## 4 Simulation Results and Analyses
### Results and Discussion
In addition, it can be seen from this figure that several other algorithms fall into local convergence prematurely, and our PA-Jaya algorithm can avoid it well by adopting a multiple-subpopulation structure. This parallel structure helps our PA-Jaya algorithm keep searching for the optimal solution during the iteration. Figure 4: Relationship between the number of iterations and the total system transmission rate. Moreover, as shown in Figure 5, with the lower BER required by the system, the constraint requirements decrease, so the total system transmission rate increases. It also indicates that the system can effectively adapt to different BER constraints in the power allocation model. For the same reason, the solution performance of our PA-Jaya algorithm is also better than those of SA, GA, PSO, DE, ICO, and the traditional Java algorithms. Figure 6 shows that with the increase of the tolerable interference threshold, the transmission power allowed to be used by the secondary users increases, and, therefore, the total  
Figure 5: Relationship between the bit error rate (BER) and the total system transmission rate. Figure 6: Relationship between the interference threshold and the total system transmission rate. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Generation** & **GA** & **PSO** & **ICO** & **SA** & **DE** & **Jaya** & **PA-Jaya** \\ \hline
0 & 28.35 & 25.83 & 26.68 & 15.75 & 26.15 & 26.90 & 27.04 \\
20 & 31.53 & 31.19 & 27.37 & 32.45 & 30.24 & 37.23 & 38.43 \\
40 & 32.76 & 32.45 & 29.23 & 32.45 & 34.02 & 38.02 & 40.01 \\
60 & 32.76 & 33.08 & 32.19 & 36.86 & 36.23 & 38.27 & 41.27 \\
80 & 32.76 & 33.08 & 33.74 & 36.86 & 37.49 & 38.40 & 41.90 \\
100 & 32.76 & 33.08 & 34.30 & 36.86 & 37.80 & 38.78 & 42.21 \\
120 & 32.76 & 33.08 & 34.50 & 36.86 & 38.12 & 39.00 & 42.53 \\
140 & 32.76 & 33.08 & 34.50 & 37.80 & 38.12 & 39.22 & 42.84 \\
160 & 32.76 & 33.08 & 34.52 & 37.80 & 38.12 & 39.34 & 42.84 \\
180 & 32.76 & 33.08 & 34.52 & 37.80 & 38.12 & 39.53 & 42.84 \\
200 & 32.76 & 33.08 & 34.52 & 37.80 & 38.12 & 39.63 & 42.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal solutions of the total system transmission rate during iterations. transmission power of the system increases. Meanwhile, with the increase of the main user's tolerable interference threshold, our algorithm achieves better performance for the same reason mentioned above. Subsequently, with the same settings for the simulation parameters, we used the proposed PA-Jaya algorithm to solve the multi-user OFDM CRN problem with the user proportional rate constraints mentioned in Section 2.2. Here, the interference generated by the secondary users to the primary users is ignored, and the proportional rate between different users is randomly generated. We set the number of secondary users as 16, with the purpose of better comparing the result of our PA-Jaya algorithm and others in the case of a large number of secondary users, which is slightly different from the former simulation setup. In Figure 7, the comparison is clearly shown, where \"Alpha\" indicates the user proportionality factor required in this model. Figures 8 and 9 show the comparison of these different algorithms in the user fairness indicator (\(\mathbf{I}\)) and channel capacity with different user numbers, respectively. From the above simulation results, it is obvious that as the number of users increases, it becomes more difficult for the ROOT-FINDING algorithm to make full use of channel resources, so the channel capacity cannot achieve its optimal value. Meanwhile, the LINEAR algorithm has difficulty meeting the user fairness requirement. Our proposed PA-Jaya algorithm makes a compromise between these two aspects by adopting the fitness function and minimum user fairness constraint set. PA-Jaya achieves a similar result to LINEAR in terms of capacity. It is also shown that with the PA-Jaya algorithm, we can better weigh the capacity and fairness indicator, thus obtaining better results than the other algorithms.