## 3 Design of a Digital Satellite-Borne Base Station for Quick Functional Reconfiguration
### Network Model
#### 4.2.2 System Modeling  
In order to achieve dynamic load balancing of the network and reduce system migration costs while still meeting user needs, migration cost is defined in this study as a function of migration overhead and the number of migrations.  
\[min\ \{\lambda\cdot f(cost)+\mu\cdot f(times)\} \tag{3}\]  
\[subject\ to:\ \sum a_{ij}=1\ a_{ij}\in\{0,1\} \tag{4}\]  
\[\frac{\sum_{i\in V^{VNF}}n_{i}^{core}\cdot a_{ij}}{N_{u_{j}}^{core}}<T_{1},\ \ \forall w\in V^{Node} \tag{5}\]  
\[\frac{\sum_{i\in V^{VNF}}n_{i}^{mem}\cdot a_{ij}}{N_{u_{j}}^{mem}}<T_{2},\ \ \forall w\in V^{Node} \tag{6}\]  
\[\frac{\sum_{i\in V^{VNF}}c_{i}\cdot a_{ij}}{\sum_{k\
eq j\in V^{Node}}C_{jk}}<T _{3},\ \ \forall w\in V^{Node} \tag{7}\]  
\[\sum_{l\in V^{VNF}}n_{l}^{core}\cdot a_{lk}+n_{i}^{core}\cdot b_{ik}<T_{1} \cdot N_{u_{k}}^{core} \tag{8}\]  
\[\sum_{l\in V^{VNF}}n_{l}^{mem}\cdot a_{lk}+n_{i}^{mem}\cdot b_{ik}<T_{2} \cdot N_{u_{k}}^{mem} \tag{9}\]  
\[\sum_{l\in V^{VNF}}c_{l}\cdot a_{lk}+c_{i}\cdot b_{ik}<T_{3}\cdot\sum_{k\
eq j \in V^{VNF}}C_{jk} \tag{10}\]  
\[a_{ij}\cdot b_{ik}\cdot\Delta\tau_{jk+}+\tau_{0}<\tau_{max},v_{i}\in f_{n} \tag{11}\]  
where \(T_{1},\ T_{2},\ \ T_{3}\) denote the occupancy thresholds of computing, storage and network resources in the network nodes, respectively, and the migration computation is initiated right away if the usage rate of any resource exceeds its predetermined threshold \(T_{k}\), \(T_{k}\in(0,1)\), \(k=1,2,3\). Each resource has a different \(T_{k}\), which is determined by the system's demand. \(\tau_{max}\) denotes the maximum latency limit of the SFC to which the migrated \(v_{i}\) belongs. Since the VNF to be migrated can be used by multiple service chains, the different SFCs have different requirements for latency, so we need to ensure that the latency of the VNF to be migrated is less than the minimum of the maximum latency received in all service chains to which the VNF belongs; the Boolean variable \(b_{ik}=\{0,1\}\) denotes whether the virtual network function \(v_{i}\) is moved to network node \(n_{k}\). A flag variable \(b_{ik}\) of 1 indicates that the virtual network function \(v_{i}\) is migrated to node \(n_{k}\), otherwise it is 0.  
The objective Function (3) represents the migration cost and is a function related to the migration overhead and the number of migrations, where \(\lambda\) and \(\mu\) are adjustable weighting factors. Constraint (4) guarantees that each virtual network function must and will be deployed on only one network node. Constraints (5)-(7) guarantee that the occupancy rate of computing, storage, and network resources of each network node will not exceed its pre-defined threshold. Constraints (8)-(10) ensure that the occupancy rate of computing, storage, and network resources of the destination network node will not exceed its pre-defined threshold after the virtual network function is migrated. Constraint (11) ensures that the sum of the additional delay due to virtual function migration and the original delay is less than the upper limit \(\tau_{max}\) of the delay of the service function chain to which it belongs.