# Analysis of the Efficiency of Slope Evacuation for Dense Crowds in Urban Street-Type Public Spaces
## 5 Results
### Effect of Street Width on Evacuation Times  
The extreme points of each curve appear in Figure 7 when the slope is 7\({}^{\circ}\), which proves that the safety of evacuation is weaker at this slope and needs to be further researched and investigated. Therefore, in this section, we simulate and analyze the effect of street width changes on evacuation results when the slope is 7\({}^{\circ}\). The variation in evacuation time for different street widths is shown in Figure 9.  
Figure 8: Variation in evacuation times for different crowd densities.  
The width-time curves for each different crowd density were obtained by fitting according to the equation \(y=Ae^{R_{0}x}+y_{0}\), as shown in Table 6. The goodness-of-fit \(R^{2}\) of each curve is greater than 0.99, which shows that the evacuation time decreases exponentially as the street width increases.  
According to Figure 9, when the street width is less than 2 m, the total evacuation time of the crowd increases exponentially, and the evacuation effect decreases significantly. When the street width is 2 m to 6 m, as the width increases, the evacuation time of the crowd under different densities decreases, while the differences among them gradually narrow. When the street width is greater than 6 m, the curves tend to be flat, and the enhancement effect of the continued increase in the width on the evacuation efficiency is nearly zero. The reasons include physiological and psychological aspects:  
1. Physiologically, when the width of the slope reaches a certain level, the pedestrian movement speed stabilizes at its limit value, and the flow rate of the evacuation channel reaches saturation. This phenomenon is consistent with the typical flow rate model: \(Q=\rho v\), where \(Q\) is flow rate, \(\rho\) is density, and \(v\) is speed. At a certain crowd density, when pedestrian speeds reach their limit values, further increasing the street width does not significantly increase the flow rate, so the change in evacuation time stabilizes.
2. Psychologically, in a wider evacuation channel, the overall crowd density and congestion decrease. Pedestrians will feel more relaxed without a sense of urgency. The effect of crowd flow may be influenced by other factors, such as the composition of the crowd, the direction of walking, the interaction between the crowd, and environmental factors.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitted Equation** & & \multicolumn{2}{c}{\(y=Ae^{R_{0}x}+y_{0}\)} \\ \hline
**Density** & \(\rho\) **= 0.50** & \(\rho\) **= 1.00** & \(\rho\) **= 1.50** & \(\rho\) **= 2.00** \\ \hline \(y_{0}\) & 305.61306 \(\pm\) 7.71701 & 319.06594 \(\pm\) 13.5872 & 345.00357 \(\pm\) 16.78607 & 373.99962 \(\pm\) 18.79928 \\ \(A\) & 2655.19675 \(\pm\) 168.95449 & 3127.81417 \(\pm\) 249.12569 & 3702.67456 \(\pm\) 303.17803 & 4261.36783 \(\pm\) 328.76825 \\ \(R_{0}\) & \(-\)1.10557 \(\pm\) 0.05901 & \(-\)1.02977 \(\pm\) 0.07287 & \(-\)1.02344 \(\pm\) 0.07483 & \(-\)1.00995 \(\pm\) 0.07032 \\ \(R^{2}\) & 0.99735 & 0.99518 & 0.99484 & 0.99529 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Fitting result.  
Figure 9: Effect of street width on evacuation time.