## 3 System Performance Analysis
### Outage Probability  
**Theorem 1**. : _The outage events at coupled IoT relays \(R_{1}\) and \(R_{2}\) occur if the minimum of instantaneous bit-rate thresholds as given by (18) and (19) for, respectively, transmission block time period \(t=\{odd,even\}\) could not reach IoT devices' threshold \(\mathcal{R}\). In other words, the OP at IoT relay \(R_{r}\) (\(r=\{1,2\}\)) are given as follows:_  
\[OP_{R_{r}}(t)=1-\text{Pr}\left\{\underset{i=\{N,\ldots,1\}}{\min}\left\{R_{ R_{r}-x_{i}}^{T_{1}^{(i)}}\right\}\geq\mathcal{R}\right\}, \tag{40}\]  
_where \(\mathcal{R}\) is pre-defined IoT devices' bit-rate threshold._By applying CDF over Rayleigh distribution, we obtain OP at coupled IoT relays in closed-form as follows:  
\[OP_{R_{r}}(t)=1-\exp\biggl{(}-\frac{\gamma}{(1-\lambda_{R_{r}}) \beta\rho\sigma_{S,R_{r}}}\biggr{)}, \tag{41}\] \[s.t. \beta_{i}=\left[ \begin{array}{cc}\alpha_{i}-\gamma\sum\limits_{j =1}^{i-1}\alpha_{j},&\quad\text{if $i>1$,}\\ \alpha_{1},&\quad\text{if $i=1$,}\end{array} \right.\] (42) \[\beta=\min\limits_{i=\{N,\ldots,1\}}\{\beta_{i}\}, \tag{43}\]  
where SINR threshold \(\gamma=2^{2\mathcal{R}}-1\), either \(r=1\) for transmission block \(t=\) odd or \(r=2\) for transmission block \(t=\) even. **Lemma 1**. : _OP at coupled IoT relays may be improved by equipping multi-antennas at IoT hub \(S\) and relays by the number of antennas \(A_{S}>1\) and \(A_{\mathcal{R}}>1\), respectively. By adopting transmit antenna selection (TAS) at IoT hub \(S\) and selection combining (SC) at coupled relays, OP at coupled IoT relay in multi-input multi-output (MIMO) scenario could be obtained as shown in (A2)._  
See Appendix A for proof. From expression (40), the study presents an algorithm for Monte Carlo simulation at IoT relay \(R_{r}\) in transmission block \(t\) as Algorithm 2. ```
0: Initialize the parameters as distances \(d_{S,R_{r}}\) and \(d_{R_{r},D_{n}}\) (\(r=\{1,2\}\) and \(n=\{N,\ldots,1\}\)), path-loss exponent factor \(\varepsilon\), PA factors \(\alpha_{i}\) as (9), randomly generate \(1\times 10^{6}\) samples for each fading channel over Rayleigh distribution;
0: Simulation (Sim) results of OP at the relay \(R_{r}\) (\(r=\{1,2\}\)). 1: Calculate SINR at IoT relay \(R_{r}\) by applying (12)-(15);
2: Calculate achievable bit-rate at IoT relay \(R_{r}\) by applying (16) or (17);
3: Find the minimum of achievable data rate \(\min\limits_{i=\{N,\ldots,n\}}\left\{R_{R_{r}-x_{i}}^{T_{1}^{(i)}}\right\}\) by applying (18) or (19);
4: Initialize variable \(count\gets 0\);
5:for\(l=1\) to \(1\times 10^{6}\) samples do
6:if\(\min\limits_{i=\{N,\ldots,1\}}\left\{R_{R_{r}-x_{i}}^{T_{1}^{(l)}}\right\}\geq \mathcal{R}\)then
7:\(count\gets count++\);
8:endif
9:endfor
10:return OP at the IoT relay \(R_{r}\) in transmission block \(t=\{odd,even\}\) as given \(OP_{R_{r}}(t)=1-count\Big{/}10^{6}\). ```  
**Algorithm 2** The algorithm for investigation OP at IoT relay \(R_{r}\) for \(r=\{1,2\}\) in transmission block \(t=\{odd,even\}\)  
**Theorem 2**. : _The outage event at IoT device \(D_{n}\) (\(n\in N\)) in transmission block \(t\) occurs, on one hand, if IoT relay \(R_{r}\) cannot decode at least a message \(x_{i}\) (\(x_{i}\in\{x_{N},\ldots,x_{n}\}\)) in the received signal as (5) for \(t=\) odd or (8) for \(t=\) even.