## 3 Outage Probabilities  
### Outage Probabilities  
Consider an OFDMA-based D2D system in a circular cell with radius \(R\). Suppose that there are \(M\) sub-channels, and one (and only one) D2D or cellular device transmits for each sub-channel at a time. When considering the transmission on the \(n^{th}\) sub-channel, the other \(M-1\) sub-channels produce other-channel interference to the \(n^{th}\) sub-channel. Let the interference effect resulting from in-band emission from the \(m^{th}\) sub-channel to the \(n^{th}\) sub-channel be \(\lambda_{m,n}\), where \(\lambda_{m,n}=1\) if \(m=n\) and \(0<\lambda_{m,n}<<1\) otherwise. The amount of interference from other sub-channels is determined by \(\lambda_{m,n}\)\((m\
eq n)\). We considered relay-based D2D transmission consisting of a source (transmitter), a destination (receiver), and a relay on the \(n^{th}\) sub-channel, as shown in Figure 7. In the first transmission phase, the source transmits data and in the second transmission phase, the relay passes the received data to the destination. Let the other-channel interferer \(1,m\) and \(2,m\) be the interferers on the \(m^{th}\) sub-channel \((m\
eq n)\) for the first (source-to-relay) and the second (relay-to-destination) transmission phases, respectively. Let the positions of the source, relay, destination, other-channel interferer \(1,m\), and other-channel interferer \(2,m\) be \((x_{S},y_{S})\), \((x_{R},y_{R})\), \((x_{D},y_{D})\), \((x_{O1,m},y_{O1,m})\), and \((x_{O2,m},y_{O2,m})\), respectively. \(d_{SD}=\sqrt{\left(x_{S}-x_{D}\right)^{2}+\left(y_{S}-y_{D}\right)^{2}}\) represents the distance between the source and the destination, \(d_{SR}=\sqrt{\left(x_{S}-x_{R}\right)^{2}+\left(y_{S}-y_{R}\right)^{2}}\) represents the distance between the source and the relay, \(d_{RD}=\sqrt{\left(x_{R}-x_{D}\right)^{2}+\left(y_{R}-y_{D}\right)^{2}}\) is the distance between the relay and the destination,  
Figure 6: Listen-before-talk (LBT) with different delays. Figure 7: Interference from other sub-channels. \(d_{O1R,m}=\sqrt{\left(x_{O1,m}-x_{R}\right)^{2}+\left(y_{O1,m}-y_{R}\right)^{2}}\) represents the distance between other-channel interferer \(1,m\) and the relay, \(d_{O1D,m}=\sqrt{\left(x_{O1,m}-x_{D}\right)^{2}+\left(y_{O1,m}-y_{D}\right)^{2}}\) is the distance between the other-channel interferer \(1,m\) and the destination, and \(d_{O2D,m}=\sqrt{\left(x_{O2,m}-x_{D}\right)^{2}+\left(y_{O2,m}-y_{D}\right)^{2}}\) is the distance between the other-channel interferer \(2,m\) and the destination. \(P_{S}^{TX}\) is the transmission power of the source, \(P_{R}^{TX}\) is the transmission power of the relay, \(P_{O1,m}^{TX}\) is the transmission power of the other-channel interferer \(1,m\), and \(P_{O2,m}^{TX}\) is the transmission power of the other-channel interferer \(2,m\). Consider the transmission from the source to the destination. The received signal power at the destination, dented as \(P_{SD}^{RX}\), can be represented as:  
\[P_{SD}^{RX}=K_{1}P_{S}^{TX}d_{SD}^{-\alpha}, \tag{1}\]  
where \(K_{1}\) is a constant and \(\alpha\) is the path-loss exponent parameter between two devices. The received interference power from the other-channel interferer \(1,m\) at the destination is denoted as:  
\[P_{O1D,m}^{RX}=K_{1}\lambda_{m,n}P_{O1,m}^{TX}d_{O1D,m}^{-\alpha}, \tag{2}\]  
and the SIR at the destination considering the \(m^{th}\) channel interferer only, denoted as \(SIR_{SD,m}\), can be written as follows:  
\[SIR_{SD,m}=\frac{P_{SD}^{RX}}{P_{O1D,m}^{RX}}=\frac{P_{S}^{TX}d_{SD}^{-\alpha }}{\lambda_{m,n}P_{O1,m}^{TX}d_{O1D,m}^{-\alpha}}. \tag{3}\]  
Assuming that \(\alpha\) is large and, thus, the nearest interferer dominates the amount of interference, we can only consider the interference from the nearest interferer, or more precisely, we can consider the \(m^{th}\) channel interferer, where \(m=\underset{m^{\prime}}{\operatorname{argmax}}\big{\{}\lambda_{m^{\prime},n} d_{O1D,m^{\prime}}^{-\alpha}\big{\}}\).