## 3 Design of a Digital Satellite-Borne Base Station for Quick Functional Reconfiguration
### Network Model
#### 4.2.5 Migration Destination Node Selection Algorithm  
The performance needs of the VNF to be migrated and the end-to-end performance limitations of the functional service chain in which the VNF is placed must both be taken into account when choosing the migration destination node. When the constraints are met, the destination node is chosen based on which node has the most residual resources and the least amount of delay. The TOPSIS algorithm is used to choose the destination node since choosing the best node entails making a multi-objective decision.  
In the first step, nodes whose additional latency meets the constraints are selected according to the end-to-end performance requirements of the functional chain of the service in which the VNF is to be migrated.  
\[a_{ij}\cdot b_{ik}\cdot\Delta\tau_{jk+}+\tau_{0}<\tau_{\max},v_{i}\in f_{n} \tag{13}\]  
In the second step, the node that meets the performance requirements of the VNF and will not be overloaded after the VNF moves in is selected from among the nodes selected according to their performance requirements.  
In the third step, the TOPSIS algorithm is used to calculate the migration index of the nodes [22].  
First, the positive ideal solution BestNode is calculated, where each property of the positive ideal solution is the best value among all alternatives.  
\[T_{i+}=\min\{T_{ki}|k\in N_{3}\},i\in\{1,2,3\} \tag{14}\]  
\[\tau_{+}=\min\{\tau_{k}|k\in N_{3}\} \tag{15}\]  
The minimum utilization of each resource in all of the nodes to be selected as the occupancy rate of the ideal solution \((T_{1+},T_{2+},T_{3+})\) is calculated; at the same time, it is also desired to optimize the additional delay caused by the migration, to derive the minimum additional delay \(\tau_{+}\) for all of the nodes to be selected as the additional delay caused by the migration of the ideal virtual network function.  
The negative ideal solution Worst Node is calculated, where each property of the negative ideal solution is the worst value among all alternatives.  
\[T_{i-}=\max\{T_{ki}|k\in N_{3}\},i\in\{1,2,3\} \tag{16}\]  
\[\tau_{-}=\max\{\tau_{k}|k\in N_{3}\} \tag{17}\]The maximum utilization of each resource in all nodes to be selected as the occupancy of the negative ideal solution \((T_{1-},T_{2-},T_{3-})\) is calculated, while the additional time delay of the negative ideal solution is \(\tau_{-}\).  
The Euclidean distance \(\theta_{ok+}\) of the node \(w_{k}\) from the BestNode and the Euclidean distance \(\theta_{ok-}\) from the WorstNode are calculated, and the proximity \(\theta_{ok}\) of the node to the BestNode is obtained.  
\[\theta_{ok+}=\sqrt{\sum_{i=1}^{3}[\gamma_{i}(T_{ki}-T_{i+})]^{2}+[\gamma_{4}( \tau_{k}-\tau_{+})]^{2}} \tag{18}\]  
\[\theta_{ok-}=\sqrt{\sum_{i=1}^{3}[\gamma_{i}(T_{ki}-T_{i-})]^{2}+[\gamma_{4}( \tau_{k}-\tau_{-})]^{2}} \tag{19}\]  
\[\theta_{ok}=\frac{\theta_{ok-}}{\theta_{ok-}+\theta_{ok+}} \tag{20}\]  
where \(\gamma_{i}\) denotes the weight of each indicator and is a predefined value; \(\theta_{ok}\) denotes the proximity of the node to the positive ideal solution, with larger \(\theta_{ok}\) indicating that the node is closer to the ideal solution.  
According to the migration index \(\theta_{ok}\) of each node, the one with the largest migration index is selected as the migration point.