predictions differ from ground truth by roughly 7-8% on average.  
The high \(\mathbf{R^{2}}\) value (0.938) demonstrates that the model explains about 93.8% of the variation in the target variable. Metrics like queue length accuracy (QLA = 0.912) and **waiting time accuracy (WTA = 0.934)** confirm its capacity to capture queueing dynamics, which is critical in planning for restroom facilities.  
Additionally, the model's fixture utilization accuracy (FUA = 0.951) and **peak** demand accuracy (PDA = 0.902) reflect the robustness of its estimates during surge periods. Gender-specific accuracy (GSA) also remains high (0.943 for males; 0.937 for females), confirming that the model can respond effectively to different gender demands.  
Figure 9 compares predicted fixture counts to actual values, with scattered blue data points representing individual predictions and the red diagonal line indicating perfect prediction (where predicted equals actual values). The tight clustering of points around the diagonal line demonstrates the model's consistent accuracy across different operational scenarios. The distribution of points shows minimal deviation from the ideal prediction line, with the majority of predictions falling within a narrow band around the diagonal. This near-diagonal relationship quantifies the model's reliable performance across various fixture requirements, from small installations (0-10 fixtures) to larger facilities (40-50 fixtures).  
As shown in Table 4, MAPE stays below 8% for a wide range of building sizes and occupancy profiles, though it inches upward as building scale and occupant load increase--an expected outcome given more complex user behaviors. These findings overall attest to the model's reliability and versatility.  
\begin{table}
\begin{tabular}{c c} \hline
**Metric** & **Value** \\ \hline MSE & 0.342 \\ RMSE & 0.585 \\ MAE & 0.463 \\ MAPE & 0.072 \\ R\({}^{2}\) & 0.938 \\ QLA & 0.912 \\ WTA & 0.934 \\ FUA & 0.951 \\ PDA & 0.902 \\ GSA (Male) & 0.943 \\ GSA (Female) & 0.937 \\ \hline \end{tabular} (for detailed queueing simulations and balking probabilities under normal conditions, see Appendix 6). Detailed hourly occupancy metrics and usage patterns are presented in Table A1, while Table A2 provides balking probabilities under normal operating conditions  
\end{table}
Table 3: Model performance on synthetic test data.  
Figure 9: Predicted vs. actual number of fixtures.  
Figure 10 presents a bar chart summarizing the model's performance across the key metrics mentioned, underscoring particularly strong outcomes in fixture utilization (FUA) and gender-specific accuracy (GSA).  
Figure 11 is a heatmap illustrating MAPE (%) based on building size and occupancy level. Despite a slight performance dip in larger, busier buildings, accuracy remains stable with only marginal variations.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Building Size** & **Low Occupancy** & **Medium Occupancy** & **High Occupancy** \\ \hline Small (\textless{}500 occupants) & 6.82\% & 7.15\% & 7.43\% \\ Medium (500–2000 occupants) & 7.04\% & 7.31\% & 7.58\% \\ Large (\textgreater{}2000 occupants) & 7.29\% & 7.52\% & 7.79\% \\ \hline \end{tabular}
\end{table}
Table 4: MAPE by building size and occupancy level.  
Figure 10: Performance metrics of the deep learning model on test data.