# Analysis of Large Scale Aerial Terrestrial Networks with mmWave Backhauling

Nour Kouzayha  Hesham ElSawy  Hayssam Dahrouj  Khlod Alshaikh  Tareq Y. Al-
Naffouri  and Mohamed-Slim Alouini  This work has been presented in part in
[1] at IEEE International Conference on Communications (ICC’2020), Dublin,
Ireland.N. Kouzayha, H. Dahrouj, T. Y. Al-Naffouri, and M.-S. Alouini are with
the Division of Computer, Electrical and Mathematical Sciences and
Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi
Arabia (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>slim.alouini@kaust.edu.sa).H. ElSawy is with
the Electrical Engineering Department, King Fahd University of Petroleum and
Minerals (KFUPM), Dhahran, Saudi Arabia. (e-mail:
hesham.elsawy@kfupm.edu.sa).K. Alshaikh is an enterprise solutions architect
at Udacity (e-mail: khlod.alshaikh@gmail.com).

###### Abstract

Service providers are considering the use of unmanned aerial vehicles (UAVs)
to enhance wireless connectivity of cellular networks. To provide
connectivity, UAVs have to be backhauled through terrestrial base stations
(BSs) to the core network. In particular, we consider millimeter-wave (mmWave)
backhauling in the downlink of a hybrid aerial-terrestrial network, where the
backhaul links are subject to beamforming misalignment errors. In the proposed
model, the user equipment (UE) can connect to either a ground BS or a UAV,
where we differentiate between two transmission schemes according to the
backhaul status. In one scheme, the UEs are served by the UAVs regardless of
whether the backhaul links are good or not. In the other scheme, the UAVs are
aware of the backhaul links status, and hence, only the subset of successfully
backhauled UAVs can serve the UEs. Using stochastic geometry, the performance
of the proposed model is assessed in terms of coverage probability and
validated against Monte-Carlo simulations. Several insights are provided for
determining some system parameters including the UAVs altitude and required
number and the beamforming misalignment error of the backhaul link. The
obtained results highlight the impact of the UAVs backhaul link on the UE
experience.

###### Index Terms:

UAV, mmWave, terrestrial BS, backhaul unaware transmission, backhaul aware
transmission, coverage probability, stochastic geometry.

##  I Introduction

Unmanned aerial vehicles (UAVs) are receiving significant interest to enhance
the wireless coverage of cellular networks [2]. UAVs are used when terrestrial
cellular systems get damaged, in hard to reach areas, and wherein there is an
occasional need for supplementary coverage. One of the main features of UAVs
is the existence of line-of-sight (LOS) links, which can improve the signal
quality compared to non-line-of-sight (NLOS) links. The LOS transmission is
considered as one of the key factors that can boost capacity in beyond $5$G
(B$5$G) ultra dense networks. The placement of UAVs at elevated altitudes can
effectively avoid obstacles, thus enabling millimeter wave (mmWave) based
technologies such as multiple-input-multiple-output (MIMO) transmissions [3].
In addition, UAVs can be further integrated with emerging network
architectures such as Internet of things [4, 5] and multiple edge computing
[6]. In this work, we consider a hybrid aerial-terrestrial cellular network
where UAVs are deployed above a built-up dense urban area to assist
terrestrial networks and are backhauled through mmWave links.

###  I-A Related Work

The foreseen potential of UAVs’ to support massive wireless connectivity has
attracted significant research efforts to model, analyze, and design UAVs
networks. Instances of such research efforts include developing air-to-air and
air-to-ground communications channel models [7, 8], optimal placement and
resource allocation for UAVs [9, 10], trajectory planning [11, 12, 4],
charging ground sensors using UAVs radio frequency (RF) signals [13, 14] and
evaluating the performance of UAV-assisted cellular networks [15, 16, 17].

To best analyze the performance of UAV networks, stochastic geometry becomes
indispensable to account for the mutual interference between the active
aerial-terrestrial links [18]. In [19], the authors use stochastic geometry to
provide an expression of the coverage probability for a UAV network under
guaranteed LOS conditions. However, the model in [19] overlooks the effect of
LOS/NLOS components of the aerial channel. A probabilistic LOS/NLOS
propagation model for aerial channels is adopted in [15] for UAV-assisted
cellular networks. Such LOS/NLOS assumption leads to more accurate results and
design insights at the expense of more involved analysis. In [20], the authors
use stochastic geometry to model a network where tethered UAVs are deployed to
assist traffic offloading under practical challenges.

Terrestrial BSs can be backhauled via wired or wireless links. In contrast,
wireless backhauling is mandatory for UAVs to support their mobility. The UAV
backhaul links can be provided by terrestrial BSs using sub-$6$ GHz
technologies, mmWave technologies [21], or free space optics (FSO) [22]. For
instance, the authors in [23] consider point-to-point FSO backhaul links for
UAVs. Laser beams are used in [24] to boost UAVs operations with both power
and backhauling capabilities. Across the RF spectrum, backhauling at the
mmWave range is shown to be superior to sub-$6$ GHz backhauling due to the
larger bandwidth and the active beam steering capabilities. In this regard,
3GPP envisions an integrated access and backhaul architecture for small cell
BSs in which the same infrastructure and spectral resources are used by the
macro BSs on the access links and the small BSs on the backhaul links [25].
This new architecture introduces new design challenges such as the proper
partitioning of resources between the access and the backhaul links [26, 27].
As for UAV networks, the authors in [21] utilize mmWave backhauling, where
issues related to beam tracking, LOS blockage and UAV discovery are
investigated.

Despite UAV communications multiple promises, modeling and analyzing the
impact of UAV backhauling remains relatively unexplored. The main focus in the
literature is mostly on the UAV access links while assuming guaranteed UAV
backhaul. The work in [28] is one of few exceptions that explicitly accounts
for the UAV backhaul. However, this work assumes that backhauling BSs cannot
directly serve UEs, which underutilizes the role of terrestrial BSs.
Furthermore, the UAV network in [28] is modeled as an infinite Poisson Point
Process (PPP) and the backhauling impact is only incorporated via simulations.
The backhaul effect is also considered in [29], where a single UAV is
introduced to maximize the throughput in a dense urban environment. Hence, the
work in [29] does not account for the possible coexistence of multiple UAVs
and the subsequent mutual interference between them. Up to the authors’ best
knowledge, the joint consideration of the access and mmWave backhaul links in
a large-scale hybrid aerial-terrestrial cellular network has not been
investigated in literature, and so this paper addresses the details and
intricacies of this problem from a stochastic geometry perspective.

###  I-B Contribution and Organization

In this paper, a hybrid aerial-terrestrial network is considered, where UAVs
are used to assist BSs in a dense urban environment in providing coverage to
UEs. The UAVs are backhauled through mmWave links, which are subject to
beamforming misalignment errors. Using stochastic geometry, we aim to evaluate
the performance of the network and the effect of its parameters, while
highlighting the impact of mmWave backhauling on UE coverage. To the best of
the authors’ knowledge, this work is the first to jointly consider the access
and mmWave backhaul links in a large-scale hybrid aerial-terrestrial network,
where the UE can be either served by a BS or by a UAV. The main contributions
of this paper can be summarized as follows:

  1. 1.

System model: We consider a hybrid aerial-terrestrial network with a finite
number of backhaul-enabled UAVs. The UAV network is modeled as a binomial
point process (BPP), which fits more realistic use cases than infinite PPPs.
The developed mathematical framework explicitly accounts for the LOS/NLOS
mmWave links along with possible beamforming misalignment errors.

  2. 2.

Comparison between backhaul aware/unaware transmission models: The proposed
model differentiates between backhaul aware and backhaul unaware transmission
models. For the unaware scenario, the serving UAV transmits data to UE without
considering its backhaul link quality. In the backhaul aware scheme, the
serving UAV is aware of the backhaul status and does not communicate with the
UE in case of backhaul link outage. The obtained results show that prior
knowledge of the backhaul link quality relieves parts of the network
interference, and hence, improves the overall coverage probability.

  3. 3.

Design guidelines and insights: The analytical results, validated using Monte-
Carlo simulations, highlight the fundamental impact of the backhaul link
quality on the UE coverage. Based on the proposed framework, we investigate
the impact of various parameters including the intensity of terrestrial BSs,
the fraction of backhaul-enabled BSs, the altitude and number of UAVs, as well
as the beamforming misalignment errors. The results reveal that the number and
altitude of UAVs must be chosen carefully to optimize the UE experience.

TABLE I: Notations Summary. Notation |  Description
---|---
$D_{u}$, $r_{u}$, $v_{0}$ |  Disk in which the UAVs are distributed, Radius of $D_{u}$, Distance from the UE to origin
$N_{u}$, $h_{u}$, $\lambda_{g}$, $h_{g}$ |  Number and height of UAVs, Density and height of terrestrial BSs with reference to ground
$\delta_{b}$, $\lambda_{b}$ |  Fraction and density of terrestrial backhaul-enabled BSs
$\Phi_{u}$, $\Phi_{g}$, $\Phi_{g_{l}}$, $\Phi_{g_{n}}$ |  Set of UAVs, terrestrial BSs, LOS BSs and NLOS BSs respectively
$\kappa_{u_{l}}(\cdot)$, $\kappa_{u_{n}}(\cdot)$, $\kappa_{b_{l}}(\cdot)$, $\kappa_{b_{n}}(\cdot)$ |  LOS and NLOS probabilities of the access link and backhaul link, respectively
$P_{u}$, $P_{g}$, $P_{b}$ |  Transmit power of UAVs, BSs on the access link, BSs on the backhaul link, respectively
$\eta_{g}$, $\eta_{l}$, $\eta_{n}$ |  Path-loss exponent parameter for terrestrial BS, LOS UAV, or NLOS UAV, respectively
$m_{l}$, $m_{n}$ |  Nakagami-m fading parameter for aerial LOS link, or aerial NLOS link, respectively
$\Omega_{g,x_{i}}$, $\Omega_{u_{l},y_{j}}$, $\Omega_{u_{n},y_{j}}$, $\Omega_{b_{l},x_{i},y_{j}}$, $\Omega_{b_{n},x_{i},y_{j}}$ |  Small scale fading gain between UE and $i$-th BS, $j$-th LOS or NLOS UAV on the access link and between $j$-UAV and $i$-th LOS or NLOS BS on the backhaul link, respectively
$s_{g,x_{i}}$, $s_{u_{l},y_{j}}$, $s_{u_{n},y_{j}}$, $s_{b_{l},x_{i},y_{j}}$, $s_{b_{n},x_{i},y_{j}}$ |  Horizontal distance between the projections of the UE and the $i$-th BS or the $j$-th LOS or NLOS UAV and between the $i$-th LOS or NLOS BS and the $j$-th UAV, respectively
$z_{g,x_{i}}$, $z_{u_{l},y_{j}}$, $z_{u_{l},y_{j}}$, $z_{b_{l},x_{i},y_{j}}$, $z_{b_{n},x_{i},y_{j}}$ |  Distance between the UE and the $i$-th BS or the $j$-th LOS or NLOS UAV and between the $i$-th LOS or NLOS BS and the $i$-th UAV, respectively
$P^{r}_{g,x_{i}}$, $P^{r}_{u_{l},y_{j}}$, $P^{r}_{u_{n},y_{j}}$, $P^{r}_{b_{l},x_{i},y_{j}}$, $P^{r}_{b_{n},x_{i},y_{j}}$ |  Received power from the $i$-th BS or the $j$-th LOS/NLOS UAV at the UE on the access link, or from $i$-th LOS/NLOS BS to the $j$-th UAV on the backhaul link, respectively
$x_{g}$, $y_{u_{l}}$, $y_{u_{n}}$, $x_{b_{l}}$, $x_{b_{n}}$ |  Distances between the UE and its serving BS, LOS or NLOS UAV on the access link and between the UAV and its serving LOS or NLOS BS on the backhaul link, respectively
$C_{l}$, $C_{n}$, $\sigma_{b}^{2}$ |  LOS and NLOS near-field path loss and noise power for backhaul connection
$G_{s}^{(\mathrm{max})}$, $G_{s}^{(\mathrm{min})}$, $\theta_{s}$ |  Antenna parameters for BSs ($s=g$) and UAVs ($s=u$) for backhaul connection
$\varepsilon_{g}$, $\varepsilon_{u}$, $\sigma_{g}^{2}$, $\sigma_{u}^{2}$ |  Additive beam-steering errors and corresponding variances of BS and UAV antennas
$\hat{I}_{g}$, $\hat{I}_{u_{l}}$, $\hat{I}_{u_{n}}$ |  Interference from all BSs or UAVs except serving BS, LOS or NLOS UAV, respectively
$I_{u}$, $I_{g_{l}}$, $I_{g_{n}}$ |  Interference from all UAVs or BSs if UE associates to a BS, a LOS or NLOS UAV, respectively
$A_{g}$, $A_{u_{l}}$, $A_{u_{n}}$, $A_{b_{l}}$, $A_{b_{n}}$ |  Association probabilities between the UE and a BS, a LOS or a NLOS UAV for backhaul unaware and between the UAV and a LOS or a NLOS BS for backhaul, respectively
$S_{l}(\cdot)$, $S_{n}(\cdot)$, $S(\cdot)$ |  Conditional backhaul probabilities given that the UAV is associated with a LOS BS, or a NLOS BS and overall backhaul probability, respectively
$P_{cov,g}$, $P_{cov,u_{l}}$, $P_{cov,u_{n}}$ |  Conditional coverage probabilities given that the UE is associated with a terrestrial BS, a LOS UAV, or a NLOS UAV for backhaul unaware transmission, respectively
$\tilde{A}_{g}$, $\tilde{A}_{u_{l}}$, $\tilde{A}_{u_{n}}$, $\tilde{A}_{f}$ |  Association probabilities between the UE and a terrestrial BS, a LOS UAV, or a NLOS UAV and service failure probability for backhaul aware transmission, respectively
$\tilde{P}_{cov,g}$, $\tilde{P}_{cov,u_{l}}$, $\tilde{P}_{cov,u_{n}}$ |  Conditional coverage probabilities given that the UE is associated with a terrestrial BS, a LOS UAV, or a NLOS UAV for backhaul aware transmission, respectively
$\tilde{N_{u}}$, $I_{\tilde{u}}$ |  Number of UAVs with successful backhaul links for backhaul aware transmission, Interference from UAVs with successful backhaul links
$P_{cov}$, $\tilde{P}_{cov}$ |  Overall coverage probabilities for backhaul unaware and aware transmission, respectively
$\tau_{a}$, $\tau_{b}$ |  $\mathrm{SIR}$ threshold on the access link, $\mathrm{SINR}$ threshold on the backhaul link

