# On the Capacity of Optical Backbone Networks
## 4 Link and Network Capacity
### Network Capacity  
Network capacity can be defined as the maximum amount of data that a network can handle per unit of time. This capacity depends on various network properties such as the physical and logical topology (traffic profile), optical reach, link capacity, node structure, etc. Physical topology describes the interconnection pattern of nodes and is typically known in advance. Nodes are considered simultaneously to be the source and destination of traffic. A starting point in the network capacity evaluation is the definition of the traffic demand profile. This profile is defined by the traffic matrix \(T=\left[t_{ij}\right]\), where each entry \(t_{ij}\) represents a traffic demand, or in other terms, the volume of traffic flowing from a source node \(i\) to a destination node \(j\), with \(i,j\in V\). In this analysis, it is assumed that the traffic profile is uniform and equal among all node pairs, which corresponds to  
\[t_{ij}= \begin{cases}1&i\
eq j\\ 0&i=j\end{cases} . \tag{15}\]  
Note that this traffic profile describes a full-mesh logical topology in the sense that each node is logically connected to every other node within the network [39]. Another important point in the network capacity evaluation is the link characterization. The link \((i,j)\in E\) can be described by two attributes: (1) length \(l(i,j)\), which is a multiple of the span length \(L_{s}\) and equal to \(l(i,j)=L_{s}N_{s}\), as explained above; (2) capacity \(c(i,j)\), determined by the number of optical channels \(N_{ch}\) available in the links, given by \(c(i,j)=N_{ch}\). As already seen, this number is limited by the bandwidth \(B_{WDM}\) and the symbol rate \(R_{s}\). For each traffic demand, it is necessary to find a path in the physical topology between each pair of nodes. This process is known as routing. Since there are multiple paths between each pair of nodes, the objective is to determine the shortest path using a heuristic like [PERSON]'s algorithm. The shortest path corresponds to the one that minimizes the total path length, defined as the sum of the lengths of all the links traversed by the path. However, in this case, the routing is constrained by the capacity \(c(i,j)\), leading to the concept of the capacitated routing (CR) problem [40]. The objective of this problem is to maximize the number of allocated traffic demands while minimizing the blocking ratio in a network with limited link capacity. In this work, the CR problem is solved through to the following steps:  
1. **Compute the shortest paths:** * Use [PERSON]'s algorithm to find the shortest path for each source-destination pair in the network; * The total path length is considered as a metric for determining the shortest paths. 2. **Order the traffic demands:** * Apply a specific sorting strategy (e.g., shortest first, longest first, largest first) to order traffic demands \(t_{ij}\). 3. **Route the demands:** * Route the demands through the precomputed shortest paths obtained in Step 1; * The routing is conducted according to the orderings considered in Step 2. 4. **Update residual capacities:** * Whenever a demand is routed, update the residual capacities of all the links traversed by the demand; * Residual capacity is defined as the difference between the link capacity and its load (number of demands already routed through the link). 5. **Path selection and blocking:** * First, attempt to use the shortest path obtained in Step 1 for each traffic demand; * Check the values of residual capacities of all links on the path. If there is enough capacity, use the path; * If the residual capacities do not allow for using the precomputed path, find an alternative shortest path;If the computing of an alternative path is not possible because there is not enough capacity, the traffic demand will be blocked. Furthermore, we assume that each path (also denoted as the lightpath, as referred to before) computed using the CR approach is physically established using an optical channel with a specific wavelength.