Optimization of Ground Station Energy Saving in LEO Satellite Constellations for Earth Observation Applications ^†

Abstract

Orbital Edge Computing (OEC) capability on board satellites in Earth Observation (EO) constellations would surely enable a more effective usage of bandwidth, since the possibility to process images on board enables extracting and sending only useful information to the ground. However, OEC can also help to reduce the amount of energy required to process EO data on Earth. In fact, even though energy is a valuable resource on satellites, the on-board energy is pre-allocated due to the presence of solar panels and batteries and it is always generated and available, regardless of its actual need and use in time. Instead, energy consumption on the ground is strictly dependent on the demand, and it increases with the increase in EO data to be processed by ground stations. In this work, we first define and solve an optimization problem to jointly allocate resources and place processing within a constellation-wide network to leverage in-orbit processing as much as possible. This aims to reduce the amount of data to be processed on the ground, and thus, to maximize the energy saving in ground stations. Given the NP hardness of the proposed optimization problem, we also propose the Ground Station Energy-Saving Heuristic (GSESH) algorithm to evaluate the energy saving we would obtain in ground stations in a real orbital scenario. After validating the GSESH algorithm by means of a comparison with the results of the optimal solution, we have compared it to a benchmark algorithm in a typical scenario and we have verified that the GSESH algorithm allows for energy saving in the ground station up to 40% higher than the one achieved with the benchmark solution.

1. Introduction

One of the most important limitations Earth Observation (EO) satellites have to face is related to the transmission of a huge amount of information to the ground in a limited amount of time in the moments during which there is a visible Ground Station (GS). In fact, nowadays, satellites typically acquire images when flying over regions of interest, store them into on-board memories, and then downlink all the gathered images to a GS when in an appropriate visibility region. To accomplish the delivery of this high amount of data (e.g., notice that a Sentinel-2 satellite produces 160 Mb of data for each second of flight over a region of interest [1]), it would be then necessary to increase either the downlink data rate or the number of GSs, both being unscalable with the increase of collected data. In particular, the data rate for transmissions from a satellite is constrained by the power available on board, which is in turn constrained by the dimensions and masses of solar panels and batteries. However, a third way to face these limitations can be given by the Orbital Edge Computing (OEC) [2], which allows the leveraging of satellites in EO constellations to process data on board. This approach is being also studied in the context of Mega-LEO constellations [3,4] in Non-Terrestrial Networks (NTNs) [5] or Space–Air–Ground Integrated Networks (SAGINs), which are being designed to support applications dedicated to 5G [6] and 6G [7]. The beneficial impact of such a solution in the EO context is surely related to the fact that not all data transmitted to the ground are actually useful to the application. For example, in a fire detection application [8], we are only interested in communicating the presence of a fire in a forest, without having to make the full image available on the ground to then be elaborated. Thus, an on-board selection of the information to be transmitted thanks to on-board data processing could better exploit the limited bandwidth, since only useful information is transmitted [9,10,11,12], and also contributes to lowering the time needed to deliver information to the ground station, in particular when Inter-Satellite Links (ISLs) are available [13,14,15]. In a complementary manner, an efficient use of on-board processing resources could also enable the transmission of a larger amount of data to the ground under the same bandwidth constraints. In order to obtain these benefits, several strategies have been identified to decide where data will be processed (satellite or GS) [16,17,18]. However, these solutions assume that processing can only be carried out by the satellite acquiring the information, while further advantages can be obtained by considering that idle satellites can be leveraged to elaborate information acquired by another satellite, in case this does not have enough resources to deal with the processing.

A further important benefit of OEC can be found in the savings we can obtain in energy consumption on GSs when data are processed in orbit. In particular, it can be noticed that, once a satellite is in orbit, energy generation always happens (e.g., solar panels always generate power when exposed to solar radiation), and the amount of generated energy is only dependent on the exposure of solar panels to the sun. In other words, we allocate in advance an amount of energy by providing for a certain worst-case energy budget, and the satellite always makes it available, regardless of its actual use, and since it is generally generated through solar panels, it is completely sustainable energy. Satellite-based ground stations play a crucial role in the communication and navigation systems that we rely on daily. These stations receive and transmit signals from satellites orbiting the Earth, enabling a wide range of applications, from weather forecasting and disaster management to global positioning and television broadcasting. As demand for satellite-based services continues to grow, so does the need for GSs. However, the environmental impact and sustainability of these facilities must be carefully considered to ensure a balance between technological advancement and ecological preservation. One of the primary environmental concerns associated with satellite-based ground stations is their energy consumption. GSs require a significant amount of power to operate, particularly for tasks such as signal amplification and data processing. This energy demand can contribute to greenhouse gas emissions, particularly if the electricity used is generated from fossil fuels. To mitigate this impact, many ground station operators are turning to renewable energy sources, such as solar panels and wind turbines, to power their facilities. By harnessing clean energy, these stations can reduce their carbon footprint and contribute to global efforts to combat climate change. The energy consumption of GSs is related to the actual energy request, being correlated to the amount of data to be processed, and that energy is generated on demand. In particular, the higher the processing on ground, the higher the ground station energy consumption, and the more energy has to be produced to satisfy this need. Furthermore, energy available on ground is typically given by mixed resources; thus, it could be only in part renewable, if not completely non-renewable. Even in case that GSs are provided with totally renewable energy, any saving on the use of this energy can be made available on the energy grid.

Starting from these premises, this paper proposes and investigates processing and bandwidth resource allocation strategies, based on the optimization problem and a heuristic, aiming at minimizing the energy consumption on GS by deciding when and where data shall be processed (i.e., in orbit or on ground and, in the former case, on which satellite of the constellation) and, jointly, determining the route the information has to follow. In particular, the contributions stemming from this work are as follows:

• We propose a new modeling of the dynamic topology related to the orbital environment, based on a time-evolving graph where each node represents a satellite dealing with a service either unprocessed or processed at a certain time, simplifying the formalization of the optimization problem.
• We formalize and solve an optimization problem aiming at jointly allocating processing and bandwidth resources to minimize the energy consumption on GS (i.e., to maximize the amount of data processed on board satellites), by taking into account constraints on energy, bandwidth, storage and processing capacity; furthermore, the proposed strategy provides for the leveraging of task offloading, in such a way that if a satellite which acquired an image has not enough resources to process it, the image can be delivered to another satellite which will be then in charge to process it, by taking advantage of the satellite network obtained through ISLs.
• We propose a heuristic mimicking the optimization problem, allowing for obtaining the desired allocation in a real orbital case study, where the complexity of the optimization problem’s solution would not allow us to obtain results in reasonable time, and we validate it by comparing the obtained results to the ones related to the optimization problem’s solution in a simplified, but still representative, scenario.
• By leveraging the proposed heuristic, we compare the energy saving in ground station consumption, with respect to having all images processed on the ground, in case our strategy is applied or a benchmark strategy not providing the possibility of task offloading to other satellites is considered.

The remainder of this paper is structured as follows. Related works are discussed in Section 2; in Section 3, we introduce the reference scenario and the problem statement; Section 4 is devoted to the definition of the network and service model; the optimization problem is formalized in Section 5, while the proposed heuristic is presented in Section 6; Section 7 shows the numerical results aiming at evaluating the proposed strategy to two benchmark solutions, and finally, Section 8 concludes the paper with the main remarks.

2. Related Works

The work proposed in this paper can be classified into several research areas. Since we are exploring possibilities offered by a satellite network, this paper is related to NTNs and their integration with other networks to obtain the SAGINs. However, since we are aiming to endow satellites with on-board processing capabilities, this research also falls into the research field related to the extension of edge computing capability to satellites, and since we are exploring the specific application to EO, it can be also framed in the context of the exploitation of a mega-constellation to process EO data in orbit, with a particular focus on research related to energy aspects in this specific application.

