## 5 Model Sample Validation
### Validity and Sensitivity Analysis
#### 5.2.2 Comparative Analyses of Fluctuations  
This section analyzes the fluctuations of the two models relative to the efficiency model based on changes in cycle length and green time ratio.  
1. Cycle Length Overall, the cycle lengths mostly increased. Concerning the models, the efficiency-fairness model significantly outperformed the fairness model, with mean change ratios of 0.013 and 0.88, respectively, representing a difference exceeding 67 times. Furthermore, the efficiency-fairness model ranged from a maximum of 0.04 to a minimum of 0.62 for the fairness model, a difference exceeding 15 times, which is shown in Figure 9.  
Regarding saturation levels, under low saturation conditions, the efficiency-fairness model exhibited better performance. The mean change ratio for saturation levels 0.1 to 0.4 was 0.0035, whereas for saturation levels 0.5 to 0.8, it was as high as 0.026. In terms of trends, with changes in saturation levels, the two models showed different trends. While the fairness model initially exhibited an increasing and then decreasing trend in change ratio and increasing saturation levels with significant fluctuations, the efficiency-fairness model showed fluctuations without a clear trend with changes in saturation levels. The figure below illustrates the change ratio curves for cycle lengths for both models.  
2. Green time ratio Additionally, to visually represent the changes in phase green time ratio for these two models relative to the efficiency model, this section selected the two phases with the lowest and highest average delay in the efficiency model for comparison.  
Overall, for phases with low green time ratios, the ratios further decreased, while for phases with high green time ratios, the ratios increased. Concerning the models, the efficiency-fairness model generally outperformed the fairness model. The mean change ratios for phases with low green time ratios were \(-\)0.48 and \(-\)0.51 for the efficiency-fairness and fairness models, respectively, while for phases with high green time ratios, the mean change ratios were 0.57 and 1.27, respectively. Only for phases with low green time ratios and saturation levels of 0.7 and 0.8 did the efficiency-fairness model slightly outperform the fairness model.  
Regarding saturation levels, although the efficiency-fairness model exhibited slightly higher change ratios under low saturation conditions, its fluctuation was significantly smaller than under high saturation conditions. For saturation levels of 0.1 to 0.4, the mean change ratios for both low and high green time ratios were 0.47 and 0.7, respectively, with standard deviations of 0.01 and 0.08. For saturation levels of 0.5 to 0.8, the mean  
Figure 9: Two–model cycle change ratio.  
change ratios were 0.484 and 0.446, respectively, with standard deviations of 0.19 and 0.43. In terms of trends, the change rate for phases with low saturation levels increased with saturation levels, while for phases with high saturation levels, the change rate decreased with saturation levels. The figures below illustrate the change ratio curves for green time ratios for both models, which are shown in Figures 10 and 11.  
In summary, the efficiency and fairness model proposed in this paper not only balances efficiency and fairness simultaneously but also has minimal impact on efficiency. Furthermore, the changes to the timing schemes in the efficiency model are much smaller compared to the fairness model. Therefore, it can be concluded that the proposed model in this paper is valid and effective. Sensitivity analysis indicates that the efficiency and fairness model is more effective in low saturation conditions.