###  I-C Paper Notations and Organization

The subscripts $\\{\cdot\\}_{u}$, $\\{\cdot\\}_{g}$, and $\\{\cdot\\}_{b}$
refer to UAV, ground BS and backhaul, respectively. The subscripts
$\\{\cdot\\}_{l}$, $\\{\cdot\\}_{n}$ refer to LOS and NLOS, respectively.
$\tilde{\\{\cdot\\}}$ denotes the backhaul aware transmission scheme.
$\mathbb{P}\\{\cdot\\}$ denotes probability, $\mathbb{E}[\cdot]$ denotes
expectation, $\mathcal{L}_{x}(\cdot)$ denotes the Laplace transform of a
random variable $x$, and
${}_{2}F_{1}(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{x^{n}}{n!}$
denotes the Gauss hyper geometric function. Other notations are listed in
Table I.

The rest of the paper is organized as follows. Section II describes the system
model. The analysis of the backhaul unaware and aware transmission scenarios
are presented in Sections III and IV, receptively. Numerical results and
design insights are discussed in Section V and validated using Monte-Carlo
simulations. Finally, the paper is concluded in Section VI.

##  II System Model

###  II-A Network Model

UAVs are deployed to improve the downlink (DL) coverage of a one-tier cellular
network, where UEs can be served by either terrestrial BSs or UAVs. The BSs
provide wireless access to the UEs, and mmWave backhaul links to UAVs. Fig. 2
presents the proposed network model. Since we consider a DL scheme, the UAVs
transmit to UEs and receive from BSs for backhauling purposes. The UAVs are
considered as simple relays with no queuing capabilities. On the other hand,
the BSs are assumed to have saturated buffers which always have packets for
their served UEs, where UEs can be directly served from the BS or through a
UAV relay. The analysis of spatio-temporal models with non-saturated BSs
buffers and queuing UAVs is beyond the scope of this paper and is left as a
future research direction.

Figure 1: Proposed system model

Figure 2: Sectored-pattern antenna.

The ground BSs are spatially distributed according to a PPP
$\Phi_{g}=\\{\textbf{x}_{i}\\}$, where $\textbf{x}_{i}$ is the location of the
$i$-th BS. We assume that the density of the ground BSs is $\lambda_{g}$ and
that they are all at the same height $h_{g}$ and transmit with the same power
$P_{g}$ to UEs. A fraction $\delta_{b}$ of the terrestrial BSs is equipped
with backhauling capabilities and transmit to the UAVs with the same power
$P_{b}$ for backhaul. Thus, the density of backhaul-enabled BSs $\lambda_{b}$
is equal to $\delta_{b}\lambda_{g}$. We assume that $N_{u}$ UAVs hover at an
altitude $h_{u}$ and are distributed uniformly in a finite disk $D_{u}$ with
radius $r_{u}$ forming a BPP $\Phi_{u}=\\{\textbf{y}_{j}\\}$, where
$\textbf{y}_{j}$ refers to the location of the $j$-th UAV. Without loss of
generality, we assume that the BPP is centered around
$\textbf{o'}=(0,0,h_{u})$ and that all the UAVs transmit with the same power
$P_{u}$ to UEs. For the access link, the BSs and UAVs are equipped with single
antennas, so as to serve single antenna UEs on the sub-$6$ GHz band. On the
other hand, dedicated antenna arrays at the BSs and UAVs are used to align the
mmWave transmissions for backhaul. Using separate antennas for mmWave backhaul
and access at the terrestrial BSs has two main benefits. Firstly, it allows
the BSs to steer their antennas to align with those of the UAVs, and hence, to
maximize the directionality gain. Secondly, it reduces the interference on
both access and backhaul links as they operate on different frequency bands.
The analysis is conducted for a UE positioned at an arbitrary distance
$\textbf{v}_{0}=(v_{0},0,0)$ from the origin $\textbf{o}=(0,0,0)$ on the
ground. All frequency resources are universally reused across the network.

###  II-B Channel Model

####  II-B1 Terrestrial Access Channel (BS-UE)

Due to the nature of terrestrial communication, the fading channel of the BS-
UE access link consists of a large-scale fading modeled using a distance-
dependent path-loss with path-loss exponent $\eta_{g}$, and a small-scale
Rayleigh fading with exponential distribution and unit mean. The signal power
received at the UE from the $i$-th BS located at $x_{i}$ can thus be expressed
as
$P_{g,x_{i}}^{r}=P_{g}(s_{g,x_{i}}^{2}+h_{g}^{2})^{-\eta_{g}/2}\Omega_{g,x_{i}}$,
where $\Omega_{g,x_{i}}$ is the small-scale fading,
$s_{g,x_{i}}=\sqrt{z_{g,x_{i}}^{2}-h_{g}^{2}}$ is the horizontal distance
separating the UE and the projection of the $i$-th BS on the ground and
$z_{g,x_{i}}=||\textbf{x}_{i}-\textbf{v}_{0}||$ is the actual distance.

####  II-B2 Aerial Access (UAV-UE) and Backhaul (BS-UAV) Channels

Both the aerial access and backhaul channels are affected by obstacles in the
environment which break the LOS links. According to the ITU recommendation
report [30], the probability of a LOS link between a transmitter and a
receiver, with heights $h_{\mathrm{TX}}$ and $h_{\mathrm{RX}}$, respectively,
is given by [30]:

$\small
P_{\mathrm{LOS}}(r)=\prod_{n=0}^{m}\left[1-\exp\left(-\frac{h_{\mathrm{TX}}-\frac{(n+\frac{1}{2})\left(h_{\mathrm{TX}}-h_{\mathrm{RX}}\right)^{2}}{m+1}}{2\gamma^{2}}\right)\right]$
(1)

where $r$ denotes the horizontal distance between the transmitter and the
receiver and $m=\lfloor\frac{r\sqrt{\alpha\beta}}{1000}-1\rfloor$, where
$\alpha$, $\beta$ and $\gamma$ are environment related parameters given in
Table I in [31]. As we can see from (1), the LOS probability is not a
continuous function of the horizontal distance which makes the analysis
intractable. As the UE is considered at the ground level, we use the
approximation proposed in [32] to simplify the LOS probability on the access
link between a UAV located at a distance $r$ and the UE by a modified Sigmoid
function given as:

$\small\kappa_{u_{l}}(r)=\frac{1}{1+a\exp(-b[\frac{180}{\pi}\arctan(h_{u}/\sqrt{r^{2}-h_{u}^{2}})-a])},$
(2)

where $\sqrt{r^{2}-h_{u}^{2}}$ is the Euclidean horizontal distance separating
the projections of the UAV and the UE and $a$ and $b$ are constant values that
depend on the environment and are given in Table I of [33]. The NLOS
probability is $\kappa_{u_{n}}=1-\kappa_{u_{l}}$.

The LOS probability model in (2) used for the UAV-UE channel was initially
developed and approximated for UEs positioned at low height levels from the
ground. This model is unsuitable for the BS-UAV aerial channel as both the BSs
and UAVs are positioned above the ground level. We use the approximation
proposed in [34] to simplify (1) as a more tractable exponential function,
representing the LOS probability between a UAV and a BS separated by a
horizontal distance $r$ on the backhaul link as:

$\small\kappa_{b_{l}}(r)=-c\exp\left(-d\arctan\left(\frac{|h_{u}-h_{g}|}{r}\right)\right)+e,$
(3)

where $r$ is the horizontal distance separating the projections of the BS and
the UAV, $c$, $d$ and $e$ are parameters that depend on the environment and
height of terrestrial BSs and are given in Table II of [34]. The NLOS
probability on the backhaul link is given as
$\kappa_{b_{n}}=1-\kappa_{b_{l}}$.

The aerial fading channel is characterized using the combination of two
components: (a) a large-scale fading modeled using a distance-dependent path-
loss and (b) a small-scale Nakagami-$m$ fading modeled using a gamma-
distributed random variable. We consider different path-loss exponents
($\eta_{l}$ for LOS and $\eta_{n}$ for NLOS) and fading parameters for the LOS
and the NLOS links ($m_{l}$ for LOS and $m_{n}$ for NLOS). The power received
at the reference UE from the $j$-th UAV located at $\textbf{y}_{j}$ is
$P^{r}_{u_{\zeta},y_{j}}=P_{u}z_{u_{\zeta},y_{j}}^{-\eta_{\zeta}}\Omega_{u_{\zeta},y_{j}}$,
where $z_{u_{\zeta},y_{j}}=||\textbf{y}_{i}-\textbf{v}_{0}||$ is the distance
separating the UE from the $i$-th UAV, $\Omega_{u_{\zeta},y_{j}}$ is the
small-scale fading, $\eta_{\zeta}$ is the path-loss exponent, and
$\zeta\in\\{l,n\\}$ indicates whether the $j$-th UAV has a LOS link or NLOS
link with the UE.

UAVs are connected to ground BSs through mmWave backhaul links. Although
mmWave channels share basic propagation characteristics like power law path-
loss with sub-$6$ GHz channels, they also have critical distinctions. For
instance, beamforming is important to compensate for the high propagation loss
in mmWave frequencies as well as the consumed power in analog to digital
conversion in large antenna arrays. Thus, the BSs and UAVs must steer their
antennas to maximize the directionality gain. However, aligning the antennas
is subject to beam-steering errors, which affects the transmission on the
backhaul link. We approximate the array patterns of the UAVs and the BSs
antennas by the model shown in Fig. 2 and given as

$\small
G_{s}(\varphi)=\begin{cases}G_{s}^{(\text{max})},&|\varphi|\leq\theta_{s}\\\
G_{s}^{(\text{min})},&|\varphi|>\theta_{s}\end{cases}\vspace{-0.2cm}$ (4)

where $\varphi\in[-\pi,\pi)$ is the angle of the boresight direction,
$G_{s}^{(\text{max})}$, $G_{s}^{(\text{min})}$ and $\theta_{s}$ are the gains
of the main and side lobes and the beamwidth for the BSs and UAVs
$(s\in\\{g,u\\})$, respectively.

For the desired link, the directivity gain is
$G_{b,0}=G_{g}^{(\text{max})}G_{u}^{(\text{min})}$ in the absence of the
beamforming misalignment errors modeled explicitly in Section III-C2. For the
interfering BSs, the beams are assumed to be randomly oriented with respect to
each other, and the steering angles are distributed uniformly in $[-\pi,\pi)$.
Since (4) produces either gains $G_{s}^{(\text{max})}$ and
$G_{s}^{(\text{min})}$, $s\in\\{g,u\\}$ over all possible input angles, the
resulting gain distribution of a BS or UAV is given as

$\small
f_{G_{s,I}}(g)=\frac{\theta_{s}}{2\pi}\delta\left(g-G_{s}^{(\text{max})}\right)+\left(1-\frac{\theta_{s}}{2\pi}\right)\delta\left(g-G_{s}^{(\text{min})}\right)\vspace{-0.1cm}$
(5)

where $\delta(\cdot)$ is the Dirac delta function, and $s\in\\{g,u\\}$.

