## 4 Network and Service Modeling  
In this section, we translate the scenario presented in Section 3 into a formal model, which will be the basis for the mathematical proposal of the optimized allocation strategy. Table 1 summarizes the considered parameters for the satellite communication infrastructure model, whereas the graph-based network and service model sets and parameters are reported in Table 2. This work focuses on an EO application in which satellites acquire images. For each acquired image, a processing service is requested. In general, this service can be made up of different tasks and these tasks could require different processing capacities. As a matter of example, this work focuses on an EO application in which satellites acquire images. For each acquired image, a processing service is requested. In general, this service can be made up of different tasks and these tasks could require different processing capacities. For example, in this work, we consider a service dedicated to active fire detection [8] that requires two different tasks: one related to the extraction of useful features from the acquired image, and the second accomplishing the actual classification based on the extracted features. In our proposal, tasks can be processed either by a satellite or by a GS. \begin{table}
\begin{tabular}{l l} \hline \hline
**Set or Parameter** & **Description** \\ \hline \(N_{sat}\) & number of satellites \\ \(N_{op}\) & number of orbital planes \\ \(h_{p}\) & orbit altitude \\ \(i_{p}\) & orbit inclination \\ \(\Omega_{p}\) & orbit right ascension of the ascending node \\ \(i\) & index in \([0,\ldots,N_{sat}-1]\) representing a satellite \\ \(p_{i}\) & orbital plane occupied by the \(i\)-th satellite, with \(i\in[0,\ldots,N_{sat}-1]\) \\ \(\mathbf{r}_{i}(t)\) & position vector of the \(i\)-th satellite, with \(i\in[0,\ldots,N_{sat}-1]\) \\ \(h_{p}\) & altitude of satellites on the \(p\)-th orbital plane, with \(p\in[0,\ldots,N_{op}-1]\) \\ \(T_{p}\) & period of motion of satellites on the \(p\)-th orbital plane, with \\  & \(p\in[0,\ldots,N_{op}-1]\) \\ \(R_{E}\) & Earth’s radius \\ \(\mu_{E}\) & Earth’s gravitational constant \\ \(G\) & antenna gain \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Satellite communication infrastructure model parameters. Figure 2: An example of the different uses of the energy available on board a satellite. In case (**a**), no processing happens in orbit and energy is requested on the ground, while, in case (**b**), the satellite processes the acquired image. In both cases, when reaching position C, the satellite has fully charged batteries, because the satellite is sunlit. \begin{table}
\begin{tabular}{l l} \hline \hline
**Set or Parameter** & **Description** \\ \hline \(c\) & speed of light \\ \(v_{tx}\) & carrier frequency \\ \(P\) & transmission power \\ \(R_{i_{s},j_{d}}\) & transmission data rate between \(i\)-th and \(j\)-th satellite, with \\ \(B\) & transmission bandwidth \\ \(k_{B}\) & [PERSON]’s constant \\ \(T_{s}\) & system noise temperature \\ \(N_{GS}\) & number of ground stations \\ \(g\) & index in \([0,\ldots,N_{GS}]\) representing a ground station \\ \(\mathbf{r}_{GS_{g}}(t)\) & position vector of the \(g\)-th ground station, with \(g\in[0,\ldots,N_{GS}-1]\) \\ \(T_{GS}\) & ground station rotation period, i.e., Earth’s sidereal day \\ \(T\) & repeat cycle time \\ \(El_{min}\) & minimum elevation angle \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._