Figure 9: Network Topology 2
with increasing number of receivers. However, we would like to note that in all three models the algorithm is able to minimize packet losses within a reasonable amount of time, leading to better network performance. Moreover, we see that three paths are sufficient to effectively distribute the traffic, which suggests that a limited number of overlay nodes are able to substantially improve the network performance.
A final remark is that in NM-III setup, the optimal $x_{o,d}^s$ values turn out to be unequal for different destinations for a given overlay node as discussed earlier. For instance, under this setup, simulation results show that $x_{9,16}^1$ is roughly $0.026 \cdot r_s$, while $x_{9,10}^1$ is $0.122 \cdot r_s$.
Figure 9 represents the second topology we consider. It is a close approximation of Sprint's backbone topology as reported in [22]. It is of interest to analyze how our routing algorithm performs under these conditions since, as mentioned in Section 1, recent findings suggest that many ISPs are in the process of increasing the node connectivity of their networks. (As a comparison, the average node degree of Sprint's topology is 5.0769 while it was 3.1667 in the first topology.)
Again all links have a bandwidth of 20 Mbps. This time, we have 3 sources that simultaneously send multicast traffic, and each source has 18 receivers. Specifically, $S = {1, 9, 22}$ and $D^1 = {2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 19, 20, 21, 22, 23, 25}$, $D^9 = {1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 16, 17, 18, 21, 22, 23, 24}$ and $D^{17} = {1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 20, 21, 23, 26}$. Nodes 10 and 23 are selected as additional overlay nodes, i.e. $O^1 = {1, 10, 23}$, $O^9 = {9, 10, 23}$ and $O^{22} = {22, 10, 23}$. Similar to previous simulation setup, each source-destination pair has three paths including the min-hop path starting at the source node. Each source generates Poisson traffic with an average rate of 10 Mbps.
Figures 10 and 11 present the results. Once again NM-I model suffers from high link stress and performs worse than DVMRP. On the other hand, since the number of receivers is relatively large, the convergence of the algorithm under NM-III and NM-IIa models are again slower than that under NM-IIb model. The optimal cost values of NM-IIa and NM-IIb models are again close (11.7612 vs. 12.5875), demonstrating that much of the benefits can be observed with the simpler network design NM-IIb. Finally, we again see that three paths per receiver is sufficient to successfully distribute the traffic.
In the last set of experiments, we have used another source model to represent VBR