# Statistical Analysis and Modeling for Optical Networks
## 5 Statistical Analysis and Modeling for Optical Networks
### Dimensioning in Optical Networks
#### 5.3.1 Estimation of Node Related Parameters  
In modern optical networks, nodes function as both sources and sinks of information while also serving as key signal processing hubs. They facilitate traffic routing between links, and in large OTNs, network topologies are often designed to be survivable--meaning each node is connected to at least two links. Survivability is quantified through nodal degree, which represents the number of connections per node. A survivable OTN topology ensures a nodal degree of at least two for all nodes. Extensive research has been conducted on nodal degrees across various network types, particularly in complex networks like the Internet, where nodal degree analysis provides critical insights into network structure, stability, and resilience. Similarly, in OTNs, nodal degree statistics offer valuable information about network features beyond survivability. The mean nodal degree, for instance, helps define network structure; when the nodal degree equals two, the network forms a simple ring. Networks with central nodes, where a single node connects to most others, typically exhibit higher mean nodal degrees. The overall statistical distribution of nodal degrees characterizes network connectivity patterns. Historically, OTNs followed a Poisson distribution for nodal degrees, but recent measurements indicate a shift towards scale-free characteristics, reflecting evolving network architectures and dynamic connectivity patterns.  
The nodal degree of a node in an optical network is the number of links connected with that node. The average nodal degree is quantified as 2L/N, where L and N denote the number of links and the number of nodes in the network, respectively. Estimation of node parameters in optical networks from network statistics is based on utilizing observed traffic patterns, connection demands, and performance measures. Based on connection establishment/release rates, holding times, and routing paths, node processing abilities, switching rates, and buffer sizes can be estimated [30]. Statistical analysis of latency and blocking probabilities provides insights into node latency contributions and contention levels [1]. In addition to that, analysis of signal degradation along routes assists in inferring node-caused impairments, such as crosstalk and filtering [2]. Sophisticated statistical techniques, such as machine learning and optimization algorithms, can be utilized for relating observations across networks to specific node properties, so that even in advanced, dynamic optical networks, accurate parameter estimation is possible. Restoration and protection in optical networks ensure resilience against failures. Protection is proactive, using pre-established backup paths like \(1+1\) (dedicated) or \(1\):\(1\) (shared) protection, switching traffic instantly upon failure [1]. Restoration is reactive, dynamically finding alternate paths post-failure, using protocols like GMPLS and ASON [2]. Techniques include optical-layer (wavelength rerouting) and network-layer (IP/MPLS rerouting) recovery. Fast-switching mechanisms, like Automatic Protection Switching and self-healing rings, enhance reliability [2]. Machine learning and SDN improve adaptive restoration, optimizing resource utilization. Hybrid approaches balance speed and cost, ensuring high availability in critical applications like telecom, data centers, and smart grids.