## 3 System Model and Problem Formulations
### The Modeling of the Solution Using the Original GA  
In Section 3, the constraints of the problems of load balancing were proposed. A local optimal solution must be achieved to solve the problems of most other cases of load balancing around the world. At the same time, the local optimal solution should be further improved to adapt to the density variances of DoU in the spatial-temporal domain. In addition, the periodic movement of satellites can provide solution optimization with more efficient prior information. Based on the above considerations, the optimal solution was modeled by GA. The overall design of the original GA is shown in Table 1.  
The specific modeling is described as follows:  
* Step 1: Modelling of Fitness Function: The fundamental purpose of load balancing for S-IoT-N is to improve the performance of the throughput and QoS. In serving period \(T_{i}\), the throughput of all Sat-Beams \(B_{j}\) should be maximized, and the average waiting time of all users should be minimized. The fitness function is defined as \[fitness=\frac{k_{1}*Tra_{total}}{k_{2}*T\_delay_{ave}}\] (14) where \(Tra_{total}\) and \(T\_delay_{ave}\) are the global throughput of S-IoT-N and the average waiting time of all users. * Step 2: Modelling of the Solution: As is shown in Figure 5, we assume that in the observed area, there are \(k\) Sat-Beams \(B_{ij}\), and the corresponding PDF of DoU is \(p\left(B_{ij}\right)\). A solution \(A_{ij}\) to the problem of load balancing, which achieves the optimization objective, is set as a single optimal solution output by GA. \(A_{ij}\) are encoded in the form of natural numbers, which is a matrix composed of \(k\) natural numbers, and \(a_{m}\) denotes the \(m\)-th access scheme of the \(m\)-th Ser-Beam. Under the possible values of each solution \(a_{m}\), the influence factors are the density variances of the user \(\Delta p_{g}\left(B_{ij},T_{i}\right)\) of the serving satellite and the difference in the density of adjacent satellite users \(\Delta p_{g}\left(B_{ij}\,,B_{i+\Delta i\,j+\Delta j}\,,T_{i}\right)\) within the period \(T_{i}\) instead of traditional random influence for the schemes to which Sat-Beams have access to. This can achieve a certain optimization direction in a specific scenario. Under the above optimization direction, we have the solutions \(A_{ij}^{1}\,,\,A_{ij}^{2}\,,\,A_{ij}^{3}\,,\,\cdots\,,A_{ij}^{n}\), which are the initial populations of GA optimization. The offspring individuals generated by the crossover mutation in each generation and the parent individuals form a new group and \(N\) individuals \(A_{ij}^{1}\,,\,A_{ij}^{2}\,,\,A_{ij}^{3}\,,\,\cdots\,,\,A_{ij}^{N}\) are selected using the tournament method to form the next generation populations. The new populations participate in the evolution of the next generation. * Step 3: Modelling the Genetic Cross: According to the individual pairing principle of matching each other, two individuals with similar fitness, \(A_{ij}^{x}\) and \(A_{ij}^{y}\), are selected. We judge whether the crossover occurs according to the crossover probability \(P_{acr}\). A floating-point number is randomly generated in the interval \([0,1]\) as the judgment factor \(\alpha\). If \(\alpha\leq P_{acr}\), performs the crossover operation,