Accordingly, the directivity gain of an interfering link is a discrete random
variable denoted as $G_{b,I}$, with a probability mass function (PMF) that can
be formulated as $f_{G_{b,I}}(g)=\left(f_{G_{g,I}}\otimes
f_{G_{u,I}}\right)(g)$, where $f_{G_{g,I}}(\cdot)$ and $f_{G_{u,I}}(\cdot)$
are the PMFs of the antenna gains of the interfering BS and the corresponding
UAV. Finally, based on (4) and (5), the directivity gain of the interfering
links takes the values $G_{k}$ with probability $p_{k}$ ($k\in\\{1,2,3,4\\}$)
defined in Table II, where $c_{g}=\frac{\theta_{g}}{2\pi}$ and
$c_{u}=\frac{\theta_{u}}{2\pi}$. $\theta_{g}$ and $\theta_{u}$ are the
beamwidth of the BS and the UAV antennas.

TABLE II: Probability mass function of $G_{b,I}$ and $G_{b,0}$. k | 1 | 2 | 3 | 4
---|---|---|---|---
$G_{k}$ | $G_{g}^{(\mathrm{max})}G_{u}^{(\mathrm{max})}$ | $G_{g}^{(\mathrm{max})}G_{u}^{(\mathrm{min})}$ | $G_{g}^{(\mathrm{min})}G_{u}^{(\mathrm{max})}$ | $G_{g}^{(\mathrm{min})}G_{u}^{(\mathrm{min})}$
$p_{k}$ | $c_{g}c_{u}$ | $c_{g}(1-c_{u})$ | $(1-c_{g})c_{u}$ | $(1-c_{g})(1-c_{u})$
$p_{k,0}$ | $F_{\lvert\varepsilon_{g}\rvert}(\frac{\theta_{g}}{2})F_{\lvert\varepsilon_{u}\rvert}(\frac{\theta_{u}}{2})$ | $F_{\lvert\varepsilon_{g}\rvert}(\frac{\theta_{g}}{2})\bar{F}_{\lvert\varepsilon_{u}\rvert}(\frac{\theta_{u}}{2})$ | $\bar{F}_{\lvert\varepsilon_{g}\rvert}(\frac{\theta_{g}}{2})F_{\lvert\varepsilon_{u}\rvert}(\frac{\theta_{u}}{2})$ | $\bar{F}_{\lvert\varepsilon_{g}\rvert}(\frac{\theta_{g}}{2})\bar{F}_{\lvert\varepsilon_{u}\rvert}(\frac{\theta_{u}}{2})$

The received power from the $i$-th BS located at $\textbf{x}_{i}$ at the
$j$-th UAV located at $\textbf{y}_{i}$ is given as
$P^{r}_{b,x_{i},y_{j}}=P_{b}G_{b}C_{\xi}\left(s_{b_{\xi},x_{i},y_{j}}^{2}+\Delta_{h}^{2}\right)^{-\eta_{\xi}/2}\Omega_{\xi,x_{i},y_{j}}$,
where $P_{b}$ is the BS transmit power on the backhaul link,
$\Delta_{h}=|h_{u}-h_{g}|$ is the difference between the BSs and the UAVs
heights, $\Omega_{b_{\xi},x_{i},y_{j}}$ is the small-scale fading, $C_{\xi}$
is the path-loss intercept and $\eta_{\xi}$ is the path-loss exponent, where
$\xi\in\\{l,n\\}$ indicates whether the $i$-th BS has a LOS link or a NLOS
link with the $j$-th UAV.

###  II-C Association Strategy and Performance Metrics

This paper studies the performance of a hybrid aerial-terrestrial network
where both BSs and UAVs are used to serve UEs. That is, each UE connects to
the BS or the UAV that offers the maximum average received signal strength.
Note that the UE can either connect to a LOS or a NLOS UAV. The coverage
probability, defined as the probability that the signal-to-interference ratio
($\mathrm{SIR}$) exceeds a threshold $\tau_{a}$, is the main performance
metric. Each UAV selects its backhauling BS based on the minimum path-loss
association rule. Hence, the BS that provides the backhaul is the best in
terms of signal strength. Each UAV and its backhauling BS steer their beams to
align their transmissions in order to strengthen the intended link and
mitigate interference. Such beam steering along with the high bandwidth and
severe path-loss of mmWave frequencies relieve the interference dominance and
make the backhaul links sensitive to the ambient noise. Thus, it is more
adequate to assess the backhaul links through the signal-to-interference-and-
noise-ratio ($\mathrm{SINR}$) rather than the $\mathrm{SIR}$. A backhaul
transmission is considered successful if and only if the $\mathrm{SINR}$
exceeds a threshold $\tau_{b}$, which defines the backhaul probability.

We differentiate between two schemes, namely, backhaul unaware and backhaul
aware transmission schemes. In the first scheme, the serving UAV transmits
directly to the UE regardless of whether the backhaul link is successful or
not. Thus, no further post processing is required when the UAV receives
packets from the BS and forwards them to the UE. The aerial coverage in the
unaware scheme requires two conditions: (i) $\mathrm{SIR}>\tau_{a}$: The
received $\mathrm{SIR}$ at the UE from its serving UAV must exceed a threshold
$\tau_{a}$, (ii) $\mathrm{SINR}>\tau_{b}$: The received $\mathrm{SINR}$ at the
serving UAV from the BS to which it connects for backhaul support must exceed
a threshold $\tau_{b}$.

In the backhaul aware scenario, the serving UAV is aware of the backhaul
status and checks the integrity of the packets before transmitting them to the
UE. This will impose a processing burden on the UAV side but will improve the
probability of coverage and limits the aerial interference to the subset of
UAVs with successful backhaul links only. An active transmission in the aware
scheme implies that the condition ($\mathrm{SINR}>\tau_{b}$) is already
satisfied. Otherwise, the UE halts operation and goes to a service failure
status in this specific time slot. Thus, the backhaul aware transmission
success is only subject to the $\mathrm{SIR}$ condition
($\mathrm{SIR}>\tau_{a}$) of the access link.

##  III Backhaul unaware transmission

In this section, we derive the coverage probability for backhaul unaware
transmission. In this scenario, the association rule is only dependent on the
quality of the UE-UAV/BS connection. However, a UAV can serve the UE only if
it has a successful backhaul link with a ground BS.

###  III-A Association Probabilities

To get coverage, the UE can associate to either a terrestrial BS or a UAV. The
set of UAVs is divided into two sub-processes according to the LOS/NLOS status
of the access links. According to the association rule, each UE connects to
the BS or the UAV that offers the maximum average received power. The serving
BS/UAV is not necessarily the nearest to the UE due to the difference in path-
loss parameters and transmit powers. However, within a particular set, the
path-loss parameters and transmit powers are the same for all links.
Therefore, for a specific set, the closest BS/UAV provides a larger average
received power than that provided by any other in this set. Thus, the UE is
served by the closest BS, LOS UAV or NLOS UAV. The association rule implies an
exclusion region on the locations of the nearest interfering device and
therefore on the locations of all the other interfering devices in each set
for each association type. Specifically, when the UE associates with a
terrestrial BS located at a horizontal distance $x_{g}$, all the LOS UAVs are
further than $E_{g_{l}}(x_{g})$. Similarly, all the NLOS UAVs are further than
$E_{g_{n}}(x_{g})$.

Figure 3: Exclusion regions $E_{g_{l}}(x_{g})$ and $E_{g_{n}}(x_{g})$ on the
LOS and NLOS UAVs.

Figure 4: Top view snapshot of BSs, LOS/NLOS UAVs and projections of
exclusions regions.

Fig. 4 visualizes the exclusion regions on the LOS and NLOS UAVs when the UE
associates to a terrestrial BS. Fig. 4 presents a top view snapshot on the
network showing the exclusion regions on the LOS and NLOS UAVs locations as
red dashed circles. Similarly, the exclusion regions $E_{u_{l}}(x_{u_{l}})$
and $E_{ln}(x_{u_{l}})$ are created based on the respective locations of the
terrestrial BSs and the NLOS UAVs when the UE associates with a LOS UAV where
$x_{u_{l}}$ is the distance to the serving UAV. Finally, the exclusions
regions $E_{u_{n}}(x_{u_{n}})$ and $E_{nl}(x_{u_{n}})$ are created based on
the respective locations of the BSs and the LOS UAVs, when the UE associates
to a NLOS UAV with $x_{u_{n}}$ being the distance to the serving NLOS UAV. The
respective exclusion regions expressions of $E_{g_{l}}(x)$, $E_{g_{n}}(x)$,
$E_{u_{l}}(x)$, $E_{u_{n}}(x)$, $E_{ln}(x)$, and $E_{nl}(x)$ are given as:

$\small
E_{g_{\zeta}}(x)=\left(\frac{P_{u}}{P_{g}}\right)^{\frac{1}{\eta_{\zeta}}}\left(x^{2}+h_{g}^{2}\right)^{\frac{\eta_{g}}{2\eta_{\zeta}}},\>E_{u_{\zeta}}(x)=\sqrt{\left(\frac{P_{g}}{P_{u}}\right)^{\frac{2}{\eta_{g}}}x^{\frac{2\eta{\zeta}}{\eta_{g}}}-h_{g}^{2}},\>E_{\zeta\bar{\zeta}}(x)=x^{\frac{\eta_{\zeta}}{\eta_{\bar{\zeta}}}},$
(6)

where $\zeta\in\\{l,n\\}$, $\bar{\zeta}=n$ if $\zeta=l$, and $\bar{\zeta}=l$
if $\zeta=n$.

The association probabilities are defined as the probabilities that the UE
associates to either a LOS UAV, a NLOS UAV, or a BS, which are characterized
in the following lemma.

######  Lemma 1.

_Denote by $A_{u_{l}}$, $A_{u_{n}}$ and $A_{g}$ the probabilities that the UE
at $\textbf{v}_{0}$ is served by a LOS UAV, a NLOS UAV or a ground BS,
respectively. $A_{u_{l}}$, $A_{u_{n}}$ and $A_{g}$ are then given as follows_

$\small
A_{u_{\zeta}}=N_{u}\int_{h_{u}}^{w_{p}}f_{W}(r)\kappa_{u_{\zeta}}(r)\exp\left(-\pi\lambda_{g}E^{2}_{u_{\zeta}}(r)\right)\left(\int_{r}^{w_{p}}f_{W}(w)\kappa_{u_{\zeta}}(w)\mathrm{d}w+\int_{E_{\zeta\bar{\zeta}}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{\bar{\zeta}}}(w)\mathrm{d}w\right)^{N_{u}-1}\mathrm{d}r,$
(7) $\small
A_{g}=2\pi\lambda_{g}\int_{0}^{E_{u}}r\exp\left(-\pi\lambda_{g}r^{2}\right)\left(\int_{E_{g_{l}}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}}\mathrm{d}r,$
(8)

_where $\zeta\in\\{l,n\\}$, $\bar{\zeta}=n$ if $\zeta=l$, and $\bar{\zeta}=l$
if $\zeta=n$.
$E_{u}=\mathrm{max}\left(E_{u_{l}}(w_{p}),E_{u_{n}}(w_{p})\right)$ and
$f_{W}(\cdot)$ is the probability density function (PDF) of the distance $w$
from an arbitrary UAV to the reference UE and is given in Lemma $2$ of [19]
as_

$\small f_{W}(w)=\begin{cases}f_{W_{1}}(w)=\frac{2w}{r_{u}^{2}},&h_{u}\leq
w\leq w_{m}\\\ f_{W_{2}}(w)=\frac{2w}{\pi
r_{u}^{2}}\arccos\left(\frac{w^{2}+v_{0}^{2}-d^{2}}{2v_{0}\sqrt{w^{2}-h_{u}^{2}}}\right),&w_{m}\leq
w\leq w_{p}\end{cases}$ (9)

_where $r_{u}$ is the radius of the disk of UAVs,
$w_{m}=\sqrt{(r_{u}-v_{0})^{2}+h_{u}^{2}}$ and
$w_{p}=\sqrt{(r_{u}+v_{0})^{2}+h_{u}^{2}}$._

###### Proof:

_See Appendix -A._ ∎

Note that the upper limit of the outer integral in (8) is not $\infty$ and is
equal to $E_{u}=\mathrm{max}\left(E_{u_{l}}(w_{p}),E_{u_{n}}(w_{p})\right)$,
where $E_{u_{l}}(\cdot)$ and $E_{u_{n}}(\cdot)$ are given in (6). Since
$w_{p}$ is the farthest distance between any UAV and the UE, the signal
received from the nearest BS is lower than that of any LOS UAV or NLOS UAV if
this BS is located further than $E_{u_{l}}(w_{p})$ or $E_{u_{n}}(w_{p})$.

