# Routing Path Assignment for Joint Load Balancing and Fast Failure Recovery in IP Network
## 4 Numerical Results and Discussions
### Utilization on the Most Congested Link  
The results obtained from Problem LBO is a lower bound to Problem LB&NFP and Problem LB&LFP; we use the normalized index to evaluate the utilization on the most congested link. For node protection cases, the normalized index is the objective function value of Problem LB&NFP divided by the objective function value of LBO. For the link protection case, the normalized index is the objective function value of Problem LB&LFP divided by the objective function value of LBO. Table 1 shows the normalized values of problem LB&LFP and LB&NFP. In case of LB&LFP, the network 1 normalized congestion value is 1.33, and the network 4 value is 1.02. As the size of the network increases, the normalized congestion value starts to decrease and makes the network recovery faster for any single node and link failure. In case of LB&NFP, all networks depicts similar trends as in LB&LFP; the network 1 normalized index is 1.44 and network 4 is 1.15. Therefore, the larger the network, the smaller the congestion. When a node failure event happens, the network consumes more bandwidth as compared to a link failure event. As shown in Figure 8, normalized congestion values of LB&NFP are higher than LB&LFP, because in a node failure event, multiple links are removed from the network. Hence, the congestion values of LB&NFP are upper bound to LB&LFP. We further test our optimization models on randomly generated networks. Table 2 shows the normalized congestion values in the instance of LB&LFP. A normalized congestion value of 1.62 exists in the network with node 10 and degree 4, and a value of 1.16 exists in the network with node 10 and degree 6. Congestion numbers decrease as the degree in the network grows. The normalized congestion values of randomly generated networks for LB&LFP are shown in Figure 9. The \(x\)-axis of the graph displays the network's degree, while the \(y\)-axis shows the normalized congestion values. The red curve in the graph represents a network with ten nodes. The normalized index for the Node 10 network is 1.62 when the degree is four, 1.46 when the degree is five, and 1.16 when the degree is six. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Benchmark Networks** & **u(LB\&LFP)/u(LBO)** & **u(LB\&NFP)/u(LBO)** \\ \hline Network 1 (NSF Network) & 1.33 & 1.44 \\ \hline Network 2 (EON Network) & 1.20 & 1.30 \\ \hline Network 3 (Cost239 Network) & 1.10 & 1.21 \\ \hline Network 4 (USA Network) & 1.04 & 1.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Normalized congestion values for the four well-known benchmark networks. Figure 8: Normalized congestion values for well-known networks. As shown in the graph, the normalized index is 1.62 when the degree is four, 1.46 when the degree is five, and 1.16 when the degree is six. As a result, as the network's degree grows, the normalized congestion value drops. From node 12 through node 20, the other networks follow the same pattern. The normalized congestion values for LB&LFP are shown in Table 3. It has the same pattern as LB&LFP. A normalized congestion value of 1.94 exists in the network with node 16 and degree 4. The normalized index is 1.58 when the degree is five, and it is 1.42 when the degree is six. The size of the network grows as the degree of the network grows, and the network's congestion decreases. As a result, the recovery of a network after a failure is sped up. Figure 10 shows the normalized congestion (u) values of randomly constructed LB&NFP networks. When the degree of the node 18 network is four, the normalized value is 1.77, whereas when the degree is six, the value is 1.18. It indicates that when the network's degree grows, the normalized congestion (u) values drop, following the same trend as LB&LFP.