## 4 Network and Service Modeling
### Graph-Based Modeling of the Dynamic Topology
Please note that, even though the capacity of a transmission link is generally expressed in Mbps, in this modeling we prefer to indicate it in Mb, since this allows for an elegant formulation of the optimization problem which will be introduced further on. Finally, the use of this link is associated with an energy cost \(\gamma^{i,p}_{i_{s},p}\) (in J/Mb) representing the amount of energy spent in transmitting 1 Mb of data, given by: \[\gamma^{i,p}_{i_{s},p}=\frac{P}{R_{i_{s},i_{d}}}\] (7) In other words, when a generic amount \(x\) (in Mb) of data are transferred from the \(i_{s}\)-th physical node to the \(i_{d}\)-th one, the former spends an amount of energy \(\varepsilon_{tx}\) given by (in J): \[\varepsilon_{tx}=x\cdot\gamma^{i_{t},p_{d}}_{i_{s},p_{s}}\] (8) while we assume that no energy consumption happens on the receiving node. For this reason, since in the considered model \(vGS\) never sends data to the satellites, but it only receives the data, it is straightforward that data transmission will not contribute to energy consumption on ground. * If \(i_{s}=i_{d}=i\), \(p_{s}=p_{d}=p\), the edge represents the storage of the information on a node during the full \(t\)-th time cycle (i.e., a memory link), and it is associated with a capacity: \[C^{t}_{\{i,p\},\{i,p\}}=M_{i}\] (9) representing the memory amount available on the \(i\)-th node; we assume that no energy is needed to store data, i.e., the use of this link is associated with an energy cost: \[\gamma^{i,p}_{i,p}=0\] (10) This holds for any node. Thus, it follows that storage in memory on the ground will not contribute to the energy consumption of ground stations. * If \(i_{s}=i_{d}=i\), \(p_{s}=0\), \(p_{1}=1\), the edge represents the service processing accomplished by the node during the \(t\)-th time cycle (i.e., a processing link), which is associated with a processing capacity defined as the amount of data in Mb which can be processed in a time cycle; its expression is the following:  
\[C_{\{i,0\},\{i,1\}}^{t}=\Gamma_{i}\cdot\tau \tag{11}\]  
since \(\Gamma_{i}\) represents the processing capacity of the node in Mbps. Again, the processing capacity is expressed in Mb instead of Mbps to obtain an elegant formalization of the optimization problem presented further on. The use of this link is associated with a unit energy cost (in J/Mb):  
\[\gamma_{i,0}^{j,1}=k\cdot n_{cyc}\cdot v_{i}^{2} \tag{12}\]  
representing the amount of energy to be spent to process 1 Mb of data, which depends on the processor effective capacitance coefficient (\(k\)), on the number of CPU cycles to process 1 bit (\(n_{cyc}\)), and on the clock frequency of the processor on the \(i\)-th physical node (\(v_{i}\)) [18]. It follows that the amount of energy that a \(i\)-th node spends to process an amount \(x\) in Mb of data is given by (in J):  
\[\varepsilon_{proc}=x\cdot\gamma_{i,0}^{j,1} \tag{13}\]  
Since any node, either satellite or \(vGS\), can process data, this amount of energy contributes to the energy consumption of ground stations. * All combinations of \(i_{s}\), \(i_{d}\), \(p_{s}\), and \(p_{d}\) not mentioned before represent edges not included in the graph, since they have no logical meaning. In the optimization problem formulation, they will be considered to have no associated capacity and no energy cost. An example of the proposed graph representation is reported in Figure 4, where \(N_{S}=3\) physical nodes are considered and the repetition period is organized in \(T=5\) time cycles. The graph is made up of two layers, each associated with the transmission and storage of data related to unprocessed (\(p\)=0) and processed (\(p\)=1) tasks, respectively. Each layer is divided in turn into \(T=5\) regions, each representing a time cycle. In the proposed example, we report the nodes and links involved in the transmission, the storage, and the processing of a task originated by the node \(n_{1}^{0,0}\), i.e., by the physical node 1 during the time cycle 0. The acquired image to be processed is first transmitted to the node \(n_{0}^{0,0}\).