###  III-B UE-BS Conditional Coverage Probability

The conditional coverage probability is the probability that the received
$\mathrm{SIR}$ is higher than the threshold $\tau_{a}$ given the association
status. When the UE associates to a BS, this probability is expressed as
$P_{cov,g}=\mathbb{P}\left[\mathrm{SIR}\geq\tau_{a}|s=g\right]$ where
$\mathbb{P}\left[s=g\right]=A_{g}$ is the BS association probability. Due to
the universal frequency reuse, the aggregate interference $I_{agg,g}$ includes
all non-serving BSs (denoted as $\hat{I}_{g}$) and all UAVs (denoted as
$I_{u}$). $P_{cov,g}$ is given in Lemma 2.

######  Lemma 2.

_The conditional coverage probability $P_{cov,g}$ given that the UE connects
to a BS is_

$\small
P_{cov,g}=\int_{0}^{E_{u}}\mathcal{L}_{\hat{I}_{g}}(s_{1})\mathcal{L}_{I_{u}}(s_{1})f_{X_{g}}(x_{g})\mathrm{d}x_{g},$
(10)

_where
$s_{1}=\frac{\tau_{a}(x_{g}^{2}+h_{g}^{2})^{\frac{\eta_{g}}{2}}}{P_{g}}$, and
$E_{u}=\mathrm{max}\left(E_{u_{l}}(w_{p}),E_{u_{n}}(w_{p})\right)$.
$\mathcal{L}_{\hat{I}_{g}}(s_{1})$ and $\mathcal{L}_{I_{u}}(s_{1})$ are the
Laplace transforms of the aggregate interference of the interfering BSs and
all the UAVs. $f_{X_{g}}(x_{g})$ is the PDF of the conditional distance
$x_{g}$ to the serving BS._

###### Proof:

_The conditional coverage probability $P_{cov,g}$ is calculated as_

$\displaystyle
P_{cov,g}=\mathbb{P}\left[\mathrm{SIR}\geq\tau_{a}|s=g\right]=\mathbb{P}\left[\frac{P_{g}(x_{g}^{2}+h_{g}^{2})^{-\eta_{g}/2}\Omega_{g,0}}{I_{agg,g}}\geq\tau_{a}|s=g\right]$
(11)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{X_{g}}\left[\mathbb{E}_{I_{agg,g}}\left[\exp\left(-\frac{\tau_{a}(\hat{I}_{g}+I_{u})}{P_{g}(x_{g}^{2}+h_{g}^{2})^{-\eta_{g}/2}}\right)\right]\right]\stackrel{{\scriptstyle(b)}}{{=}}\int_{0}^{E_{u}}\mathcal{L}_{\hat{I}_{g}}(s_{1})\mathcal{L}_{I_{u}}(s_{1})f_{X_{g}}(x_{g})\mathrm{d}x_{g},$

_where
$s_{1}=\frac{\tau_{a}(x_{g}^{2}+h_{g}^{2})^{\frac{\eta_{g}}{2}}}{P_{g}}$, (a)
follows from the exponential distribution of $\Omega_{g,0}$ and from the
expression of the interference $I_{agg,g}=\hat{I}_{g}+I_{u}$, and (b) follows
from the independence of $\hat{I}_{g}$ and $I_{u}$ and from the Laplace
transform definition. The upper limit of the integral in (11)
$E_{u}=\mathrm{max}\left(E_{u_{l}}(w_{p}),E_{u_{n}}(w_{p})\right)$ is the
maximum distance from the UE to its serving BS considering the exclusion
regions of the LOS and NLOS UAVs. $E_{u_{l}}(\cdot)$ and $E_{u_{n}}(\cdot)$
are presented in (6)._ ∎

To obtain the expression of $P_{cov,g}$, Laplace transforms of interference
terms and the PDF of the distance to serving BS must be computed. Lemma 3 and
Lemma 4 present these results.

######  Lemma 3.

_The Laplace transform of the interference $\hat{I}_{g}$ of all BSs except the
serving BS is_

$\small\mathcal{L}_{\hat{I}_{g}}(s_{1})=\exp\left[\frac{-2\pi\lambda_{g}s_{1}P_{g}\left(x_{g}^{2}+h_{g}^{2}\right)}{\left(\eta_{g}-2\right)\left(\left(x_{g}^{2}+h_{g}^{2}\right)^{\frac{\eta_{g}}{2}}+s_{1}P_{g}\right)}{{}_{2}}F_{1}\left(1,1;2-\frac{2}{\eta_{g}};\frac{1}{1+\frac{\left(x_{g}^{2}+h_{g}^{2}\right)^{\frac{\eta_{g}}{2}}}{s_{1}P_{g}}}\right)\right].$
(12)

_where ${{}_{2}}F_{1}$ denotes the Gauss hyper geometric function. The Laplace
transform of the interference $I_{u}$ from the UAVs when the UE associates to
a BS is_

$\displaystyle\mathcal{L}_{I_{u}}(s_{1})$
$\displaystyle=\left[\frac{1}{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}\left(\int_{E_{g_{l}}(x_{g})}^{w_{p}}\left(1+\frac{s_{1}P_{u}v^{-\eta_{l}}}{m_{l}}\right)^{-m_{l}}\right.\right.$
(13) $\displaystyle\times
f_{W}(v)\kappa_{u_{l}}(v)\mathrm{d}v+\int_{E_{g_{n}}(x_{g})}^{w_{p}}\left(1+\frac{s_{1}P_{u}v^{-\eta_{n}}}{m_{n}}\right)^{-m_{n}}f_{W}(v)\kappa_{u_{n}}(v)\mathrm{d}v\bigg{)}\bigg{]}^{N_{u}},$

_where $E_{g_{l}}(x_{g})$ and $E_{g_{n}}(x_{g})$ are the minimum distances at
which the LOS and NLOS UAVs are placed when the UE associates to a BS at
$x_{g}$. $f_{W}(\cdot)$ and $\kappa_{u_{l}}(\cdot)$ are given in (9) and (2).
_

###### Proof:

_See Appendix -B_ ∎

The PDF of the conditional distance from the UE to the serving BS $x_{g}$ is
provided in Lemma 4.

######  Lemma 4.

_The PDF of the distance $x_{g}$ from the UE to its serving BS is given as_

$\small
f_{X_{g}}(x_{g})=\frac{1}{A_{g}}2\pi\lambda_{g}x_{g}\exp\left(-\pi\lambda_{g}x_{g}^{2}\right)\left(\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}},$
(14)

_where $E_{g_{l}}(\cdot)$ and $E_{g_{n}}(\cdot)$ are given in (6). $A_{g}$ and
$f_{W}(\cdot)$ are given in Lemma 1. _

###### Proof:

_See Appendix -C._ ∎

###  III-C UE-UAV Conditional Coverage Probability

For a UAV to serve a UE, its backhaul link should be successful. The
conditional coverage probability given that the UE associates to a LOS/NLOS
UAV is the joint probability of two events and can be expressed as
$P_{cov,u_{\zeta}}=\mathbb{P}\left[\mathrm{SIR}\geq\tau_{a},\mathrm{SINR}\geq\tau_{b}\right|s=u_{\zeta}]$,
where the first term corresponds to the UAV-UE link and the second term to the
BS-UAV backhaul. Since the UE and the UAV connect to different BSs, and for
mathematical tractability, we approximate
$P_{cov,u_{\zeta}}\approx\mathbb{P}\left[\mathrm{SIR}\geq\tau_{a}|s=u_{\zeta}\right]\times\mathbb{P}\left[\mathrm{SINR}\geq\tau_{b}|s=u_{\zeta}\right]$.
Such approximation is further validated numerically in Section V. Note that,
due to the different propagation environments considered for access and
backhaul, and the different used probabilistic LOS models, the backhaul
probability is independent of the UE association and the access coverage.

####  III-C1 Backhaul Probability

We start by providing an expression for the backhaul probability
$S(\tau_{b})=\mathbb{P}\left[\mathrm{SINR}\geq\tau_{b}\right]$. The analysis
is performed for a reference UAV located at height $h_{u}$ and at the center
of the disk $D_{u}$. The set of backhaul-enabled BSs is divided into two sub-
processes: the LOS BSs set $\Phi_{g_{l}}$ with density
$\lambda_{b}\kappa_{b_{l}}(r)$ and the NLOS BSs set $\Phi_{g_{n}}$ with
density $\lambda_{b}\kappa_{b_{n}}(r)$, where $r$ is the UAV-BS distance and
$\kappa_{b_{l}}(\cdot)$ is given in (3). Since the UAV connects to the BS that
provides the strongest signal strength, the serving BS can only be either the
nearest BS in $\Phi_{g_{l}}$ or in $\Phi_{g_{n}}$. Lemma 5 provides the
probabilities of connecting to a LOS or a NLOS BS.

######  Lemma 5.

_The probability $A_{b_{\xi}}$ that a UAV connects to a BS with LOS/NLOS
backhaul link is_

$\small
A_{b_{\xi}}=\int_{0}^{\infty}e^{-2\pi\lambda_{b}\int_{0}^{E_{b_{\xi}}(x)}\kappa_{b_{\bar{\xi}}}(t)t\mathrm{d}t}f_{s_{b_{\xi}}}(x)\mathrm{d}x,\vspace{-0.2cm}$
(15)

_where $\bar{\xi}=n$ if $\xi=l$ and $\bar{\xi}=l$ if $\xi=n$.
$E_{b_{\xi}}(x)=\sqrt{\left(\frac{C_{\bar{\xi}}}{C_{\xi}}\right)^{\frac{2}{\eta_{\bar{\xi}}}}(x^{2}+\Delta_{h}^{2})^{\frac{\eta_{\xi}}{\eta_{\bar{\xi}}}}-\Delta_{h}^{2}}$,
$\Delta_{h}=|h_{u}-h_{g}|$ and
$f_{s_{b_{\xi}}}(x)=2\pi\lambda_{b}x\kappa_{b_{\xi}}(x)e^{-2\pi\lambda_{b}\int_{0}^{x}\kappa_{b_{\xi}}(r)r\mathrm{d}r}$.
_

###### Proof:

_See Appendix -D._ ∎

The PDF of the distances $x_{b_{l}}$ and $x_{b_{n}}$ from the UAV to the
serving BS in $\Phi_{g_{l}}$ and $\Phi_{g_{n}}$ are provided in Lemma 6.

######  Lemma 6.

_Given that a UAV connects to a BS in $\Phi_{g_{\xi}}$ to get backhaul, the
PDF of the distance $x_{b_{\xi}}$ to the serving BS is_

$\small
f_{X_{b_{\xi}}}(x)=\frac{f_{s_{b_{\xi}}}(x)}{A_{b_{\xi}}}e^{-2\pi\lambda_{b}\int_{0}^{E_{b_{\xi}}(x)}\kappa_{b_{\bar{\xi}}}(t)t\mathrm{d}t},$
(16)

_where $\xi\in\\{l,n\\}$ indicates if the UAV connects to a LOS or a NLOS BS
for backhaul support. $\bar{\xi}=n$ if $\xi=l$ and $\bar{\xi}=l$ if $\xi=n$.
$A_{b_{\xi}}$, $f_{s_{b_{\xi}}}(\cdot)$ and $E_{b_{\xi}}(\cdot)$ are given in
Lemma 5. _

###### Proof:

_See Appendix -E._ ∎

Finally, the backhaul probability is presented in Theorem 1.

