## 3 RACH Success Probability and Power Consumption Model for LEO Satellite-Based NB-IoT Networks
### The Successful Probability of Preamble Detection  
Based on the interference model, the SINR of the LEO satellite receiving a preamble signal can be expressed as the following:  
\[SINR=\frac{p_{s}|h_{0}|^{2}C_{0}R_{0}^{-\alpha}}{I_{m}+\sigma^{2}} \tag{5}\]  
where \(I_{m}\) is the cumulative interference from all other terminals in the interference area \(A_{I}\) and can be given by:  
\[I_{m}=\sum_{m=1}^{M_{I}}p_{s}|h_{m}|^{2}G_{0}R_{m}^{-\alpha} \tag{6}\]  
where \(M_{I}\) is a random variable representing the terminal causing interference, \(|h_{m}|^{2}\) is the channel power gain of Rayleigh fading, and \(p_{s}\) is the transmission power of the interfering terminal. If the SINR is higher than the LEO satellite demodulation threshold \(\gamma_{th}\), the LEO satellite can successfully detect the preamble. The probability of successful preamble detection can be expressed as follows:  
\[ \begin{split} p_{c}&=\mathbb{P}(SINR\geq\gamma_{th })\\ &=E_{r}[f\{SINR\geq\gamma_{th}|r_{0}\}]\\ &=\int\limits_{r\text{min}}^{r\text{max}}f_{R}(r)f(SINR\geq\gamma _{th}|r_{0})dr_{0}\\ &=\int\limits_{r\text{min}}^{r\text{min}}f_{R}(r)f(\frac{p_{s}|h_ {0}|^{2}G_{0}r_{0}^{-\alpha}}{\sigma^{2}+I_{m}}\geq\gamma_{th}|r_{0})dr_{0}\\ &=\int\limits_{r\text{min}}^{r_{0}}\frac{r_{0}}{2R_{c}(R_{c}+r_{ \text{min}})}f[G_{0}\geq\gamma_{th}r_{0}^{\alpha}|h_{0}|^{2}{p_{s}}^{-1}\big{(} \sigma^{2}+I_{m}\big{)}|r_{0}]dr_{0}\end{split}  \tag{7}\]  
By applying the Laplace transform to the set of interfering terminals, the expression for the successful probability of preamble detection is given by the following:  
\[ \begin{split}& f[G_{0}\geq\gamma_{th}r_{0}^{\alpha}|h_{0}|^{2}{p_ {s}}^{-1}\big{(}\sigma^{2}+I_{m}\big{)}|r_{0}]\\ &=E_{I_{m}}\Big{[}f[G_{0}\geq\gamma_{th}r_{0}^{\alpha}|h_{0}|^{2}{ p_{s}}^{-1}\big{(}\sigma^{2}+I_{m}\big{)}|r_{0},I_{m}]\Big{]}\\ &=E_{I_{m}}\Big{[}\exp\Bigl{(}-\gamma_{th}r_{0}^{\alpha}|h_{0}|^{ 2}{p_{s}}^{-1}\big{(}\sigma^{2}+I_{m}\big{)}\Big{)}|r_{0}\Big{]}\\ &=e^{-\gamma_{th}r_{0}^{\alpha}|h_{0}|^{2}{p_{s}}^{-1}\sigma^{2}} L_{I_{m}}(\gamma_{th}r_{0}^{\alpha}{p_{s}}^{-1})\end{split}  \tag{8}\]  
Substituting Equation  
where \(L_{I_{m}}(\gamma_{th}r_{0}^{\alpha}p_{s}{}^{-1})\) represents the Laplace transform of the interference terminals. And, let \(s=\gamma_{th}r_{0}^{\alpha}p_{s}{}^{-1}\) to solve the Laplace transform of the interference terminals. \[ \begin{split} L_{I_{m}}(s)&=E_{I_{m}}\left(e^{-sI_{ m}}\right)\\ &=E_{I_{m}}\left(\exp\left(-\sum\limits_{M=1}^{M_{I}}sp_{s}G_{0}R _{m}^{-\alpha}\right)\right)\\ &=E_{G_{0}D_{m}}\left[\prod\limits_{M\in M_{I}}\exp(-sp_{s}G_{0}R _{m}^{-\alpha})\right]\end{split}  \tag{10}\]  
Since \(G_{0}\) is a constant variable, we can derive the following:  
\[L_{I_{m}}(s)=E_{D_{m}}\left[\prod\limits_{n\in N_{I}}E_{G_{0}}\exp(-sp_{s}G_{0 }R_{m}^{-\alpha})\right] \tag{11}\]  
Let \(R_{m}=x\) represent the distance between the interference terminals and the LEO satellite. By utilizing the PDF from stochastic geometry theory [22], we can obtain the following result:  
\[ \begin{split} L_{I_{m}}\left(\gamma_{th}r_{0}^{\alpha}p_{s}{}^{- 1}\right)&=\exp\left[-2\pi\lambda\int_{r0}^{r_{\max}}\frac{1}{1+ \left(\gamma_{th}r_{0}^{\alpha}\right)^{-1}x^{\alpha}}xdx\right]\\ &=\exp\left[-2\pi\lambda\int_{r0}^{r_{\max}}\frac{\gamma_{th}}{ \gamma_{th}+\left(x/r_{0}\right)^{\alpha}}xdx\right]\end{split}  \tag{12}\]  
By substituting Equation (12) into Equation (9), we can obtain the expression for the probability of successful preamble detection:  
\[p_{c}=\int\limits_{r\min}^{r\max}\frac{r_{0}\cdot e^{-\gamma_{th}r_{0}^{\alpha }|I_{0}|^{2}p_{s}^{-1}\sigma^{2}}}{2R_{e}(R_{e}+r_{\min})}\exp\left(-2\pi \lambda\int_{r0}^{r_{\max}}\frac{\gamma_{th}}{\gamma_{th}+\left(x/r_{0} \right)^{\alpha}}xdx\right)dr_{0} \tag{13}\]  
As has been noted, we have derived the probability of successful preamble detection for NB-IoT via LEO satellite-based networks based on stochastic geometry theory. Moreover, one of the techniques used in the NB-IoT CE mechanism is the repetition of the preamble.