## 3 System Model
### Optimization Problem Formulation  
The optimization problem aims to determine the best assignment vector \(x_{\hat{h}}(t)\) for aerials-clusters at each time \(t\), intending to maximize their long-term accumulative data rates. This objective should be accomplished while ensuring that cluster coverage is fair over the time horizon, given that \(N<M\). The constrained resources of the aerials' batteries and LEO-Sat's bandwidth should also be considered. Mathematically, this optimization problem can be expressed in the following manner:  
\[\max_{x_{\hat{h}}(t),P_{\hat{h}\hat{s}}(t),B_{S\hat{h}}(t)}\frac{1}{T}\sum_{t= 1}^{T-1}\sum_{j=1}^{N}\sum_{i=1}^{M}\ x_{\hat{h}}(t)\min\left(\eta_{S\hat{h}}(t ),\ \Psi_{\hat{h}}(t)\right)\]  
s.t  
\[\text{C1}: x_{\hat{h}}(t)\in\{0,1\} \tag{7a}\] \[\text{C2}: \sum_{j=1}^{N}\sum_{i=1}^{M}\ x_{\hat{h}}(t)\ \leq N\] (7b) \[\text{C3}: \sum_{i=1}^{M}\ x_{\hat{h}}(t)=1\ \forall j\] (7c) \[\text{C4}: x_{\hat{h}}(t)\Gamma_{\hat{h}}(t)\leq\Gamma_{jR}(t)\forall j\] (7d) \[\text{C5}: \frac{1}{T}\sum_{t=0}^{T-1}\sum_{j=1}^{N}x_{\hat{h}}(t)\geq\delta _{i}\forall i\] (7e) \[\text{C6}: \sum_{j=1}^{N}\sum_{i=1}^{M}x_{\hat{h}}(t)B_{S\hat{h}_{j}}(t) \leq B_{Smax}\] (7f) \[\text{C7}: P_{\hat{h}\hat{s}}(t)\leq P_{Vmax} \tag{7g}\]  
where  
\[\delta_{i}=\frac{K_{i}}{\sum_{i=1}^{M}K_{i}}, \tag{8}\]  
\[\Gamma_{\hat{h}}(t)=P_{f}T_{f_{\hat{h}}}(t)+(P_{h}+P_{\hat{t}\hat{h}}S(t))T_{h_ {\hat{h}_{j}}}, \tag{9}\]  
where \(T\) is the time horizon, \(x_{\hat{h}_{j}}(t)\in\{0,1\}\) is an assignment binary indicator which equals one if cluster \(i\) is chosen to be covered by aerial \(j\) at time \(t\) otherwise zero. \(\eta_{S\hat{h}_{j}}(t)\) given in (6) is the assigned capacity of the LEO-aerial link to cover cluster \(i\) by aerial \(j\) at time \(t\). Also, \(\Psi_{\hat{h}_{j}}(t)\) given in (4) is the total data rate between aerial \(j\) and cluster \(i\). In (7), the minimum of the aerial-UD link and the aerial-LEO link data rates is taken because UD data will be relayed to LEO-Sat via its coverage aerial. Thus, the total link speed between UD and LEO-Sat will be the minimum between both links as both will exist simultaneously. The constraint C2, which refers to the total number of chosen clusters, is less than or equal to \(N\), i.e., the total number of aerials, as some aerials' batteries may run out of energy during the coverage and need re-charging. The third constraint, C3, reflects that at a time \(t\), each aerial \(j\) covers only one cluster \(i\). The fourth constraint, C4, refers to the energy consumption of aerial \(j\), \(\Gamma_{\hat{h}_{j}}(t)\), needed to assist cluster \(i\) at time \(t\) should not overcome its remaining full battery capacity \(\Gamma_{jR}(t)\) at time \(t\). \(\Gamma_{\hat{h}_{j}}(t)\) is defined in (9), where it considers flying power \(P_{f}\), hovering power \(P_{h}\), and TX communication power \(P_{\hat{h}\hat{s}}(t)\). Both \(P_{f}\) and \(P_{h}\) are assumed to be constants. However, \(P_{\hat{h}\hat{s}}(t)\) is a function of time, and it should be optimized based on the current aerial \(j\) traffic need. \(T_{f_{\hat{h}_{j}}}(t)\) indicates the flying time of aerial \(j\), which equals the division of the horizontal distance between its current position and target location in cluster \(i\) at time \(t\), over the aerial speed. \(T_{h_{\hat{h}_{j}}}\) indicates the hovering time of aerial \(j\) over cluster \(i\), which is assumed to be constant. The constraint C5 ensures fairness among clusters based on their UDs densities, as given in (8). Constraint C6 means that the sum of the allocated LEO-aerial bandwidths should not exceed the maximum allowable bandwidth ofthe LEO-Sat, i.e., \(B_{Smax}\). Finally, constraint C7 indicates that the TX communication power of aerial at time \(t\) must not exceed its maximum allowable TX power \(P_{Vmax}\).