######  Theorem 1 (Backhaul Probability).

_The backhaul probability $S(\tau_{b})$ can be derived as_

$\small
S(\tau_{b})=A_{b_{l}}S_{l}(\tau_{b})+A_{b_{n}}S_{n}(\tau_{b}),\vspace{-0.3cm}$
(17)

_where $S_{l}(\tau_{b})$ and $S_{n}(\tau_{b})$ are the conditional backhaul
probabilities given that the UAV is connected to a LOS/NLOS BS. $A_{b_{l}}$
and $A_{b_{n}}$ are the association probabilities. $S_{\xi}(\tau_{b})$ is
given as_

$\displaystyle
S_{\xi}(\tau_{b})\approx\sum_{q=1}^{m_{\xi}}(-1)^{q+1}{m_{\mathrm{\zeta}}\choose
q}\int_{0}^{\infty}e^{-\frac{q\gamma_{\xi}\left(x^{2}+\Delta_{h}^{2}\right)^{\frac{\eta_{\xi}}{2}}\tau_{b}\sigma_{b}^{2}}{P_{b}C_{\xi}G_{b,0}}-Q_{\xi}(q,x)-V_{\xi}(q,x)}f_{X_{b_{\xi}}}(x)\mathrm{d}x,$
(18)

_where $f_{X_{b_{\xi}}}(x)$ is given in (16) and_

$\displaystyle Q_{\xi}(q,x)$
$\displaystyle=2\pi\lambda_{b}\sum_{k=1}^{4}p_{k}\int_{x}^{\infty}H\left(m_{\xi},\frac{q\gamma_{\xi}\bar{G}_{k}\tau_{b}(x^{2}+\Delta_{h}^{2})^{\frac{\eta_{\xi}}{2}}}{m_{\xi}(t^{2}+\Delta_{h}^{2})^{\frac{\eta_{\xi}}{2}}}\right)\times\kappa_{b_{\xi}}(t)t\mathrm{d}t,$
(19) $\displaystyle V_{\xi}(q,x)$
$\displaystyle=2\pi\lambda_{b}\sum_{k=1}^{4}p_{k}\int_{E_{b_{\xi}}(x)}^{\infty}H\left(m_{\bar{\xi}},\frac{qC_{\bar{\xi}}\gamma_{\xi}\bar{G}_{k}\tau_{b}(x^{2}+\Delta_{h}^{2})^{\frac{\eta_{\xi}}{2}}}{m_{\bar{\xi}}C_{\xi}(t^{2}+\Delta_{h}^{2})^{\frac{\eta_{\bar{\xi}}}{2}}}\right)\times\kappa_{b_{\bar{\xi}}}(t)t\mathrm{d}t,$
(20)

_where $H(m,x)=1-1/(1+x)^{m}$, For $\xi\in\\{l,n\\}$, $\bar{\xi}=n$ if $\xi=l$
and $\bar{\xi}=l$ if $\xi=n$, $m_{\xi}$ and
$\gamma_{\xi}=m_{\xi}(m_{\xi}!)^{-\frac{1}{m_{\xi}}}$ are the Nakagami-m small
scale fading parameters; for $k\in\\{1,2,3,4\\}$,
$\bar{G}_{k}=\frac{G_{k}}{G_{b,0}}$, $G_{k}$ and $p_{k}$ are defined in Table
II._

###### Proof:

_See Appendix -F._ ∎

####  III-C2 Beamforming Alignment Errors

To optimize the backhaul connectivity between the UAV and the BS, the antennas
must be perfectly aligned. However, perfect alignment in mmWave systems is
challenging due to the required high directivity that leads to a narrow
beamwidth of the main lobe. In the absence of beam-steering errors, the
maximum directivity gain offered by directional beamforming is
$G_{g}^{(\text{max})}G_{u}^{(\text{min})}$, where $G_{g}^{(\text{max})}$ and
$G_{u}^{(\text{min})}$ are the gains of the main lobes of the BS and the UAV
antennas, respectively. To account for the beamforming alignment error of the
desired link, a beam steering error model similar to [35] is considered. We
denote by $\varphi_{s}^{*}$, $s\in\\{g,u\\}$, the angles corresponding to
error-free beam-steering and by $\varepsilon_{s}$ the additive beam-steering
errors. $\varepsilon_{g}$ and $\varepsilon_{u}$ are assumed to be randomly
distributed, independent of each other and to have a symmetric distribution
around $\varphi_{g}^{*}$ and $\varphi_{u}^{*}$, respectively. Let
$F_{|\varepsilon_{s}|}(x)=\mathbb{P}(|\varepsilon_{s}|\leq x)$ be the
cumulative distribution function (CDF) of $|\varepsilon_{\mathrm{s}}|$ for
$s\in\\{g,u\\}$, and let
$\bar{F}_{|\varepsilon_{s}|}(x)=\left(1-F_{|\varepsilon_{s}|}(x)\right)$.
Then, the PMF of the directivity gain of the desired link $G_{b,0}$ is
$f_{G_{b,0}}(g)=\left(f_{G_{g,0}}\otimes f_{G_{u,0}}\right)(g)$, where
$f_{G_{g,0}}(.)$ and $f_{G_{u,0}}(.)$ are the PMFs of the directivity gains of
the serving BS and the corresponding UAV, respectively and $\otimes$ is the
convolution operator. $f_{G_{s,0}}(g)$ can be written as:

$\small
f_{G_{s,0}}(g)=F_{|\varepsilon_{\mathrm{s}}|}\left(\frac{\theta_{s}}{2}\right)\delta\left(g-G_{s}^{(\text{max})}\right)+\bar{F}_{|\varepsilon_{s}|}\left(\frac{\theta_{s}}{2}\right)\delta\left(g-G_{s}^{(\text{min})}\right)\vspace{-0.2cm}$
(21)

where $\delta(\cdot)$ is the Dirac delta function, and $G_{s}^{(\text{max})}$,
$G_{s}^{(\text{min})}$ and $\theta_{s}$ are the respective gains of the main
and side lobes and the beamwidth of the BS and UAV antennas, for
$s\in\\{g,u\\}$, as illustrated earlier in (4). Similarly to the directivity
gain of the interfering links, the directivity gain of the desired link
$G_{b,0}$ is a discrete random variable that takes the values $G_{k}$ with
probability $p_{k,0}$, where $G_{k}$ and $p_{k_{0}}$ are defined in Table II.
The backhaul probability can be expressed as

$\displaystyle
S(\tau_{b})=\mathbb{E}_{G_{b,0}}\left[S(\tau_{b},G_{b,0})\right]=\int_{0}^{\infty}S(\tau_{b},g)f_{G_{b,0}}(g)\mathrm{d}g$
(22)
$\displaystyle=F_{|\varepsilon_{g}|}(\theta_{g}/2)F_{|\varepsilon_{u}|}(\theta_{u}/2)S(\tau_{b},G_{g}^{(\text{max})}G_{u}^{(\text{max})})+F_{|\varepsilon_{g}|}(\theta_{g}/2)\bar{F}_{|\varepsilon_{u}|}(\theta_{u}/2)S(\tau_{b},G_{g}^{(\text{max})}G_{u}^{(\text{min})})$
$\displaystyle+\bar{F}_{|\varepsilon_{g}|}(\theta_{g}/2)F_{|\varepsilon_{u}|}(\theta_{u}/2)S(\tau_{b},G_{g}^{(\text{min})}G_{u}^{(\text{max})})+\bar{F}_{|\varepsilon_{g}|}(\theta_{g}/2)\bar{F}_{|\varepsilon_{u}|}(\theta_{u}/2)S(\tau_{b},G_{g}^{(\text{min})}G_{u}^{(\text{min})}).$

####  III-C3 Conditional Coverage Probability and Distance Distribution

The UE can connect to a LOS or a NLOS UAV depending on the association rule.
The coverage probability $P_{cov,u_{\zeta}}$ when the UE associates to a LOS
UAV ($\zeta=l$) or a NLOS UAV ($\zeta=n$) is given in Lemma 7.

######  Lemma 7.

_The conditional coverage probability $P_{cov,u_{\zeta}}$ given that UE
connects to a $\zeta$-UAV is_

$\small
P_{cov,u_{\zeta}}=S(\tau_{b})\int_{h_{u}}^{w_{p}}\sum_{k=0}^{m_{\zeta}-1}\frac{(-s_{2_{\zeta}})^{k}}{k!}\left[\frac{\partial^{k}}{\partial
s_{2_{\zeta}}^{k}}\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})\mathcal{L}_{\hat{I}_{u_{\zeta}}}(s_{2_{\zeta}})\right]f_{Y_{u_{\zeta}}}(y_{u_{\zeta}})\mathrm{d}y_{u_{\zeta}},$
(23)

_where
$s_{2_{\zeta}}=\frac{m_{\zeta}\tau_{a}y_{u_{\zeta}}^{\eta_{\zeta}}}{P_{u}}$
and $f_{Y_{u_{\zeta}}}(y_{u_{\zeta}})$ is the PDF of the conditional distance
to the serving $\zeta$-UAV with $\zeta\in\\{l,n\\}$.
$\mathcal{L}_{\hat{I}_{u_{\zeta}}}(s_{2_{\zeta}})$ and
$\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})$ are the Laplace transforms of the
aggregate interference of all the UAVs except the serving UAV
$\hat{I}_{u_{\zeta}}$ and of all the BSs $I_{g_{\zeta}}$._

###### Proof:

_See Appendix -G._ ∎

As noted in (23), the Laplace transforms of the interference terms
$\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})$ and
$\mathcal{L}_{\hat{I}_{u_{\zeta}}}(s_{2_{\zeta}})$ and the PDF of the
conditional distance to the serving UAV $f_{Y_{u_{\zeta}}}(y_{u_{\zeta}})$
must be computed to obtain the final expression of $P_{cov,u_{\zeta}}$. Lemma
8 presents these Laplace transforms as follows.

######  Lemma 8.

_The Laplace transform of the interference $I_{g_{\zeta}}$ of all BSs when the
UE associates to a $\zeta$-UAV with $\zeta\in\\{l,n\\}$ is given as_

$\small\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})=\exp\left[\frac{-2\pi\lambda_{g}s_{2_{\zeta}}P_{g}\left(E_{u_{\zeta}}^{2}(y_{u_{\zeta}})+h_{g}^{2}\right)}{\left(\eta_{g}-2\right)\left(\left(E_{u_{\zeta}}^{2}(y_{u_{\zeta}})+h_{g}^{2}\right)^{\frac{\eta_{g}}{2}}+s_{2_{\zeta}}P_{g}\right)}{{}_{2}}F_{1}\left(1,1;2-\frac{2}{\eta_{g}};\frac{1}{1+\frac{\left(E_{u_{\zeta}}^{2}(y_{u_{\zeta}})+h_{g}^{2}\right)^{\frac{\eta_{g}}{2}}}{s_{2_{\zeta}}P_{g}}}\right)\right].$
(24)

_where $E_{u_{\zeta}}(y_{u_{\zeta}})$ is given in (6) and
${{}_{2}}F_{1}(\cdot)$ is the Gauss hyper geometric function. The Laplace
transform of the interference $\hat{I}_{u_{\zeta}}$ from all UAVs except the
serving UAV is given as_

$\displaystyle\mathcal{L}_{\hat{I}_{u_{\zeta}}}(s_{2_{\zeta}})$
$\displaystyle=\left[\frac{1}{\int_{y_{u_{\zeta}}}^{w_{p}}f_{W}(w)\kappa_{u_{\zeta}}(w)\mathrm{d}w+\int_{E_{\zeta\bar{\zeta}}\left(y_{u_{\zeta}}\right)}^{w_{p}}f_{W}(w)\kappa_{u_{\bar{\zeta}}}(w)\mathrm{d}w}\left(\int_{y_{u_{\zeta}}}^{w_{p}}\left(1+\frac{s_{2_{\zeta}}P_{u}v^{-\eta_{\zeta}}}{m_{\zeta}}\right)^{-m_{\zeta}}\right.\right.$
(25) $\displaystyle\times
f_{W}(v)\kappa_{u_{\zeta}}(v)\mathrm{d}v+\int_{E_{\zeta\bar{\zeta}}(y_{u_{\zeta}})}^{w_{p}}\left(1+\frac{s_{2_{\zeta}}P_{u}v^{-\eta_{\bar{\zeta}}}}{m_{\bar{\zeta}}}\right)^{-m_{\bar{\zeta}}}f_{W}(v)\kappa_{u_{\bar{\zeta}}}(v)\mathrm{d}v\bigg{)}\bigg{]}^{N_{u}-1}.$

###### Proof:

_When the UE associates to a $\zeta$-UAV, the BSs are located further than
$E_{u_{\zeta}}(y_{u_{\zeta}})$, the interfering $\zeta$-UAV are located
further than $y_{u_{\zeta}}$ and the $\bar{\zeta}$-UAV exist beyond
$E_{\zeta\bar{\zeta}}(y_{u_{\zeta}})$, where $y_{u_{\zeta}}$ is the distance
from the UE to the serving UAV. Thus,
$\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})$ is obtained by following a
similar approach to Lemma 3 while replacing the lower limit of the integral in
(34) by $E_{u_{\zeta}}(y_{u_{\zeta}})$. The interference from all UAVs except
the serving UAV can be expressed as $\sum_{y_{i}\in\Phi_{u}\backslash
y_{u_{\zeta}}}I_{u,y_{j}}$, where $I_{u,y_{j}}$ is the interference from the
$j$-th UAV and can be obtained by following a similar approach to Lemma 3 as
$\mathcal{L}_{\hat{I}_{u,y_{j}}}(s_{2_{\zeta}})=\left(\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{2_{\zeta}}I_{u,y_{j}}\right)\right]\right)^{N_{u}-1}$.
The final expression of $\mathcal{L}_{\hat{I}_{u_{\zeta}}}(s_{2_{\zeta}})$ is
obtained by replacing the lower limits of the integrals by $y_{u_{\zeta}}$ and
$E_{\zeta\bar{\zeta}}(y_{u_{\zeta}})$ and by plugging
$f_{V_{u_{l}}}(v,y_{u_{\zeta}})$ and
$f_{V_{u_{n}}}(v,E_{\zeta\bar{\zeta}}(y_{u_{\zeta}}))$ in
$\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{2_{\zeta}}I_{u,y_{j}}\right)\right]$
in (36)._ ∎

Next, we derive the PDF of the distance from the UE to the serving UAV
$y_{u_{\zeta}}$ in Lemma 9.