Starting from NTNs and SAGINs, in [5], the authors recognized some concerns associated with the integration of LEO mega-constellations and ground networks, such as latency, jitter, unstable routing, and limited network reachability. To address these challenges, the authors proposed an optimization-based solution with the objective of integrating the two network segments while ensuring minimal latency and guaranteeing stable routing. Ref. [19] also focuses on latency and proposes a solution to ensure real-time communications leveraging mega-constellations by appropriately allocating flows and selecting cloud or satellite relay servers. In [20], a deep reinforcement learning-based strategy for traffic offloading is proposed, to overcome the limits of traditional strategies that deal with traffic offloading in highly dynamic topology and traffic scenarios such as the orbital one. Another application of Artificial Intelligence (AI) in this context is given by the authors of [21], focusing on the leveraging of AI techniques in 6G networks to integrate terrestrial networks and NTNs while improving the energy efficiency of maritime networks. Instead, ref. [11] is dedicated to the evaluation of the performance of federated learning techniques applied to LEO constellations for 6G. In line with these efforts, ref. [22] tackles the routing problem in LEO satellite constellations by modeling it as a multi-agent Partially Observable Markov Decision Process (POMDP), and proposes a distributed Q-learning-based approach that operates with only local information. Their results demonstrate low-latency performance comparable to centralized benchmarks, enhanced traffic support under high load, and reduced reliance on the ground segment. Complementarily, ref. [23] addresses learning in constrained LEO environments by introducing FedAAC, a resource-aware federated learning strategy that adaptively tunes aggregation frequency and model compression. Their approach improves training time, energy consumption, and communication overhead, demonstrating its suitability for Orbital Edge Computing scenarios with intermittent connectivity.

As far as edge computing application to satellite networks is concerned, this technology has the potential to be used for two purposes: delivering services to terrestrial mobile users through satellite connectivity and facilitating Earth Observation (EO) missions. In the former context, ref. [24] proposes a study of the architectures, technologies, and challenges necessary to extend Mobile Edge Computing to SAGINs. Instead, under the application viewpoint, satellite edge computing to support Internet of Things (IoT) has been investigated in [14], focusing on the related architecture and scheduling strategies, while a framework based on an edge computing capability-endowed SAGIN to support Internet of Vehicles has been proposed in [25], whose authors leverage a deep imitation learning strategy to obtain a task offloading and caching strategy aiming at optimizing both the time needed to complete the task and the use of satellite resources. A collateral research area related to this topic focuses on the application of Network Function Virtualization (NFV) in satellite networks. Within this domain, the authors of [26] present a linear programming problem to address service provision and resource allocation challenges in a Low Earth Orbit (LEO) network, with the objective to minimize the utilization of Inter-Satellite Links (ISLs) due to their inherent instability. Building on similar challenges, ref. [27] investigates satellite edge computing (SEC) as a paradigm that integrates Mobile Edge Computing (MEC) with satellite–terrestrial networks, leveraging LEO satellites as edge servers to reduce latency. Specifically, the authors formulate a delay-aware service-chaining problem in which Virtualized Network Functions (VNFs) must be optimally placed on LEO satellites and traffic routes selected to minimize the end-to-end delay. They propose both an approximation algorithm for single-request scenarios and an online algorithm for dynamic environments with uncertain delays, demonstrating improved latency performance through their SEC framework.

Recently, there is an increasing interest in the application of mega-constellations, potentially endowed with on-board processing capability, to support EO. In particular, the $\mathsf{\Phi }$-Sat-1 Mission [9] by the European Space Agency demonstrated possibility of processing data on board of EO satellite to select and downlink useful information only by applying AI techniques, moving the next research frontier to the investigation on leveraging the combination of networks made by a high number of satellites in mega-constellations and the possibility to process data on board. Under this point of view, the authors of [28] introduced a solution to obtain lower EO data delivery latency by taking advantage of mega-constellations, while the authors of [13] improved the timeliness of EO data by leveraging load balancing in EO constellations endowed with in-orbit processing capability. Instead, the authors of [29] presented an optimal strategy to jointly allocate resource and place processing in satellite networks dedicated to EO with the aim of minimizing a total operating cost related to the transmission, processing, and storage of data. Another highly investigated aspect is related to machine learning techniques applied in this orbital scenario. In particular, the authors of [30] give a state-of-the-art presentation on federated techniques in mega-LEO constellations, a topic also investigated in papers [12,31,32] focusing on different strategies to improve the in-orbit training of machine learning to be applied to EO. In a similar direction, ref. [33] proposes a two-timescale optimization framework that combines service deployment and task scheduling in satellite edge computing, using a deep reinforcement learning approach to improve energy efficiency, load balancing, and delay performance while controlling deployment overhead through heuristic techniques. Whereas, in our previous work [34], we propose a communication strategy to enable a fully distributed learning approach for training a deep learning model in orbit, taking advantage of the potential formation of a satellite network through the availability of ISLs within and across orbital planes.

Finally, since on-board data processing requires energy, the authors of [18] propose an algorithm to minimize the energy consumption on satellites while guaranteeing desired latency, while the algorithm proposed by authors of [35] aims at optimizing the use of satellite batteries to extend their operation life. However, this may lead to an increased latency in service processing, at the expense of possible requirements on the Quality of Service in terms of timeliness. Additionally, ref. [36] proposes a hardware-level solution based on Stacked Intelligent Metasurfaces (SIM) to reduce energy consumption on LEO satellites by enabling lightweight, low-power beamforming directly in the electromagnetic domain, thus alleviating the computational load of traditional processing schemes.

To the best of our knowledge, the optimal resource allocation and processing placement in an OEC-endowed EO satellite constellation to save energy on the ground has not been extensively discussed in the literature. In [37], we first focused on the fact that satellites have an amount of extra energy being pre-allocated on board, regardless of their actual use, which can be leveraged to save energy on the ground station, where the energy consumption is instead depending on the actual energy demand. In particular, we preliminary evaluated the energy consumption in a ground station obtained when resources and processing are allocated within a satellite constellation by means of a simple heuristic. This work has been developed in preparation of [38]; in particular, we define and solve an optimization problem, we propose a new heuristic and validate it against the optimal strategy, and finally, we apply it to conduct a wider analysis in a real constellation scenario.

3. Reference Scenario and Problem Statement

In this work, we will focus on the application of OEC to EO missions. In this scenario, we have several satellites arranged in different orbital planes. All satellites belonging to the same constellation—that is, to the same EO mission—are assumed to have identical characteristics in terms of processing resources, energy availability, and storage capacity. By taking advantage of ISLs, it is possible to enable communication between pairs of satellites, even displaced on different orbital planes. Notice that in this way it is possible to obtain satellite networks, enabling the data exchange between satellites and the possibility to offload task processing. Finally, we will consider each satellite to have the ability to process data on board thanks to a processing capacity, to store data in an on-board memory, and to have an amount of energy to be dedicated to OEC applications. Moreover, in this work, we consider EO tasks that can be processed either on board the satellite or at a GS, with no difference in functionality or output.

In EO missions, services are related to the processing of images acquired by satellites. In particular, every time a satellite flies over a region of interest, it acquires an image which can be processed by the acquiring satellite itself, can be kept in its memory, or can be sent to another node (either a satellite or a ground station). In case data are sent to another node, the receiving node can in turn process the received data, store it in its memory, or send it to another node, and so forth. It is important to underline that in EO missions, just as the service source is always a satellite, the destination is always on Earth; specifically, it is any of the ground stations capable of receiving data, without any distinction among them. In addition, this process is completely known a priori due to the deterministic motion of the satellites that lead to a deterministic EO service. A clarification of the described scenario is given by the example shown in Figure 1, where the satellite A flies over a region of interest and acquires images. A can then transfer data to the ground station $G{S}_1$ passing through satellite B, thanks to the ISL A-B, leading to a lower delay than the current state-of-the-art operations providing for the satellite A storing the acquired information in its memory until reaching the point $A^{\prime }$, when it has the possibility of transferring data to ground station $G{S}_2$. It is also important to underline that, in the scenario, we consider in this work any of the nodes A, B, and $G{S}_1$ can process the data. This is particularly important in case, for example, the bandwidth on link B-$G{S}_1$ is enough to transmit processed data, but not to carry unprocessed data. In this case, by processing on either A or B, the link B-$G{S}_1$ would be able to carry data and, thus, would again allow us to obtain a reduction in the delay in information delivery with respect to the state-of-the-art operations.

