## 4 Network and Service Modeling
### Graph-Based Modeling of the Dynamic Topology  
The main difficulty in obtaining an abstract model of the presented scenario is related to the fact that both satellites and Earth move, leading to a dynamical topology. However, this motion is periodic, and for this reason a cyclostationary behavior can be assumed for both topology and service generation. In particular, by calling \(T_{c}\) the repetition period of the satellites and Earth relative motion, we can consider this \(T_{c}\) period to be discretized in \(T\) cycles, each having \(\tau\) duration. This leads to the definition of a graph \(\mathcal{G}=(\mathcal{N},\mathcal{E})\), where \(\mathcal{N}\) represents the node set, while \(\mathcal{E}\) is the edge set. In particular, to deal with the dynamical behavior of the topology, nodes in the graph are not simply a representation of a physical node (i.e., a satellite or GS), but of a physical node at a certain time cycle, which only deals with services in a specific processing state (i.e., either unprocessed or processed). Let us clarify this assumption. The physical nodes composing our topology are satellites and GSs. As far as GSs are concerned, since we assume that all GSs are the same and thus there is no preference on which GS images shall reach, we only consider a virtual GS (\(vGS\)) node in our topology, representing any of the GSs enabled to receive data from the constellation. Thus, the satellites and the vGS lead us to a topology consisting of \(N_{S}\) physical nodes. This translates into a graph whose nodes \(n^{i}_{t,p}\in\mathcal{N}\), with \(i\in[0,\ldots,N_{S}-1]\), \(t\in[0,\ldots,T-1]\), and \(p\in\{0,1\}\) representing not only the \(i\)-th physical node but also its functional state with respect to service processing. Specifically, \(p=0\) denotes nodes handling unprocessed data, while \(p=1\) denotes nodes handling already processed data, at time slot \(t\). This modeling abstraction enables us to capture both the temporal evolution of the system and the processing state of services within each node. By splitting each physical node into two state-specific virtual nodes, we avoid the need to explicitly define virtual links for modeling data reduction due to processing. Instead, we define _processingedges_ connecting the two states of the same node, which simplifies the graph structure and facilitates the formulation of optimal resource allocation strategies. This approach supports a more tractable yet expressive representation of the dynamic orbital environment and its impact on service placement decisions. We arbitrarily assume that the \(vGS\) is represented by the \(i\)-th node when \(i=N_{S}-1\). Furthermore, each physical \(i\)-th node is endowed with a maximum energy \(\varepsilon_{i}\) during each time cycle (expressed in J). As far as edges are concerned, \(e^{t}_{\{i,p_{i}\},\{i,p_{d}\}}\in\mathcal{E}\), with \(i_{s}\), \(i_{d}\in[0,\ldots,N_{S}-1]\), \(t\in[0,\ldots,T-1]\), \(p_{s}\), \(p_{d}\in[0,1]\), represents a link between two nodes. Each edge is associated with a capacity \(C^{t}_{\{i,p_{s}\},\{i,p_{d}\}}\) expressed in Mb. In particular, the following link types can be distinguished:  
* If \(i_{s}\
eq i_{d}\), \(p_{s}=p_{d}=p\), the edge represents a transmission link, associated with a capacity representing the maximum amount of information that can be transferred during the actual visibility time between the two physical nodes in the \(t\)-th time cycle as defined in the previous section. After having determined the actual visibility time, the capacity is given by: \[C^{t}_{\{i_{s},p\},\{i_{d},p\}}=R_{i_{s},i_{d}}\cdot\hat{\tau}^{t}_{i_{s},i_{d}}\] (6) Thus, the capacity of the transmission link between two different physical nodes in a certain time cycle represents the maximum amount of data in Mb that can be transferred from the sending node to the receiving one within a time cycle.