## 4 Results and Discussion
### Wavelength Usage with 1 + 1 OCh Protection  
This subsection studies the impact of 1 + 1 OCh protection on wavelength requirements. The objective is to determine the wavelength increment factor, \(F_{\lambda}\), which is defined as the ratio of the number of wavelengths required with protection \((N_{\lambda p})\) to the number of wavelengths required without protection \((N_{\lambda})\). Additionally, deriving a prediction formula for \(F_{\lambda}\) would be particularly useful, as it helps estimate the extra wavelengths needed for protection without requiring complex simulations. To achieve this, one can begin by obtaining a prediction formula for \(N_{\lambda p}\), using a procedure similar to the one previously applied to derive \(\hat{N}_{\lambda}\), but accounting for 1 + 1 OCh protection. In this context, we obtained Figure 4, which depicts the number of required wavelengths \((N_{\lambda p})\) as a function of algebraic connectivity with the number of nodes and the average node degree as parameters. The data set considered is the same as in Section 4.2. The results show that the number of wavelengths \(N_{\lambda p}\) is still correlated with algebraic connectivity. However, compared with Figure 3, the power-law relationships are less evident. Therefore, the linear trend lines in the log-log graph do not fit the data points \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Network** & \(N\) & \(L\) & \(<\)\(\delta\)\(>\) & \(a_{G}\) & \(N_{\lambda}\) & \(N_{\lambda p}\) & \(F_{\lambda}\) & \(\hat{F}_{\lambda}\) & \(RE3\)**(\%)** \\ \hline SANReN & 7 & 7 & 2.00 & 0.75 & 8 & 21 & 2.63 & 2.62 & 0.2 \\ BERN & 10 & 11 & 2.20 & 0.46 & 15 & 45 & 3.00 & 2.61 & 13.1 \\ ComputeServe & 11 & 14 & 2.25 & 0.64 & 19 & 46 & 2.42 & 2.44 & 0.9 \\ COST239 & 11 & 26 & 4.73 & 2.15 & 9 & 16 & 1.78 & 2.09 & 17.4 \\ Polska & 12 & 18 & 3.00 & 0.71 & 14 & 43 & 3.07 & 2.36 & 23.2 \\ Finland & 12 & 19 & 3.17 & 0.88 & 18 & 37 & 2.06 & 2.29 & 11.6 \\ Google B4 & 12 & 19 & 3.17 & 0.4 & 20 & 37 & 1.85 & 2.54 & 37.4 \\ NSFNET & 14 & 21 & 3.00 & 0.73 & 24 & 57 & 2.38 & 2.26 & 5.0 \\ DTAG & 14 & 23 & 3.29 & 0.43 & 21 & 51 & 2.43 & 2.42 & 0.4 \\ Atlanta & 15 & 22 & 2.93 & 0.43 & 37 & 89 & 2.41 & 2.37 & 1.4 \\ GBN & 17 & 26 & 3.06 & 0.3 & 41 & 76 & 1.85 & 2.39 & 29.1 \\ NLRN & 19 & 23 & 2.42 & 0.16 & 63 & 142 & 2.25 & 2.51 & 11.2 \\ EON & 19 & 37 & 3.89 & 0.55 & 30 & 61 & 2.03 & 2.13 & 5.0 \\ ARPANET & 20 & 32 & 3.20 & 0.3 & 41 & 113 & 2.76 & 2.27 & 17.6 \\ Italia21 & 21 & 36 & 3.43 & 0.24 & 68 & 101 & 1.49 & 2.30 & 54.9 \\ BT-UK & 22 & 35 & 3.18 & 0.25 & 58 & 148 & 2.55 & 2.25 & 11.7 \\ GEANT & 22 & 36 & 3.27 & 0.42 & 42 & 109 & 2.60 & 2.11 & 18.9 \\ UBN & 24 & 43 & 3.58 & 0.27 & 64 & 140 & 2.19 & 2.16 & 1.1 \\ PBN & 26 & 36 & 2.77 & 0.14 & 96 & 228 & 2.38 & 2.29 & 3.5 \\ US26 & 26 & 42 & 3.23 & 0.2 & 80 & 178 & 2.23 & 2.19 & 1.6 \\ Euro28 & 28 & 41 & 2.93 & 0.71 & 98 & 220 & 2.24 & 2.18 & 2.9 \\ CONUS30 & 30 & 36 & 2.40 & 0.08 & 123 & 333 & 2.71 & 2.35 & 13.3 \\ GEANT2 & 32 & 52 & 3.25 & 0.22 & 118 & 218 & 1.85 & 2.02 & 9.1 \\ COST266 & 37 & 57 & 3.08 & 0.16 & 180 & 317 & 1.76 & 2.01 & 14.2 \\ Janos-us-ca & 39 & 61 & 3.13 & 0.11 & 162 & 358 & 2.21 & 2.08 & 5.7 \\ Germany50 & 50 & 88 & 3.52 & 0.18 & 204 & 392 & 1.92 & 1.89 & 1.8 \\ CONUS60 & 60 & 79 & 2.63 & 0.05 & 543 & 1040 & 1.92 & 2.29 & 19.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Estimation of wavelength requirements for real-world networks with and without protection. as well as they do in the unprotected case. Despite this, we still used these trend lines to derive the following prediction formula:  
\[\hat{N}_{\lambda p}=10^{aN^{2}+bN+c}\times a_{G}^{d} \tag{8}\]  
where \(a=-3\times 10^{-4}\), \(b=3.64\times 10^{-2}\), \(c=1.018\) and the power law slope \(d=-0.7\).