# Spectrum Allocation Using Integer Linear Programming and Kerr Optical Frequency Combs
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Request**} & \multirow{2}{*}{
\begin{tabular}{c} **Distance** \\ **(km)** \\ \end{tabular} } & \multirow{2}{*}{**BW (Gbit/s)**} & \multicolumn{3}{c}{**Lines**} \\ \cline{4-6}  & & & **16-QAM** & **32-QAM** & **64-QAM** \\ \hline U1 & 7 & 215 & 3 & 3 & 3 \\ U2 & 5 & 38 & 1 & 1 & 1 \\ U3 & 4 & 8 & 1 & 1 & 1 \\ U4 & 5 & 234 & 3 & 3 & 3 \\ U5 & 9 & 177 & 3 & 3 & 3 \\ U6 & 8 & 176 & 3 & 3 & 3 \\ U7 & 7 & 36 & 1 & 1 & 1 \\ U8 & 3 & 174 & 3 & 3 & 3 \\ U9 & 2 & 12 & 1 & 1 & 1 \\ U10 & 8 & 26 & 1 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Case study of assignment requests. Figure 3: Spectrum assignment for the first case study. By varying these parameters with discrete uniform distributions, we effectively model a wide range of demands and operational scenarios for PON networks, simulating both current and future conditions in terms of bandwidth requirements. The selection of these ranges is based on the typical operational characteristics of PON networks and projections of future bandwidth demand growth. We propose the BBR as a key metric to validate the performance of our resource allocation method in optical networks. The BBR quantifies the likelihood that the network rejects requests when the required bandwidth exceeds the available resources, enabling us to evaluate the network's capacity to meet demand without overloading its capabilities. To calculate the BBR, we analyze the ratio between the total rejected bandwidth and the total requested bandwidth, which helps identify bottlenecks and optimize network performance. By considering both the number of rejections, the number of requests, and the requested bandwidth, this metric provides a comprehensive view of resource management efficiency, which is crucial to ensure the network can adapt to growing traffic demands without compromising service quality. Equation (14) details the mathematical model used to calculate the BBR. \[BBR=\sum_{i=1}^{N}\frac{BWR_{i}}{BWS_{i}} \tag{14}\]  
In Equation (14), BWR represents the total rejected bandwidth, and BWS represents the total requested bandwidth. These metrics allow us to verify the network's performance. Figure 4 presents a three-dimensional graphical representation where the \(Y\)-axis corresponds to the bandwidth, the \(X\)-axis represents the number of requests, and the \(Z\)-axis illustrates the BBR index. This visualization enhances our understanding of the system's performance by showing how the BBR varies depending on bandwidth and the number of requests. The proposed method was evaluated across 1400 request scenarios to analyze network performance and calculate the Blocking to Bandwidth Ratio (BBR). The results demonstrated the method's effectiveness, as most requests exhibited low blocking probabilities. Figure 4 illustrates the BBR performance in relation to bandwidth and the number of clients, showing that the method performs adequately up to 110 clients, achieving up to 15 THz of simultaneous bandwidth utilization. Figure 4 also highlights scenarios where the BBR exceeds a value of 1, reaching up to 1.8. These scenarios are marked with ellipses indicating the highest rejection values,  
Figure 4: Number of rejections vs. bandwidth and number of requests. The highest BBR values are highlighted with red ellipses, indicating critical points in the analysis. Additionally, a dotted red line marks the point where the allocation method begins to show a decline. depicted in yellow and green. A dashed line marks the point, around 110 requests, where the allocation method begins to deteriorate, leading to a noticeable rise in the BBR. From this point onward, bandwidth rejections increase significantly, with maximum values occurring as the number of clients continues to grow. The rise in BBR between 110 and 200 clients is attributed to limitations in the number of available carriers.