## 3 System Model Based on Time Switching and NOMA
### Outage Probability, Throughput and Sum-Throughput  
According to Equation (1), the received signal to noise ratio (SNR) at \(IoT_{R}\) is given by:  
\[\hat{\gamma}_{IoT_{R}}=\frac{P_{s}|h_{IoT_{R}}|^{2}}{\sigma_{IoT_{R}}^{2}}= \hat{\delta}|h_{IoT_{R}}|^{2} \tag{7}\]  
where \(\hat{\delta}\triangleq\frac{P_{s}}{\sigma_{IoT_{R}}^{2}}\) represents the transmit signal-to-noise ratio (SNR) from the source. According to Equation (4), the received SNR with \(x_{IoT_{R}}\) and \(x_{s}\) at the receiving source user is given by:  
\[\hat{\gamma}_{src}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}\hat{P}_{IoT_{R}}|h_{ src}|^{2}}{\phi_{1}\hat{P}_{IoT_{R}}|h_{src}|^{2}+\sigma_{src}^{2}} \tag{8}\]  
\[\hat{\gamma}_{src}=\frac{\phi_{1}\hat{P}_{IoT_{R}}|h_{src}|^{2}}{\sigma_{src} ^{2}} \tag{9}\]  
where \(\hat{\gamma}_{src}^{x_{IoT_{R}}\to x_{s}}\) is the SNR required at \(x_{s}\) to decode and cancel \(x_{IoT_{R}}\). The received SNR at IoT user associated with symbol \(x_{IoT_{R}}\) is given by:  
\[\hat{\gamma}_{IoT_{nc}}=\frac{\phi_{2}\hat{P}_{IoT_{R}}|h_{IoT_{nc}}|^{2}}{\phi _{1}\hat{P}_{IoT_{R}}|h_{IoT_{nc}}|^{2}+\sigma_{IoT_{nc}}^{2}} \tag{10}\]  
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 can be given as:  
\[\hat{P}_{Out_{S}}=Pr(min(\hat{\gamma}_{IoT_{R}},\hat{\gamma}_{src})\leq\hat{ \psi}) \tag{11}\]  
where \(\hat{\psi}=2^{R}-1\) is the lower threshold for SNR i.e., outage probability. Similarly, the outage probability of the IoT relay node \(IoT_{R}\) can be given as:  
\[\hat{P}_{Out_{IoT_{R}}}=Pr(min(\hat{\gamma}_{src}^{x_{IoT_{R}}\to x_{s}}, \hat{\gamma}_{IoT_{nc}})\leq\hat{\psi}) \tag{12}\]  
The throughput of the source node can be given as:  
\[Thr_{S}=\frac{(1-\hat{P}_{out_{S}})(1-\alpha)R}{2} \tag{13}\]  
where \(R\) is the transmission rate in bits per second per hertz. The throughput of the IoT relay node \(IoT_{R}\) can be given as:  
\[Thr_{IoT_{R}}=\frac{(1-\hat{P}_{Out_{IoT_{R}}})(1-\alpha)R}{2} \tag{14}\]  
Therefore, the sum-throughput of the whole system using TS and NOMA can be given as:  
\[Thr=T\hat{Thr}_{S}+T\hat{Thr}_{IoT_{R}}=\frac{(1-\hat{P}_{Out_{S}})(1-\alpha)R }{2}+\frac{(1-\hat{P}_{Out_{IoT_{R}}})(1-\alpha)R}{2} \tag{15}\]  
**Theorem 1**. : _The outage probability and throughput of the source node using TS and NOMA can be expressed as:_  
\[\hat{P}_{Out_{S}}=1-2\sqrt{\frac{\lambda_{h}\lambda_{g}x_{0}}{k}}K_{1}\Big{(}2 \sqrt{\frac{\lambda_{h}\lambda_{g}x_{0}}{k}}\Big{)}+\sum_{n=0}^{\infty}\frac{(-1 )^{n}}{n!}(\lambda_{h}x_{0})^{n+1}E_{n+2}\Big{(}\frac{\lambda_{g}}{k}\Big{)} \tag{16}\]  
\[\hat{\gamma}_{src}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}\hat{P}_{IoT_{R}}|h_{ src}|^{2}}{\phi_{1}\hat{P}_{IoT_{R}}|h_{src}|^{2}+\sigma_{src}^{2}} \tag{8}\]0  
_where, \(x_{0}=\frac{\hat{\Psi}}{\hat{\xi}}\), \(k=\frac{2\alpha\eta\phi_{1}}{(1-\alpha)}\), \(K_{1}(. )\) 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 given in Appendix A. **Theorem 2**. : _The outage probability and throughput of the IoT relay node using TS and NOMA can be expressed as:_  
\[\hat{\gamma}_{src}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}\hat{P}_{IoT_{R}}|h_{ src}|^{2}}{\phi_{1}\hat{P}_{IoT_{R}}|h_{src}|^{2}+\sigma_{src}^{2}} \tag{8}\]1  
\[\hat{\gamma}_{src}^{x_{IoT_{R}}\to x_{s}}=\frac{\phi_{2}\hat{P}_{IoT_{R}}|h_{ src}|^{2}}{\phi_{1}\hat{P}_{IoT_{R}}|h_{src}|^{2}+\sigma_{src}^{2}} \tag{8}\]2  
_where, \(d=\frac{\hat{\psi}}{(\phi_{2}-\phi_{1}\hat{\psi})! }\), \(l=\frac{2\alpha\eta P_{s}}{(1-\alpha)}\)_  
Proof. : The detailed proof is given in Appendix B. Combining Equations (17) and (19), we finally get the analytical equation for the sum-throughput of the proposed system using TS and NOMA.