## 2 System Model
* TCP: To simulate the TCP distribution, we again assume a homogeneous Poisson point process with an intensity \(\Lambda_{\text{gue}}\) within \(\text{AR}_{\text{sim}}\). For each point of the underlying point process, a Poisson number of points with mean \(\mu>0\), we simulate two independent zero-mean normal variables with a variance \(\sigma^{2}\) corresponding to the Cartesian coordinates. The considered underlying point process is also called the parent (point) process and its points are the centers of the cluster disks. The subsequent point process on all the disks is called the daughter (point) process and each daughter point is scattered around the origin based on a normal distribution with variance \(\sigma^{2}\). In the assumed public safety scenario, \(N_{\text{gue}}\) are the total number of GUEs scheduled in AG-HetNet. Consider an arbitrary GUE \(n\), such that the nearest distance from any macrocell of interest (MOI) and UABS' cell of interest (UOI) is given by \(d_{on}\) and \(d_{un}\), respectively. With a fixed antenna height, the location of wireless node MBS is modeled using a 2D Poisson point process (PPP) with densities \(\Lambda_{\text{mbs}}\). On the other hand, UABS is deployed on a fixed hexagonal grid at a fixed altitude. The densities and deployment heights of each wireless node in the AG-HetNet under consideration are specified in the Table defined in Section 4. Finally, assuming a Nakagami-m fading channel, the reference symbol received power from UOI and MOI can be given as  
\[\text{Pow}^{\prime}_{\text{uabs}}(d_{un}) =\frac{\text{Pow}_{\text{uabs}}A_{\text{3 DBF}}(\phi,\theta)\text{ F}}{10\%(d_{un})/10},\] \[\text{Pow}^{\prime}_{\text{mbs}}(d_{on}) =\frac{\text{Pow}_{\text{mbs}}A_{\text{3 DBF}}(\phi,\theta)\text{ F}}{10\%(d_{un})/10}, \tag{1}\]  
where the variables \(\varphi(d_{on})\) and \(\varphi(d_{un})\) are the path loss, respectively, observed from MBS and UABS in dB. Additionally, the random variable \(F\) accounts for Nakagami-m fading, whose probability density function is given by [4]  
\[f_{N}(\psi,w)=\frac{w^{w}\psi^{w}}{\Gamma(w)}\exp(-w\psi), \tag{2}\]  
where \(w\) is the shaping parameter, \(\psi\) is the channel amplitude, and \(\Gamma(w)\) is the standard Gamma function given as \(\Gamma(w)=\int_{0}^{\infty}\exp(-u)u^{w-1}\text{d}u\). Through the shaping parameter \(w\), the received signal power can be approximated to variable fading conditions. The value \(w>1\) approximates to Rician fading along line-of-sight (LOS) and \(w=1\) approximates to Rayleigh fading along NLOS. In Equation (1), we define \(A_{\text{3 DBF}}(\phi,\theta)\) as the transmitter antenna's 3 DBF element, where \(\theta\) and \(\phi\) is the zenith and azimuth of the spherical angles and unit vectors. By restating the definition of \(A_{\text{3 DBF}}(\phi,\theta)\) given in [4, 38], the equation is as follows  
\[A_{\text{3 DBF}}(\phi,\theta)=G_{\text{3 DBF,max}}-\min\big{\{}- (A_{\text{Hor}}(\phi)+A_{\text{Ver}}(\theta)),\,A_{m}\big{\}}, \tag{3}\] \[G_{\text{3 DBF,max}}=8\;\text{dBi},\;A_{m}-30\;\text{dB},\]where \(A_{\rm{Hor}}(\phi)\) is the antenna horizontal element, whereas \(A_{\rm{Ver}}(\theta)\) is the antenna vertical element of the radiation pattern and is given by  
\[A_{\rm{Hor}}(\phi)=-\min\left[12\bigg{(}\frac{\phi}{\phi_{\rm{3 dB}}}\bigg{)}^{2 },A_{m}\right],\ \phi_{\rm{3 dB}}=65^{\circ}, \tag{4}\]  
\[A_{\rm{Ver}}(\theta)=-\min\left[12\bigg{(}\frac{\theta-\theta_{\rm{tilt}}}{ \theta_{\rm{3 dB}}}\bigg{)}^{2},SLAV\right],\ \theta_{\rm{tilt}}=90^{\circ}, \tag{5}\]  
\[SLAV=30,\ \theta_{\rm{3 dB}}=65^{\circ}.\]  
3 DBF can further assist the interference coordination process and has the potential to significantly improve the signal-to-interference ratio (SIR) of the desired signal by calibrating MBS (\(\rm{Pow}_{\rm{mbs}}\)) and UABS (\(\rm{Pow}_{\rm{uabs}}\)) [39]. Thus, by regulating the transmission power, we achieve significant improvements to SIR for the GUEs located at the cell edge or the GUEs in the range-expanded regions.