# Link-State Aware Hybrid Routing in the Terrestrial-Satellite Integrated Network +
## 3 System Model
### Space-Time Topology Model
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Definitions** \\ \hline \(S\) & The set of satellites \(s_{i}\) \\ \(G\) & The set of TSEs \(g_{i}\) \\ \(UT\) & The set of UTs \(UT_{i}\) \\ \(D\) & The total number of time intervals in path plan \\ \(\Delta\tau\) & The duration of each time interval \\ \(G(t)\) & The directed graph composed of T layers \\ \(V\) & The vertices set of \(G\) \\ \(E(t)\) & The links set of \(G\) \\ \(E_{l}(t)\) & The links set of spatial link \\ \(E_{b}(t)\) & The links set of temporal link \\ \(d(v_{i}^{l},v_{j}^{l})\) & The distance of link \((v_{i}^{l},v_{j}^{l})\) \\ \(v_{p}\) & The transmission speed of radio signal \\ \(r(v_{i}^{l},v_{j}^{l})\) & The capacity of link \((v_{i}^{l},v_{j}^{l})\) \\ \(b_{i}^{l}(k)\) & The data volume in storage of \(v_{i}\) for \(UT_{k}\) during the t-th time interval \\ \(\theta(v_{i}^{l},v_{j}^{l},k)\) & The capacity of link \((v_{i}^{l},v_{j}^{l})\) for \(UT_{k}\) data \\ \(\omega(v_{i}^{l},v_{j}^{l})\) & The connection state of link \((v_{i}^{l},v_{i}^{l})\) \\ \(T_{p}(v_{i}^{l},v_{j}^{l})\) & The propagation delay \\ \(T_{q}(v_{i}^{l},v_{j}^{l})\) & The queuing delay \\ \(T_{u}(v_{i}^{l},v_{j}^{l})\) & The transmission delay \\ \(E_{gu}(t)\) & The communication opportunities for UTs to TSE in each time interval \\ \(E_{sg}(t)\) & The communication opportunities for satellite to TSE in each time interval \\ \(E_{ss}(t)\) & The communication opportunities for satellite to satellite in each time interval \\ \(C(v_{i}^{l},v_{j}^{l})\) & The maximum capacity of \((v_{i}^{l},v_{j}^{l})\) \\ \(r_{\min}\) & The minimum rate requirement of data transmission \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations and definitions. dynamic links can imagine a \"band\" (of message transmission duration) going from left to right as we travel from source to destination. Occasionally this band has to \"slide\" toward the right when there is no link available for transmission immediately to the chosen next hop node. For example, in Figure 2, after receiving the message from \(s_{1}^{t}\), \(s_{2}^{1}\) could not forward it immediately to \(s_{3}^{1}\), but had to carry it for some time until the link \((s_{2}^{2},s_{3}^{2})\) became active. Since the ground stations always have massive storage, we only focus on the limited buffer size for satellites. The set of temporal links is denoted as \(E_{b}(t)=\{(v_{i}^{t},v_{i}^{t+1})|v_{i}^{t}\in V_{s}\cup V_{g},1\leq t\leq D-1\}\), \(E(t)=E_{i}(t)\cup E_{b}(t)\). By defining the space-time graph G, any communication operation in the time-evolving network can be simulated on this directed graph. This guarantees that the packet can be delivered between any two nodes in the network over the period \(D\). Since the antenna or laser terminal of the TSL and TSE requires more complex capture, tracking, and aiming mechanisms, it is difficult to install multiple antennas at the same time on the satellite platform. Therefore, by using the rotation-limited beam or fixed beam, a satellite can set up ISLs with four neighbors at most, with two in its own orbit and the other two in the neighboring orbits [24; 25], so as to realize the optimal constellation design, such as Iridium, SpaceX, Telesat, Kuiper, and Hongyun [5; 6]. It can also establish a TSL with the TSEs in its coverage, where all TSEs are represented by a single ground antenna for simplicity. In this way, the four ISLs would be established at the same time, while one TSL is established by one satellite at one time. Considering the intermittent connectivity of ISLs and TSLs, the hybrid routing mechanism of TSIN is modeled as an integer linear programming problem, aimed at link selection by minimizing the network delay. The network delay usually includes processing delay, queuing delay, transmission delay, and propagation delay. Wherein, the processing delay is related to hardware performance which is almost negligible.