# Link-State Aware Hybrid Routing in the Terrestrial-Satellite Integrated Network +
## 3 System Model
### Problem Formulation  
Constrained by the link intermittent connectivity, ISLs and TSLs have time-varying capacity and connection. In this case, there are three basic types of constraints, i.e., link connection constrain, data flow balance constraints, and communication velocity constraints. (1) Link Connection Constraints: To model the contention caused by dynamic satellites, a set of boolean variables is introduced. \[\omega(v_{i}^{t},v_{j}^{t})=\{0,1\},\forall(v_{i}^{t},v_{j}^{t})\in E_{sg} \cup E_{ss}, \tag{2}\]  
which takes the value of 1 if link \((v_{i}^{t},v_{i}^{t})\) is active, and otherwise. Due to the limitation of antenna pointing, most LEO constellation systems can establish inter-satellite links with four satellites around them, such as Iridium and Hongyun, each satellite has two neighbors on the same orbit and two neighbors on the adjacent orbit, i.e.,  
\[C1:\sum_{(v_{i}^{t},v_{j}^{t})\in E_{ss}}\omega(v_{i}^{t},v_{j}^{t})\leq 4. \tag{3}\]  
Since the ISL is bidirectional, \(\omega(v_{i^{\prime}}^{t},v_{j}^{t})\) need to satisfy:  
\[C2:\omega(v_{i}^{t},v_{j}^{t})=\omega(v_{j}^{t},v_{i}^{t}),\forall(v_{i}^{t},v _{i}^{t})\in E_{ss}. \tag{4}\]  
Meanwhile, the TSE can only be established by one satellite at one time, i.e.,  
\[C3:\sum_{(v_{i}^{t},v_{j}^{t})\in E_{sg}}\omega(v_{i}^{t},v_{j}^{t})\leq 1, \forall v_{i}^{t}\in V_{s},v_{j}^{t}\in V_{g}. \tag{5}\]  
(2) Data Flow Balance Constraints: to balance the change in the storage occupancy of a satellite against the incoming data, we define \(b_{i}^{t}(k)\) as the data volume of mission flow \(UT_{k}\) in the storage of node \(v_{i}\) at the end of the time interval \(t\); thus, the amount of all incoming flows and all outgoing flows at a node must satisfy the flow conservation. The data flow balance constraint can be expressed as:  
\[C4:\sum_{(v_{i}^{t},v_{j}^{t})\in E_{1}}\vartheta(v_{i}^{t},v_{j}^{t},k)+b_{i }^{t}(k)=\sum_{(v_{i}^{t},v_{j}^{t})\in E_{1}}\vartheta(v_{j}^{t},v_{i}^{t},k )+b_{i}^{t-1}(k). \tag{6}\]  
Furthermore, the aggregate data in a link should not exceed the maximum amount of data that can be transmitted by this link, thus we have:  
\[C5:\sum_{k}\vartheta(v_{i}^{t},v_{j}^{t},k)\leq C(v_{i}^{t},v_{j}^{t}). \tag{7}\]  
(3) Communication Velocity Constraints: the transmission velocity needs to satisfy the minimum rate requirement of the data transmission for UT, thus we have:  
\[C6:r(v_{i}^{t},v_{j}^{t})\geq r_{\min}. \tag{8}\]  
Therefore, combining the restrictions of link connection, data flow balance, and communication velocity, the hybrid routing problem can be formulated to minimize the end-to-end delay and be expressed as:  
Based on the celestial sphere coordinate system, the coordinates of a satellite on an orbital sphere at time \(t\) is expressed as:  
\[\left\{ \begin{array}{l}x(t)=r\times\cos(\phi_{s}(t)\times\cos(\lambda_{s}(t)), \\ y(t)=r\times\cos(\phi_{s}(t)\times\sin(\lambda_{s}(t)),\\ z(t)=r\times\sin(\phi_{s}(t)).\end{array} \right. \tag{12}\]  
The time-varying distance of the interplane link between satellite \(s_{i}\) and \(s_{j}\) is \(d(s_{i}^{t},s_{j}^{t})\), which is derived as:  
\[d(s_{i}^{t},s_{j}^{t})=\sqrt{(x_{i}(t)-x_{j}(t))^{2}+(y_{i}(t)-y_{j}(t))^{2}+( z_{i}(t)-z_{j}(t))^{2}}. \tag{13}\]  
Combining (12) and (13), the ISLs length is derived as a time-dependent parameter, which can be expressed as:  
\[d(s_{i}^{t},s_{j}^{t})=r\sqrt{f(s_{i}^{t},s_{j}^{t})}, \tag{14}\]  
\[f(s_{i}^{t},s_{j}^{t})=2[1-\cos\phi_{i}(t)\times\cos\phi_{j}(t)\times\sin( \lambda_{i}(t)+\lambda_{j}(t))-\sin\phi_{i}(t)\times\sin\phi_{j}(t)]. \tag{15}\]  
To explore the time-varying features of ISL, the ISL distance of the Iridium and Hongyun constellation is compared. As shown in Table 3, the simulated satellite orbit parameters and the propagation delay of ISLs are presented. The comparison results show that the intraplane ISL length increases with the increasing of orbital height, and decreases with more satellites distributed in the orbit plane.