## 4 Brown-Field Migration Aware Routing and Spectrum Assignment Algorithm
### Subsection Probabilistic Migration Label Model  
On the basis of the fixed-grid optical networks deployed, in order to increase network flexibility and meet the needs of service carrying, some nodes will be selected as upgrade nodes. Since the traffic distribution and topology of the optical network are different, the probability of upgrading fixed-grid nodes to flex-grid nodes in the network is also different. The upgrade probability of each node is different, due to their various traffic load, degree, economic factors and so on. The different upgrade nodes have noticeable differences in performance. The value of node upgrade probability contains two aspects, one is the self-influence of a node which represents the impact of traffic load. The other is the inter-influence of the nodes, which is closely related with the connectivity. Considering the interaction between nodes, the probabilities of different nodes can affect each other, especially in the case of adjacent nodes because the high bandwidth and spectrum-efficient channel is embodied only between two adjacent flex-grid nodes. Different calculation strategies of node upgrade probability can be obtained by adjusting the weight coefficient of the intra-influence and the inter-influence of the node, based which the node upgrade probability can be calculated [32]. In this paper, we introduced the concept of upgrade probability to describe the importance of upgrading each node in the network to support flex-grid technology, and proposed a migration-aware routing and spectrum allocation algorithm in such a scenario.  
Suppose that there are N nodes in the network, where \(p_{i}\) is defined as the upgrade probability of the fixed-grid node i, which is calculated by taking self-influence and inter-influence into consideration. The basic of the proposed _brown-field migration aware routing and spectrum assignment (BMA-RSA_) algorithm is the process of labeling-setting. The label of each node is the value derived from the probability of upgrade and distance from source to the current node. The definition of Equation (1) includes \(dis_{src,j}\) and \(p_{j}\), where \(\alpha\) and \(\beta\) are  
Figure 2: Migration scenario with traffic survivability issue when two nodes will be selected to be upgraded to flex-grid.  
used as the influenced parameter of the distance and the upgrade probability, respectively. Note that \(dis_{src,j}\) is a normalized value, which is denoted by the distance between the source node to the current node over the distance of the longest link in the network topology.  
\[Label\{j\}=\alpha\times dis_{src,j}+\beta\times p_{j}, \tag{1}\]  
During the process of label-setting, a NodeSet maintained by _BMA-RSA_ exists. Initially, only the source node is included in the NodeSet, and on each iteration, _BMA-RSA_ traverses all the nodes outside the NodeSet, selects and adds the node which has the lowest label value to the NodeSet. As shown in Equation (2), the label of the adjacent node is upgraded if the value of the label is reduced because of the expansion of the path from the newly assigned node to those nodes. Note that \(dis_{j,k}\) is also a normalized value.  
\[Label\{k\}=Label\{j\}+\alpha\times dis_{j,k}+\beta\times p_{k}, \tag{2}\]