## 3 System Model
### Link Transmission Model  
Define \(R^{i}_{m,k,t}\) as the data rate between the IoT device \(ID_{m}\) and the satellite \(RS_{k}\) at time \(t\):  
\[R^{i}_{m,k,t}=x^{i}_{m,k,t}B\log_{2}\left(1+\frac{P^{i}_{m}G_{m}G^{r}_{k}L^{i }_{m,k,t}L^{pt}_{m,k,t}}{\sigma^{2}}\right) \tag{1}\]  
Here, \(P^{i}_{m}\) is the transmission power, \(G_{m}\) and \(G^{r}_{k}\) are antenna gains, \(L^{i}_{m,k,t}\) is path loss, \(L^{pt}_{m,k,t}\) is rain attenuation, and \(\sigma^{2}\) is noise power. The antenna gains and free space path loss are critical components in the link budget analysis. They determine the signal-to-noise ratio (SNR) at the receiver, which in turn affects the achievable data rate as per [PERSON]'s capacity formula. Higher antenna gains improve the SNR, while larger pathlosses deteriorate it. According to Equation (23) in [45], \(L^{i}_{m,k,t}=\left(4\pi d^{i}_{m,k,t}f/c\right)^{2}\) where \(c\) is the speed of light, \(d^{i}_{m,k,t}\) is the distance, and \(f\) is the frequency.  
Let \(R^{g}_{k,t}\) represent the link rate between the relay satellite \(RS_{k}\) and the ground station in time slot \(t\). It is given by the following:  
\[R^{g}_{k,t}=x^{g}_{k,t}B\log_{2}\left(1+\frac{P^{TS}_{k}C^{TS}_{k}G^{TS}_{k}L^ {S}_{k,t}L^{pt}_{k,t}}{\sigma^{2}}\right) \tag{2}\]  
where \(P^{TS}_{k}\) is the relay satellite's transmission power to the ground station, \(G^{TS}_{k}\) is the transmission antenna gain, \(G^{g}_{t}\) is the receiving antenna's gain, \(L^{pt}_{k,t}\) denotes the rain attenuation, and \(L^{g}_{k,t}\) indicates the free space loss, which is defined as follows: \(L^{g}_{k,t}=\left(4\pi d^{g}_{k,t}f/c\right)^{2}\) where \(d^{g}_{k,t}\) is the distance between \(RS_{k}\) and the ground station during time slot \(t\).  
Let \(R^{o}_{n,k}\) represent the link rate between the observation satellite \(OS_{n}\) and the relay satellite \(RS_{k}\), which is given by Equation (4) in [46]:  
\[R^{o}_{n,k}=x^{o}_{n,k}\frac{G^{o}_{n}G^{G}_{k}P_{n}L^{o}_{n,k}}{k_{s}T_{s}E_{ b}/N_{0}} \tag{3}\]  
where \(G^{o}_{n}\) is the observation satellite's transmission gain, \(G^{r}_{k}\) is the relay satellite's receiving gain, \(P_{n}\) is the power of \(OS_{n}\), \(L^{o}_{n,k}=\left(4\pi d^{o}_{n,k}f/c\right)^{2}\) is the free space loss between \(OS_{n}\) and \(RS_{k}\), \(k_{s}\) is the Boltzmann constant, \(T_{s}\) is the system noise temperature, \(E_{b}\) is the energy per bit, and \(N_{0}\) is the noise power spectral density. \(d^{o}_{n,k}\) is the distance between \(OS_{n}\) and \(RS_{k}\).  
\(R^{r}_{k,k^{\prime}}\) specifies the rate between the relay satellites \(RS_{k}\) and \(RS_{k^{\prime}}\), which is given by the following:  
\[R^{r}_{k,k^{\prime}}=x^{r}_{k,k^{\prime}}\frac{G^{r}_{k}G^{r}_{k^{\prime}}P^{ r}_{k}L^{r}_{k,k^{\prime}}}{k_{s}T_{s}E_{b}/N_{0}} \tag{4}\]  
where \(G^{r}_{k}\) and \(G^{r}_{k^{\prime}}\) are the relay satellite \(k\)'s transmission gain and \(k^{\prime}\)'s receiving gain, \(P^{r}_{k}\) is the power for inter-satellite transmission, \(L^{r}_{k,k^{\prime}}=\left(4\pi d^{o}_{k,k^{\prime}}f/c\right)^{2}\) is the free space loss between \(RS_{k}\) and \(RS_{k^{\prime}}\). \(d^{r}_{k,k^{\prime}}\) is the distance between \(RS_{k}\) and \(RS_{k^{\prime}}\).