## 3 Problem Statement  
Our paper presents details regarding a new mathematical formulation for QoS of nanosatellites, which was formulated to account for the behavior of constellations. The proposed formulation extends on the limitations presented in the previous work of [PERSON] et al. [9], whose constraints--despite accounting for the energy-aware behavior of the object--only considered a single nanosatellite. In this regard, it is possible to explore the Pareto-frontier achieved from a trade-off between individual and collective tasks (i.e., constellation tasks). In other words, the model aims at achieving broad applicability, in order to serve as a tool for planning multi-satellite space missions. The main additions to the original model are listed below:  
* Model expanded for multiple satellites;
* Separation between individual and collective (constellation) tasks;
* Possibility of redundancy;
* Analysis of the Pareto-frontier for the trade-off between individual and constellation tasks.  
Table 1 details the notation of the sets and variables. Model inputs are defined in Table 2. The following parts of this section discuss the objective functions, model constraints, and an auxiliary irradiation model we adopted for simulating energy availability in CubeSats.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \multicolumn{2}{l}{_General_} \\ \hline \(S\) & the number of subsystems. \\ \(J\) & the number of tasks/jobs. \\ \(T\) & the size of \(\mathcal{T}\). \\ \hline \multicolumn{2}{l}{_Sets_} \\ \hline \(\mathcal{S}\) & Set of subsystems (satellites). \(\mathcal{S}=\{s\mid s\in\mathbb{N},s\leq S\}\). \\ \(\mathcal{J}\) & Set of jobs (or tasks). \(\mathcal{J}=\{j\mid j\in\mathbb{N},j\leq J\}\). \\ \(\mathcal{T}\) & Set of units of time. \(\mathcal{T}=\{t\mid t\in\mathbb{N},t\leq\mathbb{T}\}\). \\ \hline \multicolumn{2}{l}{_Subsets_} \\ \hline \(\mathcal{C}\) & Set of constellation tasks. \(\mathcal{C}\subset\mathcal{J}\). \\ \(\mathcal{I}\) & Set of individual tasks. \(\mathcal{I}\subseteq(\mathcal{J}\setminus\mathcal{C})\). \\ \(\Xi\) & Set of simultaneous tasks. \(\Xi\subset\mathcal{C}\). \\ \(\mathcal{R}\) & Set of tasks with encouraged redundancy. \(\mathcal{R}\subset(\mathcal{C}\setminus\Xi)\). \\ \hline \multicolumn{2}{l}{_Model variables_} \\ \hline \(x_{s,j,t}\) & Decision variable. Equals 1 when, in subsystem \(s\), job \(j\) is running at instant \(t\); else, equals 0. \(x_{s,j,t}\in\{0,1\}\). \\ \(\phi_{s,j,t}\) & Auxiliary variable. Equals 1 when, in subsystem \(s\), job \(j\) is starting its execution at instant \(t\); else, equals 0. \(\phi_{s,j,t}\in\{0,1\}\). \\ \(a_{s,j,t}\) & Auxiliary variable. Equals 1 when, in subsystem \(s\), job \(j\) is ending its execution at instant \(t\); else, equals 0. \(a_{s,j,t}\in\{0,1\}\). \\ \(\beta_{s,s^{\prime},j,t}\) & Auxiliary variable. Equals 1 when, in subsystems \(s\) and \(s^{\prime}\), job \(j\) is running at instant \(t\); else, equals 0. \(\beta_{s,s^{\prime},j,t}\in\{0,1\}\). \\ \(b_{s,t}\) & Auxiliary variable. Equals 1 when, in subsystem \(s\), the battery is being charged at instant \(t\); else, if the battery is being discharged, it equals 0. \(b_{s,t}\in\{0,1\}\). \\ \(i_{s,t}\) & Battery current, in Ampere, for subsystem \(s\), at instant \(t\). \(i_{s,t}\in\mathbb{R}\). \\ \(SoC_{s,t}\) & State of charge (SoC) of the battery at instant \(t\) and subsystem \(s\). \(SoC_{s,t}\in\mathbb{R}^{+},\leq 1\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Sets and model variables.