# Statistical Analysis and Modeling for Optical Networks
## 5 Statistical Analysis and Modeling for Optical Networks
### Dimensioning in Optical Networks
#### 5.3.2 Statistical Modeling of Links  
Links in optical networks serve as crucial connectors between nodes via fiber channels, with link lengths representing the fiber distances over which information travels. The total fiber length in a network is the sum of all link lengths, influencing several link-dependent parameters, such as the number of amplifiers required, the selection of modulation and demodulation schemes at nodes, the necessity of compensation techniques, and the number of preamplifiers needed. Despite their significance, statistical models for OTN link lengths were previously unavailable. The link length statistical model developed in [28] fulfils that gap and helps estimate link-dependent parameters just from the coverage area and the number of nodes. The developed model for the USA100 (a Pan-American optical network with 100 nodes) optical network is in Figure 5. It shows that the link lengths of optical networks follow GEV distribution. In traffic routing and forwarding, the shortest path lengths between nodes are critical, particularly in transparent OTNs, where they determine appropriate modulation-demodulation schemes and compensation methods at nodes. It has been observed that shortest path lengths influence various network estimations, aiding both data and control plane operations. However, like link lengths, statistical models for shortest path lengths in OTNs were historically lacking, highlighting a gap in network analysis and optimization.  
Typically, the link length statistics in optical networks are estimated from their average link lengths [3]. The average link lengths can be estimated from the coverage areas of the optical networks. The coverage areas have been defined in three different ways: convex area; exact area; and geographical area [28]. The convex area is the area of the convex polygon that takes all the nodes of the optical network into consideration (an example is shown in Figure 6). The exact area is the area confined by the outer-most links of the optical network and covered by the topology of the optical network. The geographical area is the area of the countries or states in which the optical network is confined. From the measurements, and subsequent analysis, it was confirmed that the convex area is the most accurate areas for the estimation of the average link lengths of the optical networks.  
The link lengths of optical networks follow GEV distribution. The parameters of the GEV distribution of optical networks can be estimated from their average link lengths [28]. The average link length (\(<\!l\!>\)) of an optical network can be estimated from its convex area (\(A_{C}\)) with good accuracy, as shown below.  
\[<l>=\frac{0.97\sqrt{A_{c}}}{\sqrt{N}-1} \tag{2}\]  
Figure 5: Link length distribution of USA100 (histogram) fitted with GEV distribution. This distribution can be used for the estimation of the link-related parameters of optical networks [28].  
The GEV distribution has three parameters: \(\alpha\), \(\beta\), and \(\xi\). The three parameters can be estimated from the average link length, as shown below.  
\[\alpha=0.6577<l>+8.67 \tag{3}\]  
\[\beta=0.441<l>-12.37 \tag{4}\]  
\[\xi=\frac{0.0887<l>-1.557}{0.5297<l>-13.927} \tag{5}\]  
With the knowledge of these parameters, the probability distribution function (PDF) of the GEV distribution of optical networks can be estimated using the following expression.  
\[f(l,\alpha,\ \beta,\ \xi)=\frac{1}{\beta}\left[1+\frac{\xi(l-\alpha)}{\beta} \right]^{(-1/\xi)-1}exp\left\{-\left[1+\frac{\xi(l-\alpha)}{\beta}\right]^{-1/ \xi}\right\} \tag{6}\]  
From this PDF of the GEV distribution of link lengths, several link-dependent parameters of the optical networks can be estimated with very good accuracy. The convex area of the optical networks, which is a prime parameter for these estimations, can be estimated from the network topology and its circumferential ellipse [32].