## 2 System Model
### Channel Model  
The multi-packet reception (MPR) channel model used in this paper is a generalized form of the packet erasure model. A block fading channel model is considered here with Rayleigh fading, i.e., the fading coefficients \(h_{ji}\) remain constant during one timeslot, but change independently from one timeslot to another, based on a circularly symmetric complex Gaussian distribution with zero mean and unit variance. The noise is assumed to be additive white Gaussian with zero mean and unit variance. With \(p_{j}^{t}\), we denote the transmission power of node \(j\), and \(d_{ji}\) is the distance between transmitter \(j\) and receiver \(i\), with \(a\) being the path loss exponent. Let \(\mathcal{D}_{j,i}^{\mathcal{M}}\) denote the event that node \(i\) is able to decode the packet transmitted from node \(j\), given a set of active transmitters denoted by \(\mathcal{M}\). For the topology presented in Figure 1 for example, \(\mathcal{D}_{1,R}^{\{1,2\}}\) denotes the event that the relay \(R\) can decode the information from the node \(1\) when nodes \(1\) and \(2\) are active (\(\mathcal{M}=\{1,2\}\)). The transmission rates of node \(1\) and \(2\) are fixed at \(R_{1}\) and \(R_{2}\). Note that we use \(R\) for the relay and \(R_{i}\) to denote the transmission rate of node \(i\). When only \(j\) is active, the event \(\mathcal{D}_{j,i}^{\{j\}}\) is defined as  
\[\mathcal{D}_{j,i}^{\{j\}}\triangleq\Big{\{}R_{j}\leq\log_{2}\!\left(1+|h_{ji} |^{2}d_{ji}^{-a}p_{j}^{t}\right)\Big{\}}, \tag{1}\]  
which is equivalent to \(\mathcal{D}_{j,i}^{\{j\}}=\Big{\{}2^{R_{j}}-1\leq|h_{ji}|^{2}d_{ji}^{-a}p_{j}^ {t}\Big{\}}\). For convenience, we define \(\text{SNR}_{ji}\triangleq|h_{ji}|^{2}d_{ji}^{-a}p_{j}^{t}\) and \(\gamma_{j}\triangleq 2^{R_{j}}-1\). The probability that the link \((j,i)\) is not in outage when only \(j\) is active is given by [23]  
\[\Pr\!\left(\mathcal{D}_{j,i}^{\{j\}}\right)=\Pr\!\left\{\text{SNR}_{ji}\geq \gamma_{j}\right\}=\exp\!\left(-\frac{\gamma_{j}d_{ji}^{a}}{p_{j}^{t}}\right). \tag{2}\]Let us consider the case that, the relay node \(R\) treats interference from node 2 as noise, when both nodes 1 and 2 are active. The event \(\mathcal{D}_{1R}^{\{1,2\}}\) is given by  
\[\mathcal{D}_{1,R}^{\{1,2\}}\triangleq\bigg{\{}R_{1}\leq\log_{2}\!\left(1+\frac {|h_{1R}|^{2}d_{1R}^{-a}p_{1}^{t}}{1+|h_{2R}|^{2}d_{2R}^{-a}p_{2}^{t}}\right) \bigg{\}}, \tag{3}\]  
which is equivalent to  
\[\mathcal{D}_{1,R}^{\{1,2\}}=\bigg{\{}\gamma_{1}\leq\frac{|h_{1R}|^{2}d_{1R}^{- a}p_{1}^{t}}{1+|h_{2R}|^{2}d_{2R}^{-a}p_{2}^{t}}\triangleq\text{SINR}_{1R} \bigg{\}}. \tag{4}\]  
The probability that the link \((1,R)\) is not in outage when both nodes 1 and 2 are active is given by [23]:  
\[\text{Pr}^{IAN}\Big{(}\mathcal{D}_{1,R}^{\{1,2\}}\Big{)}=\text{Pr}\{\text{ SINR}_{1R}\geq\gamma_{1}\}= \tag{5}\] \[=\exp\!\left(-\frac{\gamma_{1}d_{1R}^{a}}{p_{1}^{t}}\right)\! \left[1+\gamma_{1}\frac{p_{2}^{t}}{p_{1}^{t}}\Big{(}\frac{d_{1R}}{d_{2R}} \Big{)}^{a}\right]^{-1}.\]  
Let us consider the case that the relay node R deploys successive interference cancelation (SIC), when both nodes 1 and 2 are active. If the relay R knows the codebook of the node 2, it can perform SIC by first decoding the message sent by 2, removing its contribution (interference) to the received signal, and then decoding the message coming from node 1.