# On the Capacity of Optical Backbone Networks
## 4 Link and Network Capacity
### Network Capacity
In other words, a channel \(i=\{p_{i},\lambda_{i}\}\in S\) has an associated path \(p_{i}\) and wavelength \(\lambda_{i}\) and belongs to the set of optical channels required to implement a logical full-mesh topology \(S=\{1,\,2,\ldots,N(N-1)\}\). In this case, one can apply the concepts of channel capacity introduced in Section 3 to compute the total network capacity, which can be written as [17]  
\[C_{net}=\sum_{i\in S}C_{ch,i}, \tag{16}\]  
where \(C_{ch,i}\) is the capacity of channel \(i\), which, according to (10) and (12), becomes  
\[C_{ch,i}=2R_{s}\text{log}_{2}(1+SNR_{i}), \tag{17}\]  
with \(SNR_{i}\) being the \(SNR\) of channel \(i\). The \(SNR_{i}\) can be readily evaluated using (6), assuming that the optical nodes (ROADMs) are ideal and, as a result, do not affect the calculations. In this context, the number of spans for optical channel \(i\) is denoted as \(n_{s,i}=L_{i}/L_{s}\), with \(L_{i}\) representing the length of path \(p_{i}\). To avoid calculating the \(SNR_{i}\) and to reduce computation time, we can take advantage of the analysis undertaken in Section 3 and use the optical reach to obtain the channel's capacities. For instance, by knowing the lengths of the different paths and utilizing the data from Table 4, we can obtain the capacities of the different channels at two span lengths (80 and 100 km). These capacities are referred to as real capacities, an alternative to ideal or Shannon capacities derived using the spectral efficiencies of the Gaussian constellations shown in Figure 4.  
An additional important metric for network analysis is the network-wide average channel capacity, defined as [14]  
\[\overline{C}_{ch}=\sum_{i\in S}C_{ch,i}/\sum_{i\in S}\gamma_{i}, \tag{18}\]  
where \(\gamma_{i}\) denotes the expected utilization ratio of channel \(i\). For the sake of simplicity, it is assumed that \(\gamma_{i}=1\) for all \(i\in S\). As a result, the sum in the denominator of (18) equals the total number of paths in the network, which, for a full-mesh logical topology, amounts to \(N(N-1)\). With this simplification, the network capacity for a full-mesh logical topology reduces to  
\[C_{net}=\overline{C}_{ch}\times N(N-1). \tag{19}\]  
For illustrative purposes, we considered the three physical network topologies shown in Appendix A: the COST network (\(N=11\), \(K=26\), and \(\overline{L}=462.6\) km), the NSFNET (\(N=14\), \(K=21\), and \(\overline{L}=1211.3\) km), and UBN (\(N=24\), \(K=43\), and \(\overline{L}=993.2\) km), with \(\overline{L}\) being the average link length. The other parameters considered in the analysis are the ones in Table 1. Tables 7 and 8 give \(\overline{C}_{ch}\) and \(C_{net}\), along with the average path lengths for these networks, considering the real capacities and Shannon capacities, respectively [41]. Note that \(\overline{C}_{ch}\) was computed using the reach values referred to above, while \(C_{net}\) was obtained using Equation (19). As seen, the average channel capacities in the COST239 network are larger than those in the other networks due to the network's shorter link and path lengths. In fact, the paths in the NSFNET and UBN networks are, on average, 233% and 339% longer than those in the COST239 network, respectively. On the other hand, the UBN network offers a significantly larger capacity compared to the other two networks, despite having the smallest value of \(\overline{C}_{ch}\). This is attributed to its capability to support a larger number of optical channels. In practice, the UBN network accommodates 552 optical channels, whereas the NSFNET supports 182 and COST239 only supports 110. Another conclusion we can obtain from these results is that Shannon network capacities outperform the actual network capacities by a factor of about 50%. This is expected, as the Shannon capacity serves as an upper band for the real values. Following the previously mentionedt trends, the results show that for a span length of \(L_{s}=80\) km, performance is superior compared to \(L_{s}=100\) km.