# Location-Based Relay Selection in Full-Duplex Random Relay Networks
## 3 Outage Probability of an FDRRN  
In this section, the outage probability of an FDRRN is formulated. We begin by defining the instantaneous signal-to-interference ratio (SIR) for the \(i\)th relay hop, where the transmitter and receiver nodes are denoted as \(\mathbf{x}\) and \(\mathbf{y}\), respectively, as follows:  
\[\gamma_{i}=\frac{P_{t}H_{\mathbf{xy}}D_{\mathbf{xy}}^{-\alpha}}{I_{SI}+\sum \limits_{Z\in\mathbf{\Pi}_{i}/\{\mathbf{X},\mathbf{Y}\}}P_{t}H_{Z\mathbf{Y}} D_{Z\mathbf{Y}}^{-\alpha}},\ \ \ \forall i\in\{1,2\}, \tag{1}\]  
where \(\alpha\) is the path-loss exponent, \(P_{t}\) is the transmission power of the transmit node, and \(I_{SI}\) means the residual SI after performing imperfect SIC. In addition, \(H_{\mathbf{xy}}\) and \(D_{\mathbf{xy}}\) denote the Rayleigh fading channel gain and the distance between the transmitter node \(\mathbf{x}\) and receiver node \(\mathbf{y}\), respectively.  
Based on Equation (1), the success probability of a typical link for the \(i\)th relay hop can be given by [37]  
\[p_{s,i}(\tau)=\exp\biggl{\{}-\frac{I_{SI}D_{i}^{\alpha}\tau}{P_{t}}\biggr{\}} \exp\Bigl{\{}-v_{\alpha}D_{i}^{2}\tau^{2/\alpha}\lambda_{i}\Bigr{\}}, \tag{2}\]  
where \(\tau\) is the SIR threshold, \(D_{i}\) is the distance of the \(i\)th relay hop, and \(v_{\alpha}\stackrel{{\Delta}}{{=}}(2\pi/\alpha)\Gamma(2/\alpha) \Gamma(1-2/\alpha)\) with the gamma function \(\Gamma(x)=\int_{0}^{\infty}y^{x-1}e^{-y}dy\).  
Figure 1: An example of an FDRRN. Red circles and squares are the transmitter and receiver nodes, respectively. In addition, black circles with dotted lines and solid lines are the candidate and selected relay nodes, respectively. Also, the dotted red line denotes SI in a relay node.  
Proof.: The successful transmission of the SIR in (1) for a Rayleigh fading channel can be expressed as  
\[\begin{split}\mathbb{P}(\gamma_{i}\geq\tau)&=\mathbb{P} \bigg{\{}H_{\mathbf{xy}}>\frac{D_{XY}^{\mathbf{x}}\tau}{P_{t}}(I_{SI}+I_{i}) \bigg{\}}\\ &=\mathbb{E}_{I_{i}}\bigg{\{}\exp\bigg{\{}\frac{D_{XY}^{\mathbf{x }}\tau}{P_{t}}(I_{SI}+I_{i})\bigg{\}}\bigg{\}},\end{split} \tag{3}\]  
where \(I_{i}=\sum\limits_{Z\in\mathbf{\Pi}_{i}/\big{\{}\mathbf{X},\mathbf{Y}\big{\}}} P_{t}H_{Z\mathbf{Y}}D_{Z\mathbf{Y}}^{-\mathbf{x}}\), the sum of the interference power from the transmitter nodes for the \(i\)th relay hop. Hence, by applying the Laplace transform for a general Poisson shot noise process with independent, identically distributed marks [34], we can obtain (2).  
Note that we assume that only the relay node has FD capability, so the residual SI at the receiver node of the second hop is zero. Consequently, the SIR and the probability of successful transmission for the second hop in (1) and (2) can be obtained with \(I_{SI}=0\).  
Under the DF relay protocol, an outage event happens when the SIR of either the first or second hop link fails to meet the required SIR threshold. Hence, when a typical transmitter node sends its data to the receiver node through a relay node located at \(\mathbf{x}_{\text{r}}\), the outage probability of a two-hop FDRRN can be expressed as follows:  
\[p_{\text{o}}(\mathbf{x}_{\text{r}})=1-p_{\text{s},1}(\tau)p_{\text{s},2}(\tau). \tag{4}\]  
By substituting (2) into (4), we can express the outage probability of an FDRRN as follows:  
\[p_{\text{o}}(\mathbf{x}_{\text{r}})=1-\exp\bigg{\{}-\frac{I_{SI}D_{1}^{\mathbf{ x}}\tau}{P_{t}}-v_{\text{a}}\tau^{2/\text{a}}\Big{(}D_{1}^{2}\lambda_{1}+D_{2}^{2} \lambda_{2}\Big{)}\bigg{\}}. \tag{5}\]  
From (5), we can guess that the outage probability of an FDRRN is influenced by both network interference and the distance of each relay hop, which is determined by the location of the selected relay node. Based on this observation, we propose a relay selection algorithm for FDRRNs in the next section.