# Cooperative Admission Control with Network Coding in 5G Underlying D2D-Satellite Communication
## 2 System Model  
We considered a single cell for simplicity with the base station at the center of the cell. The same can also be used for multiple cells. Internet of things (IoT's) and mobile users are distributed stochastically around the base station with distance \(\mathsf{d}_{cell_{1}}\). The model is shown in the Figure 1. The smart devices are equipped with both terrestrial and satellite antennas. D2D users act as D2D transmitters when the mode is assigned as D2D with other receivers in close proximity as per SINR calculation. The D2D receivers are statistically distributed around the transmitter using the same frequency as used by cellular links. Let \(l\) represents all the links in our complete network model, i.e., SAT, mobile, and D2 Dlinks. The \(R\) represents all the users selected in the network known as admission users and includes D2D \(X_{d}^{l}\) users, \(X_{c}^{l}\) cellular users, and \(X_{s}^{l}\) satellite. The channel gain \(h\) for \(l^{th}\) links is modeled as:  
\[h=h^{*}\beta G_{e}(\frac{d_{o}}{d})^{\eta} \tag{1}\]  
where \(d\) is distance between transmitter to receiver and \(d_{o}\) is antenna far field reference distance. \(\eta\) is path loss factor and \(h^{*}\) is exponential random channel.  
The \(P_{l_{d}}^{l}\) and \(P_{l_{c}}^{l}\) is the optimum power allocated by our algorithm for transmission in D2D and cellular modes, respectively. The SINR calculation of terrestrial links represented as  
\[SINR_{(l_{d},l_{c})}=\frac{|h_{(l_{d}/l_{c})}|^{2}P_{(l_{d}/l_{c})}^{l}}{N_{t}+ \sum_{l=1}^{l-(l_{d}-1/l_{c}-1)}|h_{(l-(l_{d}/l_{c})}|^{2}P_{l-(l_{d}/l_{c})}}. \tag{2}\]  
Same terrestrial frequency resources are allocated to SAT links. The SINR of satellite users \(X_{s}\) selected in \(l\) links are mathematically represented as  
\[SINR_{(l_{s})}=\frac{|h_{l_{s}}|^{2}P_{l_{s}}}{N_{t}+\sum_{l=1}^{l-s}|h_{(l-l_{ s})}|^{2}P_{(l-l_{s})}}. \tag{3}\]  
The gain of the SAT-terrestrial link includes a space environment for calculating power received by satellite. The \(P_{l_{s}}^{l}\) is optimum power allocated by terrestrial user for \(l_{s}\) links. The maximum output power of base station and satellite links denoted by \(P_{enb}^{max}\) and \(P_{SAT}^{max}\). An integer indicator is defined for mode selection, which is defined as.  
\[\begin{split}\
u^{l}=-1,&\quad\quad\text{SAT}\\ 0,&\quad\quad\text{D2D}\\ 1,&\quad\quad\text{cellular}.\end{split} \tag{4}\]  
Figure 1: Network Model.  
The capacity of the network is defined mathematically as  
\[\begin{split}\Delta_{T}^{l}=&\sum_{0}^{l}-(1+v^{l})*(1-v ^{l})log(1+(SINR)_{X_{d}}^{l})+\\ & 1/2*(1+v^{l})*(v^{l})log(1+(SINR)_{X_{c}}^{l})+\\ &(1/2)*v^{l}*(v^{l}-1)log(1+(SINR)_{X_{s}}^{l}).\end{split} \tag{5}\]  
The Coarraza-Vatalaro channel model is used, a combination of Rice and lognormal distribution, and best suits for terrestrial-LEO earth orbit communication. The pdf of Rician distribution conditioned on shadowing \(s\) is given as:  
\[P(r/s)= 2(K+1)\frac{r}{s^{2}} \tag{6}\] \[\exp\left\{-(K+1)\frac{r^{2}}{s^{2}}-K\right\}I_{o}(2\frac{r}{s} \sqrt{K(K+1)}). \tag{7}\]  
where K is Rician factor and I\({}_{o}\) is Bessel function with zero order. Two link losses are considered, i.e., free space loss (FSL) and atmospheric loss [24]. FSL in link distance d given as:  
\[F_{(frespacloss)}=20*log\frac{4\pi d}{\times}. \tag{8}\]