# Energy-Efficient Controller Placement in Software-Defined Satellite-Terrestrial Integrated Network
## 3 System Model and Problem Formulation
### Software-Defined STIN Architecture  
The software-defined STIN architecture is depicted in Figure 2. It consists of three planes: the management plane, data plane, and control plane. The entities in the management plane are deployed on the terrestrial network to realize mobility and handover management functions. The ground contains infrastructure such as base stations, gateways, vehicles, ships, computers, as well as mobile phones. The terrestrial network communicates with the satellite network via satellite gateways in the management plane. The LEO satellites are responsible for carrying the switch function and forwarding data packets in the data plane. It is assumed that each satellite can host the control unit. Several LEO satellites are selected as controllers to achieve the distributed control of the STIN, and they are only responsible for communicating with the other LEO satellites under their coverage instead of all LEO satellites in the space network.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Ref.**} & \multirow{2}{*}{**Locations**} & \multirow{2}{*}{**Algorithms**} & \multicolumn{4}{c}{**Metrics**} \\ \cline{3-6}  & & & **Latency** & **Cost** & **Reliability** & **Load** \\ \hline
[35] & Air & K-means & ✓ & & & \\
[25] & Ground & SA & ✓ & & ✓ & \\
[30] & Satellites & ILP & ✓ & & & \\
[31] & Ground, Satellites & GA & ✓ & & & ✓ \\
[32] & Satellites & AA & ✓ & & & ✓ \\
[33] & Satellites & PSO & ✓ & & ✓ & ✓ \\
[34] & Satellites & SA & & & & ✓ \\ Proposed & Satellites & SA & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the proposed scheme with state-of-the-art methods.  
The network topology of the STIN in each time slot is defined as a graph \(G=(V,E)\), where \(V\) represents the set of the network nodes (LEO satellites and satellite gateways), and \(E\) represents the set of links, i.e., ISLs and user data links (UDLs). Let \(S\subseteq V\) denote the set of satellite nodes, \(SG\subseteq V\) denote the set of satellite gateways, and \(C\subseteq S\) denote the set of satellite controllers.  
For the communication between the satellites selected as SDN controllers and the potential controller set, SDN switches collect the status instruction via the existing ISLs. Almost every satellite has four ISLs [36]: two intra-plane ISLs which link to the adjacent satellites in the same orbit plane and two inter-plane ISLs which link to the satellites in the left- or right-neighbor orbit plane. However, the satellites which rotate in opposite directions in two adjacent orbit planes only have three ISLs: two intra-plane ISLs and one inter-plane ISL. The length of the intra-plane ISLs is constant, which is computed as:  
\[L_{intra}=\sqrt{2}R_{o}\sqrt{1-cos(\frac{360^{\circ}}{M})} \tag{1}\]  
where \(R_{o}\) is the radius of the orbit, and \(M\) is the number of satellites in an orbit.  
The length of the inter-plane ISLs is constantly varying due to the movements of the satellites, which is computed as:  
\[L_{inter}=\sqrt{2}R_{o}\sqrt{1-cos(\frac{360^{\circ}}{N})cos(lat)} \tag{2}\]  
where \(lat\) refers to the latitude, and \(N\) is the number of orbits. The length of inter-plane ISLs decreases as the latitude increases. Once the satellites move into the polar region, the inter-plane ISLs break.  
Furthermore, we assume that the speed of transmitting flows is equal to the speed of light. That is to say, the length of ISLs also determines the latency in the STIN. Although  
Figure 2: The overview of the software-defined STIN architecture.  
controllers are the same as satellite switches physically, they are separated from each other in function. The position of controllers determines the way to process traffic flows, and the corresponding ISLs are regarded as control paths.