## 3 System Model
### Satellite Coverage and Transmission Model
#### 3.3.1 LEO Satellite Coverage Model
\right.. \tag{7}\]  
According to Equation (7), the average coverage time \(E(t_{c})\) of LEO satellite to ground users is as follows [29]:  
\[E(t_{c})=\frac{2\cos\gamma_{\max}}{\omega^{\gamma}\gamma_{\max}}\int_{0}^{ \gamma_{\max}}\frac{x\tan x}{\sqrt{\cos^{2}x-\cos^{2}\gamma_{\max}}}dx. \tag{8}\]  
The elevation \(\theta_{u}^{l}\) used between \(u\) and LEO satellite \(l\) is  
\[\theta_{u}^{l}=arc\tan\left(\frac{\cos\Theta-R_{e}/(R_{e}+h)}{\sin\Theta} \right), \tag{9}\]  
where  
\[cos\Theta=\cos(u_{lo}-l_{lo})\cos u_{la}\cos l_{la}+\sin u_{la}\sin l_{la}. \tag{10}\]  
In Equation (10), \(u_{lo}\) and \(u_{la}\) represent the longitude and latitude of the user, respectively, while \(l_{lo}\) and \(l_{la}\) represent the longitude and latitude of LEO satellite, respectively. When the LEO satellite is flying around the equator, \(l_{la}\) = 0, the longitude of the user and the satellite is the same, \(u_{lo}\) = \(l_{lo}\), the elevation is the maximum, and Equation (10) can be simplified into  
\[cos\Theta=\cos u_{la}. \tag{11}\]  
Therefore, within the average coverage time \(E(t_{c})\), the maximum \(\theta_{u\max}^{l}\) of \(\theta_{u}^{l}\) is  
\[\theta_{u\max}^{l}=arc\tan(\frac{\cos u_{la}-R_{e}/(R_{e}+h)}{\sin u_{la}}). \tag{12}\]  
To ensure that elevation increases monotonously, we set \(\Omega\) as the elevation of LEO satellite from the horizon to the user. The relationship between \(\Omega\) and \(\theta_{u}^{l}\) is as follows:  
\[\Omega=\left\{ \begin{array}{c}\theta_{u}^{l}\ \theta_{u}^{l}\leq\theta_{u, \max}^{l}\\ 2\ast\theta_{u,\max}^{l}-\theta_{u}^{l}\ \theta_{u}^{l}>\theta_{u,\max}^{l} \end{array} \right.. \tag{13}\]  
The maximum value of \(\Omega\) is \(\Omega_{\max}=2\ast\theta_{u,\max}^{l}\). In this model, the smaller \(\Omega\) is, the longer the LEO satellite coverage time will be. LEO satellites have more time to transmit, cache, and compute information with users. The larger \(\Omega\) is, the shorter the coverage time of LEO satellite to users will be, and the less time it will take for the LEO satellite to transmit, cache, and compute information with users. Because there are many LEO satellites in the SIN, we cannot determine which LEO satellite is connected to the user, nor can we determine the elevation of user \(u\) and satellite \(l\) at the next moment. Therefore, we set the elevation angle of user \(u\) and satellite \(l\) to the random variable \(\Omega_{u}^{l}\). The value range of \(\Omega_{u}^{l}\) can be divided into \(Y^{\prime}\) segments: \(\Omega_{0}^{*}\leq\Omega_{u}^{l}\leq\Omega_{1}^{*}\), \(y_{0}\): \(\Omega_{1}^{*}\leq\Omega_{u}^{l}\leq\Omega_{2}^{*}\), \(y_{1}\): \(\ldots\) ; \(\Omega_{Y^{\prime}-1}^{*}\leq\Omega_{u}^{l}\leq\Omega_{u,\max}^{l}\), \(y_{Y^{\prime}-1}\). Each segment conforms to a Markov chain model and has \(Y^{\prime}\) segments, that is, \(y=\{y_{0},y_{1},\ldots,y_{Y^{\prime}-1}\}\). The elevation of user \(u\) and LEO satellite \(l\) at time \(t\) is expressed as \(w_{u}^{l}(t)\), where \(t\in\{0,1,2,\ldots,T-1\}\). We have a total of T time slots, representing the total time from the user's application to the user's receiving and processing information. Based on a certain transition probability, \(w_{u}^{l}(t)\) transfers from one state to another. The probability of transition from state \(\overline{S11}\) to state \(\overline{S12}\) is expressed as \(\kappa_{\overline{S11512}}(t)\). We can get a \(Y^{\prime}\times Y^{\prime}\) dimensional elevation state transition probability matrix between user \(u\) and a LEO satellite \(l\) as follows:  
\[\kappa_{u}^{l}(t)=\left[\kappa_{\overline{S11512}}(t)\right]_{Y^{\prime}\times Y ^{\prime\prime}} \tag{14}\]where \(\kappa_{\overline{511512}}(t)=\Pr\!\left(w_{u}^{l}(t+1)=\overline{S12}\middle|\!w_{u} ^{l}(t)=\overline{S11}\right)\), \(\overline{S11},\overline{S12}\in y\).