# Intelligent Beam-Hopping-Based Grant-Free Random Access in Secure IoT-Oriented Satellite Networks
## 3 Beam-Hopping-Based Grant-Free Random Access
### Resource-Hopping Mechanism
The chance of a target device experiencing a resource collision with active devices, using an activation probability \(v\) and a quantity of available orthogonal resources \(R\) during a time slot, is expressed as \[\beta_{\text{random}}\left(K,v\right)=\binom{G-1}{K}\left(\frac{v}{R}\right)^{K} \left(1-\frac{v}{R}\right)^{G-1-K} \tag{7}\]  
This paragraph posits that a target device is functional and utilizes a certain orthogonal resource within a designated time window. Consequently, our focus is solely on the remaining \((G-1)\) devices. The activation is characterized by the binomial distribution \(\binom{G-1}{S}v^{G}(1-v)^{G-1-g}\), representing the probability that \((G-1-g)\) devices are active while \((G-1-g)\) devices remain inactive. This likelihood is shown as a percentage. Furthermore, \(\binom{g}{K}(1/R)^{K}(1-[1/R])^{g-K}\) represents the likelihood that \(K\) active devices among \(g\) active devices share the same resource as the target device, while the remaining \((g-k)\) devices do not. \[\beta_{\text{random}}\left(K,v\right) =\sum_{g=K}^{G-1}\binom{g-1}{g}v^{g}(1-v)^{G-1-g}\binom{g}{K} \left(\frac{1}{R}\right)^{K}\left(1-\frac{1}{R}\right)^{g-K} \tag{8}\] \[=\binom{v}{R}^{K}\sum_{g=K}^{G-1}\binom{G-1}{g}\binom{g}{K}\left( v-\frac{v}{R}\right)^{g-K}(1-v)^{G-1-g}\] \[=\binom{g-1}{K}\left(\frac{v}{R}\right)^{K}\sum_{g=K}^{g-1}\binom {g-1-K}{g-K}\left(v-\frac{v}{R}\right)^{g-K}(1-v)^{g-1-g}\] \[=\binom{g-1}{K}\left(\frac{v}{R}\right)^{K}\left(1-\frac{v}{R} \right)^{g-1-K}\]  
From (8), the probability that \((L_{device0},\ldots,L_{deviceG-1})\) resources are collided by \(G\) interfering devices, respectively, is given by  
\[\left[Q^{l}\right]_{i,j}=\left(l\times i+j\right)\text{ mod }R \tag{3}\]0 \[\left[Q^{l}\right]_{i,j}=\left(l\times i+j\right)\text{ mod }R \tag{3}\]1  
where \(L_{device0}+\ldots+L_{deviceG-1}=R\). The expected amount of resources collided by \(K\) interfering devices for a specified activation probability \(v\) is also expressed as  
\[\left[Q^{l}\right]_{i,j}=\left(l\times i+j\right)\text{ mod }R \tag{3}\]2  
(3) Group-Divided Hopping (GDH) Pattern: Analogous to orthogonal hopping, once the device accidentally chooses the single preamble, the hopping scheme that serves as data transmission is the same, hence preventing any resource clashes. Nevertheless, as each group employs the same method that filled the time-frequency slot, and as there is a likelihood that an individual device engages in a BH-GFRA with interfering devices from a different group using the activation probability \(v\) as an example, there is no difference in the resource-hopping matrix between any of the resource groups. Due to the fact that all devices in each group are operational, additional devices that select the same preamble are subject to collisions with two devices that disrupt the data transmission throughout the process. Also, the other group will not experience the same collisions due to the GDH hopping scheme. A complete packet containing \(R\) resources is subjected to interference from \(K\) devices from various groups with a probability of \(\delta_{\text{group}}\left(K,v\right)\), which is described as  
\[\left[Q^{l}\right]_{i,j}=\left(l\times i+j\right)\text{ mod }R \tag{3}\]3