# Statistical Analysis and Modeling for Optical Networks
## 5 Statistical Analysis and Modeling for Optical Networks
### Dimensioning in Optical Networks
#### 5.3.4 Statistical Modeling of Shortest Path Lengths  
Statistical modeling of shortest path lengths in optical networks is important for understanding and optimizing network performance. This analysis involves characterizing the distribution of distances between node pairs, which significantly impacts factors like routing, latency, resource allocation, and overall network efficiency. Researchers employ  
Figure 7: Circumferential ellipse (shown in red color) of the BREN network in Bulgaria [32; 77]. In this case, the circumferential ellipse touches the topology only at four points. The major axis of this circumferential ellipse is AB, the mirror axis is CD, and their intersection point O is the center. statistical techniques to analyze real optical transport networks, often finding that distributions, like the Johnson S\({}_{\text{B}}\) distribution, effectively model these path lengths [33; 90]. These models enable the estimation of key network parameters, such as average path lengths and variations, from fundamental network information like node locations. This allows for the prediction of path-length-dependent system behaviors, including the selection of appropriate modulation formats, enhancing network design and planning. By statistically analyzing shortest path lengths, engineers can improve network reliability, predict potential bottlenecks, and optimize routing algorithms, ultimately leading to more robust and efficient optical communication systems. The shortest path lengths in optical networks are found to follow the Johnson S\({}_{\text{B}}\) distribution [33; 90]. This distribution is one of the transformed distributions of the 'Johnson System' family [58; 72; 73]. It is also a bounded distribution. Johnson S\({}_{\text{B}}\) is similar to the normal distribution in many aspects. It can be regarded as a modified lognormal distribution [33]. This distribution has four parameters (\(\gamma,\delta,\lambda,\zeta\)), and its PDF is shown below. \[f\left(z;\gamma,\delta,\lambda,\zeta\right)=\frac{\delta}{\lambda\sqrt{2\pi} \ z(1-z)}exp\left[-\frac{1}{2}\left\{\gamma+\delta log\left(\frac{z}{1-z} \right)\right\}^{2}\right] \tag{7}\]  
Its parameters can be estimated from the convex area of the optical networks (\(A_{c}\)) and their number of nodes (\(N\)). These estimations are shown below. \[\lambda= \begin{cases}\begin{array}{c}10.67\big{(}\sqrt{A_{c}}\big{)}^{0.756 4},\text{ }if\text{ }\sqrt{A_{c}}<1000\\ 0.0016A_{c}-4.076\sqrt{A_{c}}+5600,\text{ }if\text{ }\sqrt{A_{c}}\geq 1000\end{array} \end{cases}  \tag{8}\]  
\[\delta= \begin{cases}\begin{array}{c}cosec\left(\frac{t+30}{40}\right)+0.25 cos(62.46t)-0.1 cos(112.62t),\text{ }if\text{ }t\left(=\frac{\sqrt{A_{c}}}{N}\right)<70\\ 0.8 sin\left(\frac{t+126.5}{162}\right)+0.17 cos(50.258t)+0.12 cos(149.226t),\text { }if\text{ }t\text{ }\geq 70\end{array} \end{cases}  \tag{9}\]  
\[\xi= \begin{cases}\begin{array}{c}0.1\sqrt{A_{c}},\text{ }if\text{ }\delta<0.82\\ -0.11\sqrt{A_{c}},\text{ }if\text{ }\delta\geq 0.82\end{array} \end{cases}  \tag{10}\]  
\[\gamma= \begin{cases}\begin{array}{c}0.12+|1.02-0.06t|+0.03 sin(4t),\text{ }if\text{ }t\big{(}=\frac{\sqrt{A_{c}}}{N}\big{)}<29\\ 0.235+sinc(2t-112.4),\text{ }if\text{ }29\leq t\leq 96\\ 0.24-\frac{1}{t-95}+0.45 sinc(2t-253),if\text{ }t>96\end{array} \end{cases}  \tag{11}\]  
The Johnson S\({}_{\text{B}}\) distribution is well-suited for modeling shortest path lengths in optical networks due to its ability to capture both unimodal and bimodal probability density functions, frequently observed in these networks. Notably, bimodality arises in networks with large convex areas and sparse node distribution. We have demonstrated that the mean, median, and standard deviation of shortest path lengths can be linearly estimated from these convex areas. This estimation, along with the upper bound of the maximum path length, is achievable without complete network topology knowledge, as the convex area can be calculated directly from node coordinates.