## 4 System Model Based on Power Splitting and NOMA
### Outage Probability, Throughput and Sum-Throughput  
According to Equation (22), the received signal to noise ratio (SNR) at \(IoT_{R}\) node is given by:  
\[\gamma_{IoT_{R}}=\frac{(1-\varepsilon)P_{s}|h_{IoT_{R}}|^{2}}{\sigma_{IoT_{R} }^{2}}=(1-\varepsilon)\delta|h_{IoT_{R}}|^{2} \tag{27}\]  
where \(\delta\triangleq\frac{P_{s}}{\sigma_{IoT_{R}}^{2}}\) represents the transmit signal-to-noise ratio (SNR) from the source. According to Equation (25), the received SNR with \(x_{IoT_{R}}\) and \(x_{s}\) at the receiving source user is given by:  
\[\gamma_{s_{src}}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}P_{IoT_{R}}|h_{s_{src}} |^{2}}{\phi_{1}P_{IoT_{R}}|h_{s_{src}}|^{2}+\sigma_{s_{src}}^{2}} \tag{28}\]  
\[\gamma_{s_{src}}=\frac{\phi_{1}P_{IoT_{R}}|h_{s_{src}}|^{2}}{\sigma_{s_{src}} ^{2}} \tag{29}\]  
where \(\gamma_{s_{src}}^{x_{IoT_{R}}\to x_{s}}\) is the SNR required at the receiving source user to decode and cancel \(IoT_{R}\) information i.e., \(x_{IoT_{R}}\). The received SNR at the receiving IoT user node associated with symbol \(x_{IoT_{R}}\) is given by:  
\[\gamma_{IoT_{src}}=\frac{\phi_{2}P_{IoT_{R}}|h_{IoT_{src}}|^{2}}{\phi_{1}P_{IoT _{R}}|h_{IoT_{src}}|^{2}+\sigma_{IoT_{src}}^{2}} \tag{30}\]As we can see from Figure 2, the data transmission is break down into two separate hops which are independent of each other. Hence, the outage occurs only if source to \(IoT_{R}\) path and \(IoT_{R}\) to corresponding destination path fails to satisfy the SNR constraint. Therefore, the outage probability of the source node can be given as:  
\[P_{Out_{S}}=Pr(min(\gamma_{IoT_{R}},\gamma_{s_{max}})\leq\psi) \tag{31}\]  
where \(\psi=2^{R}-1\) is the lower threshold for SNR i.e., outage probability, \(R\) being the target data rate. Similarly, the outage probability of the \(IoT_{R}\) node can be given as:  
\[P_{Out_{IoT_{R}}}=Pr(min(\gamma_{s_{max}}^{\times_{IoT_{R}}\to x_{s}}, \gamma_{IoT_{Tex}})\leq\psi) \tag{32}\]  
The throughput of the source node can be given as:  
\[Thr_{S}=\frac{(1-P_{Out_{S}})R}{2} \tag{33}\]  
where \(R\) is measured in bits per second per hertz. The throughput of the IoT relay node can be given as:  
\[Thr_{IoT_{R}}=\frac{(1-P_{Out_{IoT_{R}}})R}{2} \tag{34}\]  
The factor \(1/2\) in Equations (33) and (34) is originated by the predicament that the two transmission phases are involved in the system. Therefore, the sum-throughput of the whole system can be given as:  
\[Thr=Thr_{S}+Thr_{IoT_{R}}=(1-P_{Out_{S}})\frac{R}{2}+(1-P_{Out_{IoT_{R}}}) \frac{R}{2} \tag{35}\]  
**Theorem 3**. : _The outage probability and throughput of the source node using PS and NOMA can be expressed as:_  
\[P_{Out_{S}}=1-2\sqrt{\frac{\lambda_{h}\lambda_{g}(1-\varepsilon)x_{0}}{a}}K_{ 1}\bigg{(}2\sqrt{\frac{\lambda_{h}\lambda_{g}(1-\varepsilon)x_{0}}{a}}\bigg{)} +\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}(\lambda_{h}x_{0})^{n+1}E_{n+2}\big{(} \frac{(1-\varepsilon)\lambda_{g}}{a}\big{)} \tag{36}\]  
\[\gamma_{s_{src}}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}P_{IoT_{R}}|h_{s_{src}} |^{2}}{\phi_{1}P_{IoT_{R}}|h_{s_{src}}|^{2}+\sigma_{s_{src}}^{2}} \tag{28}\]0  
_where \(x_{0}=\frac{\psi}{(1-\varepsilon)\psi}\), \(a=\varepsilon\eta\phi_{1}\), \(K_{1}(. )\) is a first-order modified Bessel function of the second kind, and \(E_{n}(a)=\int_{y=1}^{\infty}y^{-n}e^{-ay}dy\) is the exponential integral of order \(n\)._  
Proof. : The detailed proof is formulated in Appendix C.  
**Theorem 4**. : _The outage probability and throughput of the IoT node using PS and NOMA can be expressed as:_  
\[\gamma_{s_{src}}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}P_{IoT_{R}}|h_{s_{src}} |^{2}}{\phi_{1}P_{IoT_{R}}|h_{s_{src}}|^{2}+\sigma_{s_{src}}^{2}} \tag{28}\]1  
\[\gamma_{s_{src}}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}P_{IoT_{R}}|h_{s_{src}} |^{2}}{\phi_{1}P_{IoT_{R}}|h_{s_{src}}|^{2}+\sigma_{s_{src}}^{2}} \tag{28}\]2  
_where \(c=\frac{\psi}{(\phi_{2}-\phi_{1}\psi)\psi}\), \(b=\eta\delta\varepsilon\)._  
Proof.