## 7 Numerical Results
### Results  
The first analysis we propose is a comparison between the energy saving on GSs obtained by solving the optimization problem and by applying the GSESH algorithm. The results are shown in Figures 5 and 6 for \(\
abla=0.1\%\) (i.e., \(\varepsilon_{Sat}/\tau=1.7\) W, or, equivalently, \(\varepsilon_{Sat}=1020\) J) and \(\
abla=0.2\%\) (i.e., \(\varepsilon_{Sat}/\tau=3.4\) W, or, equivalently, \(\varepsilon_{Sat}=2040\) J), respectively, and \(\alpha\in[0,\ldots,16]\). The available energy values have been chosen to evaluate the behaviour of the optimal solution when a small amount of energy is available on board with respect to the number of services to be processed. In this first analysis, the number of services is small with respect to a real orbital case, but it is high enough to consider a simple, yet not trivial scenario. It can be noticed that the energy-saving curve obtained by applying the proposed heuristic follows the same trend of the optimal one, differingfrom the optimal result in the value by not more than 7.10% when \(\
abla=0.1\%\). However, this difference between the two curves even drops to zero with a small increase in the amount of energy available on board, i.e., when \(\
abla=0.2\%\). It is interesting to note the particular behaviour both the curves follow. First, it can be noticed that there is an interval of \(\alpha\) values where the energy-saving percentage increases, until it reaches a maximum and then it starts decreasing, finally reaching the same value that would be obtained by processing all the data on the ground. This is due to the fact that, by increasing \(\alpha\), we linearly increase the number of tasks which can be processed, but since the on-board CPU clock frequency also increases linearly with \(\alpha\) and the processing energy consumption increases quadratically with the clock frequency, the energy consumption for processing a task increases quadratically with \(\alpha\). For this reason, there is a first interval of \(\alpha\) where the energy consumption on ground stations decreases because the number of services each satellite can process increases. Notice that the slope of the energy-saving curves changes because of the energy margin which limits the number of services that can be handled by satellites. However, after a certain value of \(\alpha\) (for example, in the case shown in Figure 5, after \(\alpha=6\)) even though a satellite has an increased computational capacity, the energy required to process a task is such that the satellite does not have enough energy to process all the \(\alpha\) services it would be able to elaborate; thus, the number of services processed on board decreases because of energy constraints. For instance, in the case shown in Figure 5, we obtain the same performance for \(\alpha=4\) and \(\alpha=7\). In fact, even though when \(\alpha=7\), each satellite has enough computational capacity to process seven services during a time cycle, due to the limited energy available on board, it can only process around four services; thus, we obtain the same performance of a less powerful CPU having \(\alpha=4\), since with \(\alpha=7\) there is a higher power consumption when elaborating every single task, which leads to a higher consumption of the limited amount of energy available on board. Thus, due to constraints on the energy available on board, after a certain value of \(\alpha\), the energy saving in ground stations decreases because a higher number of tasks are processed on ground, until the region of \(\alpha\) values is such that only one full service can be processed in space due to limited on-board energy, where the energy consumption in ground stations reaches a plateau. Finally, after a certain value of \(\alpha\) (in the case of Figure 5, for \(\alpha\geq 15\)), the energy that would be consumed on board to process even a single task is higher than the energy available on board; thus, no service is processed on board and tasks are all offloaded to the ground stations, thus leading to no energy saving on the ground.