# Performance Analysis of D2D Communication with Retransmission Mechanism in Cellular Networks
## 4 Queuing Model and Feedback Strategy
### Stochastic Geometry Analysis for Service Probability  
While maintaining a minimum level of quality of service, the servers' probability of the D2D data queue was defined as the probability of the received SINR above a certain threshold \(\eta\), which also accounted for the interference between the backlogged links. Using stochastic geometry, the transmission success probability \(p_{suc}\) was characterized via the following lemma. Figure 2: Feedback queuing model. **Lemma 1**. : _In the depicted underlying D2D network, the transmission success probability \(p_{suc}\) was the complementary cumulative distribution function (CCDF) of the SINR of D2D links, which is expressed as:_  
\[ \begin{array}{l}p_{suc}=E_{r}\left[P\left[SINR_{m}>\eta\left|r\right.\right] \right]\\ =e^{-\frac{\eta}{P_{D}\left[\alpha_{m},m\right]^{-\alpha_{D}}}\cdot\sigma_{N^{ 2}}}\cdot\mathcal{L}_{I_{C}}\left(\frac{\eta}{P_{D}\left|d_{m,m}\right|^{- \alpha_{D}}}\right)\cdot\mathcal{L}_{I_{D}}\left(\frac{\eta}{P_{D}\left|d_{m,m }\right|^{-\alpha_{D}}}\right)\end{array}  \tag{3}\]  
_where \(\mathcal{L}_{I_{C}}\left(\cdot\right)\) and \(\mathcal{L}_{I_{D}}\left(\cdot\right)\) are the Laplace transforms of \(I_{C}\) and \(I_{D}\) given by:_  
\[\mathcal{L}_{I_{C}}\left(\frac{\eta}{P_{D}\left|d_{m,m}\right|^{-\alpha_{D}}} \right)=\exp\left(-2\pi\lambda_{B}\int\limits_{R}^{\infty}\left(1-E_{S_{l,m}} \left[\frac{1}{1{+}sP_{C}\omega s_{l,m}d_{l,m}^{-\alpha_{C}}}\right]\right) vdv\right) \tag{4}\]  
_and:_  
\[\mathcal{L}_{I_{D}}\left(\frac{\eta}{P_{D}\left|d_{m,m}\right|^{-\alpha_{D}}} \right)=\exp\left(-2\pi\rho\lambda_{D}\int\limits_{0}^{\infty}\left(1-E_{S_{m \alpha}}\left[\frac{1}{1{+}sP_{D}v^{-\alpha_{D}}}\right]\right)vdv\right) \tag{5}\]  
Proof of Lemma 1.: Following [9], we can denote the Laplace transform of the total interference \(I\) measured at the origin, as for the benefit of the QoS performance of D2D communications, the servers' probability is defined as the probability of the received SINR above a certain threshold, which also accounts for the interference between the backlogged links. The successful transmission probability is expressed as:  
\[ \begin{array}{l}P\left[SINR_{m}>\eta\right]=E_{r}\left[P\left[SINR_{m}>\eta \left|r\right.\right]\right]\\ =\int_{0}^{R}P\left[SINR_{m}>\eta\left|r\right.\right]f_{r}\left(r\right)dr\\ =\int_{0}^{R}P\left[\frac{P_{D}\sigma_{m,m}\left|d_{m,m}\right|^{-\alpha_{D}}} {I_{C}+I_{D}+\sigma_{N^{2}}}>\eta\left|r\right.\right]\frac{2r}{R^{2}}dr\\ =\int_{0}^{R}\frac{2}{R^{2}}P\left[g_{m,m}>\frac{\eta}{P_{D}\left|d_{m,m}\right| ^{-\alpha_{D}}}\left(I_{C}+I_{D}+\sigma_{N}{}^{2}\right)\right]rdr\end{array}  \tag{6}\]  
where \(f_{r}\left(r\right)\) is the PDF of the distance from the typical D2D user to the served BS, which is equal to \(\frac{2r}{R^{2}},r\in\left(0,R\right)\). Because \(g_{m,m}\sim\exp\left(1\right)\), by following the same derivation given in [40], the coverage probability is expressed as:  
\[\begin{array}{l}P\left[g_{m,m}>\frac{\eta}{P_{D}\left|d_{m,m}\right|^{- \alpha_{D}}}\left(I_{C}+I_{D}+\sigma_{N}{}^{2}\right)\right]=E_{I_{C}I_{D}} \left[P\left[g_{m,m}>\frac{\eta}{P_{D}\left|d_{m,m}\right|^{-\alpha_{D}}} \left(I_{C}+I_{D}+\sigma_{N}{}^{2}\right)\left|r\right.I_{C},I_{D}\right] \right]\\ =e^{-\frac{\eta}{P_{D}\left|d_{m,m}\right|^{-\alpha_{D}}}\cdot\sigma_{N^{2}}} \cdot\mathcal{L}_{I_{C}}\left(\frac{\eta}{P_{D}\left|d_{m,m}\right|^{-\alpha _{D}}}\right)\cdot\mathcal{L}_{I_{D}}\left(\frac{\eta}{P_{D}\left|d_{m,m}\right| ^{-\alpha_{D}}}\right) \tag{7}\]  
and:  
\[