From the described scenario, it follows that by designing appropriate strategies to decide how each satellite has to deal with each image at every moment (i.e., to determine whether it shall process it, store it, or transmit it), it would be possible to optimize desired metrics. All of the solutions proposed in the literature aim to minimize the energy consumed in the satellite for on-board processing, but this choice may lead to a waste of the photovoltaic energy produced. Indeed, once the batteries are fully charged, the excess energy produced by the photovoltaic panels is dissipated or otherwise not used. In contrast, the GS operates on an on-demand energy usage model, which depends on the number of tasks being processed. For this reason, the objective of the paper is to propose a resource allocation strategy that exploits all of the energy produced in the satellite. The resulting advantage is the possibility of saving energy in the ground station due to the reduction in the number of tasks processed. We highlight the advantages in the example reported in Figure 2. In particular, in both cases (a) and (b), we have a satellite first acquiring an image of a predefined region, then moving on its orbit until reaching a position B where the satellite is sunlit and the battery can be charged by means of solar panels, and finally reaching a position C where the acquired information can be downlinked to the ground. In case (a), no processing occurs in orbit. In particular, the satellite acquires the image, and when it reaches position B it has half its battery charge. However, from B to C, the satellite is sunlit, and during the movement from B to C, the battery gets fully charged. Finally, when the satellite reaches position C, the acquired image is downlinked and processed by the ground station. Instead, in case (b), the satellite processes the acquired image when it reaches position B. For this reason, the on-board battery has a smaller charge with respect to position B in the former case. However, in the path from B to C, the satellite receives enough solar radiation to charge the battery even though the initial energy is lower. Thus, battery is again fully charged when the satellite sends the already processed image to the GS in position C, and at the same time we saved energy on the ground. From this example, it follows that, even though on-board energy is a limited and valuable resource, it is renewed whenever the satellite is sunlit, and for this reason it is not important to minimize its usage, but to optimize it in such a way that by leveraging, when possible, the energy available on satellites, we can reduce the consumption on the ground. Obviously, the usage of extra energy available on board the satellites may lead to an increased number of battery charge–discharge cycles, and this may reduce the operative life of each satellite. However, it is difficult to quantify the cost of this operative life reduction and to compare it to the savings resulting from a smaller use of on-ground energy, since it is strictly related to the specific mission. Furthermore, a more complex problem which takes into account this cost can be also proposed, but it is beyond the scope of this work, since we want to first study the energy savings which can be obtained by leveraging OEC and satellite networks, and to provide some insight on how processing capacity available on board the satellites has to be chosen to maximize the energy savings on the ground. Finally, in terms of operating cost, it has already been discussed in the literature that, following appropriate resource allocation and processing placement strategies, the total operating cost when data are processed on board is smaller than when processing happens on ground, since a reduced amount of information has to be transferred [29].

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. Satellite communication infrastructure model parameters.

Set or Parameter	Description
$N_{sat}$	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 $[0,\dots ,N_{sat}-1]$ representing a satellite
$p_i$	orbital plane occupied by the i-th satellite, with $i\in [0,\dots ,N_{sat}-1]$
${\mathbf{r}}_i\left(t\right)$	position vector of the i-th satellite, with $i\in [0,\dots ,N_{sat}-1]$
$h_p$	altitude of satellites on the p-th orbital plane, with $p\in [0,\dots ,N_{op}-1]$
$T_p$	period of motion of satellites on the p-th orbital plane, with $p\in [0,\dots ,N_{op}-1]$
$R_E$	Earth’s radius
${\mu }_E$	Earth’s gravitational constant
G	antenna gain
c	speed of light
${\nu }_{tx}$	carrier frequency
P	transmission power
$R_{i_s,j_d}$	transmission data rate between i-th and j-th satellite, with $i,j\in [0,\dots ,N_{sat}-1],i\ne j$
B	transmission bandwidth
$k_B$	Boltzmann’s constant
$T_s$	system noise temperature
$N_{GS}$	number of ground stations
g	index in $[0,\dots ,N_{GS}]$ representing a ground station
${\mathbf{r}}_{G{S}_g}\left(t\right)$	position vector of the g-th ground station, with $g\in [0,\dots ,N_{GS}-1]$
$T_{GS}$	ground station rotation period, i.e., Earth’s sidereal day
T	repeat cycle time
$E{l}_{min}$	minimum elevation angle

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. Network and service model sets and parameter.

Set or Parameter	Description
$\mathcal{G}=\left(\mathcal{N},\mathcal{E}\right)$	graph made by node set $\mathcal{N}$ and edge set $\mathcal{E}$
$T_c$	cyclostationary period for both topology and service generation
$\tau$	discrete time-cycle duration
T	number of time cycle of duration $\tau$ in a cyclostationary period
$N_S$	number of physical nodes (either satellites or $vGS$)
i	index in $[0,\dots ,N_S-1]$ representing the physical node
t	index in $[0,\dots ,T-1]$ representing the time cycle
p	index in $[0,1]$ representing the processing state (0 standing for unprocessed, 1 processed)
$n_i^{t,p}$	node in the set $\mathcal{N}$
$e_{\{i_s,p_s\},\{i_d,p_d\}}^t$	edge in $\mathcal{E}$ between nodes $n_{i_s}^{t,p_s}$ and $n_{i_s}^{t,p_s}$
${\mathsf{\Gamma }}_i$	processing capacity associated to the i-th physical node (in Mbps)
${\gamma }_{i_s,p_s}^{i_d,p_d}$	energy cost to be paid for a Mb of data crossing $e_{\{i_s,p_s\},\{i_d,p_d\}}^t$ edge (in J/Mb)
$vGS$	virtual ground station represented by the $i=N_S-1$ physical node
$C_{\{i_s,p_s\},\{i_d,p_d\}}^t$	capacity associated to the $e_{\{i_s,p_s\},\{i_d,p_d\}}^t$ edge
$M_i$	data storage capacity associated to the i-th physical node (in Mb)
${\epsilon }_i$	energy available at the beginning of each time cycle on the i-th physical node (in J)
${\widehat{\tau }}_{i_s,i_d}^t$	actual visibility time between $i_s$-th and $i_d$-th physical nodes during t-th time cycle
k	processor effective capacitance coefficient
$n_{cyc}$	number of CPU cycles to process 1 bit
${\nu }_i$	clock frequency of the processor on the i-th physical node
$\mathsf{\Sigma }$	set of all generated services
$f^h$	service in $\mathsf{\Sigma }$, with $h\in \left[0,\dots ,N_T-1\right]$
$N_T$	total number of generated services
$f_s^h$	$f^h$ service source satellite index
$f_0^h$	$f^h$ service pre-processing size
$f_1^h$	$f^h$ service post-processing size
$f_t^h$	$f^h$ service generation time cycle
$f_d^h$	number of time cycles after generation within which $f^h$ service shall be delivered to the $vGS$

. 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.

4.1. 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 ${\mathsf{\Omega }}_p$, for each $p\in \{0,\dots ,N_{op}-1\}$. The i-th satellite (with $i\in \{0,\dots ,N_{Sat}-1\}$) is assigned to the orbital plane $p_i=\lfloor i·N_{op}/N_{Sat}\rfloor$, and its spatial coordinates at time t are denoted by ${\mathbf{r}}_i\left(t\right)$ 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:

$$T_p=2\pi \sqrt{{\displaystyle \frac{{(R_E+h_p)}^3}{{\mu }_E}}},$$

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:

$$d(i,j,t)=∥{\mathbf{r}}_i\left(t\right)-{\mathbf{r}}_j\left(t\right)∥.$$

(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:

$$d(i,j,t)\le d_{\max }(i,j),$$

(2)

where $d_{\max }(i,j)$ denotes the maximum communication range under ideal channel conditions. Assuming an AWGN channel, it can be approximated by:

$$d_{\max }(i,j)={\displaystyle \frac{Gc}{4\pi {\nu }_{tx}}}\sqrt{{\displaystyle \frac{P}{\left(2^{R_{i_s,j_d}/B}-1\right)k_B{T}_{s}B}}},$$

(3)

with G the antenna gain, c the speed of light, ${\nu }_{tx}$ the carrier frequency, P the transmit power, $R_{i,j}$ 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 via geometric analysis rather than by evaluating (2) at runtime. The following conditions must be satisfied for stable connectivity between adjacent satellites:

$$\begin{array}{cc}\hfill & 2(R_E+h_p)sin\left({\displaystyle \frac{2\pi N_{op}}{N_{Sat}}}\right)\le d_{\max }(i,j)\hfill \\ \hfill & \mathrm{and}(R_E+h_p)cos\left({\displaystyle \frac{2\pi N_{op}}{N_{Sat}}}\right)>R_E,\hfill \end{array}$$

(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 ${\mathbf{r}}_{G{S}_g}\left(t\right)$ due to Earth’s rotation. Their motion follows a sidereal day period of $T_{GS}=86164$ s. As a result, the entire system is periodic with a global period $T=\mathrm{lcm}(T_{GS},T_0,\dots ,T_{N_{op}-1})$.