######  Lemma 9.

_The PDF of the distance $y_{u_{\zeta}}$ to the serving $\zeta$-UAV
($\zeta\in\\{l,n\\}$) is given as_

$\displaystyle f_{Y_{u_{\zeta}}}(y_{u_{\zeta}})$
$\displaystyle=\frac{N_{u}}{A_{u_{\zeta}}}f_{W}(y_{u_{\zeta}})\kappa_{u_{\zeta}}(y_{u_{\zeta}})\exp\left(-\pi\lambda_{g}E_{u_{\zeta}}(y_{u_{\zeta}})^{2}\right)\left(\int_{y_{u_{\zeta}}}^{w_{p}}f_{W}(w)\kappa_{u_{\zeta}}(w)\mathrm{d}w\right.$
(26)
$\displaystyle\left.+\int_{E_{\zeta\bar{\zeta}}(y_{u_{\zeta}})}^{w_{p}}f_{W}(w)\kappa_{u_{\bar{\zeta}}}(w)\mathrm{d}w\right)^{N_{u}-1},$

_where $f_{W}(\cdot)$ and $A_{u_{\zeta}}$ are given in Lemma 1.
$E_{u_{\zeta}}(\cdot)$ and $E_{\zeta\bar{\zeta}}(\cdot)$ are given in (6)._

###### Proof:

_This proof follows a similar approach to Lemma 4._ ∎

###  III-D Overall Coverage Probability

After deriving the association probabilities, the conditional distance
distributions and the conditional coverage probabilities, the overall coverage
probability is obtained through the law of total probability as
$P_{cov}=A_{u_{l}}P_{cov,u_{l}}+A_{u_{n}}P_{cov,u_{n}}+A_{g}P_{cov,g}$.

##  IV Backhaul Aware Transmission

For the backhaul aware transmission scenario, the UE activity is dependent on
the BS-UAV instantaneous backhaul link in case of aerial coverage. That is,
the UAV is aware of the backhaul status as it checks the integrity of packets
received from the BS before transmitting them to the UE. If the backhaul
$\mathrm{SINR}$ satisfies the threshold, the UAV can serve the UE
successfully. When the backhaul is down, the UAV refrains its transmission to
the UE. The UE does not change its association,111Changing association based
on instantaneous link status may lead to undesirable excessive handovers known
as the ping pong effect. but rather goes to a service failure event in this
time slot. As result, the backhaul aware transmission scheme has the following
two main advantages, i) the established DL transmissions are subject to the
access link $\mathrm{SIR}$ only, since UAVs have the ability to refrain from
transmitting corrupted packets, and ii) the interference is relieved because
only successfully backhauled UAVs contribute to it, thereby improving the
overall coverage probability. Such advantages come at the cost of extra
processing at the UAVs because they check the validity of the packets before
transmitting to UE. To characterize the coverage probability of the backhaul
aware scheme, we follow the same steps of Section III. In particular, we start
by deriving the transmission probabilities, then we find the conditional
coverage probabilities, then we obtain the distance distributions, and finally
we provide the unified expression for the coverage probability.

###  IV-A Transmission Probabilities

In the context of backhaul aware transmission, we define the transmission
probability as the probability that the UE is being served by a BS or a
successfully backhauled LOS/NLOS UAV. Note that in the case where the UAV-UE
access link is better than the BS-UE link and the UAV backhaul link is not
successful, the UE is considered in a service failure state and fails to
connect. Hence, the service failure event here refers to the event where the
UE is not served in the DL in a time slot not as the event that DL is
established and a decoding error occurs. It is important to note that an
established DL connection is different from UE to UAV association, where the
latter is independent from the backhaul link state. When the UE is served by a
terrestrial BS, the transmission probability is expressed as
$\tilde{A}_{g}=\mathbb{P}[s=g]$. Note that the transmission probability when
the UE connects to a BS is the same as the association probability to a BS for
the unaware transmission scenario and is presented in (8). For a UAV to
transmit to the UE, the backhaul link of this UAV should be successful. The
UAV transmission probability is the joint probability of two events and can be
expressed as
$\tilde{A}_{u_{\zeta}}=\mathbb{P}[s=u_{\zeta},\mathrm{SINR}>\tau_{b}]$, where
the first term corresponds to the UE-UAV association rule
$\left(\zeta\in\\{l,n\\}\right)$ and the second term to the BS-UAV backhaul
link. The independence assumption is also considered and validated in Section
V. Thus, the UAV transmission probability can be expressed as
$\tilde{A}_{u_{\zeta}}=\mathbb{P}[s=u_{\zeta}]\times\mathbb{P}[\mathrm{SINR}>\tau_{b}]$,
where $\mathbb{P}[s=u_{\zeta}]=A_{u_{\zeta}}$ is the UAV association
probability for the backhaul unaware scenario derived in Lemma 1 and
$\mathbb{P}[\mathrm{SINR}>\tau_{b}]=S(\tau_{b})$ is the backhaul probability
given in (17). The service failure probability is defined as
$\tilde{A}_{f}=\mathbb{P}[s=u_{l},\mathrm{SINR}<\tau_{b}\|s=u_{n},\mathrm{SINR}<\tau_{b}]$
and derived as

$\displaystyle\tilde{A}_{f}$
$\displaystyle=\mathbb{P}\left[s=u_{l}\right]\mathbb{P}\left[\mathrm{SINR}<\tau_{b}\right]+\mathbb{P}\left[s=u_{n}\right]\mathbb{P}\left[\mathrm{SINR}<\tau_{b}\right]=A_{u_{l}}\left(1-S(\tau_{b})\right)+A_{u_{n}}\left(1-S(\tau_{b})\right),$
(27)

where $A_{u_{l}}$ and $A_{u_{n}}$ are derived in Lemma 1 and $S(\tau_{b})$ in
Theorem 1.

###  IV-B Conditional and Overall Coverage Probabilities

The conditional coverage probability given that the UE is served by a BS is
expressed as $\tilde{P}_{cov,g}=\mathbb{P}[\mathrm{SIR}\geq\tau_{a}|s=g]$. Due
to the dependency of the UAV transmission on the quality of the backhaul link,
the aggregate interference for the backhaul aware scenario $\tilde{I}_{agg,g}$
includes the interference from only the UAVs with successful backhaul links
denoted as $I_{\tilde{u}}$ and from all BSs except the serving BS denoted as
$\hat{I}_{g}$. An expression for $\tilde{P}_{cov,g}$ is given in Lemma 10.

######  Lemma 10.

_The conditional coverage probability given that the UE is served by a BS for
the backhaul aware scenario is given as_

$\small\tilde{P}_{cov,g}=\int_{0}^{E_{u}}\mathcal{L}_{\hat{I}_{g}}(s_{1})\mathcal{L}_{I_{\tilde{u}}}(s_{1})f_{X_{g}}(x_{g})\mathrm{d}x_{g},$
(28)

_where
$s_{1}=\frac{\tau_{a}(x_{g}^{2}+h_{g}^{2})^{\frac{\eta_{g}}{2}}}{P_{g}}$,
$E_{u}=\mathrm{max}\left(E_{u_{l}}(w_{p}),E_{u_{n}}(w_{p})\right)$,
$\mathcal{L}_{\hat{I}_{g}}(s_{1})$ is given in (12) and $f_{X_{g}}(x_{g})$ in
(14). $\mathcal{L}_{I_{\tilde{u}}}(s_{1})$ is the Laplace transform of the
aggregate interference from the UAVs that have successful backhaul links with
ground BSs._

###### Proof:

_This proof follows a similar approach to Lemma 2. _ ∎

A lower bound for the Laplace transform $\mathcal{L}_{I_{\tilde{u}}}(s_{1})$
is derived in Lemma 11.

######  Lemma 11.

_The Laplace transform of the aggregate interference $I_{\tilde{u}}$ from all
the UAVs that have a successful backhaul link when the UE is served by a BS
can be lower bounded by_

$\displaystyle\mathcal{L}_{I_{\tilde{u}}}(s_{1})$
$\displaystyle\geq\left[\frac{1}{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}\left(\int_{E_{g_{l}}(x_{g})}^{w_{p}}\left(1+\frac{s_{1}P_{u}v^{-\eta_{l}}}{m_{l}}\right)^{-m_{l}}\right.\right.$
(29) $\displaystyle\times
f_{W}(v)\kappa_{u_{l}}(v)\mathrm{d}v+\int_{E_{g_{n}}(x_{g})}^{w_{p}}\left(1+\frac{s_{1}P_{u}v^{-\eta_{n}}}{m_{n}}\right)^{-m_{n}}f_{W}(v)\kappa_{u_{n}}(v)\mathrm{d}v\bigg{)}\bigg{]}^{N_{u}\times
S(\tau_{b})},$

_where $E_{g_{l}}(\cdot)$ and $E_{g_{n}}(\cdot)$ are given in (6),
$f_{W}(\cdot)$ and $w_{p}$ are given in Lemma 1. _

###### Proof:

_We denote $\tilde{N_{u}}$ the number of UAVs that have successful backhaul
links. $I_{\tilde{u}}$ is given as
$\sum_{y_{j\in\tilde{\Phi_{u}}}}I_{u,y_{j}}$ where $I_{u,y_{j}}$ denotes the
interference from the $j$-th UAV and $\tilde{\Phi_{u}}$ is the set of UAVs
with successful backhaul links. The Laplace transform of $I_{\tilde{u}}$ can
be calculated as_

$\displaystyle\mathcal{L}_{I_{\tilde{u}}}(s_{1})=\mathbb{E}_{I_{\tilde{u}}}\left[e^{-s_{1}I_{\tilde{u}}}\right]=\mathbb{E}_{I_{\tilde{u}}}\left[\exp\left(-s_{1}\sum_{j=1}^{\tilde{N_{u}}}I_{u,y_{j}}\right)\right]\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{\tilde{N_{u}}}\left[\prod_{j=1}^{\tilde{N_{u}}}\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]\right]$
(30)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathbb{E}_{\tilde{N_{u}}}\left[\left(\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]\right)^{\tilde{N_{u}}}\right]\stackrel{{\scriptstyle(c)}}{{\geq}}\left(\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]\right)^{\mathbb{E}\left[\tilde{N_{u}}\right]},$

_where (a) follows from the independent and identically distributed (iid)
fading gains and from their independence of the interferers distances in the
expression of the interference $I_{u,y_{j}}$. (b) follows from the iid
distribution of the interferers UAVs. (c) follows from the Jensen’s inequality
that gives a lower bound for $\mathcal{L}_{I_{\tilde{u}}}(s_{1})$. The number
of UAVs that have successful backhaul links $\tilde{N_{u}}$ follows a binomial
distribution with backhaul probability $S(\tau_{b})$ and mean
$\mathbb{E}[\tilde{N_{u}}]=N_{u}S(\tau_{b})$. The remaining proof follows the
procedure adopted in Lemma 3._ ∎

Similar to the UE-BS conditional coverage probability, the conditional
coverage probability given that the UE is served by a $\zeta$-UAV is expressed
as
$\tilde{P}_{cov,u_{\zeta}}=\mathbb{P}[\mathrm{SIR}\geq\tau_{a}|s=u_{\zeta}]$.
The aggregate interference $\tilde{I}_{agg,u_{\zeta}}$ includes the
interference from the UAVs with successful backhaul links except the serving
UAV denoted as $\hat{I}_{\tilde{u_{\zeta}}}$ and from all BSs denoted as
$I_{g_{\zeta}}$. An expression for $\tilde{P}_{cov,u_{\zeta}}$ is given in the
following lemma.

