## 4 Network and Service Modeling
### Modeling of the Satellite Communication Infrastructure  
Figure 3 illustrates the considered satellite infrastructure. The system comprises a constellation of \(N_{Sat}\) satellites organized into \(N_{op}\) circular orbital planes. Each orbit is characterized by its altitude \(h_{p}\), inclination \(i_{p}\), and right ascension of the ascending node \(\Omega_{p}\), for each \(p\in\{0,\ldots,N_{op}-1\}\). The \(i\)-th satellite (with \(i\in\{0,\ldots,N_{Sat}-1\}\)) is assigned to the orbital plane \(p_{i}=\lfloor i\cdot N_{op}/N_{Sat}\rfloor\), and its spatial coordinates at time \(t\) are denoted by \(\mathbf{r}_{i}(t)\) in the Earth-Centered Inertial (ECI) frame. Each satellite's motion is periodic, with a period \(T_{p}\) dependent on the orbital altitude, given by:  
\[T_{p}=2\pi\sqrt{\frac{(R_{E}+h_{p})^{3}}{\mu_{E}}},\]  
where \(R_{E}\) and \(\mu_{E}\) represent the Earth's radius and gravitational constant, respectively.  
The Euclidean distance between two satellites, \(i\) and \(j\), at time \(t\) is defined as:  
\[d(i,j,t)=\big{\|}\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t)\big{\|}. \tag{1}\]  
This measure is critical to determine whether an ISL is feasible. Communication between two nodes is possible if their separation remains below a maximum threshold:  
\[d(i,j,t)\leq d_{\max}(i,j), \tag{2}\]  
where \(d_{\max}(i,j)\) denotes the maximum communication range under ideal channel conditions. Assuming an AWGN channel, it can be approximated by:  
\[d_{\max}(i,j)=\frac{Gc}{4\pi\
u_{tx}\sqrt{\frac{P}{\left(2^{R_{i},i_{d}/B}-1 \right)k_{B}T_{s}B}}}, \tag{3}\]  
with \(G\) the antenna gain, \(c\) the speed of light, \(\
u_{tx}\) the carrier frequency, \(P\) the transmit power, \(R_{ij}\) the target data rate, \(B\) the bandwidth, \(k_{B}\) Boltzmann's constant, and \(T_{s}\) the system noise temperature.  
As illustrated in Figure 3, the ISLs can be classified into two categories: intra-orbital (red) and inter-orbital (green). Intra-orbital links connect neighboring satellites on the same orbital path. Because such satellites maintain a constant angular velocity and relative spacing, these links exhibit a time-invariant topology. Their availability can be determined  
Figure 3: An overview of the considered satellite communication infrastructure.  
via geometric analysis rather than by evaluating (2) at runtime. The following conditions must be satisfied for stable connectivity between adjacent satellites:  
\[\begin{split}& 2(R_{E}+h_{p})\sin\biggl{(}\frac{2\pi N_{op}}{N_{Sat}} \biggr{)}\leq d_{\max}(i,j)\\ &\text{and}\ (R_{E}+h_{p})\cos\biggl{(}\frac{2\pi N_{op}}{N_{Sat}} \biggr{)}>R_{E},\end{split} \tag{4}\]  
with \(i\) and \(j=(i+1)\mod(N_{Sat}/N_{op})\) being neighboring nodes in the same orbit. The first inequality ensures line-of-sight connectivity within radio range, while the second guarantees that the signal path remains unobstructed by the Earth.  
Conversely, inter-orbital ISLs involve satellites on different orbital planes, whose relative positions evolve over time. Therefore, these links are only intermittently available, depending on dynamic geometrical conditions. However, since orbital parameters are deterministic, link availability can be evaluated a priori using expressions (2) and (3).  
On the ground segment, we consider \(N_{GS}\)GSs, each with time-varying position \(\mathbf{r}_{GS_{g}}(t)\) due to Earth's rotation. Their motion follows a sidereal day period of \(T_{GS}=86164\) seconds. As a result, the entire system is periodic with a global period \(T=\text{lcm}(T_{GS},T_{0},\ldots,T_{N_{op}-1})\).  
A communication link between satellite \(i\) and GS \(g\) is feasible if the elevation angle exceeds a minimum threshold \(El_{\min}\), defined via:  
\[\frac{\pi}{2}-\arccos\left(\frac{\mathbf{r}_{GS_{g}}(t)\cdot\mathbf{r}_{i}(t)} {\|\mathbf{r}_{GS_{g}}(t)\|\|\mathbf{r}_{i}(t)\|}\right)\geq El_{\min}, \tag{5}\]  
where \(\cdot\) denotes the dot product. In the case of multiple simultaneous visibility conditions, only one GS is assumed to be connected to the satellite at a given time.