A communication link between satellite i and GS g is feasible if the elevation angle exceeds a minimum threshold $E{l}_{\min }$, defined via:

$$\begin{array}{c}\hfill {\displaystyle \frac{\pi }{2}}-arccos\left({\displaystyle \frac{{\mathbf{r}}_{G{S}_g}\left(t\right)·{\mathbf{r}}_i\left(t\right)}{\parallel {\mathbf{r}}_{G{S}_g}\left(t\right)\parallel \parallel {\mathbf{r}}_i\left(t\right)\parallel }}\right)\ge E{l}_{\min ,}\end{array}$$

(5)

where · 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.

4.2. 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 $\mathcal{G}=\left(\mathcal{N},\mathcal{E}\right)$, 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 $n_{t,p}^i\in \mathcal{N}$, with $i\in [0,\dots ,N_S-1]$, $t\in [0,\dots ,T-1]$, 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 processing edges 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 ${\epsilon }_i$ during each time cycle (expressed in J).

As far as edges are concerned, $e_{\{i_s,p_s\},\{i_d,p_d\}}^t\in \mathcal{E}$, with $i_s,i_d\in [0,\dots ,N_S-1],$$t\in [0,\dots ,T-1],p_s,p_d\in [0,1]$, represents a link between two nodes. Each edge is associated with a capacity $C_{\{i_s,p_s\},\{i_d,p_d\}}^t$ expressed in Mb. In particular, the following link types can be distinguished:

• If $i_s\ne i_d$, $p_s=p_d=p$, 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:

  $$C_{\{i_s,p\},\{i_d,p\}}^t=R_{i_s,i_d}·{\widehat{\tau }}_{i_s,i_d}^t$$

  (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. 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 ${\gamma }_{i_s,p}^{i_d,p}$ (in J/Mb) representing the amount of energy spent in transmitting 1 Mb of data, given by:

  $${\gamma }_{i_s,p}^{i_d,p}={\displaystyle \frac{P}{R_{i_s,i_d}}}$$

  (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 ${\epsilon }_{tx}$ given by (in J):

  $${\epsilon }_{tx}=x·{\gamma }_{i_s,p_s}^{i_d,p_d}$$

  (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:

  $$C_{\{i,p\},\{i,p\}}^t=M_i$$

  (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:

  $$C_{\{i,0\},\{i,1\}}^t={\mathsf{\Gamma }}_i·\tau$$

  (11)

  since ${\mathsf{\Gamma }}_i$ 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):

  $${\gamma }_{i,0}^{i,1}=k·n_{cyc}·{\nu }_i^2$$

  (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 (${\nu }_i$) [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):

  $${\epsilon }_{proc}=x·{\gamma }_{i,0}^{i,1}$$

  (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 $n_0^{0,0}$. Then, physical node 0 stores data in its memory until time cycle 2, when it processes the information, thus, a layer change occurs. Finally, the processed image is kept in memory until the last time cycle, when data are transmitted to the node $n_2^{4,1}$, associated to the vGS. All the links previously described are shown in the given example. In particular:

• Transmission links are inserted between couple of nodes in a region of a same layer, in case their distance is short enough to allow for communication; we report the two transmission links $e_{\{1,0\},\{0,0\}}^0$ and $e_{\{0,1\},\{2,1\}}^4$ in the example of Figure 4;
• Memory links are inserted between nodes related to the same physical node in two different regions of a same layer, representing the data storage during the entire time cycle associated to the originating region; for the sake of clarity, even though this link type is always present in case a node has enough memory to store at least a task, we only report the four involved storing links in the example of Figure 4;
• Processing links are inserted between the same physical node belonging to the same region on two different layers, representing the data processing; again, even though this link type is always present in case a node has enough processing capacity to elaborate at least a task in a time cycle, for the sake of clarity we only report processing link $e_{\{0,0\},\{0,1\}}^2$ in the example of Figure 4.

To summarize, the proposed modeling provides for each satellite to consume energy when it transmits or processes data. Instead, the only contribution to the energy consumption on ground is given by the processing that takes place on ground stations, i.e., by the energy spent to cross processing links associated to the $vGS$ in the proposed model.

4.3. Services

As far as service generation is concerned, each satellite takes images of the Earth while flying over determined regions, and we suppose, without loss of generality, that each image requires a single processing task to extract the information useful to the application. For this reason, from now on, the terms task and service will be used as synonyms. Image processing also reduces the data size. We represent the set of the $N_T$ services generated in a cyclostationary period with:

$$\mathsf{\Sigma }={\left\{f^h\right\}}_{h\in [0,\dots ,N_T-1]}$$

(14)

where each service $f^h$ is represented by the tuple:

$$f^h=\{f_s^h;f_0^h;f_1^h;f_t^h;f_d^h\}$$

(15)

including the source node, the pre-processing size in Mb, the post-processing size in Mb, the service generation time cycle and the maximum number of time cycles within which the service shall reach the ground station, respectively. Finally, we assume that all images need the same processing task, which requires $n_{cyc}$ CPU cycles for each bit to be processed.

5. Optimization Problem

Following the approach proposed in the literature, we will evaluate the proposed solutions by means of fluidic instead of packet-based models. For this reason, we assume that the bytes of an unprocessed or processed image are routed through the same network path. The network and service modeling presented in Section 4 leads to a straightforward formalization of the optimal resource allocation problem to minimize the energy consumption on the ground due to data processing, under energy, bandwidth, storage, and processing capacity constraints. In particular, since each node in the graph on which the problem is defined represents a physical node in a certain time, the proposed formulation allows one to consider the dynamic environment. Furthermore, since in our modeling we divided nodes dealing with unprocessed and processed tasks, it is possible to obtain an Integer Linear Programming (ILP) formulation able to take into account the data reduction because of processing without having to introduce virtual links and additional constraints, which would increase the problem complexity.

The optimization problem is characterized by the variables $y_{\{i_s,p_s\},\{i_d,p_d\}}^{t,h}$ ($t\in [0,\dots ,T-1$, $h\in [0,\dots ,N_T-1$, $i_s\in [0,\dots ,N_S-1]$, $i_d\in [0,\dots ,N_S-1]$, $p_s,p_d\in \{0,1\}$) as defined in (16). The variables are defined for each task and characterize which links of the graph are busy to transmit/process/store the task from the satellite acquiring the image to the virtual ground station. In particular for the h-th task, the variable $y_{\{i_s,p_s\},\{i_d,p_d\}}^{t,h}$ assumes a value of one if the link $e_{\{i_s,p_s\},\{i_d,p_d\}}^t$ is used to transmit/process/store the bytes of the h-th task, otherwise its value is zero.

$$y_{\{i_s,p_s\},\{i_d,p_d\}}^{t,h}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}h\text{-}\mathrm{th}\mathrm{task}\mathrm{occupies}\mathrm{the}\mathrm{link}\mathrm{from}\mathrm{node}(i_s,p_s)\mathrm{to}\mathrm{node}(i_d,p_d)\hfill \\ \hfill & \mathrm{during}t\text{-}\mathrm{th}\mathrm{time}\mathrm{cycle}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(16)

It can be noticed that the joint routing and processing placement problem can be defined as finding the path for each service globally optimizing the desired metric on the graph previously introduced, since in that formulation links can be related to either transmission/storage or processing. In particular, recalling that the only contribution to energy consumption on the ground is given by data processing, we can express the ground station energy consumption as follows:

$${\epsilon }_{GS}^{tot}=\sum _{h=0}^{N_T-1}\sum _{t=0}^{T-1}{y}_{\{N_S-1,0\},\{N_S-1,1\}}^{t,h}{f}_0^h{\gamma }_{N_S-1,0}^{N_S-1,1}$$

