## 3 System Model and Problem Formulations
### User Model and Distribution  
The DoU are analyzed in two scopes: the global scope, and the local scope. In the global scope, the user distributions are relatively permanent and follow global population distributions. This can provide useful prior knowledge. The densities of users vary greatly between each F-Ser-Beam. In the local scope, since the coverage of F-Sat-Beams is much larger than those of cells of mobile communication networks, there are very weak mutual influences between non-adjacent Ser-Beams in S-IoT-N. Therefore, the handovers between adjacent Ser-Beams based on received signal-power are not considered in our scenarios. We are concerned with the problem of which Ser-Beams the users should access for load balancing. The solutions were further improved to adapt to different density variances around the world. Next, we proposed to model the density variances of users under different moving directions, which is shown in Figure 3.  
First, we built a two-dimensional satellite to ground grid map (SGGM). In an SGGM, a plane projection \(S_{u}\) of the surface of the earth is generated. \(S_{u}\) can be divided into \(a\times b\) grids according to the configured satellite constellation, where \(a\) is the number of grids in the latitudinal direction, and \(b\) is the number of grids in the longitudinal direction. It can be deduced as:  
\[a=\Big{[}a_{0}\times\frac{N_{s0}}{N_{s1}}\times\frac{H_{01}}{H_{00}}\Big{]} \tag{1}\]  
\[b=\Big{[}b_{0}\times\frac{N_{s0}}{N_{s1}}\times\frac{H_{01}}{H_{00}}\Big{]} \tag{2}\]  
Figure 2: The network framework of S-IoT-N.  
where \(a_{0}\) and \(b_{0}\) are the number of grids in the directions of latitude and longitude in the case of the referenced satellite constellation, respectively. \(N_{s0}\) and \(H_{o0}\) are the satellite number and the orbital altitude of the referenced satellite constellation. \(N_{s1}\) and \(H_{o1}\) are the satellite number and orbit height of the observed satellite constellation. The observed grid \(G_{(a,b)}\) is associated with an actual latitude and longitude, which is \(\left(x_{(Ga_{1},Gb_{1})},y_{(Ga_{1},Gb_{1})}\right)\), \(\left(x_{(Ga_{2},Gb_{2})},y_{(Ga_{2},Gb_{2})}\right)\), \(\left(x_{(Ga_{1},Gb_{2})},y_{(Ga_{1},Gb_{2})}\right)\), and \(\left(x_{(Ga_{2},Gb_{1})},y_{(Ga_{2},Gb_{1})}\right)\).  
Second, the probability density functions (PDF) of DoU \(p_{0}(x,y)\) of each grid were obtained by combining the SGGM and global IoT device distribution. The value of PDF is shown in the second part of Figure 3. If the number \(G_{(a,b)}\) of observed satellite grids is greater than the number \(G_{(a0,b0)}\) of referenced satellite grids, that is, \(a_{1}>a_{0}\), \(p_{1}(x,y)\) can be exploited directly. If the number \(G_{(a,b)}\) of observed satellite grids is less than the number \(G_{(a0,b0)}\) of referenced satellite grids, that is, \(a_{0}>a_{1}\), the densities \(p_{1}(x,y)\) can be gained by obtaining the average of the densities \(p_{0}(x,y)\) of the surrounding grids. Next, the new PDF of DoU \(p_{g}\left(G_{(a,b)}\right)\) with different specifications of the SGGM was obtained.  
Third, the PDF of DoU densities \(p_{g}\left(G_{(a,b)}\right)\) were further normalized. We designed 6 levels of normalized densities for \(p_{g}\left(G_{(a,b)}\right)\), which are \(\left\{Q_{1},Q_{2},Q_{3},Q_{4},Q_{5},Q_{6}\right\}\); \(Q_{1}:p_{g}\left(G_{(a,b)}\right)\in[0:0.02]\),\(Q_{2}:p_{g}\left(G_{(a,b)}\right)\in[0.02:0.06]\), \(Q_{3}:p_{g}\left(G_{(a,b)}\right)\in[0.06:0.1]\), \(Q_{4}:p_{g}\left(G_{(a,b)}\right)\in[0.1:0.2]\), \(Q_{5}:p_{g}\left(G_{(a,b)}\right)\in[0.2:0.5]\), and \(Q_{6}:p_{g}\left(G_{(a,b)}\right)\in[0.5:1]\).  
Finally, the PDF variances with moving SSPs were obtained. Under the observation of SSP movement directions, the differences \(\Delta p_{g}(x,y)\) in PDF during a Sat-Beam serving period \(T_{d}\) from one grid to another grid are expressed as  
\[\Delta p_{g}\left(G_{(a,b)}\right)=p_{g}\left(G_{(a,b_{1})}\right)-p_{g}\left( G_{(a_{i+o}a_{T_{d}})},y_{i+i\circ t_{d}}\right)\text{.} \tag{3}\]