## 3 System Performance Analysis
### Outage Probability at Devices  
**Theorem 2**. : _The outage event in an odd or even transmission block at devices \(D_{n}\) for \(n=\{1,\ldots,N\}\) occurs when, on the one hand, the relays \(R_{1}\) and \(R_{2}\) for odd and even transmission blocks, respectively, cannot successfully decode at least data symbol \(x_{i}\) for \(i=\{N,\ldots,n\}\) from the best received signal \(\max\limits_{[A_{0}\times A_{1}]}\Biggl{\{}\left|\mathbf{Y}_{R_{1}}^{\left( \Upsilon_{1}^{(odd)}\right)}\right|\Biggr{\}}\) as (5) or \(\max\limits_{[A_{0}\times A_{1}]}\Biggl{\{}\left|\mathbf{Y}_{R_{2}}^{\left( \Upsilon_{1}^{(even)}\right)}\right|\Biggr{\}}\) as (20). On the other hand, the coupled relays can successfully decode all data symbol \(x_{i}\) for \(i=\{N,\ldots,n\}\), but device \(D_{n}\) cannot successfully decode at least data symbols \(x_{i}\) for \(i=\{N,\ldots,n\}\) from the best received signal \(\max\limits_{[A_{1}\times A_{2}]}\Biggl{\{}\left|\mathbf{Y}_{n}^{\left( \Upsilon_{2}^{(odd)}\right)}\right|\Biggr{\}}\) as (11) for an odd transmission block or \(\max\limits_{[A_{1}\times A_{2}]}\Biggl{\{}\left|\mathbf{Y}_{n}^{\left( \Upsilon_{2}^{(even)}\right)}\right|\Biggr{\}}\) as (24) for an even transmission block._Therefore, the OP at devices in an odd or even transmission block is expressed as follows:_  
\[ \begin{split}\mathrm{OP}_{n}^{(\theta)}&=\sum_{i=N}^{n} \left\{\Pr\Biggl{\{}\max_{[A_{0}\times A_{1}]}\Biggl{\{}\mathbf{R}_{\mathcal{R}-x_{ i}}^{\left(\tau_{1}^{(\theta)}\right)}\Biggr{\}}<R_{i}^{*}\Biggr{\}}\underbrace{ \prod_{k=N}^{i+1}\Pr\Biggl{\{}\max_{[A_{0}\times A_{1}]}\Biggl{\{}\mathbf{R}_{ \mathcal{R}-x_{k}}^{\left(\tau_{1}^{(\theta)}\right)}\Biggr{\}}\geq R_{k}^{*} \Biggr{\}}}_{\text{for }i<N}\right\}\\ &+\sum_{i=N}^{n}\left\{\Pr\Biggl{\{}\max_{[A_{1}\times A_{2}]} \Biggl{\{}\mathbf{R}_{\mathcal{R}-x_{i}}^{\left(\tau_{2}^{(\theta)}\right)}\Biggr{\}} <R_{i}^{*}\Biggr{\}}\underbrace{\prod_{k=N}^{i+1}\Pr\Biggl{\{}\max_{[A_{1} \times A_{2}]}\Biggl{\{}\mathbf{R}_{\mathcal{R}-x_{k}}^{\left(\tau_{2}^{(\theta)} \right)}\Biggr{\}}\geq R_{k}^{*}\Biggr{\}}}_{\text{for }i<N}\right.\\ &\times\prod_{t=N}^{n}\Pr\Biggl{\{}\max_{[A_{0}\times A_{1}]} \Biggl{\{}\mathbf{R}_{\mathcal{R}-x_{t}}^{\left(\tau_{1}^{(\theta)}\right)}\Biggr{\}} \geq R_{t}^{*}\Biggr{\}}\Biggr{\}}\end{split}  \tag{36}\]  
_From (36), the OP at user \(D_{n}\) is obtained in closed-form as follows:_  
\[ \begin{split}\mathrm{OP}_{n\in N}^{(\theta)}&=\sum_{i=N }^{n}\left(\underbrace{\prod_{a_{0}}^{A_{0}}\prod_{a_{1}}^{A_{1}}\left(1-\exp \left(-\frac{\gamma_{i}^{*}}{(1-\lambda)\beta_{i}\rho_{S}\sigma_{R}^{2}} \right)\right)}_{\mathcal{Q}_{i}}\underbrace{\prod_{k=N}^{i+1}(1-Q_{k})}_{ \text{for }i<N}\right)\\ &+\sum_{i=N}^{n}\left(\underbrace{\prod_{a_{1}}^{A_{2}}\prod_{a_ {2}}\left(1-\exp\left(-\frac{\gamma_{i}^{*}}{\beta_{i}\rho_{R}\sigma_{R}^{2}} \right)\right)}_{\mathcal{K}_{i}}\underbrace{\prod_{k=N}^{i+1}(1-K_{k})}_{ \text{for }i<N}\prod_{t=N}^{n}(1-Q_{t})\right)\end{split}  \tag{37}\]  
_where \(\sigma_{n}^{2}=E\Bigl{\{}\left|\mathbf{H}_{n}\right|^{2}\Bigr{\}}\). Note that \(\theta=\text{odd}\) and \(\mathcal{R}=R_{1}\), or \(\theta=\text{even}\) and \(\mathcal{R}=R_{2}\)._  
See Appendix C for proof. Similar to (30), note that Expression (36) is challenging in Monte Carlo simulations if the network has a large number of devices. Therefore, the OP at devices \(D_{n}\) can be rewritten as follows:  
\[\mathrm{OP}_{n}^{(\theta)}=1-\prod_{i=N}^{n}\Pr\Biggl{\{}\max_{[A_{0}\times A_ {1}]}\Biggl{\{}\mathbf{R}_{\mathcal{R}-x_{i}}^{\left(\tau_{1}^{(\theta)}\right)} \Biggr{\}}\geq R_{i},\max_{[A_{1}\times A_{2}]}\Biggl{\{}\mathbf{R}_{n-x_{i}}^{ \left(\tau_{2}^{(\theta)}\right)}\Biggr{\}}\geq R_{i}\Biggr{\}}. \tag{38}\]  
From (38), we obtain the expression of OP at devices \(D_{n}\) in closed-form as follows:  
\[