(17)

For this reason, we can introduce the objective function as follows, since maximizing the energy saving on ground stations is equivalent to minimize the energy consumption on the ground:

$$min{\epsilon }_{GS}^{tot}$$

(18)

The proposed graph also simplifies the formalization of the constraints, which can be written as reported in (20)–(24). In their definition, we leveraged the Kronecker’s delta notation with the following meaning:

$${\delta }_{x,y}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}x=y\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(19)

$$\begin{array}{c}\hfill \sum _{h=0}^{N_T-1}\left[(1-{\delta }_{p_s,p_d})y_{\{i_s,p_s\},\{i_d,p_d\}}^{t,h}{f}_{p_s}^h+{\delta }_{p_s,p_d}\sum _{p=0}^1{y}_{\{i_s,p\},\{i_d,p\}}^{t,h}{f}_{p_s}^h\right]\le C_{\{i_s,p_s\},\{i_d,p_d\}}^t,\\ \hfill \forall i_s,i_d\in [0,\dots ,N_S-1],\forall p_s,p_d\in \{0,1\},\forall t\in [0,\dots ,T-1]\end{array}$$

(20)

$$\begin{array}{c}\hfill \sum _{h=0}^{N_T-1}\sum _{i_d=0}^{N_S-1}\sum _{p_s=0}^1\sum _{p_d=0}^1{y}_{\{i_s,p_s\},\{i_d,p_d\}}^{t,h}{f}_{p_s}^h{\gamma }_{i_s,p_s}^{i_d,p_d}\le {\epsilon }_{i_s}^t,\\ \hfill \forall i_s\in [0,\dots ,N_S-1],\forall t\in [0,\dots ,T-1]\end{array}$$

(21)

$$\begin{array}{c}\hfill (1-{\delta }_{f_t^h,f_d^h})·y_{\{f_s^h,0\},\{f_s^h,0\}}^{f_t^h,h}+y_{\{f_s^h,0\},\{f_s^h,1\}}^{f_t^h,h}+\sum _{\begin{array}{c}{i}_d=0\\ i_d\ne f_s^h\end{array}}^{N_S-1}{y}_{\{f_s^h,0\},\{i_d,0\}}^{f_t^h,h}=1,\\ \hfill \forall h\in [0,\dots ,N_T-1]\end{array}$$

(22)

$$\begin{array}{c}\hfill (1-{\delta }_{i,f_s^h}{\delta }_{t,f_t^h}{\delta }_{p,0}-{\delta }_{i,N_S-1}{\delta }_{t,f_t^h+f_d^h}{\delta }_{p,1})·\\ \hfill \{\left[(1-{\delta }_{t,f_t^h})·y_{\{i,p\},\{i,p\}}^{(t-1)modT,h}+{\delta }_{p,1}·y_{\{i,0\},\{i,1\}}^{tmodT,h}+\sum _{\begin{array}{c}a=0\\ a\ne i\end{array}}^{N_S-1}{y}_{\{a,p\},\{i,p\}}^{tmodT,h}\right]-\\ \hfill \left[(1-{\delta }_{t,f_t^h+f_d^h})·y_{\{i,p\},\{i,p\}}^{tmodT,h}+{\delta }_{p,0}·y_{\{i,0\},\{i,1\}}^{tmodT,h}+\sum _{\begin{array}{c}b=0\\ b\ne i\end{array}}^{N_S-1}{y}_{\{i,p\},\{b,p\}}^{tmodT,h}\right]\}=0\\ \hfill \forall i\in [0,\dots ,N_S-1],\forall h\in [0,\dots ,N_T-1],\\ \hfill \forall t\in [f_t,\dots ,f_t^h+f_d^h],\forall p\in \{0,1\}\end{array}$$

(23)

$$\begin{array}{c}\hfill (1-{\delta }_{f_d^h,0})·y_{\{N_S-1,1\},\{N_S-1,1\}}^{(f_t^h+f_d^h-1)modT,h}+y_{\{N_S-1,0\},\{N_S-1,1\}}^{(f_t^h+f_d^h)modT,h}+\sum _{i_s=0}^{N_S-2}{y}_{\{i_s,1\},\{N_S-1,1\}}^{(f_t^h+f_d^h)modT,h}=1,\\ \hfill \forall h\in [0,\dots ,N_T-1]\end{array}$$

(24)

In particular, constraint (20) ensures that the resource allocation does not exceed the capacity of each link (notice that, thanks to the proposed modeling, this expression is valid for all transmission, storage, and processing links). Constraint (21) limits the energy consumption during each time cycle in each node as the maximum energy amount available on the node at the beginning of the time cycle. Constraints (22)–(24) are related to flow conservation. Specifically, following the characteristics of the considered EO applications, constraint (22) sets the source node for each service as the one related to the satellite which acquired the image, dealing with unprocessed data, at the service generation cycle, and ensures that node has only an outgoing flow for each service. In particular, constraint (24) sets the destination node for each service as the $vGS$ dealing with the processed data in the service delivery deadline time cycle, and ensures that node has only an ingoing flow for each service. Please note that constraint (24) also ensures that all services are processed and does not prevent a service to reach the ground station before the service delivery deadline. Finally, constraint (23) simply imposes flow conservation on intermediate nodes on the path each service crosses from the source to the destination.

It could be interesting to underline further details on how the specific application of OEC to EO missions impacts the definition of the presented optimization problem. In particular, since both topology and service generation are cyclostationary, we just have to define and solve the problem in a cyclostationary period, since what happens in that period repeats all mission long. For this reason, constraints (20) and (21) shall be written only for the time cycles composing a cyclostationary period, while in constraints (22)–(24), since a service delivery deadline cycle could exceed the end of the current cyclostationary period, we leverage modulo operation to bring back all that would happen after the end of the cyclostationary period to its beginning, since cyclostationary periods always repeat the same one after another.

Finally, we discuss the complexity of the proposed formulation. In particular, it can be noticed that it is analogous to the decision multi-commodity flow problem, to which an additional constraint related to energy is added. For this reason, since the decision of the multi-commodity flow problem is NP hard [39], the same applies to our problem.

6. Heuristic

Since the presented optimal strategy is NP hard, in order to evaluate the benefits of the proposed allocation strategy in a complex scenario like a real orbital case, it is necessary to introduce a heuristic mimicking the optimization problem solution. We propose a heuristic leveraging the Dijkstra’s algorithm, as detailed in Algorithm 1. This algorithm applies on a new graph $\tilde{\mathcal{G}}=\left(\tilde{\mathcal{N}},\tilde{\mathcal{E}}\right)$, being a modified version of the graph presented in Section 4, where instead of distinguishing nodes dealing with only two processing states, we also consider a third processing state, the “in-processing” state. This is due to the fact that the introduction of the “in-processing” layer allows us to take into account that a node processing a task can transmit it in the same time cycle, during which processing happens only if the energy remaining after processing is enough to accomplish the transmission of a processed task. In fact, while this condition was verified by means of appropriate constraints in the optimization problem definition, in the case of the proposed shortest-path-based heuristic, we have to input the algorithm with a graph containing only admissible paths. For this reason, we need to update the values assumed by the parameter p defined in Section 5. While defining the heuristic, we consider $p\in \{0,1,2\}$, where $p=0$ represents a node dealing with unprocessed tasks, $p=1$ a node dealing with in-processing tasks, and $p=2$ a node handling processed tasks. In particular, while transmission and storage links remain the same as defined in Section 4, processing links are distinguished as follows:

• If $i_s=i_d=i$, $p_s=0$, $p_1=1$, the edge represents the service processing accomplished by the i-th physical node during the t-th time cycle (i.e., a processing link), with associated capacity and energy cost as discussed in the processing link definition given in Section 4;
• If $i_s=i_d=i$, $p_s=1$, $p_1=2$, the edge represents a fictitious link to bridge nodes in the processing state to nodes in the processed state, associated with an infinite capacity and no energy cost.

