## 4 Results and Discussion
### Wavelength Usage with 1 + 1 OCh Protection
By using (4) and (8), one arrives at the following equation for the estimator of the wavelength increment factor:  
\[\hat{F}_{\lambda}=\hat{N}_{\lambda p}/\hat{N}_{\lambda}=10^{\alpha N^{2}+\beta N +\gamma}\times a_{G}^{\delta} \tag{9}\]  
where \(\alpha=1.186\times 10^{-4}\), \(\beta=-1.195\times 10^{-2}\), \(\gamma=7.798\times 10^{-1}\) and \(\delta=-0.13\). To analyze the accuracy of the wavelength increment factor estimator, it makes sense to apply Formula (9) to real-world networks and compare the predicted values with those of \(F_{\lambda}\), obtained through simulations based on the procedure described in Section 3. In order to quantify this accuracy, we define the following mean relative error:  
\[RE3(\%)=\frac{\left|F_{\lambda}-\hat{F}_{\lambda}\right|}{F_{\lambda}}\times 1 00. \tag{10}\]  
The results presented in Table 1 show that \(F_{\lambda}\) ranges from 1.49 to 3.07, with a mean value of 2.26 computed across all networks. This indicates that a network with \(1+1\) OCh protection typically requires about twice as many wavelengths as in the unprotected case. It is also observed that networks with the same number of nodes, the same number of links that implies the same average node degree can have different wavelength increment factors. This happens for instance with the Finland and Google B4 networks that have a value of \(F_{\lambda}\) equal to 2.06 and 1.85, respectively. When comparing the real factor \(F_{\lambda}\) with the estimated factor \(\hat{F}_{\lambda}\), it is observed that the prediction formula tends to underestimate or overestimate this last factor for some real networks, while providing very accurate estimations for others. The highest estimation error, 54.9%, was observed on the Italia21 network, which has the lowest wavelength increment factor, while the smallest, 0.2%, occurred on the SANReN network. The outlier result for the Italia21 network may be explained by its atypically high number of nodes with degrees above the average. On the other hand, the average estimation error is 12.7%. For approximately 70% of the networks, the estimated wavelength increment factor achieves an error of less than 15%. It is worth  
Figure 4: Wavelength requirements as a function of algebraic connectivity, with the number of nodes and average node degree as parameters, for networks with \(1+1\) OCh protection. noting that, although the real wavelength increment factor varies from 1.49 to 3.07, the predicted values range between 1.89 and 2.62. Finally, Table 2 presents the computational time of the RWA algorithm for different random networks. In general, the computational time increases cubically with \(N\) and decreases with the average node degree. This indicates the algorithm performs faster in more connected networks compared to those with fewer links, for a given value of \(N\). Additionally, as expected, the average computation time is significantly higher for networks with \(1+1\) OCh protection. On average, for networks with \(10\leq N\leq 50\), the computational time is approximately 5 times higher than that of unprotected networks. Note that these simulation results were obtained on a computer with an Intel i7-8750H CPU @ 2.20 GHz (ASUSteK Computer Inc., Shanghai, China) processor and 16 GB of RAM.