## 5 Simulation Results
### Reliability Maximization with Latency Constraints
Furthermore, as shown in Figure 10, the running times of the given topologies varied from each other. It is noteworthy that the effect of the intrinsic properties of the given topologies on the running time of various algorithms is still unknown, which is one of our future research directions. Figure 11 depicts the comparison results of the overall average network reliability in different topologies when the number of gateways was two and the number of controllers four. One can observe that SAPKM outperformed SACA in multiple topologies, i.e., Nsfnet, Aarnet, ATT, and Geant, while the opposite results occurred in Agis and Chinnat. Thus, we further investigated whether SAPKM would have better average network reliability than that of SACA when the number of controllers increased. Owing to the enormous computing complexity of OEAJ, only simulation results in Agis are completely provided, and we mainly analyzed efficient algorithms on the performance and running time. To further evaluate the performance of SACA, JPKM, and SAPKM, we executed these algorithms on two different topologies, i.e., Agis and Chinnat, with \(k=3\) and \(L_{\text{cst}}=10\) ms.  
Simulation results depicted from Figures 12-15 show the effect of the number of controllers on both the overall running time and the overall average network reliability in Agis and Chinnat, respectively. Figures 12 and 13 were implemented on Agis with the number of controllers varied from 1-5, while Figures 14 and 15 were implemented on Chinnat with the number of controllers varied from 1-10. Figure 12: Comparisons of the overall running time in Agis. Figures 12 and 14 show the effect of the number of controllers on the overall running time for Agis and Chinnat. Figure 12 shows the overall running time when the number of controllers varied from 1-5. Apparently, the computational complexity of both JPKM and SAPKM was more efficient than that of SACA and OEAJ, with about 1 ms for convergence. As \(m\) grew gradually, the running time of OEAJ increased exponentially, while that of the other algorithms increased slightly. Figure 14 implied that the running time of SACA, JPKM, and SAPKM also showed the same tendency in the topology of Chinnat with \(m\) increased from 1-10. Furthermore, we can observe the increment in  
Figure 14: Comparisons of the overall running time in Chinnat. Figure 13: Comparisons of the overall average network reliability in Agis. Figure 15: Comparisons of the overall average network reliability in Chinnat. running time between Chinnet and Agis, because there were 13 more nodes in Chinnet (38 nodes) than Agis (25 nodes). Figures 13 and 15 demonstrate the impact of the number of controllers on the overall average network reliability. Conspicuously, the network reliability had a monotone increasing tendency with a growing number of controllers. In Figure 13, OEAJ achieved an optimal solution, SACA and SAPKM obtained approximate solutions, and JPKM obtained the worst performance. What is more, SAPKM had almost the same network reliability as SACA did when \(m\) ranged from 2-4. SPAKM had a slightly better performance than that of SACA when \(m\) equaled five. In Figure 15, we mainly compared the performance among SACA, JPKM, and SAPKM. It is noteworthy that SAPKM had better network reliability than that of SACA when \(m\) was greater than or equal to four, and JPKM had the same tendency when \(m\) was greater than or equal to eight.