## 5 Evaluations
### Performance Evaluation
#### 5.2.1 Performance versus Traffic Load  
We compare the execution time of the four techniques for different traffic loads by varying the number of flows while fixing the network type and network size. The number of flows ranges from 10 to 40 flows sent by 12 end systems exchanging the flows over four bridges, and the network type is SM. The maximum execution time and average execution time for 100 test cases are shown in Figure 7. The spherical marker indicates the maximum execution time and the square marker indicates the average execution time. The \(y\)-axis shows the execution time of successfully optimal cases, hence the terminated cases due to infeasibility and timeout do not contribute to those computed. Almost low execution times can be achieved by MPFRS-FC and APCRS-FC algorithms at the millisecond level, while  
Figure 6: Percentage distribution of optimal, timeout, and infeasible instances of SPRF-EtoED, LBF-EtoED, MPFRS-FC, and APCRS-FC for different network topos and different number of flows. (**a**) Topo 1: small ring. (**b**) Topo 2: small mesh. (**c**) Topo 3: medium ring. (**d**) Topo 4: medium mesh. the execution time of the SPRF-EtoED and LBF-EtoED algorithms increases significantly and steeply with the number of flows, even at small network sizes. Moreover, it can be noticed that the speed-up gain increases with a larger number of flows. Furthermore, we noticed that there is also a difference in the performance with the same amount of flows and different types of flow types. In flow group 1, \(num(F_{S1})\) is larger than flow group 2, since we set a special flow type in flow group 2. This not only affects the success rate of scheduling, but we can also see from the Figure 7b that it has a significant impact on the execution time of the SPRF-EtoED and LBF-EtoED algorithms. In test cases on a slightly larger network scale, we find that the performance gain at runtime is more pronounced. The execution time results in the MM network are shown in Figure 8. The number of flows ranges from 10 to 40 flows sent by 12 end systems exchanging the flows over 15 bridges, and the network type is MM. We set the timeout limit in this test case as 180 s (the yellow band in Figure 8b). For example, the LBF-EtoED technique takes over 180s on average to schedule 40 flows, and it cannot tackle problems of such size. Almost low execution times can also be achieved by MPFRS-FC and APCRS-FC algorithms at the millisecond level in the MM network. Comparisons of the success rate and the end-to-end delay for four flow setups, 10, 20, 30, and 40 flows in the SM and MM networks, are shown in Figures 9 and 10, respectively. SPRF-EtoED and LBF-EtoED obtained better the sum of end-to-end latency metrics, nevertheless, it can be checked in Figure 6 that the majority of unsolved cases by LBF-EtoED were terminated due to the timeout constraint, and those by SPRF-EtoED were terminated due to the infeasible constraint. When the network scale is not large, all algorithms can converge, while when the network scale and flow scale increase, SPRF-EtoED and LBF-EtoED techniques cannot solve the majority of test cases shown in Figure 10a,b. As with the SPRF-EtoED technique, too many flows try to traverse the same path at the same time, which arouses massive infeasible instances. Additionally, the slow exploration of extensive search space by the LBF-EtoED technique can be optimized by the exploration of a limited search space. In addition, LBF-EtoED effectively improves the scheduling performance, while it still cannot further improve the schedulability when the traffic load increases. LBF-EtoED achieves indistinguishable scheduling for flows with different sizes and different cycles, while it does not consider the characteristics of the flow and the combined flows scheduling on the same links. For the highly utilized test cases (40 flows), both proposed algorithms were able to solve 80% of the test cases.