######  Lemma 12.

_The coverage probability $P_{cov,u_{\zeta}}$ given that the UE is served by a
$\zeta$-UAV is_

$\small\tilde{P}_{cov,u_{\zeta}}=\int_{h_{u}}^{w_{p}}\sum_{k=0}^{m_{\zeta}-1}\frac{(-s_{2_{\zeta}})^{k}}{k!}\left[\frac{\partial^{k}}{\partial
s_{2_{\zeta}}^{k}}\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})\mathcal{L}_{\hat{I}_{\tilde{u_{\zeta}}}}(s_{2_{\zeta}})\right]f_{Y_{u_{\zeta}}}(y_{u_{\zeta}})\mathrm{d}y_{u_{\zeta}},$
(31)

_where $\zeta\in\\{l,n\\}$,
$s_{2_{\zeta}}=\frac{m_{u_{\zeta}}\tau_{a}y_{u_{\zeta}}^{\eta_{u_{\zeta}}}}{P_{u}}$.
$f_{Y_{u_{\zeta}}}(y_{u_{\zeta}})$ is given in (26) and
$\mathcal{L}_{I_{g_{\zeta}}}(s_{2_{\zeta}})$ in (24).
$\mathcal{L}_{\hat{I}_{\tilde{u_{\zeta}}}}(s_{2_{\zeta}})$ is the Laplace
transform of the aggregate interference of all the UAVs that have successful
backhaul links except the serving UAV and is given as_

$\displaystyle\mathcal{L}_{\hat{I}_{\tilde{u_{\zeta}}}}(s_{2_{\zeta}})$
$\displaystyle\geq\left(\frac{1}{\int_{y_{u_{\zeta}}}^{w_{p}}f_{W}(w)\kappa_{u_{\zeta}}(w)\mathrm{d}w+\int_{E_{\zeta\bar{\zeta}}(y_{u_{\zeta}})}^{w_{p}}f_{W}(w)\kappa_{u_{\bar{\zeta}}}(w)\mathrm{d}w}\left(\int_{y_{u_{\zeta}}}^{w_{p}}\left(1+\frac{s_{2_{\zeta}}P_{u}v^{-\eta_{\zeta}}}{m_{\zeta}}\right)^{-m_{\zeta}}\right.\right.$
(32) $\displaystyle\times
f_{W}(v)\kappa_{u_{\zeta}}(v)\mathrm{d}v+\int_{E_{\zeta\bar{\zeta}}(y_{u_{\zeta}})}^{w_{p}}\left(1+\frac{s_{2_{\zeta}}P_{u}v^{-\eta_{\bar{\zeta}}}}{m_{\bar{\zeta}}}\right)^{-m_{\bar{\zeta}}}f_{W}(v)\kappa_{u_{\bar{\zeta}}}(v)\mathrm{d}v\bigg{)}\bigg{)}^{N_{u}S(\tau_{b})-1}.$

###### Proof:

_The proof of this lemma follows a similar approach to Lemma 3 and Lemma 11._
∎

The overall coverage probability for the backhaul aware transmission scenario
can be derived through the law of total probability as
$\tilde{P}_{cov}=\tilde{A}_{u_{l}}\tilde{P}_{cov,u_{l}}+\tilde{A}_{u_{n}}\tilde{P}_{cov,u_{n}}+\tilde{A}_{g}\tilde{P}_{cov,g}$,
where $\tilde{A}_{u_{l}}$, $\tilde{A}_{u_{n}}$ and $\tilde{A}_{g}$ are given
in Section IV-A and $\tilde{P}_{cov,u_{l}}$, $\tilde{P}_{cov,u_{n}}$ and
$\tilde{P}_{cov,g}$ are given in Section IV-B.

##  V Numerical results and Discussions

In this section, we verify the analytical analysis against Monte-Carlo
simulations and highlight the impact of different system parameters. Unless it
is stated explicitly otherwise, we assume that the UE is at the origin
($v_{0}=0$) and we employ the simulation parameters listed in Table III.
Assuming dense urban environment, we set the environmental parameters
$a=0.136$ and $b=11.95$ as in [36] to determine the LOS probability on the
UAV-UE access links. The dense urban environment parameters of the LOS
probability model adopted for the BS-UAV backhaul links are given in [34] as
$h_{g}=25$ m, $c=1$, $d=0.106$ and $e=1$.

TABLE III: Simulation Parameters. Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---
($P_{g}$, $P_{b}$, $P_{u}$) | ($20$, $10$, $1$) W | ($\eta_{g}$, $\eta_{l}$, $\eta_{n}$) | ($4$ , $2.5$, $4$) | ($\theta_{g},\theta_{u}$) | ($20^{\circ}$, $20^{\circ}$)
$r_{c}$ |  $1000$ m | ($m_{l}$, $m_{n}$) | ($3$, $2$) | ($a$, $b$) | ($11.95$, $0.136$)
($h_{g}$, $h_{u}$) | ($25$, $100$) m | ($C_{l}$, $C_{n}$) | ($-69.8$, $-69.8$) dB | ($c$, $d$, $e$) | ($1$, $0.106$, $1$)
$\lambda_{g}$ |  $10$ BS/$\mathrm{km}^{2}$ | ($G_{g}^{(\text{max})}$, $G_{g}^{(\text{min})}$) | ($18$, $-2$) dB | $\sigma_{b}^{2}$ |  $4\cdot 10^{-11}$ W
$N_{u}$, $\delta_{b}$ |  $10$, $1$ | ($G_{u}^{(\text{max})}$, $G_{u}^{(\text{min})}$) | ($18$, $-2$) dB | ($\tau_{a}$, $\tau_{b}$) | ($0$, $10$) dB

$0$$100$$200$$300$$400$$500$$0$$0.2$$0.4$$0.6$$0.8$$1$$\text{Height of
UAVs}\;\;h_{u}\;\text{(m)}$ $\text{Backhaul probability}\;\;S(\tau{}_{b})$
$\tau_{b}=5$ dB$\tau_{b}=10$ dB$\tau_{b}=15$ dB$\tau_{b}=20$ dB Simulation

(a) Backhaul probability.

$0$$100$$200$$300$$400$$500$$600$$700$$800$$900$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$\text{Height
of UAVs}\;\;h_{u}\;\text{(m)}$ Association probability
$\tilde{A}_{u_{l}}+\tilde{A}_{u_{n}}$ $\tilde{A}_{g}$$\tilde{A}_{f}$Simulation

(b) Association probabilities.

$0$$100$$200$$300$$400$$500$$600$$700$$800$$900$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$\text{Height
of UAVs}\;\;h_{u}\;\text{(m)}$ $\text{Coverage probability}\;\;P_{\text{cov}}$
$\tau_{a}=-5$ dB$\tau_{a}=0$ dB$\tau_{a}=5$ dB$\tau_{a}=10$ dBSimulation

(c) Coverage probability.

Figure 5: Validation of analytical results and impact of UAVs height for
backhaul aware.

###  V-A Numerical results

We analyze the impact of the UAV height on the capability of the UAV to get a
good backhaul link in Fig. 5(a). The solid lines represent the analytical
results and the markers represent the simulation results. It is noted that the
analytical results match perfectly with the simulations, which validates the
proposed model. When a UAV hovers at higher altitude, its distance to the
serving BS increases, which causes a degradation in the received signal power.
However, a higher altitude means a higher LOS probability. For very low
altitudes above the terrestrial BSs, the LOS probability is the dominant
factor, and hence, the backhaul probability improves. As the UAV increases its
altitude, the path-loss dominates and the backhaul probability degrades. An
optimal height can be noted in Fig. 5(a), which maximizes the backhaul
probability.

In Fig. 5(b), we show the aerial and terrestrial association probabilities in
addition to the service failure probability for the backhaul aware scenario as
function of the height of the UAVs. In the low heights region, the UE tends to
associate more with UAVs and less with terrestrial BSs as the height of these
UAVs increases. This is due to the fact that, as the height increases, more
UAVs come within LOS conditions with respect to the UE. However, with a
further increase in height, the UAV association probability starts to decrease
because of the significant path-loss caused by the increasing distances
between the UE and the UAVs. At the same time, the backhaul probability
degrades which causes the increase of the service failure probability that
corresponds to a good UAV-UE access connection with bad backhauling. The
existence of UAVs in this case is harming the performance of the UE as
backhauling is not guaranteed at high heights. With a further increase in
height, the service failure probability starts to decrease because of the
significant path-loss encountered on the UAV-UE access links. In this high
height regime, the role of UAV becomes minor and the UE sticks to terrestrial
connections.

Fig. 5(c) demonstrates how the increase of the UAVs height affects the
coverage probability for different $\mathrm{SIR}$ thresholds in the backhaul
aware scenario. For low heights, the coverage probability increases as the UAV
height increases. This is due to the fact that, as the height increases, more
UAVs will have successful backhaul links. For higher altitudes, the backhaul
link deteriorates and the distances from the UE to its serving device and
interfering UAVs increase, and so the $\mathrm{SIR}$ and the coverage
probability decrease. This corresponds to the increase in the service failure
probability as seen in Fig. 5(b) which reflects the existence of some UEs with
good UAV-UE access links and bad backhaul links. With a further increase in
height, the coverage probability starts to increase again until a certain
limit after which it stabilizes. For these high heights, there is no point of
using UAVs and all the UEs are served with the terrestrial network. Fig. 5(c)
reveals that the optimal UAV height is related to the value of the
$\mathrm{SIR}$ threshold $\tau_{a}$. For higher thresholds, the UAVs must be
deployed at lower height to maximize the coverage probability.

$0$$10$$20$$30$$40$$50$$0.4$$0.45$$0.5$$0.55$$0.6$$0.65$$0.7$$0.75$$0.8$$\text{Number
of UAVs}\;N_{u}$ $\text{Coverage probability}\;\;P_{\text{cov}}$ Aware -
$\tau_{b}=10$ dBUnaware - $\tau_{b}=10$ dBAware - $\tau_{b}=15$ dBUnaware -
$\tau_{b}=15$ dBAware - $\tau_{b}=20$ dBUnaware - $\tau_{b}=20$ dB

(a) Number of UAVs.

$0$$5$$10$$15$$20$$25$$30$$0.3$$0.35$$0.4$$0.45$$0.5$$0.55$$0.6$$0.65$$0.7$$0.75$$0.8$$\text{Backhaul
threshold}\;\;\tau{}_{b}\;\text{(dB)}$ $\text{Coverage
probability}\;\;P_{\text{cov}}$
$\sigma_{s}=0$$\sigma_{s}=0.2$$\sigma_{s}=0.5$$\sigma_{s}=1$

(b) Beamforming misalignment error.

$0$$100$$200$$300$$400$$500$$600$$700$$800$$900$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$\text{Height
of UAVs}\;\;h_{u}\;\text{(m)}$ $\text{Coverage probability}\;\;P_{\text{cov}}$
$\lambda_{g}=1$ BS/km2$\lambda_{g}=5$ BS/km2$\lambda_{g}=10$
BS/km2$\lambda_{g}=20$ BS/km2

(c) Density of terrestrial BSs.

Figure 6: Coverage probability as function of number of UAV (for backhaul
unaware and aware transmissions), beamforming misalignment error and density
of BSs.

In Fig. 6(a), we illustrate how the number of UAVs impacts the coverage
probability for $\tau_{a}=0$ dB and $h_{u}=100$ m for different backhaul
thresholds $\tau_{b}$. Solid and dashed lines present the results of the
backhaul aware and unaware transmission scenarios, respectively. It is clearly
seen that increasing the number of UAVs first improves the coverage
probability but then deteriorates it for both scenarios. This is because
adding more UAVs has both a positive and a negative impact. First, a higher
UAVs number increases the average received power from the serving UAV since
the distance between the UE and the UAV is lower and the LOS probability is
higher. However, the added UAVs impose higher interference levels. Due to the
bounded region in which the UAVs are distributed and since the UE-UAV distance
is always $\geq h_{u}$ and the LOS probability $\leq 1$, the power of the
received signal reaches a maximum when a sufficient number of UAVs is
deployed. On the other side, the interference remains increasing when more
UAVs are deployed and the negative effect becomes the dominant factor. Fig.
6(a) also reveals that there exists an optimal number of UAVs that must be
deployed to optimize the UE service. Furthermore, Fig. 6(a) shows that the
awareness of the backhaul link improves the coverage probability compared to
the unaware scenario. This is due to the exclusion of the UAVs that do not
have a successful backhaul link from the interfering UAVs. As the number of
UAVs in the network increases, the gap between the aware and unaware coverage
results becomes larger. It is worth to note here that, even though a service
failure event is considered for backhaul aware transmission, its performance
remains superior to the unaware scenario for different network parameters.

To evaluate the impact of the beamforming misalignment error on the backhaul
link, we introduce the alignment error in the backhaul probability expression
in (22). We assume that the additive beam steering errors $\varepsilon_{g}$
and $\varepsilon_{u}$ for both BSs and UAVs follow a Gaussian distribution
with zero mean and variances $\sigma_{g}^{2}$ and $\sigma_{u}^{2}$,
respectively. Thus, $\lvert\varepsilon_{s}\rvert$ follows a half-normal
distribution with
$F_{\lvert\varepsilon_{s}\rvert}=\mathrm{erf}\left(\frac{x}{\sqrt{2}\sigma_{s}}\right)$
and
$\bar{F}_{\lvert\varepsilon_{s}\rvert}(x)=1-F_{\lvert\varepsilon_{s}\rvert}(x)$
where $s\in\\{g,u\\}$ and $\mathrm{erf}(\cdot)$ denotes the error function.
Fig. 6(b) presents the coverage probability as function of the threshold
$\tau_{b}$ for different values of the misalignment error variance
($\sigma_{u}=\sigma_{g}$). We can clearly notice that the quality of the
backhaul link has a direct impact on the UE experience. When the BS and UAV
antennas are not aligned on the backhaul link, the coverage probability
deteriorates. Thus, overlooking the backhaul effect can lead to misleading
coverage insights. The obtained results validate the importance of our work
compared to the works that consider guaranteed backhaul links.

