## 2 System Model and Formulation
### Formulations
\[\alpha_{N}>\ldots>\alpha_{1}, \tag{10}\] \[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]0  
In another term of NOMA theory, the SIC mechanism was adopted at coupled IoT relays. Please note that the message \(x_{N}\) owned the biggest PA factor \(\alpha_{N}\). The message \(x_{N}\) thereby is prioritized SIC first by treating the other messages \(x_{j}=\{x_{1},\ldots,x_{N-1}\}\) and AWGN \(n_{R_{r}}\) as interference. The SIC mechanism repeats until successfully decoded the last message \(x_{1}\) since it was allocated the smallest PA factor \(\alpha_{1}\). Therefore, signal to interference plus noise ratio (SINR) obtained at the IoT relay \(R_{1}\) in odd transmission block time period \(T^{(odd)}\), when \(R_{1}\) decoded the message \(x_{i}\) (\(i=\{N,\ldots,1\}\)) by treating the messages \(x_{j}=\{x_{1},\ldots,x_{i-1}\}\) and AWGN \(n_{R_{1}}\) as interference, as follows:  
\[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]1 \[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]2  
where SNR \(\rho=\frac{P}{N_{0}}\). Similarly, SINR obtained at the IoT relay \(R_{2}\) in even transmission block time period \(T^{(even)}\), when \(R_{2}\) decoded the message \(x_{i}\) (\(i=\{N,\ldots,1\}\)) by treating the messages \(x_{j}=\{x_{1},\ldots,x_{i-1}\}\) and AWGN \(n_{R_{2}}\) as difference, as follows:  
\[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]3 \[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]4  
An instantaneous bit-rate threshold is achievable after the coupled IoT relays \(R_{1}\) and \(R_{2}\) decoded the message \(x_{i}\) in the received signal given by (5) for odd transmission block time period \(T^{(odd)}\) and (8) for even transmission block time period \(T^{(even)}\) as follows:  
\[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]5 \[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]6  
Minimum of instantaneous bit-rates at relay \(R_{1}\) in odd transmission block time period \(T^{(odd)}\) and relay \(R_{2}\) in even transmission block time period \(T^{(even)}\) express as follows:  
\[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]7 \[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]8  
In the second time slot, the IoT relays \(R_{1}\) for \(T^{(odd)}_{2}\) and \(R_{2}\) for \(T^{(even)}_{2}\) retrieves, re-encodes and forwards the superimposed signals to the IoT devices. There are two feature forwarding protocols (i.e., amplify-and-forward (AF) and DF). This study employed the DF protocol at IoT relays \(R_{1}\) and \(R_{1}\) to ensure that \(R_{1}\) and \(R_{1}\) received and decoded the messages successfully. After receiving the signals as given by (5) for odd transmission block and (8) for even transmission block, the IoT relays \(R_{1}\) and \(R_{2}\) decoded the messages of devices and retrieved the messages and then forwarded them to the devices using the EH as given by (1) for odd transmission block \(T^{(odd)}\) and (4) for even transmission block \(T^{(even)}\). The received signal at IoT device \(D_{n}\) for \(n\in N\) is expressed as follows:  
\[EH_{S,R_{2}}^{T_{(odd)}^{(odd)}} =\eta\lambda_{R_{1}}P\sigma_{S,R_{2}},\]9 \[EH_{S,R_{1}}^{T_{(even)}^{(even)}} =\eta\lambda_{R_{2}}P\sigma_{S,R_{1}},\]0  
where \(n_{D_{n}}\) is AWGN at the terminal device \(D_{n}\), i.e., \(n_{D_{n}}\sim CN(0,N_{0})\). IoT device \(D_{n}\) also adopted the SIC mechanism to decode its own message \(x_{n}\).