# Spectrum Allocation Using Integer Linear Programming and Kerr Optical Frequency Combs
## 3 Results and Discussion  
To validate the proposed method, we established a reduced allocation scenario that allows us to test the assignment. This scenario can be visually verified; we anticipate that the algorithm will identify this solution. The number of lines or carriers is calculated based on the bandwidth required by each request. Table 2 details the characteristics of each client involved in the allocation scenario, including their distance in kilometers and bandwidth in Gbit/s. Additionally, the lines used are specified by modulation, with options for 16-QAM, 32-QAM, and 64-QAM, along with the number of carriers required according to the proposed multilevel QAM modulation format. A bit error rate (BER) of \(0.7\times 10^{-9}\) was set to impose a more stringent requirement. By using a lower BER threshold, we aimed to rigorously evaluate the performance, ensuring that the methodology is robust. This format provides flexibility in modulation levels, allowing various bandwidth options to be tailored to transmission distances. Figure 3 illustrates the allocation results. It can be observed that the proposed method assigns each client a modulation format that ensures the transmission distance while requiring the fewest possible carriers. The circles represent the distance limit, and no request exceeds this limit. Additionally, it is noteworthy that four client requests use super-channels to accommodate the modulation, and their allocation is contiguous to ensure the required bandwidth separation. Each user is assigned a different color in the figure to facilitate visualization. These observations support the conclusion that the method correctly allocates the requests and adapts to different types of demands. \begin{table}
\begin{tabular}{l c} \hline \hline \multicolumn{3}{c}{**Restrictions Cost Function**} \\ \hline \(\sum_{k}x_{kmu\omega}\leq 1\) & (5) \\ \hline \(\sum_{\omega}x_{kmu\omega}\leq 1\) & (6) \\ \hline \(\sum_{k}x_{kmu\omega}\leq 1\) & (7) \\ \hline \(\sum_{m}x_{kmu\omega}=n_{\omega k}*t_{k\omega}\) & (8) \\ \hline \(d_{k}x_{kmu\omega}\leq P_{\omega m}\) & (9) \\ \hline \(\sum_{m}\sum_{\omega}x_{kmu\omega}\leq\sum_{k}n_{k\omega}-1\) & (10) \\ \hline \(x_{kmu\omega}-x_{k(m-1)\omega}=y_{p}km\omega-y_{n}km\omega\) & (11) \\ \hline \(\sum_{\omega}t_{k\omega}\leq 1\) & (12) \\ \hline \(\sum_{m}y_{p}km\omega+y_{n}km\omega\ \leq 2\) & (13) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Restrictions cost function. We propose conducting tests in real network scenarios tailored to typical passive optical network (PON) situations, as validating the performance of the spectrum allocation algorithm in a practical environment is essential for assessing the network's operational capabilities. Creating a simulated database is a useful tool for modeling various bandwidth demands, such as the distance of end users to the central office, the number of subscribers, and the required transmission speeds. Testing in real environments allows for observing the system's behavior under different load conditions, the interaction between network modules, and the algorithm's ability to efficiently adapt to users' dynamic demands. This approach also facilitates the fine-tuning and refinement of the algorithm, improving its robustness and reliability, ensuring effective spectrum optimization, and meeting quality-of-service requirements under real operational conditions. To complement this approach, we utilized the database proposed in [19], which includes network scenarios where key parameters such as the number of requests, their distance from the central office, and the required network speed vary randomly. In these scenarios, the number of requests ranges from [1, 200], the distance (in kilometers) ranges from [1, 80], and the transmission speed (in Gbit/s) ranges from [1, 250], with a uniform distribution. This type of distribution ensures that each value within the defined ranges has the same probability of occurring, providing an equitable representation of various network conditions.