# SDN-Based Routing for Backhauling in Ultra-Dense Networks
## 7 Results
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Considered area** (A) & \multicolumn{2}{c}{1 km\({}^{2}\)} \\ \hline \multirow{2}{*}{**Number of cells**} & \multicolumn{2}{c}{random value distributed as a PPP with intensity \(\lambda_{s}\)} \\  & \multicolumn{2}{c}{Mean number \(S=\lambda_{s}*A\)} \\ \hline \multirow{4}{*}{**Number of edges per node**} & \multicolumn{2}{c}{Low-connected: unif. distributed in [1; 2]} \\  & \multicolumn{2}{c}{Medium-connected: unif. distributed in [1; 5]} \\  & \multicolumn{2}{c}{High-connected: unif. distributed in [2; 7]} \\ \hline \multirow{2}{*}{**Data traffic demand** (\(D_{v}^{D}\))} & macrocell & microcell & picocell and femtocell \\  & unif.dist. in [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64;  
To see how the degree of connection of the network impacts on system performance, Figure 6 reports the total amount of UTDs when the mean number of edges per node varies (i.e., low, medium and highly connected network), for both low- and medium-capacity networks. As the network connection degree increases, also the number of edges in the network increases. Hence, the data traffic has more alternative paths to reach the destination and the links' congestion decreases. This is particularly true for the proposed method that can better distribute the traffic among the possible routes. Benefits are more evident in critical situations, i.e., when the network has low-capacity links. Moreover, the proposed algorithm aims at fairly distributing UTDs among different data flows. To evaluate this aspect, we have derived the Jain index of the percentage of the data-traffic demand that is not satisfied (i.e., following the proportional fair approach). Jain index provides a measure of the fairness of a given performance \(x\), and is defined as \(\frac{(\sum_{i=1}^{N}x_{i})^{2}}{N\sum_{i=1}^{N}x_{i}^{2}}\). Figure 7 represents the fairness index of the proposed scheme as a function of the number of small cells in the area. We can see that the proposed scheme achieves a higher proportional fairness, while the other two schemes have almost the same performance. Indeed, even if the HC approach allows distribution of the traffic mainly in high-capacity links, it does not take into account the fair distribution of UTDs among the flows. The behavior is almost constant with the number of cells, there is a little increase, because there are more paths, and hence, a reduction of UTDs that can be better distributed. When the network has very high capacity, all the traffic (both data and control) is supported, with any algorithm. Nevertheless, in this case it is interesting to evaluate the average delay of the network. We can see that using a more efficient routing algorithm allows reduction of the delay because traffic is routed in less congested links thus speeding up the transmission. With a rough approximation the average delay of the control and data traffic has been estimated as  
\[\bar{\delta}^{C}=\frac{1}{\sum_{(v,w)\in\mathcal{V}^{2}}d_{v,w}^{C}}\sum_{e_{ ij}\in\mathcal{E}}\frac{\sum_{(v,w)\in\mathcal{V}^{2}}d_{v,w}^{C}p_{\mathcal{C},w,e_{ij}}^{C}}{e_{e_{ij}}-\sum_{(v,w)\in\mathcal{V}^{2}}d_{v,w}^{C}p_{\mathcal{ C},w,e_{ij}}^{C}} \tag{7}\]  
Figure 6: Averaged amount of total UTDs of data traffic vs. mean number of edges per node. The mean number of small cells is 30. \[\delta^{D}=\frac{1}{\sum_{(v,w)\in\mathcal{V}^{2}}\left(d^{C}_{v,w}+d^{D}_{v,w} \right)}\sum_{e_{ij}\in\mathcal{E}}\frac{\sum_{(v,w)\in\mathcal{V}^{2}}\left[d^{ C}_{v,w}p^{C}_{v,w,e_{ij}}+d^{D}_{v,w}p^{D}_{v,w,e_{ij}}\right]}{c_{e_{ij}}- \sum_{(v,w)\in\mathcal{V}^{2}}\left[d^{C}_{v,w}p^{C}_{v,w,e_{ij}}+d^{D}_{v,w}p^{ D}_{v,w,e_{ij}}\right]} \tag{8}\]  
Table 2 reports the average network delay of the network for the proposed and benchmark methods when the network has high capacity and UTDs is zero for both data and control traffic.