# Performance of Millimeter Wave Dense Cellular Network Using Stretched Exponential Path Loss Model
## Appendix A Derivation of SINR Coverage Probability
### Laplace Transform of Interference  
According to the thinning property of PPP the interference components can be split into independent PPPs for the LOS and NLOS BS.  
\[I_{z}=I_{LOS}+I_{NLOS} \tag{10}\] \[I_{LOS}=\sum_{z\in\Phi_{L}}|h_{L}|^{2}G_{z}I_{LOS}(D_{z})\] (11) \[I_{NLOS}=\sum_{z\in\Phi_{N}}|h_{N}|^{2}G_{z}I_{NLOS}(D_{z}) \tag{12}\]  
The LT of interference caused by LOS BSs can be derived according to the following steps:  
\[\mathcal{L}_{I_{LOS}}(\mu_{L}) =\mathbb{E}[\exp(-\mu_{L}I_{LOS})] \tag{13}\] \[=\mathbb{E}_{\Phi_{L}}\Bigg{[}\exp\left(-\mu_{L}\sum_{z\in\Phi_{L }\backslash\mathcal{B}(0,r)}|h_{z}|^{2}G_{z}D_{z}^{-\alpha_{L}}\right)\Bigg{]}\] (14) \[=\mathbb{E}_{G_{z},D_{z},h_{z}}\left[\prod_{z\in\Phi_{L} \backslash\mathcal{B}(0,r)}\exp\left(-\mu_{L}|h_{z}|^{2}G_{z}D_{z}^{-\alpha_ {L}}\right)\right]\] (15) \[=\mathbb{E}_{G_{z}}\left[\prod_{z\in\Phi_{L}\backslash\mathcal{B }(0,r)}\mathbb{E}_{D2}\left[\frac{1}{\left(1+\mu_{L}G_{z}D_{z}^{-\alpha_{L}} \right)}\right]\right]\] (16) \[=\exp\left(-2\pi\lambda\quad\mathbb{E}_{Gz}\left[\int_{r}^{\infty }\left(1-\frac{1}{1+\mu_{L}G_{z}t^{-\alpha_{L}}}\right)tp_{LOS}(t)dt\right]\right)\] (17) \[=\exp\left(-2\pi\lambda\sum_{G\in G_{z}}P_{G_{z}}\left(\int_{r}^{ \infty}\left(1-\frac{1}{1+\mu_{L}G_{z}t^{-\alpha_{L}}}\right)tp_{LOS}(t)dt \right)\right) \tag{18}\]  
where (15) is due to the independence of random variables \(G_{z}\), \(D_{z}\), \(h_{z}\) across \(\phi_{L}\). (16) is due to the MGF of \(h_{z}\). (17) is using PGFL of a PPP [51] and \(P_{G_{z}}\) in (18) is given in (7).  
The LT of interference caused by NLOS BSs can be derived according to the following steps:  
\[\mathcal{L}_{I_{NLOS}}(\mu_{L}) =\mathbb{E}[\exp(-\mu_{L}I_{NLOS})] \tag{19}\] \[=\mathbb{E}_{\theta_{L}}\left[\exp\left(-\mu_{L}\sum_{z\in\phi_{L }\backslash\mathcal{B}(0,r)}|h_{z}|^{2}G_{z}I(D_{z})\right)\right]\] (20) \[=\mathbb{E}_{G_{z},D_{z},h_{z}}\left[\prod_{z\in\theta_{L} \backslash\mathcal{B}(0,r)}\exp\left(-\mu|h_{z}|^{2}G_{z}exp\left(-\kappa \mathcal{D}_{z}^{z}\right)\right)\right]\] (21) \[=\exp\left(-2\pi\lambda\sum_{k=1}^{4}\,h_{k}\left[\int_{r}^{ \infty}\mathbb{E}_{h_{k}}\left(1-\exp\left(-\mu_{l}a_{k}|h_{k}|^{2}exp\left(- \kappa\mathcal{k}^{r}\right)\right)\right)tp_{NLOS}(t)dt\right]\right)\] (22) \[=\exp\left(\frac{2\pi\lambda}{\zeta(-\kappa)^{\frac{1}{2}}}\sum_{k =1}^{4}\,h_{k}\left[\int_{0}^{\infty r(-\kappa\mathcal{k}^{r})}\frac{\mathbb{E }_{h_{k}}(1-\exp\left(\mu_{l}a_{k}|h_{k}|^{2}u\right))}{u}\ln\left(u\right)^{ \frac{1-\kappa}{4}}p_{NLOS}\left(\frac{-ln(u)}{\kappa}\right)^{\frac{1}{4}}du \right]\right)\] (23) \[=\exp\left(\frac{2\pi\lambda}{\zeta(-\kappa)^{\frac{1}{2}}}\sum_{k =1}^{4}\,h_{k}\left[\int_{0}^{\infty r(-\kappa\mathcal{k}^{r})}\frac{\left(1- \frac{1}{1+\mu_{l}a_{k}u}\right)}{u}\ln\left(u\right)^{\frac{2-\kappa}{4}}p_{ NLOS}\left(\frac{-ln(u)}{\kappa}\right)^{\frac{1}{4}}du\right]\right)\] (24) \[=\exp\left(-\frac{2\pi\lambda}{\zeta(\kappa^{\frac{1}{2}})}\sum_{k =1}^{4}\,h_{k}\left[\int_{0}^{\infty r(-\kappa\mathcal{k}^{r})}\frac{\mu_{l}a_{ k}}{1+\mu_{l}a_{k}u}(-ln(u))^{\frac{1-\kappa}{4}}p_{NLOS}\left(\frac{-ln(u)}{ \kappa}\right)^{\frac{1}{4}}du\right]\right) \tag{25}\]  
(22) is obtained by using PGFL of PPP and by substituting \(u=\exp(-\kappa\mathcal{k}^{r})\). The integration limits from r to \(\infty\), implies that the closest interferer lies at least at a distance rfrom tagged BS. (A23) is from moment generating function of \(h_{z}\sim\exp(1)\) and \(a_{k}\) and \(b_{k}\) are constants defined in (7).