# Trade-offs between Risk and Operational Cost in SDN Failure Recovery Plan
## 3 Design Framework
These paths are presented in Table 2 with their current risks and survivability probabilities (considering the current values of \(\{p_{ij,j}\}\)), individual operational costs, and the values of the corresponding \(\eta\) in P1 or P2. In the previous example, we assumed that only one flow (between nodes \(v_{10}\) and \(v_{15}\)) required rerouting. Now, let us assume that a flow exists between any pair of nodes \(v_{i}\) and \(v_{j}\), where \(i\
eq j\), in Figure 2. Prior to the failure, the flow between nodes \(v_{i}\) and \(v_{j}\) traversed the path with the lowest risk. When the failure occurs, the risk of some of these paths increases. For this example, let us study P2 for the maximum allowed total operational cost of 250, i.e., \(\eta\) = 250. As such, the paths are selected such that the worst-case risk is maximized while the total operational cost is capped below 250. As depicted in Figure 3, the total operational cost is 249, which is smaller than \(\eta\) = 250. In addition, the lowest survivability probability is 0.79881. Because of the network topology and the distribution of the link failure probabilities, the risk of many paths is always lower than the lowest possible risk. Such flows do not require rerouting, which accounts for 599 flows in this example. Figure 4 shows the maximum allowed total operational cost \(\eta\) versus the minimum obtained survivability probability in P1 or exp \((-\tau)\) in P2. In this example, it can be observed that the smallest value of exp \((-\tau)\) = 0.8442 can be obtained for \(\eta=345\), and \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Path (R)** & \(\eta_{i}\) & \(\phi\) & \(\beta\) & \(\pi\) \\ \hline \(R_{1}\) : 8-10-12-14 & 0 \(\leq\)\(\eta_{i}\) \(<\) 3 & 0 & 0.6784 & 0.5074 \\ \(R_{1}^{\prime}\) : 8-10-11-37-14 & 3 \(\leq\eta_{i}\) \(<\) 4 & 3 & 0.2874 & 0.7502 \\ \(R_{1}^{\prime}\) : 8-6-13-15-14 & 4 \(\leq\eta_{i}\) & 4 & 0.1526 & 0.8585 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Table of Discovered Paths in the First Example. Figure 3: Number of rerouted flows, their survivability probabilities, and individual operational cost in the solution of P2 for \(\eta\) = 250. Figure 2: European Reference Network (ERnet). Copyright© IEEE. All rights reserved. Reprinted, with permission, from [32]. no smaller value for exp (\(-\tau\)) can be obtained even by increasing the maximum total operational cost beyond \(\eta=345\). As discussed before, this is because many paths are at the lowest possible risk and, therefore, do not require rerouting due to the network topology and the distribution of the link failure probabilities. In order to determine the minimum required total operational cost such that the minimum risk is guaranteed for all the paths, the following binary integer program must be solved:  
\[\text{P3}:\min_{k,\{x^{k}_{i,j}\}}\sum_{(i,j)}f^{k}_{i,j}x^{k}_{i,j}\]  
\[s.t.\left\{ \begin{array}{l}\sum\limits_{j:(i,j)\in E}x^{k}_{i,j}-\sum\limits _{j:(j,i)\in E}x^{k}_{j,i}=b^{k}_{i}\,\ \forall i,k\\ -\sum\limits_{(i,j)}x^{k}_{i,j}\log(1-p_{i,j})\ \leq\ \tau\,\forall k \end{array} \right. \tag{12}\]  
which can be simplified as:  
\[\text{P4}:\min_{\{x_{i,j}\}}\sum_{(i,j)}f_{i,j}x_{i,j}\]  
\[s.t.\left\{ \begin{array}{l}\sum\limits_{j:(i,j)\in E}x_{i,j}-\sum\limits_{j :(j,i)\in E}x_{j,i}=b_{i}\,\ \forall i\\ -\sum\limits_{(i,j)}x_{i,j}\log(1-p_{i,j})\ \leq\ \tau\end{array} \right. \tag{13}\]  
Figure 5 shows the simulation results for the aforementioned settings, i.e., P4 is solved only for the paths that are low risk given the prior values of \(\{p_{i,j}\}\) and become high risk given the current values of \(\{p_{i,j}\}\) with \(\tau=\text{e}^{-0.5}\). By solving P4 for these paths, we make the following observations. In general, 80% of the flows can be restored and achieve a survivability probability of higher than 50%. With one flow operation, 7.41% of the restorable flows can be rerouted to the low-risk paths.