# A Novel Multi-Cell Interference-Aware Cooperative QoS-Based NOMA Group D2D System
## 3 Outage Probability Analysis  
This section presents the analysis of outage probability for the \(R_{x}\) in proposed Q-CNOMA-based D2D relaying system. Note that from (5), the received SINR/signal depends on interference. As such, in order to compute the outage probability, the interference distribution is required. Therefore, before deriving the outage probability, the interference distribution is calculated using the Lemma 1, as given below.  
Figure 1: Proposed system model.  
Now with interference distribution in hand, we can proceed towards finding the outage probability. In order to write the outage event at \(R_{x}\), let us first define the following two events:  
\[\mathcal{A}_{1} =\{\text{Outage in Phase I}\} \tag{7}\] \[\mathcal{A}_{2} =\{\text{Success in Phase I}\}\text{and}\{\text{Outage in Phase II}\} \tag{8}\]  
Note that the events \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) are mutually exclusive. As such, based on (7) and (8), the overall outage probability at the \(R_{x}\) in the considered network, denoted as \(\mathds{P}_{out}\), can be written as follows:  
\[\mathds{P}_{out}=\mathds{P}_{1}+(1-\mathds{P}_{1})\mathds{P}_{2} \tag{9}\]  
where \(\mathds{P}_{1}\) and \(\mathds{P}_{2}\) represent the outage probability of Phase I and Phase II, respectively. Subsequently, the outage probabilities \(\mathds{P}_{1}\) and \(\mathds{P}_{2}\) are expressed in following Lemma and Theorem, respectively.  
**Lemma 1**.: _The outage probability at the receiver in Phase I can be obtained using the results presented in [2] and can be expressed as follows:_  
\[\mathds{P}_{1}=\sum_{l=1}^{L}b_{l}e^{\theta_{m,1}^{max}\psi_{l}}\mathcal{L}_{ l}\left(\frac{\phi_{m,1}^{max}\psi_{l}\rho}{\rho_{t}}\right) \tag{10}\]  
_where \(b_{l}=\omega\sqrt{1-\theta_{l}^{2}}(1+\theta_{l})\), \(\omega=\frac{\pi}{2}\), \(\psi_{l}=\left(\frac{c}{2}(1+\theta_{l})\right)^{\alpha}\), \(\theta_{l}=\cos\left(\frac{(2l+1)\pi}{2L}\right)\), \(c\) represents the radius of coverage area, \(\mathcal{L}_{l}\) denotes the Laplace transform of the interference \(l\), \(L\) is the complexity-accuracy trade-off parameter, \(\rho=\frac{P_{l}}{\sigma^{2}}\), \(P_{l}\) is the maximum transmission power available to interfering transmitters, \(\rho_{t}=\frac{P_{C_{T_{1}}}}{\sigma^{2}}\), \(\phi_{m,1}^{max}=\max(\phi_{l,1,}\phi_{2,1},\cdots,\phi_{m,1})\), \(\phi_{j,1}=\frac{\tau_{j,1}}{\beta_{j}-\tau_{j,1}\sum_{m,1}^{M_{1}+1}\beta_{m,1 }}\), \(\tau_{j,1}=2^{\mathds{P}_{j,1}^{M_{1}}}-1\), and \(\mathds{B}_{j,1}\)th is the targeted rate of jth user in Phase I._  
**Theorem 1**.: _The outage probability at the receiver in Phase II can be expressed as follows:_  
\[\begin{split}\mathds{P}_{2}=& 1-\exp^{-\gamma_{l}/\Omega_{2j}}+ \sum_{l=1}^{L}\sum_{k_{1}=0}^{N_{1}}\sum_{k_{2}=0}^{2N_{2}}\sum_{k_{3}=1}^{N_ {3}}\frac{b_{l}}{T}e^{-\gamma_{1}}\\ &\cdot\frac{(-1)^{k_{1}}}{k_{1}!}(\psi_{l}\eta_{2})^{k_{1}} \Gamma(1-k_{1},\hat{\gamma}_{t})\omega_{k_{2}}e^{x_{k_{3}}}\\ &\cdot(1+x_{k_{3}}\xi_{l})^{k_{1}}\Re\Big{[}\mathcal{L}_{l}(s)e^{ \left(\gamma+\frac{\pi}{l}\right)x_{k_{3}}}\Big{]}\end{split} \tag{11}\]  
_where \(\gamma_{t}=\frac{\tau_{R_{x}}}{\xi_{R_{x}}}\), \(\Omega_{2,l}\) is the variance of the Rayleigh CDF, \(\tau_{R_{x}}=2^{\mathds{P}_{R_{x}}^{K}}-1\), and \(\xi_{R_{x}}=\frac{P_{C_{T_{2}}}}{N^{2}}\); \(L\), \(N_{1}\), \(N_{2}\), and \(N_{3}\) are the complexity-accuracy trade-off parameters; \(\xi_{l}=\frac{1}{\Re^{2}}\); \(\eta_{1}=\frac{\tau_{R_{x}}}{\beta_{R_{x}}}\sum_{m_{2}=j_{2}}^{M_{2}}\beta_{m_{ 2}}\); \(\eta_{2}=\frac{\tau_{R_{x}}}{\xi_{R_{x}}P_{R_{x}}}\); \(\beta_{m_{2}}\) is the power allocation coefficient of user \(m_{2}\); \(M_{2}\) are the total number of users in Phase II, and \(\Gamma(\cdot)\) is the Gamma function._  
Proof.: Please see Appendix A.