# Performance of Millimeter Wave Dense Cellular Network Using Stretched Exponential Path Loss Model
## 2 System Model
### Path Loss and Channel Model  
The path loss on a link of length \(r\) is  
\[l(r)=\mathbb{1}l_{LOS}(r)p_{LOS}(r)+\mathbb{1}l_{NLOS}(r)(1-p_{LOS}(r)) \tag{2}\]  
Here, \(\mathbb{1}\) is a function that indicates an active link-state. Incorporating NLOS propagation in the path loss model accounts for the presence of obstacles in the path between BS and UE, which causes exponential attenuation [22]. Following the approach in [42], the blockages are modeled as an element of large-scale fading with a certain probability of occurrence. According to SEPLM, the signal attenuates over a distance \(r\) as follow [33]:  
\[l_{NLOS}(r)=\exp(-\kappa r^{\zeta}) \tag{3}\]  
where \(\kappa\), \(\zeta>0\) are path loss parameters that define several propagation environments. \(\zeta\) defines the factor with which the obstructing objects scale with respect to the path length (\(r^{\zeta}\)) and \(\kappa\) is the average multiplicative attenuation caused by these obstructions. This path loss model is particularly used to describe an urban network where signal mostly attenuates due to discrete obstructions such as buildings, infrastructures, etc. and has been validated through empirical measurements data of an urban small cell environment within the distance range \(r\in[5,350]\) m for small to medium cell radii. The scaling of obstruction has the following interpretations:  
1. \(\zeta=1\), defines the environment where the obstacles scale linearly with the path length between UE and BS. (3) can be written as \(L(r)=\exp(-\kappa r)\). This case captures obstacles distribution similar to the one defined in [22], based on random shape theory that is widely adopted for performance analysis of mmwave networks. The randomly oriented obstacles are uniformly distributed over the plane intersecting the path length \(r\) between BS and UE and scales linearly with the distance \(r\). \(\kappa\) in this case depends upon the attenuation of each blocking object.
2. \(\zeta=2\), defines the environment where the obstacle scales with \(r^{2}\). Assuming UE to be located at the centre of a disc \(B(0,r)\) and signal propagates within the disc sector extending to the BS, (3) can be written as \(L(r)=\exp(-\kappa r^{2})\) and takes the form analogous to LOS probability function in [43] expressed as \(\exp(-(\frac{r}{L})^{2})\), where \(L\) depends upon the density of large obstructing objects in the propagation environment. A larger value of \(\kappa\) signifies a sparse environment having high LOS probability with distance.
3. \(\zeta<1\) defines the case similar to the ray propagation in lattice modeling of urban areas with regular building blockage [28]. Here, \(\kappa\) depends on the properties of the considered lattice and the reflectivity of the obstacle.
4. The special values of \((\kappa,\zeta)=(0.3,2/3)\) and \((0.94,1/2)\) in SEPLM reduces to multislope path loss model [44] consistent with the one adopted in 3 GPP standardization. Accordingly, the path loss on the LOS and NLOS links \(L_{LOS}(r)\) and \(L_{NLOS}(r)\) are defined as follows: \[l(r)=\left\{\begin{array}{l}l_{LOS}(r)=r^{-\alpha_{L}}\\ l_{NLOS}(r)=\exp(-\kappa r^{\zeta})\end{array}\right.\] (4)  
Here, \(\alpha_{L}\) is the LOS path loss exponent. The probability of a link being in LOS (\(p_{LOS}(r)\)), specified by 3 GPP for outdoor urban microcellular environment is considered in this analysis [45],  
\[p_{LOS}(r)=\exp\Bigl{(}-\frac{r}{L}\Bigr{)} \tag{5}\]Similarly, the probability of the link being in NLOS state is given by:  
\[p_{NLOS}(r)=1-p_{LOS}(r)=1-\exp\Bigl{(}-\frac{r}{L}\Bigr{)} \tag{6}\]  
\(L\) is a parameter that determines the likelihood of LOS of a certain propagation environment as a function of the distance. For each link between a UE and its tagged BS and between interfering UEs and typical BS, Rayleigh fading is assumed.