Algorithm 1: Ground Station Energy-Saving Heuristic (GSESH)

Let us detail the main steps of the Ground Station Energy-Saving Heuristic (GSESH) presented in Algorithm 1. After having initialized the matrices tracking the link occupancy and node energy usage in time (Line 1), we apply the following steps for each service to be allocated. We first extract a subgraph containing only nodes related to time cycles between the service generation and delivery deadline cycles and only edges with enough capacity to host the unprocessed service and related to nodes with enough energy to accomplish its transmission and/or processing (Line 3). Subgraph extraction is obtained by applying Algorithm 2, which will be discussed in detail further on. We then apply Dijkstra’s algorithm on the extracted subgraph, in order to determine the shortest path from the node associated to the source satellite in the unprocessed state during the service generation cycle to the node representing the ground station in the processed state at service delivery deadline cycle (Line 5). While applying Dijkstra’s algorithm, distances are related to energy consumption due to crossing edges between couples of nodes, but since we are interested in minimizing the energy consumption in ground stations, while calculating path distances we only consider energy to be paid to process data on ground stations, and we zero any contribution given by crossing a link different from a processing link of the $vGS$ node. Then, if a path has been found, the path is added to the list of paths (Line 7); thus, in Lines 8–11, first the matrix tracking the link occupancy is updated to consider the hosting of the current service on the crossed links by means of the following expression:

$${\mathsf{\Lambda }}_{\{{\widehat{i}}_s,{\widehat{p}}_s\},\{{\widehat{i}}_d,{\widehat{p}}_d\}}^{\widehat{t}}={\mathsf{\Lambda }}_{\{{\widehat{i}}_s,{\widehat{p}}_s\},\{{\widehat{i}}_d,{\widehat{p}}_d\}}^{\widehat{t}}+f_{\lceil {\widehat{p}}_s/2\rceil }^h$$

(25)

second, similarly, the matrix tracking the node energy usage in time is updated to consider the hosting of the current service on nodes crossed by the service with the following expression:

$${\mathsf{\Psi }}_{{\widehat{i}}_s}^{\widehat{t}}={\mathsf{\Psi }}_{{\widehat{i}}_s}^{\widehat{t}}+{\gamma }_{{\widehat{i}}_s,{\widehat{p}}_s}^{{\widehat{i}}_d,{\widehat{p}}_d}·f_{\lceil p_s/2\rceil }^h$$

(26)

and, finally, the current service is removed from the buffer tracking the services still to be handled by the algorithm (Lines 12–14); otherwise, the current service is included into the set of rejected services, i.e., services which cannot be handled because of lack of resources (Lines 15–17). As far as the Extract Subgraph algorithm illustrated in Algorithm 2 is concerned, it first allows for extracting from the graph only the nodes which can potentially be traversed by the current service, i.e., all nodes between the service generation time cycle and the service delivery deadline one (Lines 8–9). Furthermore, this algorithm also allows for the following Dijkstra’s algorithm application dealing with the fact that links have a finite capacity and nodes have a finite amount of energy. In particular, in Lines 7–15, the first condition to be verified to include an edge between a couple of nodes in the subgraph is that the edge has enough capacity to handle the current service, by checking the following condition:

$${\mathsf{\Lambda }}_{\{i_s,p\},\{i_d,p\}}^{\widehat{t}}+f_{\lceil p/2\rceil }^h\le C_{\{i_s,p\},\{i_d,p\}}^{\widehat{t}}$$

(27)

Then, a transmission edge between two nodes in the unprocessed (processed) state is included in the subgraph only if the transmitting node has enough energy to transmit the unprocessed (processed) service plus a margin. This is checked by means of the following expression:

$${\mathsf{Ϋ}}_{i_s}^{\widehat{t}}+f_{\lceil p/2\rceil }^h·{\gamma }_{i_s,p}^{i_d,p}+{\mu }_{i_s}^{\widehat{t}}·\left(1-{\delta }_{i_d,N_S-1}\right)\le {\epsilon }_{i_s}$$

(28)

Please notice that, in the case of memory links, since we assume no energy is required to store data, this energy condition is automatically verified. Instead, a processing edge is included in the subgraph only if the node has enough energy to process the service plus a margin, and we assume that processing on a node can happen only in the first cycle during which it has enough capacity and energy to accomplish it (Lines 16–18). In particular, the availability of processing capacity is checked by means of the following expression:

$${\mathsf{\Lambda }}_{\{i_s,0\},\{i_s,1\}}^{\widehat{t}}+f_0^h\le C_{\{i_s,0\},\{i_s,1\}}^{\widehat{t}}$$

(29)

while if the following expression is verified, there is enough energy to accomplish service processing:

$${\mathsf{Ϋ}}_{i_s}^{\widehat{t}}+f_0^h·{\gamma }_{i_s,0}^{i_s,1}+{\mu }_{i_s}^{\widehat{t}}\le {\epsilon }_{i_s}$$

(30)

Finally, a fictitious edge from a node in the processing state to the processed one is included whenever the node in the processing state does not have an incoming processing edge (i.e., it is reachable only through a storage edge in the processing state), or, in case there is an incoming processing edge, when the node has enough energy to process the service and then transmit it in its processed form (i.e., with a reduced size), plus a margin (Lines 19–21), with this condition being checked by means of the following expression:

$${\mathsf{Ϋ}}_{i_s}^{\widehat{t}}+f_0^h·{\gamma }_{i_s,0}^{i_s,1}+f_1^h·\left(\underset{i_d\in [0,N_S-1]}{max}{\gamma }_{i_s,2}^{i_d,2}\right)+{\mu }_{i_s}^{\widehat{t}}\le {\epsilon }_{i_s}$$

(31)

Note that the introduction of the in-processing layer and fictitious links with respect to the model introduced in Section 4 is necessary to make the algorithm consider that, if processing happens on a node in a certain time cycle, the node may run out of energy in that time cycle; thus, it cannot transmit the service after having processed it. Instead, the introduction of a margin in the calculation of energy availability is necessary because the proposed heuristic is greedy: it tries to optimize the on-ground energy consumption for the current task without taking into account that the current resource allocation has an impact on the resources available for the other tasks still to be handled, leading to a non-optimal solution or even to the impossibility to allocate resources for all the services. For this reason, while selecting a link in the subgraph, we add it only if the energy the current service requires to cross it is small enough to allow the satellite to transmit all its remaining tasks to the ground station. In particular, the margin calculation is strictly related to the buffer tracking the services still to be handled by the algorithm in a certain time period and having a specific satellite as source. This buffer can be imagined as a list, whose entry ${\beta }_{i_s}^{\widehat{t}}\in \beta$, with $i_s\in [0,\dots ,N_S-2],\widehat{t}\in [0,\dots ,T-1]$ is a set containing services $f^h\in \mathsf{\Sigma }$, with $h\in [0,\dots ,N_T-1]$ generated by the $i_s$-th satellite and such that $f_t^h\le \widehat{t}\le f_t^h+f_d^h$. Consequently, the energy margin to be left on the $i_s$-th satellite during the $\widehat{t}$-th time cycle is given by the sum of the pre-processing size and post-processing size of all services $\phi \in {\beta }_{i_s}^{\widehat{t}}$, multiplied by the maximum unit transmission energy cost, following the expression:

$${\mu }_{i_s}^{\widehat{t}}=\sum _{\phi \in {\beta }_{i_s}^t}({\phi }_0+{\phi }_1)·\underset{i_d\in [0,N_S-1]}{max}{\gamma }_{i_s,2}^{i_d,2}$$

(32)

It is important to underline that the proposed margin calculation is empirical, and its optimization could open interesting research prospective which are out of the scope of this work.

