# Intelligent Beam-Hopping-Based Grant-Free Random Access in Secure IoT-Oriented Satellite Networks
## 3 Beam-Hopping-Based Grant-Free Random Access
### Optimization of Activation Factors by Exploiting Space-Time Distribution Constraint  
In the abovementioned system model, the construction and constraints analyses show that the random access procedure is affected by both the number of users initiating access and the effects of retransmission on users in the cycle after a collision. During the access process, the user's probability of avoiding retreat is denoted as \(\rho_{t}\). After a device collision occurs, the retransmission timing is determined using a discrete Gaussian distribution. Hence, the total number of users involved in the access process can be represented as  
\[ \begin{split} U_{u}(\mathcal{L}_{s},t)=& U_{num}( \mathcal{L}_{s},t)+U_{num}(\mathcal{L}_{s},t)*\rho_{t-1}-\\ & U_{num}(\mathcal{L}_{s},t)*\rho_{t+1}\end{split}  \tag{12}\]  
where the user's avoidance probability is set to be \(\rho\) during the access process, and after the user collision, the user is obtaining retransmission timing in the form of a discrete Gaussian distribution. The user's retransmission timing can be determined by considering the system occupancy. This paper calculates the probability of retransmission in a collision by determining the number of retransmissions. The user who needs to access it again can choose this type of preamble, leading to a more balanced distribution of resources and an overall improvement in performance. The disparate propagation delays between users and the satellite result in varying times for users to receive access message feedback. In a given moment, user feedback may be received simultaneously by multiple ROs that collided in the previous access cycle. Accordingly, this paper assumes that the number of colliding users at a given time follows a distribution with density function \(\rho_{\mathrm{T}}\). This distribution is described by a Gaussian distribution with a mean equal to the average transmission delay, \(\mu_{t}\), and a user set variance of \(\sigma^{2}\). From the aforementioned grant-free access process, it can be observed that the time window size is represented by \(t_{win}\). Consequently, the change in user access following a user collision is given by \(U_{u}(L_{s},t)\). Furthermore, the time range within which the collision user will be obtained is set within the time window of \([-t_{win}\ /2,t_{win}\ /2]\). \[U_{col}(\mathcal{L}_{s},t)=\sum_{i=\mu_{t}-t_{win}/2}^{\mu_{t}+t_{win}\ /2}U_{num}( \mathcal{L}_{s},t-T-i)*\rho_{T-i} \tag{13}\]  
Figure 3: An example of a random hopping pattern-based resource scheme. where \(U_{col}(\mathcal{L}_{s},t)\) is the number of collisions. \[ \begin{split}& U_{u}(\mathcal{L}_{s},t)=\\ & U_{\text{num}}\left(\mathcal{L}_{s},t\right)+U_{\text{num}}\left( \mathcal{L}_{s},t\right)*\rho_{t-1}-U_{\text{num}}\left(\mathcal{L}_{s},t \right)*\rho_{t+1}+U_{\text{col}}\left(\mathcal{L}_{s},t\right)\end{split}  \tag{14}\]  
By combining the previously derived collision probability for sensitive users, it can be seen that this formula is  
\[ \begin{split}& U_{u}(\mathcal{L}_{s},t)=U_{\text{num}}\left( \mathcal{L}_{s},t\right)+U_{\text{num}}\left(\mathcal{L}_{s},t\right)*\rho_{t- 1}-U_{\text{num}}\left(\mathcal{L}_{s^{\prime}},t\right)*\rho_{t+1}+\\ &\sum_{i=t_{t}-t_{\text{win}}\,/2}^{\mu_{t}+t_{\text{win}}\,/2}U_ {\text{num}}\left(\mathcal{L}_{s},t-T-i\right)*(1-\alpha_{\text{sens}})\rho_{ T-i}\end{split}  \tag{15}\]  
where \(\rho_{T}\) is described by \(f(T)=\frac{1}{\sqrt{2\pi}}\exp\Bigl{(}-\frac{(T-i)^{2}}{2\sigma^{2}}\Bigr{)}\). Accordingly, the formula for the change in the total number of users can be derived as follows:  
\[