# Performance of Millimeter Wave Dense Cellular Network Using Stretched Exponential Path Loss Model
## 3 SINR Coverage Probability and Area Spectral Efficiency
### SINR Coverage Probability  
For a given SINR threshold (T), the probability of SINR coverage denoted by \(P_{c}(T)\) is mathematically given by \(P_{c}(T)=\mathbb{P}[\text{SINR}>T]\). The SINR received at typical UE at a distance \(r\) is:  
\[\text{SINR}=\frac{|h_{0}|^{2}G_{0_{BS}}G_{0_{UE}}l(r)}{\sigma^{2}+\sum_{Z\in \phi_{Z}}I_{Z}} \tag{8}\]  
\(I_{Z}\) in the above expression represents total interference in the network and is given as:  
\[I_{Z}=\sum_{Z\in\phi_{Z}}|h_{Z}|^{2}G_{Z}l(D_{Z}) \tag{9}\]  
The channel fading power on the desired link follows an exponential distribution i.e., \(|h_{0}|^{2}\sim\exp(1)\). \(G_{0_{UE}}\) and \(G_{0_{BS}}\) are the antenna gains of typical UE and tagged BS. \(|h_{Z}|^{2}\) is the normalized fading power on interfering link. An interfering BS is located at a distance \(D_{z}\) from the tagged BS. \(l(r)\) and \(l(D_{z})\) are the path losses on the desired link and interfering links, respectively. The term \(\sigma^{2}\) represents zero mean thermal noise power. \(\phi_{Z}\) represents the point process of interfering s and \(z\) denotes the interfering BS with antenna gain \(G_{Z}\). Depending upon the propagation distance, the serving BS can either be in LOS or NLOS  
\begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Definition** \\ \hline \(\phi_{B},\phi_{UL},\phi_{Z}\) & BS, UE and Interfering UE point process \\ \(\lambda,\lambda_{U}\) & BS and UE densities \\ \(l(r)\) & Path loss at distance \(r\) \\ \(a_{L}\) & LOS path loss exponent \\ \((\kappa,\zeta)_{\zeta})\) & NLOS path loss parameter \\ \(f_{c}\) & Operating frequency \\ B & Channel Bandwidth \\ \(r_{c}\) & Cell radius, \(\sqrt{\frac{1}{\pi\lambda}}\) \\ \(\sigma^{2}\) & Noise power \\ \(f_{R}(r)\) & PDF of distance \(r\) between typical UE and tagged BS \\ \(D_{Z}\) & Distance between interfering UE and tagged BS \\ \(l(D_{z})\) & Path loss on the link \(D_{z}\) \\ \(h_{0}\) & Small-scale fading on the link from serving BS \\ \(h_{Z}\) & Small scale fading on interfering link \\ \(G_{O_{j}}\) & Maximum directivity gain, \(j\in\{UE,BS\}\) \\ \(\theta_{j}\) & Antenna HPBW, \(j\in\{UE,BS\}\) \\ \(G_{sl_{j}}\) & Side lobe gain, \(j\in\{UE,BS\}\) \\ \(G_{Z}\) & Directivity gain of interfering UE \\ \(\theta_{Z}\) & Beamwidth of interfering UE \\ \(P_{c}(T)\) & SINR coverage probability at threshold T \\ \(P_{c,s}(T)\) & SINR coverage probability, on 1 link for \\ \(l\in\{LOS,NLOS\}\) & \\ \(p_{s}(r)\) & Probability of link being in 1 condition for \\ \(l\in\{LOS,NLOS\}\) \\ \(\mathcal{L}_{I}(s)\) & Laplace Transform of interference component \\ ASE & Area spectral efficiency \\ \(L_{l}(k)\) & \(k\)th order logarithmic function \\ \hline \end{tabular}
\end{table}
Table 1: List of notations and their definition.  
to the typical UE. According to the law of total probability, the SINR coverage probability \(P_{c}(T)\) can be expressed as:  
\[P_{c}(T)=p_{LOS}(r)P_{c,LOS}(T)+p_{NLOS}(r)P_{c,NLOS}(T) \tag{10}\]  
The derived expression for the probability of SINR coverage \(P_{c}(T)\) averaged over \(\mathbb{R}^{2}\) conditioned on the nearest BS at a distance \(\mathsf{r}\) from the typical UE is given below:  
\[P_{c}(T)=2\pi\lambda\int_{0}^{\infty}\exp^{(-\mu_{L}\tau^{2})} \exp\left(-2\pi\lambda\sum\limits_{G\in\mathbb{G}_{c}}P_{G_{c}}\left(\int_{r}^ {\infty}\left(1-\frac{1}{1+\mu_{L}G_{z}+z_{L}}\right)tp_{LOS}(t)dt\right)\right)\] \[r\exp(-\pi\lambda r^{2})\gamma p_{LOS}(r)dr+2\pi\lambda\int_{0}^{ \infty}r\exp\left(-\lambda\pi r^{2}\right)\times\] \[\exp\left(-\frac{2\pi\lambda}{\zeta\left(\kappa^{\frac{1}{\kappa} }\right)}\sum\limits_{k=1}^{4}\,h_{k}\left[\int_{0}^{exp(-\kappa r^{2})}\frac{ sq_{k}}{1+s_{k}u}(-ln(u))^{\frac{2-\zeta}{1}}p_{NLOS}\left(\frac{-ln(u)}{ \kappa}\right)^{\frac{1}{4}}du\right]\right)p_{NLOS}(r)dr \tag{11}\]  
Proof.: The proof is given in Appendix A.  
HMThe integrals obtained for total coverage probability are too complex to be evaluated in the closed form. The expression can only be simplified in its closed form for special values of \(\zeta\) in the following subsection.