Algorithm 2: Extract Subgraph

Finally, the complexity of Algorithm 2 can be easily computed by considering that there are three nested loops repeating at most for $N_S-2$, T, and 2, respectively, leading to a complexity given by $\mathcal{O}\left(N_S^{2}T\right)$. Furthermore, we recall that the most efficient complexity of Dijkstra’s algorithm is given by $\mathcal{O}\left(\left|\mathcal{N}\right|{log}_2\left(\left|\mathcal{N}\right|\right)+\left|\mathcal{E}\right|\right)$. In our scenario, the number of nodes is given by $3N_{S}T$, while in calculating the maximum number of edges, we have to take into account that the maximum number of transmission links is reached when each couple of satellites can communicate during each time cycle (i.e., there are at most $N_{S}T(N_S-1)/2\sim N_S^{2}T$ links); the maximum number of memory links is given by $N_S(T-1)$, the maximum number of processing links is given by $N_S$, while the maximum number of fictitious links is given by $N_{S}T$. It follows that $\left|\mathcal{E}\right|\sim N_S^{2}T$. Finally, the two loops in Lines 8–11 repeat for the maximum path length determined by Dijkstra’s algorithm and the maximum deadline, which can be assumed to be equal to T. However, these are lower-order contributions. For this reason, the complexity of Algorithm 1 can be easily calculated to be $\mathcal{O}\left(N_T{N}_{S}T\left(N_S+{log}_2\left(N_{S}T\right)\right)\right)$, since for $N_T$ times we apply Algorithm 2 followed by a Dijkstra’s algorithm execution.

7. Numerical Results

In this section, we numerically evaluate how a strategy to jointly place processing and route information within a satellite constellation dedicated to EO can lower the energy consumption on GS due to data processing. The proposed performance investigation is related to the evaluation of the ground station energy saving, defined as the saved energy consumption percentage of the proposed solution with respect to the case in which all of the tasks are processed in the GS. The amount of services processed in orbit depends on the resource allocation and processing placement strategy applied. In particular, we first evaluate the energy saving obtained by means of an optimal strategy, and we compare it to the one obtained by leveraging the GSESH algorithm. In fact, it would be impractical to solve the optimization problem in a real orbital scenario, thus, we first validate the GSESH algorithm with respect to the optimal solution in a simpler, but not trivial, scenario, in such a way that we can then leverage the heuristic to evaluate the effectiveness of the proposed solution in a real orbital case. In this latter case study, the performance is compared to two benchmarks, the first stipulating that processing always happens on ground (Always Ground solution, AG), the second stipulating that processing can happen either on the source satellite or on ground (Always First or Ground solution, FoG) [18]. In particular, in the FoG solution, all satellites have processing capacity on board, but they can leverage it only to process images they acquired when flying over a region of interest. In other words, satellites cannot offload processing of images they acquired by another satellite, and if they do not have enough resources to process data, they shall offload it to the GS. In our solution, instead, service processing can be carried out by any satellite of the constellation, even different from the one originating it, or by a GS (i.e., if a satellite does not have enough resources to accomplish the task processing, it can offload it to another satellite with more resources, or, in the worst case, to the GS).

The following parameters will be the same for all the analysis. The time-cycle duration will be set to $\tau =10$ min, as a compromise value between the granularity of the dynamic topology representation and the complexity of running the algorithm. In concordance with what happens on Sentinel-2, we consider all services to have a pre-processing size [1] equal to

$$f_0^h=\overline{f_0}=160\mathrm{Mb},\forall h\in [0,\dots ,N_T-1]$$

(33)

Although the proposed model for EO services allows tasks to have different post-processing sizes—and a parametric analysis has already been carried out in [29]—in this paper we assume a uniform post-processing size for all tasks, set to:

$$f_1^h=\overline{f_1}=16\mathrm{Mb},\forall h\in [0,\dots ,N_T-1]$$

(34)

The number of CPU cycles to process a service will be considered to be $n_{cyc}=737.5$ cycles/bit, regardless of the service and processing node [18]. On GSs, since we assume that data centers have enough processing capacity to process EO data, we will consider unlimited processing capacity, thanks to an unlimited number of CPUs, each having a clock frequency ${\nu }_{GS}=4.5$ GHz (a common value for CPUs operating in ground computers), while we first assume that only one CPU is available on board satellites (due to limitations on available space on-board), each with a clock frequency:

$${\nu }_{Sat}=\alpha ·{\widehat{\nu }}_{Sat}$$

(35)

where:

$${\widehat{\nu }}_{Sat}={\overline{f}}_0·n_{cyc}/\tau$$

(36)

stands for the frequency needed on board of satellites to process a task during a time cycle. Thus, by setting $\alpha$, we set the maximum number of services which can be completely processed by a satellite in a time cycle. In particular, the $\alpha$ parameter influences both the processing capacity available on satellites:

$${\mathsf{\Gamma }}_{Sat}={\nu }_{Sat}/n_{cyc}=\alpha ·{\overline{f}}_0/\tau$$

(37)

and the unit-processing energy cost on satellites:

$${\gamma }_p^i=k·n_{cyc}·{\nu }_i^2=k·n_{cyc}^3·{({\overline{f}}_0/\tau )}^2·{\alpha }^2$$

(38)

where k represents the processor effective capacitance coefficient, and it will be set equal to ${10}^{-27}$ [18]. Please notice that, given this energy consumption model, by increasing $\alpha$ we linearly increase the number of services which can be processed in a time cycle, but at the same time we quadratically increase the energy consumption to process each service. While we assume that GSs have unlimited available energy, satellites nodes have a limited amount of energy available on board to be consumed for OEC-related operations (i.e., the processing and transmission of data). In particular, we assume that satellites have an amount of energy:

$${\epsilon }_{Sat}=\nabla ·\overline{P}·\tau$$

(39)

available during each time cycle, whose value is dependent on the ∇ parameter indicating the percentage of the maximum power $\overline{P}$ available on board a Sentinel-2 satellite [1]. This assumption is justified by the fact that the power available on board of satellites is always higher than the amount needed by the satellite (both for bus devices and payload instruments); then, an amount of this extra power (which is always generated, regardless of its actual consumption) can be used for OEC operations. For example, in the case of Sentinel-2 satellites, at the end of its life, each satellite has a power available equal to $\overline{P}=1700$ W, but only 1250 W is actually consumed: thus, up to around 25% of the available power can be still used to support OEC. Moving to transmission links, following [40], we will consider a transmission power $P_t=10$ W, antenna gain $G=27$ dBi, carrier frequency $\nu =26$ GHz, system noise temperature $T_s=290$ K, and minimum required Eb/N0 ratio ${Eb/N0|}_{min}=6$ dB. We also consider a data rate of $R_{ISL}=500$ Mbps for ISLs and $R_{dl}=520$ Mbps in case of downlink to the GS, as in the case of Sentinel-2 satellites [1]. Finally, the memory on each satellite is limited to $M_i=100$ Gb, $\forall i\in [0,\dots ,N_S-2]$ to support the worst case load of data in the service generation scenario described further on, while we will consider GSs to have an unlimited storage capacity, since we assume data centers with enough capacity to handle all the data produced by the constellation. The energy cost to store data in memory will be considered equal to zero.

7.1. Topology and Services

We developed a Python tool to generate Walker constellations (that is, constellations to optimize Earth coverage) [41] and obtain the position of satellites and GSs over time, which are then translated into a graph, as is described in Section 4. In this paper, we suppose to work with a constellation of 24 satellites divided into six orbital planes. Values of orbital height and inclination are chosen to be consistent with EO applications, and are, respectively, equal to 712.84 km and 98.24 deg, leading to a cyclostationary period (i.e., the repeat cycle) equal to 2 sidereal days. We assume three ground stations, placed in Matera (Italy), Kiruna (Sweden), and Kourou (French Guyana). All this translates into having $N_S=25$ actual nodes (24 satellites and 1 virtual ground station), and $T=287$ time cycles. However, although this complete graph is considered as a real orbital scenario evaluated by means of the GSESH algorithm, due to the NP completeness of the proposed optimization problem, its solution is restricted to the case that provides for $T=4$ time cycles. We consider services to be generated while a satellite flies over a user-defined region. We developed a Python tool to determine these events and obtain the following service generation. In particular, each satellite generates a task for each second of flying over the region of interest. Again, while comparing the optimal solution to the GSESH algorithm one, we will consider that services are generated when satellites flies over Italy during four time cycles, since this leads to a total number of 724 services, thus to a simplified, yet not trivial, scenario. Instead, in the real orbital scenario, we consider satellites to generate services while flying over Australia and Mexico during an entire repeat cycle, leading to a total of 111,483 services.

