## 3 System Model and Problem Formulations
### The Modeling of the Solution Using the Original GA
\begin{table}
\begin{tabular}{c c} \hline
**Genetic Algorithm Mode** & **Modeling of Satellite User Load Balancing Based on Genetic Algorithms** \\ \hline Fitness Function & The throughput of adjacent Sat-Beam \(B_{i},B_{j}\) in the satellite service period \(T_{i}\) and average waiting time for users of adjacent Sat-Beam \(B_{i},B_{j}\). \\ \hline Fitness Rules & The fitness increases with the increase of throughput and decreases with the increase of the average waiting time for users. \\ \hline Single Optimal Solution & The load balancing scheme \(A_{ij}\) of the local \(k\) Ser-Beams \(b_{ij}\) when the user density is \(p\left(b_{ij}\right)\) and the serving satellite is \(S_{ij}\). \\ \hline Solution Encoding & Natural number encoding. \\ \hline Solution Code & \(A_{ij}=\{a_{1}\,,a_{2}\,,a_{3}\,,\cdots a_{k}\}\), \(a_{m}\)\((1\leq m\leq k)\) denotes the access scheme of the \(m\)-th F-Sat-Beams. \\ \hline Factors Influencing the Solution & The densities variances \(\Delta p_{g}\left(b_{ij}\,,T_{i}\right)\) of DoU in the direction of SSPs moving in a Ser-Beam \(b_{ij}\) and serving time \(T_{i}\), differences with that of adjacent Ser-Beams \(\Delta p_{g}\left(b_{ij}\,,b_{i+\Delta i\,j}\,,T_{i}\right)\). \\ \hline Selected Set of Solutions & n load balancing schemes \(A_{ij}^{1}\,,\,A_{ij}^{2}\,,\,A_{ij}^{3}\,,\,\cdots\,,A_{ij}^{n}\) of local k Ser-Beams \(b_{ij}\) \\  & when the user density is \(p\left(b_{ij}\right)\) and the serving satellite is \(S_{ij}\) \\ \hline A Set of Solutions Selected According to Fitness & The N schemes with the highest fitness among the existing load balancing schemes \(A_{ij}^{1}\,,\,A_{ij}^{2}\,,\,A_{ij}^{3}\,,\,\cdots\,,A_{ij}^{N}\) \\ \hline The Process of Coding Crossover & Select two load balancing schemes \(A_{ij}^{x}\,,\,A_{ij}^{y}\) with similar fitness according to the crossover probability. Randomly select the points for crossover and exchange the elements of the corresponding points of the two solutions. \\ \hline The Process of Coding Mutation & Select a load balancing scheme \(A_{ij}^{x}\) based on mutation probability. The mutation points are randomly selected, and the elements of the corresponding points of \(A_{ij}^{x}\) are changed according to the selection probability in the value set. \\ \hline \end{tabular}
\end{table} 
Table 1: The solution modeling of load balancing by GA.  
randomly select l consecutive elements (\(l<k\)) in the individual \(A_{ij}^{x}\), and exchange them with the elements in the same position in \(A_{ij}^{y}\) to generate two offspring individuals. The probability of genetic crossover is defined as follows:  
\[P_{acr}=\left\{ \begin{array}{l}P_{acr}(1),fitness\leq fitness_{ave}\\ P_{acr}(0)\times\frac{fitness_{max}-fitness}{fitness_{max}-fitness_{ave}}\,else \end{array} \right. \tag{15}\]  
where \(fitness\) is the fitness of individuals with higher fitness in an individual \(A_{ij}^{x}\) and \(A_{ij}^{y}\), \(fitness_{ave}\) is the average fitness of the contemporary population, \(fitness_{max}\) is the fitness of the best individual in the contemporary population, and \(P_{acr}(1)\) and \(P_{acr}(0)\) are probability constants. * Step4: Modelling Genetic Mutation:  
For every individual, judge whether there is a genetic mutation according to the mutation probability \(P_{mut}\). A floating-point number is randomly generated in the interval \([0,1]\) as the judgment factor \(\beta\). If \(\beta\leq P_{mut}\), the mutation operation is performed, and the mutation point is randomly selected in the individual \(A_{ij}^{x}\). Next, the elements of the corresponding point of \(A_{ij}^{x}\) are changed according to the selection probability in the value set to generate a child individual. The probability of genetic mutation is defined as follows:  
\[P_{mut}=P_{mut}(0)\times\frac{t_{max}-t}{t_{max}}, \tag{16}\]  
where \(t_{max}\) is the maximum evolution generation of the genetic algorithm, \(t\) is the current evolution generation, and \(P_{mut}(0)\) is a probability constant.