## 3 Design of a Digital Satellite-Borne Base Station for Quick Functional Reconfiguration
### Network Model  
For NFV networks \(G=\left(V^{Node},E\right)\), \(V^{Node}\) denotes the collection of nodes, and \(E\) represents the collection of links between nodes. The characteristics of the nodes and links are described as follows:  
\(N_{w}^{core}\): denotes the computing resources owned by node \(w\), expressed as the number of cores.  
\(N_{w}^{mem}\): denotes the storage resources owned by node \(w\), expressed in gigabytes.  
\(C_{i,j}\): denotes the link bandwidth between node \(i\) and node \(j\). \(i,j\in V^{Node}\).  
\(V^{VNF}\): denotes the collection of virtual network functions (VNFs), which are characterized as follows:  
\(n_{y}^{core}\): denotes the necessary computing resources for the virtual network operation \(y\).  
\(n_{y}^{mem}\): denotes the necessary storage resources for the virtual network operation \(y\).  
\(c_{y}\): denotes the necessary web resources for the virtual network operation \(y\).  
\(a_{ij}\): denotes whether virtual network function \(v_{i}\) is deployed on node \(n_{j}\). \(a_{ij}=1\) means \(v_{i}\) is deployed on \(n_{j}\), otherwise 0.  
We define the utilization rate of a resource of a node as the ratio of the used resources to the total resources on the current node. The temporary condition for migration calculation is to perform the migration calculation immediately after any resource utilization rate of any resource exceeds its preset threshold \(T_{k}\), \(T_{k}\in(0,1),k=1,2,3\), and \(T_{k}\) can be different for each resource.  
For node \(w_{j}\), the occupancy rate of its computing resources is \(\frac{\sum_{i\in VNF}n_{i}^{mem}\bullet a_{ij}}{N_{w_{j}}^{mem}}\), the occupancy rate of its storage resources is \(\frac{\sum_{i\in VNF}n_{i}^{mem}\bullet a_{ij}}{N_{w_{j}}^{mem}}\), and the occupancy rate of its network resources is \(\frac{\sum_{i\in VNF}c_{i}\bullet a_{ij}}{\sum_{k\
eq j\in V^{Node}}C_{j\pm}}\). The optimization goal is to minimize the overall system downtime by reasonably selecting the migrated virtual network function and the destination node for migration.  
The set of service function chains (SFC) is \(F\), \(F=\{f_{1},f_{2},\ldots,f_{n}\}\), and the maximum delay constraint for each function chain \(f_{n}\) is \(\tau_{n}\). Therefore, when the VNF to be migrated belongs to \(f_{n}\) (assuming per VNF instance only belonging to one SFC), the sum of its additional delay \(\tau_{+}\) and the original delay \(\tau_{0}\) should be less than \(\tau_{n}\), i.e., \(\tau_{+}+\tau_{0}<\tau_{n}\).  
The migration cost is defined as a function \(F(cost,times)\) of the migration overhead \(f(cost)\) and the number of migrations \(f(times)\), i.e.,  
\[F(cost,times)=\lambda\cdot f(cost)+\mu\cdot f(times) \tag{1}\]  
where \(\lambda\) and \(\mu\) are adjustable weighting factors.  
The migration overhead \(f(cost)\) is the sum of VNF migration computation time and implementation migration time, and the number of migrations \(f(times)\) is the number of VNF migrations that occur during the system operation. The migration computation time is determined by the number of system nodes and the performance of the controller; the implementation migration time is defined as the decommissioning time of the VNF that implements the migration, and the decommissioning time of \(v_{i}\), i.e., the migration time \(T\big{(}v_{i},n_{j},n_{k}\big{)}\) for \(v_{i}\) to be migrated from node \(n_{j}\) to node \(n_{k}\) is:  
\[T\big{(}v_{i},n_{j},n_{k}\big{)}=\frac{n_{v_{i}}^{mem}}{B(v_{i})} \tag{2}\]  
where \(n_{v_{i}}^{mem}\) denotes the size of storage resources occupied by virtual function \(v_{i}\), and \(B(f_{l})\) is the transmission bandwidth occupied by virtual network function \(v_{i}\) during migration.