# Quality of Service Based NOMA Group D2D Communications
## 2 System Model
### Interference Distribution  
The reception at the probe DR from the test GT is interfered by the other GTs. The interference at probe DR is given as, \(I=\sum_{x\in\Phi_{\mathrm{GT}}\backslash x_{0}}|g_{x}|^{2}d_{x}^{-a}\), where \(g_{x}\) and \(d_{x}\) represent the Rayleigh fading channel gain and distance between probe DR and interferer at \(x_{0}\), respectively. The following lemma provides the Laplace transform of the interference at probe DR.  
**Lemma 1**. : _Consider a GPP \(\Phi_{\mathrm{GT}}\) with parent process intensity \(\lambda_{\mathrm{GT}}\) modeling spatial topology of the GTs in a Q-NOMA group D2D communications. Then, the Laplace transform of the interference at the probe \(DR\) conditioned at the location of test GT is given by:_  
\[\mathcal{L}_{1}\left(s\right)=e^{-2\pi\lambda_{\mathrm{GT}}\sum_{p=1}^{p} \Omega_{p}\frac{a\left(1-\chi_{1}\left(r_{p}\right)\right)+sr_{p}^{-a}}{1+r_{ p}^{a}}}\cdot\Lambda_{2}\left(d\right), \tag{4}\]  
_where \(\Omega_{p}=\omega_{p}e^{r_{p}}\), \(\omega_{p}=\frac{\Gamma\left(P+2\right)r_{p}}{p_{\Gamma\left(P+1\right)}\left( L_{p+1}\left(r_{p}\right)\right)^{2}}\), \(L_{p}\left(\cdot\right)\) is the Laguerre polynomial of degree \(P\), \(r_{p}\) are the roots of \(L_{p}(\cdot)\), \(X_{1}(\cdot)\) and \(\Lambda_{2}(\cdot)\) are given in Equations (A7) and (A10), respectively._  
Proof. : See Appendix A. **3. Outage Probability Analysis**  
In this section, we focus on the outage probability for the DRs in the considered Q-NOMA group D2D communication. Let \(\tau_{m}\) and \(R_{m}\) denote the SINR threshold and targeted rate of DR \(m\), respectively,where \(\tau_{m}=2^{R_{m}}-1\). Since each DR employs SIC, the outage at DR \(m\) occurs if it does not meet the targeted rate of any higher order DR \(j\), where \(1\leq j<m\). Denote \(\zeta_{m\to j}=\left\{\tilde{R}_{m,j}<R_{j}\right\}\) as the outage event at DR \(m\) due to decoding of DR \(j\), where \(\tilde{R}_{m,j}\) is the achievable rate of user \(j\) at DR \(m\). The outage event \(\zeta_{m\to j}\) can be expressed as:  
\[\zeta_{m\to j} = \left\{\tilde{R}_{m,j}<R_{j}\right\} \tag{5}\] \[= \left\{\log_{2}\left(1+\frac{h_{m}\beta_{j}P}{h_{m}P\sum_{i=j+1}^ {M}\beta_{i}+I+\sigma^{2}}\right)<R_{j}\right\}\] \[= \left\{h_{m}<\frac{\varphi_{j}(\rho I+1)}{\rho_{t}}\right\},\]  
where \(\varphi_{j}=\frac{\tau_{j}}{\beta_{j}-\tau_{j}\sum_{i=j+1}^{M}\beta_{i}}\), \(\rho=\frac{P_{t}}{\sigma^{2}}\), \(\rho_{t}=\frac{R_{\text{GT}}}{\sigma^{2}}\) is the transmit signal-to-noise ratio (SNR) and \(P_{t}\) is the maximum received interference power at the probe DR.  
Next, define \(\varphi_{m}^{\max}=\max\left\{\varphi_{1},...,\varphi_{m}\right\}\). Based on Equation (3), the outage probability at the DR \(m\) can be given as:  
\[\mathbb{P}_{m} = \Pr\left(h_{m}<\frac{\varphi_{m}^{\max}(\rho I+1)}{\rho_{t}}\right) \tag{6}\] \[= \mathbb{E}_{I}\left[F_{h_{m}}\left(\frac{\varphi_{m}^{\max}(\rho x +1)}{\rho_{t}}\right)\right],\]  
where \(F_{h_{m}}\) is the cumulative distribution function (CDF) of \(h_{m}\). Note that the set \(\left\{h_{i}\right\},i=1,...,M\), of channel gains is not ordered because the users are sorted in ascending order of their targeted rates. Since the channel gains are i.id. random variables with common CDF \(F_{h}\), Equation (4) can be re-written as:  
\[\mathbb{P}_{m}=\mathbb{E}_{I}\left[F_{h}\left(\frac{\varphi_{m}^{\max}(\rho x +1)}{\rho_{t}}\right)\right]. \tag{7}\]  
Consequently, the outage probability of DR \(m\) is provided in the following theorem. **Theorem 1**. : _The outage probability of DR \(m\) in the Q-NOMA group D2D communications is derived as:_  
\[\mathbb{P}_{m}=\sum_{l=1}^{L}b_{l}e^{-\frac{\varphi_{m}^{\max}c_{l}}{\rho_{t} }}\mathcal{L}_{I}\left(\frac{\varphi_{m}^{\max}c_{l}\rho}{\rho_{t}}\right), \tag{8}\]  
_where \(b_{l}=\omega_{l}\sqrt{1-\phi_{l}^{2}}(1+\phi_{l})\), \(\omega_{l}=\frac{\pi}{L}\), \(c_{l}=\left(\frac{R_{D}}{2}(1+\phi_{l})\right)^{\alpha}\), \(\phi_{l}=\cos\left(\frac{(2l-1)\pi}{2L}\right)\) and \(L\) is the complexity-accuracy trade-off parameter._  
Proof.