## 4 Performance Evaluations
### Experiment Results
#### 4.2.2 Performance Results of the Proposed PBFT Method Based on the Number of Child Groups  
Figure 6 presents the consensus message processing time of the proposed PBFT method, varying with the number of child groups, as the total number of participating nodes in the blockchain changes. In the experiments, we varied the maximum number of child groups from 2 to 20 and assessed the total processing time across different node counts. Our observations indicated that the ideal number of child groups shifted as the total number of nodes increased. For configurations with 120 and 160 nodes, peak performance was obtained with four child groups during the formation of the hierarchical group structure. For the setup with 200 nodes, using six child groups resulted in quicker  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameters** & **Descriptions** \\ \hline \(N_{total}\) & 120, 160, 200, 240, 280, 300 \\ \(N_{g\_node}\) & 4 \\ \(N_{cluster}\) & 4 \\ \(N_{child}\) & 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \\ \(S_{block}\) & 1 MB \\ \(L_{inter}\) & 0.063 s, 0.126 s, 0.252 s \\ \(L_{intra}\) & 0.03 s \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters used in the experiment and their settings.  
consensus processing than with four child groups. In situations with 240 and 280 nodes, the best performance was reached with eight child groups. For the 300 nodes, the best performance was recorded with 10 child groups.  
As we explained, the experimental results indicate that the most efficient number of maximum child groups, providing the fastest processing performance, varies according to the number of nodes participating in the blockchain network. This means that there is a need for dynamic adjustments in the hierarchical structure based on the total number of nodes, rather than just using the fixed maximum number of child groups to a specific value. Additionally, it was observed that as the total depth of the hierarchical structure approached one layer, efficiency decreased in terms of the processing time. When there were 120 nodes, one cluster consisted of 30 nodes. In this scenario, the total number of layers of the hierarchical structure was one when the maximum number of child groups was set to eight. For 160 nodes, one cluster contained 40 nodes, and the total layer of the hierarchical structure was one when the maximum number of child groups was set to ten. As the node count increased to 200, the total layer of the hierarchical structure had one when the maximum number of child groups was fourteen. Similarly, for 240 nodes, it was 16 child groups. For 280 and 300 nodes, it was 20 child groups.  
Figure 5: Results of the consensus message processing performance. (**a**) Total number of nodes (\(N_{total}\)) is 120. (**b**) Total number of nodes (\(N_{total}\)) is 160. (**c**) Total number of nodes (\(N_{total}\)) is 200. (**d**) Total number of nodes (\(N_{total}\)) is 240. (**e**) Total number of nodes (\(N_{total}\)) is 280. (**f**) Total number of nodes (\(N_{total}\)) is 300.  
In short, as the hierarchical structure gradually disappears, performance decreases. This happens because as the hierarchical depth converges to one, each cluster performs consensus operations more similar to the typical PBFT. Therefore, it is crucial to establish an appropriate hierarchical structure that depends on the number of nodes participating in consensus after location-based clustering to achieve the best performance.