## 3 System Model and Problem Formulations
### Problems Formulations  
In Equation (3), the PDF variances of different grids changing around the SGGM can be described by \(\Delta p_{g}(x,y)\) to provide a foundation for solving the problems of global load balancing adaption. Here, the problems should be further analyzed in depth. In actual S-IoT-N scenarios, users are not distributed uniformly in different grids, especially for non  
Figure 3: The user model and distribution of S-IoT-N.  
centered SSPs. As shown in Figure 4, two adjacent F-Sat-Beams \(B_{1}\) and \(B_{2}\) were observed. \(B_{1}\) can be further divided into several F-Ser-Beams \(\{b_{11},b_{12},\dots,\ b_{1n}\}\); \(B_{2}\) can be further divided into several F-Ser-Beams \(\{b_{21},b_{22},\dots,\ b_{2n}\}\). The Ser-Beams \(b_{edge}(B_{1},B_{2})\) around the edge of \(B_{1}\) and \(B_{2}\) are of most concern. The serving periods \(\text{T}_{n}\) of different satellites in the satellite constellation serving on the same F-sat-Beam are different, which have different start-serving-times \(t_{s}\) and over-serving-times \(t_{o}\). Figure 4 shows the trajectories of the SSPs in F-Ser-Beams \(B_{1}\) and \(B_{2}\) at the periods of \(T_{1}\) and \(T_{2}\), which are set as \(l_{11}@B1@T1\) and \(l_{21}@B2\&T1\). Next, in the period \(T_{1}\), the center-point of \(b_{1e}\in b_{edge}(B_{1},B_{2})\) is closer to the SSP trajectory \(l_{21}\) of \(B_{2}\) than SSP trajectory \(l_{11}\) of \(B_{1}\). In the period \(T_{2}\), the center-point of \(b_{1e}\) is closer to the SSP trajectory \(l_{12}\) of \(B_{1}\) than the SSP trajectory \(l_{22}\) of \(B_{2}\). Considering that the distances between the center-points of Ser-Beams and satellites are similar, the SSPs in the two Ser-Beams are similar too. Therefore, the problem of load balancing lies in determining which satellite's Ser-Beams users should access, and not the access threshold. Following this problem, it can be analyzed that the constraints of optimal load balancing relate to the density variances of DoU with the SSP movement, the number of users with different services, and the differences in DoU of adjacent F-Ser-Beams. During a specific serving period \(T_{i}\), Sat-Beam \(B_{i}\) was observed in the analysis. The maximum capacity of a satellite was assumed as \(Tra_{max}(B_{i})\), the total bandwidths as \(BW_{i}\), and the DoU density as \(p_{\bar{g}}(B_{i}\,,T_{i}\,,t),t\in[T_{i-1}\,,T_{i}]\). The Ser-Beams of \(B_{i}\) are \(b_{B_{i}}=\{b_{i1},b_{i2},\dots,\ b_{in}\}\), the area of each Ser-Beam is \(s\), and the corresponding PDF of DoU is \(p_{\bar{g}}(b_{ik}),k=1\cdots\ n\). The average arrival probability of services requests is \(\mu_{i}(t)\,t\in[T_{i-1}\,,T_{i}]\). The average service traffic is \(\varphi_{i}\). The service traffic of Ser-Beams is,  
\[Tra(b_{ik})=\iint_{S}p_{\bar{g}}(b_{ik})\times\mu_{i}(t)\times\varphi_{i}. \tag{4}\]  
The adjacent Sat-Beam of \(B_{i}\) is \(B_{j}\). The adjacent Ser-Beams are \(b_{m}\in b_{edge}\big{(}B_{i},B_{j}\big{)}\). The judgment coefficient function \(\lambda(b_{m},*)\) is defined as  
\[\lambda(b_{m},B_{i})= \begin{cases}1,\ \text{\it users in beam $b_{m}$ access to satellite $B_{i}$}\\ 0,\ \text{\it users in beam $b_{m}$ not access to satellite $B_{i}$}\end{cases}  \tag{5}\]  
Next, the traffic of the Sat-Beam is,  
\[Tra(B_{i})=\sum_{b_{ik}\in b_{B_{i}}}Tra(b_{ik})+\sum_{b_{m}\in b_{edge}(B_{ i},B_{j})}\lambda(b_{m},\ B_{i})\times Tra(b_{m})-\sum_{b_{ik}\in b_{\bar{g}}\cap b_{edge}(B_{ i},B_{j})}Tra(b_{ik}) \tag{6}\]  
We can also obtain the traffic of Sat-Beam \(B_{j}\) with \(Tra\big{(}B_{j}\big{)}\). Finally, the problem of load balancing for adjacent Ser-Beams of S-IoT-N can be formulated as:  
* Objectives: Objective 1: achieve the maximum throughput of adjacent Sat-Beams: \[\text{MAX}\ \big{[}Tra(B_{i})+\sum Tra\big{(}B_{j}\big{)}\big{]}\] (7)  
Figure 4: The case of different results at different times and F-Ser-Beams.