## 3 RACH Success Probability and Power Consumption Model for LEO Satellite-Based NB-IoT Networks
### RACH Success Probability Analysis  
As mentioned above, the traffic model of NB-IoT terminals is modeled as a Poisson distribution. On this basis, we give the probability of preamble collision and derive the RACH success probability in LEO satellite-based networks. We consider that \(N_{acc}\) NB-IoT terminals participate in the RA procedure to contend \(S\) available preambles. Therefore, the number of NB-IoT terminals \(N_{s}\) that successfully access the network in this RA procedure can be given by the following:  
\[N_{s}=N_{acc}\cdot e^{\frac{N_{acc}}{S}} \tag{15}\]In the \(i\)th RA procedure, the number of NB-IoT terminals which have the \(n\)th repetition can be denoted by the following:  
\[N_{i}=\sum\limits_{n=1}^{N_{rep}^{\max}}N_{i}[n] \tag{16}\]  
where \(N_{rep}^{\max}\) is the maximum number of repetitions. The number of NB-IoT terminals that successfully and unsuccessfully acquire access in the \(i\)th RA procedure can be derived by Equations (15) and (16), respectively, as shown in Equations (17) and (18):  
\[N_{i}^{s}=N_{i}\cdot e^{-\frac{N_{i}}{S}}=\sum\limits_{n=1}^{N_{rep}^{\max}} \left(N_{i}[n]\cdot e^{-\frac{N_{i}}{S}}\right) \tag{17}\]  
\[N_{i}^{f}=N_{i}\cdot\left(1-e^{-\frac{N_{i}}{S}}\right)=\sum\limits_{n=1}^{N_ {rep}^{\max}}\left(N_{i}[n]\cdot\left(1-e^{-\frac{N_{i}}{S}}\right)\right) \tag{18}\]  
If data packet collision occurs, the NB-IoT terminal will restart the RA procedure in the next time slot after the back-off window. Due to the different sizes of the back-off windows, the terminal starts the RA procedure in a different time slot. Without a loss of generality, we consider that the uniform back-off strategy is employed, and the probability of different terminals' back-off to a time slot is the same. The probability can be denoted as \(P\):  
\[P=\frac{1}{Len}=\left[\frac{T_{\text{RAO}}}{W_{BO}+T_{RAR}+RTD+\text{Start RAR}}\right] \tag{19}\]  
where \(Len\) is the maximum number of RA procedures that NB-IoT terminals can restart during a RAO, \(W_{BO}\) is the size of the back-off window, \(T_{RAR}\) is the size of the RAR window, \(RTD\) signifies the Round-Trip Delay (RTD), and StartRAR denotes the active timer of the waiting RAR which value is 41 ms in the NTN scenario. When \(n>1\), \(N_{i}[n]\) represents the number of NB-IoT terminals that start the \(i\)th RA procedure which can be expressed as follows:  
\[N_{i}[n]=P\cdot\sum\limits_{k}^{Len}N_{i-k}^{f}[n-1]\ \ 2\leq n\leq N_{\text{rep}}^{\max} \tag{20}\]  
From this equation, we can deduce the number of NB-IoT terminals \(N_{i}^{s}[n]\) which successfully acquire access in the \(n\)th random access procedure as follows:  
\[N_{i}^{s}[n]=N_{i}[n]\cdot e^{-\frac{N_{i}}{S}} \tag{21}\]  
Therefore, the probability of successful access can be given by the following:  
\[ \begin{array}{l}p_{u}=\frac{T_{\max}}{\sum\limits_{i=1}^{T_{\max}}N_{i}^{s}} =\frac{T_{\max}}{\sum\limits_{n=1}^{T_{\max}}N_{\text{rep}}^{\max}}N_{i}^{s}[ n]=\frac{T_{\max}}{\sum\limits_{n=1}^{T_{\max}}N_{\text{rep}}^{\max}}\sum \limits_{k=1}^{Len}P\cdot N_{i-k}^{f}[n-1]\cdot e^{-N_{i}/S}}{N_{acc}}\\ =\frac{T_{\max}}{\sum\limits_{i=1}^{T_{\max}}N_{\text{rep}}^{\max}}\sum \limits_{k=1}^{Len}\frac{W_{BO}+T_{\max}\cdot e^{\frac{\text{i}}{S}}+\text{ RTD}+41}{R_{AO}}N_{i-k}^{f}[n-1]\cdot e^{-N_{i}/S}}{N_{acc}}\end{array}  \tag{22}\]  
where \(E_{Trans}\) is the power consumption during preamble repetition and \(E_{connect}\) represents the power consumption in the back-off state and in receiving RAR. Considering that NB-IoT terminals are usually deployed in hard-to-reach areas, such as mountains and oceans, power consumption requirements are more stringent in these scenarios. Therefore, finding an efficient CE mechanism can strike a balance between power consumption and RACH success probability. This balance ensures the most appropriate relationship of RACH success probability and power consumption for different applications in LEO satellite-based IoT networks.