## 4 Proposed CB-FBC and Genetic Approach
### Optimization of \(P_{tj_{i}S}(t)\) and \(B_{Sj_{i}}(t)\)
This means that as the value of \(\chi\) becomes smaller, better fairness is maintained in the clusters' coverage. From Figure 5, the proposed approach has the lowest \(\chi\) values at all tested number of aerials. The clusters' fairness index of the \"UCB-GA\" algorithm is higher than that of the proposed approach because it has no fairness functionality. As the clusters are randomly selected in the \"Rand\" selection approach, the \(\chi\) value is almost constant irrespective of the number of aerials used, because the term \(\left(\frac{h_{i}(T)}{\sum_{i=1}^{N}h_{i}(T)}\right)\) almost has the same value for \(\forall i\), which is almost equal to \(1/M\). On the other hand, both \"UCB-GA\" and the \"Nearest\" approaches have the worst fairness performance among the schemes involved in the comparisons because aerials always select their high confidence or nearest clusters, which causes unfairness in cluster coverage. Moreover, as the number of aerials increases, the \(\chi\) values of both schemes are decreased, enhancing their fairness performance. This is because many clusters are selected at each time step, which induces better fairness performance over time horizon. As the number of aerials increments, the \(\chi\) of all schemes tends to approach that of the \"Rand\" selection. This is due to the limited number of clusters accessible for choice at every trial, and as the number of aerials is equal to the number of clusters; all schemes will merge to the same point of \"Rand\" selection. From Figure 4, at \(N=2\), the clusters selection fairness index of the proposed scheme is lower than that corresponding to \"UAB+GA\", \"Nearest\", and \"Rand\" selections by 98.5%, 98.57 %, and 95.83%, respectively. However, at \(N=14\), these values become 73.21%,74.64%, and 50.7%, respectively. Figure 6 displays the ratio of LEO-Sat bandwidth utilization for various schemes. The \"Nearest\" algorithm fully utilizes the LEO-Sat bandwidth since it distributes it equally among the LEO-aerial links. As shown in Figure 6, its bandwidth utilization ratio equals 1. On the other hand, the \"Rand\" scheme's bandwidth utilization ratio is nearly equal to 0.5 since the bandwidth of each LEO-aerial link is randomly assigned from the range \([1,B_{Smax}/N]\). As the proposed approach and \"UCB-GA\" algorithms optimize the allocated LEO-Sat bandwidth according to the aerials' traffic requirements using GA, they have the same performance that comes from utilizing the LEO-Sat bandwidth most efficiently. Initially, they achieve a low Sat bandwidth utilization ratio at fewer aerials. However, full Sat bandwidth utilization occurs as the number of aerials increases. This results from aerials' high total traffic demands, which operate the LEO-Sat at its maximum capacity. At \(N=2\), they have a lower SAT bandwidth utilization ratio than the \"Nearest\" and \"Rand\" approaches by 82%. Figure 5: Clusters selection fairness index against the number of aerials. ### Computational Complexity Analysis

The main computational complexity of the proposed approach issues from the proposed \"CB-FBC\" and the \"GA\" algorithms. The computational complexity of the envisioned \"CB-FBC\" algorithm is equal to \(\mathcal{O}\big{(}N^{2}\big{)}+\mathcal{O}(N+1)\), where \(\mathcal{O}\big{(}N^{2}\big{)}\) comes from minimizing energy consumption and \(\mathcal{O}(N+1)\) from the sorting operation and updating the parameters of the selected super arm. The computational complexity of a \"GA\" algorithm is influenced by several factors, such as the population size, the number of generations, the objective function complexity, and \"GA\" algorithm functions such as mutation, crossover, and selection. Generally, this can be expressed as follows [32]:

\[\text{Complexity of GA}=\mathcal{O}(nGP(C+UOS)) \tag{16}\]

where \(n\) is the number of decision variables that is equal to \(2N\) in the proposed \"GA\" algorithm. \(G\) and \(P\) are the number of maximum generations and the population size, which are set to 50 and 20, respectively.