## 4 Problem Modeling and Solution
### Logical Topologies Construction
The figures below compare the average packet delay achieved by different multi-topology construction methods and the single-path switching method in relation to the number of contents requested by the mobile device. Results are presented for cases where the mobile device moves to S1, S10, and S12. It can be seen from Figure 3 that when the device moves to S1, the RANK method achieves the lowest average packet delay. When the number of data flows ranges from 5 to 15, the performance of the RANK, k-shortest, and single-path methods is similar, as all of them can provide the shortest path, and the available bandwidth of the shortest path is sufficient for all data flows. When the number of data flows increases to 20, the single-path method experiences congestion, leading to a significant increase in the average packet delay. As the number of data flows continues to grow, congestion worsens, resulting in further increases in the average packet delay. In contrast, for the RANK and k-shortest methods, the average packet delay increases slowly as the number of data flows grows from 15 to 30. This is because the bandwidth of the shortest path is exhausted, and longer paths are used. However, when the number of data flows reaches 35, the k-shortest method begins to experience congestion, causing a noticeable increase in the average packet delay to 39.7 milliseconds (ms). The RANK method, on the other hand, continues to provide paths that can accommodate all data flows, avoiding congestion and keeping a lower average packet delay of 21.5 ms. Among the four methods, the AMPLE method performs the worst. It provides longer paths with a lower overall bandwidth, resulting in the highest average packet delay even when the bandwidth is sufficient. When the number of data flows reaches 15, severe congestion occurs. It is worth noting that as the number of data flows increases, congestion worsens. However, since the link queue size is fixed, the average packet delay does not increase indefinitely; instead, more packets are directly dropped. Figure 4a shows how the four methods perform when the device moves to S10. When the number of data flows is between 5 and 10, all four methods provide paths that can accommodate all data flows. Among them, the RANK, k-shortest, and single-path methods provide the shortest paths, resulting in lower average packet delays around 35 ms. In contrast, the AMPLE method provides longer paths, leading to a higher average packet delay of around 46 ms. As the number of data flows increases to 15, the single-path method experiences congestion, causing a significant increase in average packet delay. When the data flow number reaches 25, both the k-shortest and AMPLE methods also encounter congestion. However, the RANK method avoids congestion, maintaining a low average packet delay of 41.2 ms even as the data flow number grows to 35. Figure 3: Comparison of average packet delay among different methods with the increasing number of flows when the device moves to S1. Figure 4b shows the performance of the four methods when the device moves to S12, which is similar to the last case. As the number of data flows increases, the RANK method consistently maintains a lower average packet delay. When the number of data flows reaches 35, congestion occurs in the RANK method, but the level of congestion is lighter compared to the other three methods. The average packet delay rises to 65.2 ms, which is 42% lower than the second-lowest k-shortest method. Overall, regardless of the access switch the device moves to, the RANK method achieves a lower average packet delay. This is because the RANK method ranks paths to maximize transmission capacity while prioritizing shorter paths, which reduces delay. Additionally, while satisfying the flow entry capacity constraints of each node, the RANK method ensures a balance in the number of paths between each node pair.