7.2. 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 Figure 5 and Figure 6 for $\nabla =\mathrm{0.1\%}$ (i.e., ${\epsilon }_{Sat}/\tau =1.7$ W, or, equivalently, ${\epsilon }_{Sat}=1020$ J) and $\nabla =\mathrm{0.2\%}$ (i.e., ${\epsilon }_{Sat}/\tau =3.4$ W, or, equivalently, ${\epsilon }_{Sat}=2040$ J), respectively, and $\alpha \in [0,\dots ,16]$. 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, differing from the optimal result in the value by not more than 7.10% when $\nabla =\mathrm{0.1\%}$. 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 $\nabla =\mathrm{0.2\%}$. 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 $\alpha \ge 15$), 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.

After having validated the GSESH algorithm, we apply it to a complete orbital scenario to evaluate how the possibility to process data on board of satellites and to leverage the full constellation in processing in such a way that if a satellite does not have enough resource to process data can offload the elaboration to another satellite, allows to save energy on the GS. In Figure 7, we report the ground station energy saving as a function of the maximum number $\alpha$ of tasks that can be processed by a satellite during a time cycle, for $\alpha \in [0,\dots ,50]$, and $\nabla =\mathrm{1\%}$ (i.e., ${\epsilon }_{Sat}/\tau =17$ W, or, equivalently, ${\epsilon }_{Sat}=10,200$ J). We report the results of the following solutions: GSESH represents the proposed heuristic, FoG involves task processing being allowed only on the satellite which acquired the image or on the ground, while in the AG solution, data can be only elaborated on the ground. In order to better understand the energy-saving trends related to the GSESH solution, we also report the following energy-consumption components as a function of the time cycle: the energy consumed by the GSs in Figure 8; the total (processing and transmission) energy used by satellite 13 of the considered a constellation in Figure 9; the space segment total energy consumption (i.e., the sum of the total energy consumption of all the satellites in the constellation) in Figure 10. In each figure, three curves are reported for values $\alpha$ equal to 5, 12, and 20, corresponding to $\alpha$ values smaller than, equal to, and higher than the optimum value $\alpha =12$ showing the maximum ground station energy saving, as can be noticed in Figure 7.

The results reported in Figure 7 show that when there is the possibility to process data on board (i.e., in case of the GSESH algorithm or FoG benchmark solution), we save energy on GSs, and the ground station energy-saving trend follows the same behaviour as discussed before. In particular, the peak energy saving in ground stations is reached for an increased value of $\alpha$ with respect to the case seen for the heuristic validation, since in this case we have an increased amount of energy available on board (in particular, $\nabla =\mathrm{1\%}$ with respect of $\nabla =\mathrm{0.1\%}$ of the previous analysis) to handle a higher number of services. Furthermore, in this complete case-study orbital scenario, it is possible to notice that in the region where ground station energy saving decreases because of limited on-board energy, there are some $\alpha$ value regions where the energy saving remains constant. This is because of the energy margin limiting the amount of data each satellite can handle. Finally, as expected, energy saving in the ground station is zero when the AG benchmark solution is applied.

The ground station energy-saving trend is confirmed by the results shown in Figure 8, Figure 9 and Figure 10, where we can remark that: (i) the energy consumption on the GSs (Figure 8) is minimum when the processing capacity installed on board satellites is such that $\alpha =12$ (Figure 8b), i.e., for the $\alpha$ value associated to maximum energy saving on ground, while higher consumption is reached for $\alpha$ equal to 5 (Figure 8a) and 20 (Figure 8c), i.e., when satellites have limited processing capacity or in case the energy required to process all the images each satellite would be able to elaborate on board is higher than the energy available on board, respectively; (ii) both the total energy consumption on any satellite, i.e., satellite 13, (Figure 9) and the total energy consumption of the space segment (Figure 10) are minimum for $\alpha =5$ because satellites are able to process only a few (at most 5) tasks in a time cycle, while they are both maximum for $\alpha =20$. It is important to underline that the maximum energy consumption achieved by a single satellite is limited by the amount of energy available on board, as can be seen in Figure 9. Instead, the total energy consumption on both satellite 13 and the space segment is intermediate for $\alpha =12$. Finally, we separately report the space segment energy consumption due to processing and transmission, again for $\alpha =\{5,12,20\}$, in Figure 11 and Figure 12, respectively. From these figures, we can observe that the energy consumption due to on-board processing is predominant with respect to the energy amount used for data transmission in the considered realistic scenario.

We also propose an analysis aiming at studying the impact on the percentage of ground station energy saving of the amount of energy available on board satellites. In Figure 13, we show the energy consumed on GSs by varying $\alpha \in [0,\dots ,50]$ and $\nabla \in \{\mathrm{1\%},\mathrm{5\%},\mathrm{10\%},\mathrm{15\%}\}$, by applying the GSESH algorithm. It can be noticed that, by increasing the energy available on board, there is an extension of the region of $\alpha$ values where the ground station energy saving increases, and the peak ground station energy-saving percentage also increases with ∇. In particular, it can be noticed that with only the 10% of energy available on Sentinel-2 to be dedicated to OEC operations, it is possible to obtain 99.95% of ground station energy saving when $\alpha =27$, while with the 15% of energy available on Sentinel-2 we reach 100% ground station energy saving when $\alpha =28$. Please notice that drops in energy saving, which can be seen in the cases $\nabla =\mathrm{10\%}$ and $\nabla =\mathrm{15\%}$, are due to a particular interaction between the set of services and the empirically determined energy margin calculation, leading to a drop in services that can be managed by the satellites when $\alpha =23$ and $\alpha =29$, respectively.

Finally, we have evaluated how the ground station energy saving is influenced by the number of cores available on board of satellites. In particular, by increasing the number of cores but leaving $\alpha$ fixed, we linearly increase the number of services which can be processed in a time cycle, and in this case the energy consumption increases linearly, too, since the CPU clock frequency of each single core remains the same, and consequently the energy demand of each core does not change. In particular, the results reported in Figure 14, where we considered $\alpha \in [0,\dots ,50]$ and $\nabla =\mathrm{1\%}$, show that by increasing the number of cores, we increase the peak ground station energy saving, which is also reached with a lower $\alpha$ value, i.e., with less powerful cores. This is due to the fact that, at the same $\alpha$ value, when we increase the number of cores we still increase computational capacity linearly, but in this case the energy consumption increases linearly with the number of cores, since the $\alpha$ value remains the same. In other words, this involves a dual-core architecture with each core having a CPU clock frequency such that $\alpha =1$ would allow for the same computational capacity of a single-core architecture where the CPU clock frequency of the core is such that $\alpha =2$, but the dual-core architecture would consume half the energy with respect to the single-core solution, and this helps in overcoming the limitation of the energy available on board, which would prevent the complete use of the available computational capacity. However, an increase in the number of cores would increase the on-board computer complexity, and would require more space in the limited satellite volume. For this reason, we limited our analysis to a maximum of four cores.

8. Conclusions

In this paper, we evaluated how the combined application of both satellite networks thanks to Inter-Satellite Links and edge computing capabilities on board satellites can be fruitfully leveraged by EO satellite constellations to save energy on GS by elaborating images directly on board of satellites instead of on the ground. This allows us to better leverage an amount of unused energy which is preallocated on board satellites, instead of requiring extra energy on the ground. We formalized and solved an optimization problem to find the optimal energy savings, which however was determined to be NP hard. For this reason, we also proposed the Ground Station Energy-Saving Heuristic (GSESH) Algorithm to be applied to a real orbital scenario. We showed that even with a small amount of energy available on board the satellites, it is possible to obtain substantial energy savings on GSs. In particular, we have verified that the GSESH algorithm allows for energy savings in the ground station up to 40% higher than those achieved with the benchmark solution in a real scenario. However, although intuitively one might assume that the higher the computational capacity on board the satellites, the higher the ground station energy saving (since more data are processed in orbit), we showed that the limitation on the energy available on board the satellites is such that there is a CPU clock frequency value for which the maximum ground station energy saving is achieved, above which this saving starts decreasing because even though there is a higher computational capacity, the energy to be used to process every single task also increases, and the small amount of energy available on board is such that there is not enough energy to leverage the full available computational capacity. Future work will explore a more realistic modeling of solar energy dynamics on board satellites, distinguishing between direct solar input and battery storage, as well as accounting for eclipse periods. Additionally, future extensions could relax the fixed time-slot assumption by allowing for processing and transmission operations to span multiple or variable-length time slots, thus improving scheduling flexibility and resource utilization.