Figure 4: Network Topology 1
basic unicast routing to reach each destination, while in NM-IIa, NM-IIb and NM-III models it starts with a single shortest path multicast tree (e.g., DVMRP tree) rooted at each source node and gradually shift traffic to alternative trees that are rooted at overlay nodes 9 and 17.
Figures 5 and 6 illustrate the variation of total network cost and loss rate for different models. We have also computed the optimal cost values of network models NM-II and NM-III using a MATLAB optimization package. The optimal value of NM-II is 4.7979 while that of NM-III is 4.3548. As we see, the optimal value of NM-III is smaller than that of all other models, which is expected as it has the highest level of multicast functionality/intelligence. Also, our algorithm does better than DVMRP under NM-IIa and NM-IIb models as a consequence of the availability of multiple trees to distribute the traffic load. However, while under NM-I model the algorithm is able to minimize the cost to a certain level, it cannot eliminate the packet losses and has a much higher overall cost compared to DVMRP. (The cost of NM-I model decreases to around 14.197 while it is around 7.45 for DVMRP.) The reason behind this result is the lack of multicast functionality. Since we cannot create multicast trees, the only savings due to multicasting occur between the sources and overlay nodes. Once multicast packets reach the overlays, overlay nodes need to create independent unicast sessions for each destination ignoring the multicast nature of the traffic and this creates a high level of link stress in the network as multiple copies of the same packets are generated.
One important observation is that the optimal cost values of NM-II and NM-III models are close. Hence, the additional complexity of having smart routers that are able to forward packets onto each branch at a different rate, offers only a marginal benefit in this scenario.⁷ In addition, the algorithm is able to converge faster in network model NM-IIb than all other models. This is due to the fact that, as a consequence of Corollary 4.2, we only need to optimize the overlay rates $x_o^s$ instead of individual receiver rates $x_{o,d}^s$. Hence, the number of parameters to be calculated is much smaller than the other two cases (6 versus 36). This is clearly seen in Figures 7 and 8, which present the variation of cost and packet loss rates when the number of receivers is increased to 11 for both sources. Specifically, this time $D^1 = {3, 4, 7, 8, 9, 10, 11, 12, 13, 16, 17}$ and $D^5 = {1, 2, 3, 4, 8, 9, 10, 12, 15, 16, 17}$. We see that as the number of receivers increases, the convergence of NM-IIa and NM-III becomes considerably slower. On the contrary, the algorithm does not suffer from a slower convergence rate under the NM-IIb model
⁷It is still hard to draw any conclusions as this result may depend on the specific topology and source-destination pair selections.