Fig. 6(c) shows the coverage probability $P_{cov}$ as function of
$\lambda_{g}$ and the $h_{u}$ considering backhaul aware transmission. When
the UAVs hover at low altitudes, increasing the number of BSs decreases
$P_{cov}$. Higher densities of BSs result in a higher backhaul probability
since the distance separating the UAV from its serving BS decreases, which
increases the average received signal power and consequentially enhances the
LOS probability. At the same time, at lower heights, the UE tends to associate
more to UAVs, and since backhaul aware transmission is considered, more
successfully backhauled UAVs means high interference levels and lower coverage
probability. At the optimal height, the UEs are striking a balance between
good backhauling and association with UAVs. As the height further increases,
the coverage probability starts decreasing and reaches a break even point
after which adding more terrestrial BSs improves the coverage probability. At
this break even point, connecting to a UAV or a terrestrial BS would give the
same coverage performance. This analysis is valid until a certain value of the
density of BSs, below which insufficient backhauling is encountered. This is
clearly noticed for low BSs densities as the UAV has less BSs to align beam
with, and initiate backhaul connection. For high heights, the impact of the
aerial network is minimal and the UE associates with the terrestrial network.
Thus, adding more BSs would decrease the distance separating the UE from its
serving BS, which increases the average received signal power and
consequentially the coverage probability.

$0.1$$0.3$$0.5$$0.7$$0.9$$0.1$$0.3$$0.5$$0.7$$0.9$$\text{Fraction of BSs with
backhaul }\delta{}_{\text{b}}$ $\text{Backhaul probability}\;\;S(\tau_{b})$
$h_{u}=30$ m$h_{u}=100$ m$h_{u}=300$ m$h_{u}=500$ m

(a) Backhaul probability.

$0.1$$0.3$$0.5$$0.7$$0.9$$0.3$$0.4$$0.5$$0.6$$0.7$$\text{Fraction of BSs with
backhaul }\delta{}_{\text{b}}$ $\text{Coverage probability}\;\;P_{\text{cov}}$
$h_{u}=30$ m$h_{u}=100$ m$h_{u}=300$ m$h_{u}=500$ m

(b) Coverage probability.

$0$$10$$20$$30$$40$$50$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$\text{Number
of UAVs}\;N_{u}$ $\text{Coverage probability}\;\;P_{\text{cov}}$ Instan. -
$h_{u}=30$ mAware - $h_{u}=30$ mInstan. - $h_{u}=120$ mAware - $h_{u}=120$
mInstan. - $h_{u}=200$ mAware - $h_{u}=200$ m

(c) Aware vs. Instantaneous.

Figure 7: Backhaul and coverage probabilities as function of the fraction of
BSs with backhaul and comparison between backhaul aware transmission and
backhaul aware instantaneous association.

Fig. 7(a) and Fig. 7(b) show the impact of having a fraction $\delta_{b}\leq
1$ of backhaul-enabled BSs on both the backhaul and the coverage
probabilities, respectively. For low heights, adding more backhaul-enabled BSs
increases both the backhaul and the coverage probabilities. This is due to the
fact that, as more BSs provide backhauling, the distance separating the UAV
from its serving BS decreases, which increases the average received power and
consequentially enhances the backhaul probability. As this probability
increases, and since at low heights, the UE associates more to UAVs, the
coverage probability increases. Thus, at low heights, all the terrestrial BSs
must be able to provide backhaul capabilities to optimize the coverage. At
higher heights, the backhaul and coverage probabilities increase with the
increase of $\delta_{b}$ until a certain limit after which the two
probabilities stabilize. Thus, adding more backhaul-enabled BSs will not bring
any benefit and only a fraction is needed to achieve the optimal coverage. As
the UAVs height further increases, the backhaul probability starts to
deteriorate with the increase of $\delta_{b}$. This is caused by the increased
interference levels added to the high path loss and LOS probability
encountered on the backhaul links. The coverage probability, however, is not
affected with the increase of $\delta_{b}$ since at high heights, the UE tends
to associate more to the terrestrial network. The obtained results suggest
that, a good coverage probability can be achieved when a fraction of the
terrestrial BSs are backhaul-enabled. The fraction of backhaul-enabled BSs
that optimizes the coverage of UEs is highly dependent on the height of the
UAVs.

Finally, we compare the proposed backhaul aware transmission scheme with the
instantaneous backhaul aware association scenario. In the latter scenario,
when the UE-UAV access link quality is better than the UE-BS link but the UAV
does not have a successful backhaul link, the UE is not considered in a
service failure state. On the contrary, the UE changes its association to the
best terrestrial BS alternative. The instantaneous backhaul aware association
scenario models the case when UAVs are used to complement existing networks by
providing additional capacity to hotspots areas and network coverage in hard
to reach areas. In such scenario, when the UAV connection does not have any
added value, the UE operates through the terrestrial connection. The backhaul
aware transmission scheme is more applied in unexpected natural disasters. In
such scenarios, the existing networks can be damaged. Thus, if the connection
to UAV fails and the UE-BS access link quality is bad, the UE cannot connect
to the network.

Fig. 7(c) presents the comparison between backhaul aware transmission and the
simulation results of the instantaneous backhaul aware transmission scheme as
a function of the number and the height of UAVs. Note here that the analytical
derivations can be conducted by thinning the BPP of UAVs according to the
backhaul probability and then conducting the backhaul unaware association
analysis. It is clearly seen that, as the number of UAVs increases, the
coverage probability of backhaul aware instantaneous association exceeds that
of backhaul aware transmission. The gap gets smaller with the increase of the
UAVs height. Although the coverage probability is better for backhaul aware
association, this scheme suffers from excessive handovers (i.e., ping pong
effect) between UAV association and BS association. Such behavior highlights
the validity of the proposed aware transmission scheme, especially for high
UAVs altitudes.

###  V-B Discussions and Design Insights

In this section, we investigate the impacts of different system parameters to
illustrate generic design guidelines and recommendations. Our results reveal
that the height and number of assisting UAVs should be carefully adjusted to
achieve optimal system performance. Furthermore, the characteristics of the
terrestrial network must be taken into consideration while increasing the
number of deployed UAVs. An interesting observation found in Fig. 5(c) is that
an optimal height exists for the UAVs at which the coverage probability is
maximized. This result is due to the applied probabilistic LOS/NLOS
propagation model, implying that transmitting signals from higher altitude can
benefit from increased possibility of LOS links. The optimal height is the
point at which the UEs strike a balance between benefiting from the improved
quality of received signals due to LOS and maintaining a good backhaul link
between the UAVs and the terrestrial network. A trade-off can also be observed
in Fig. 6(a) regarding the number of deployed UAVs, which demonstrates that
dense UAV deployment improves the UE performance if the UAVs are hovering in
the low altitudes region and the backhaul links are guaranteed. For high
heights, sparse UAV deployment is necessary to lower the interference.

Based on the interpretation of Fig. 6(c), we can notice that, for dense
terrestrial BSs deployments, assisting the network with UAVs has a limited
contribution on the coverage of UEs. The obtained results in Fig. 6(b) reveal
the importance of considering the backhauling quality of the deployed UAVs as
a key parameter in the design of the hybrid network. The advantages of
assisting the terrestrial network with aerial BSs would not appear if these
BSs are not supported with sufficient backhauling. In addition, adding UAVs
with bad backhaul links would harm the UEs performance. Fig. 7(b) offers the
network operator the possibility of upgrading only a fraction of the existing
BS network to provide backhauling for UAVs and optimized coverage for UEs;
thereby reducing the needed infrastructure cost. Finally, supporting the UAVs
with sufficient processing capabilities to be aware of the backhaul link
quality would further improve the coverage.

##  VI Conclusion

This paper uses stochastic geometry to assess the performance of mmWave
backhauling for UAVs in a hybrid aerial-terrestrial cellular network
considering key system parameters, e.g., UAVs height and number, and
beamforming alignment error. We addressed both the backhaul unaware and the
backhaul aware transmission scenarios. After characterizing the backhaul
probability, the association and transmission probabilities and the distance
distributions, we obtained an expression for the overall coverage probability
and validated our results using Monte-Carlo simulations. Our results show that
the quality of the backhaul link has a significant impact on the UE
experience, and directly affects the deployment of UAVs for assisting
terrestrial networks.

###  -A Proof of Lemma 1

To derive the association probabilities, we follow a similar approach to [15].
Thus, the UE associates to a LOS UAV at distance $r$ if three conditions are
satisfied:

  * •

The UAV at distance $r$ is a LOS UAV with a LOS probability
$\kappa_{u_{l}}(r)$.

  * •

The remaining $(N_{u}-1)$ UAVs are either LOS UAVs located further than $r$ or
NLOS UAVs located further than $E_{ln}(r)$. The probabilities of these two
events are calculated as
$\int_{r}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w$ and
$\int_{E_{ln}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w$, where
$f_{W}(w)$ is the PDF of the distance from an arbitrary UAV to the UE. Since
the $(N_{u}-1)$ UAVs are iid, the probability of this event is
$\left(\int_{r}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{ln}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}-1}$.

  * •

The nearest BS is located further than $E_{u_{l}}(r)$ with a probability of
$\exp\left(-\pi\lambda_{g}E_{u_{l}}^{2}(r)\right)$.

Since these conditions are independent, the probability that the serving
device at distance $r$ is a LOS UAV is
$\kappa_{u_{l}}(r)\left(\int_{r}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{ln}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}-1}\exp\left(-\pi\lambda_{g}E_{u_{l}}^{2}(r)\right)$.
There are $N_{u}$ ways of choosing a UAV from iid $N_{u}$ UAVs. Thus, the
probability that the UE associates to a LOS UAV at distance $r$ is calculated
as:

$\small
N_{u}\kappa_{u_{l}}(r)\left(\int_{r}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{ln}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}-1}\exp\left(-\pi\lambda_{g}E_{u_{l}}^{2}(r)\right)\vspace{-0.1cm}$
(33)

Finally, $A_{u_{l}}$ is obtained by integrating over the feasible region
$h_{u}\leq r\leq w_{p}$. The association probabilities to a NLOS UAV and to a
BS $A_{u_{n}}$ and $A_{g}$ can be obtained through similar derivation.

###  -B Proof of Lemma 3

The aggregate interference of the interfering BSs is given as
$\hat{I}_{g}=\sum\limits_{x_{i}\in\phi_{g}\backslash
x_{g}}P_{g}(s_{g,x_{i}}^{2}+h_{g}^{2})^{-\eta_{g}/2}\Omega_{g,x_{i}}$. Thus,
$\mathcal{L}_{\hat{I}_{g}}(s_{1})$ can be derived as

$\displaystyle\mathcal{L}_{\hat{I}_{g}}(s_{1})=\mathbb{E}_{\phi_{g}}\left[e^{-\sum_{x_{i}\in\phi_{g}\backslash
x_{g}}s_{1}P_{g}(s_{g,x_{i}}^{2}+h_{g}^{2})^{-\frac{\eta_{g}}{2}}\Omega_{g,x_{i}}}\right]\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{\phi_{g}}\left[\prod_{x_{i}\in\phi_{g}\backslash
x_{g}}\mathbb{E}_{\Omega_{g,x_{i}}}\left[e^{-s_{1}P_{g}(s_{g,x_{i}}^{2}+h_{g}^{2})^{-\frac{\eta_{g}}{2}}\Omega_{g,x_{i}}}\right]\right]$
(34)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathbb{E}_{\phi_{g}}\left[\prod_{x_{i}\in\phi_{g}\backslash
x_{g}}\frac{1}{1+s_{1}P_{g}\left(s_{g,x_{i}}^{2}+h_{g}^{2}\right)^{-\frac{\eta_{g}}{2}}}\right]\stackrel{{\scriptstyle(c)}}{{=}}\exp\left[-2\pi\lambda_{g}\int_{x_{g}}^{\infty}\left(1-\frac{1}{1+s_{1}P_{g}(z^{2}+h_{g}^{2})^{-\frac{\eta_{g}}{2}}}\right)z\mathrm{d}z\right],$

where (a) follows from the iid distribution of the fading gain and its
independence of $\phi_{g}$, (b) from the exponential distribution of
$\Omega_{g,x_{i}}$ and (c) from the probability generation functional (PGFL)
of $\phi_{g}$ [18] and from replacing $s_{g,x_{i}}$ with $z$. Using (p.$315$)
from [37] and the Euler’s hyper geometric transformation, a closed form
expression is found for (34) as presented in (12).

To derive the Laplace transform of the interference from all UAVs
$I_{u}=\sum\limits_{y_{j}\in\Phi_{u}}I_{u,y_{j}}$, where $I_{u,y_{j}}$ is the
interference from the $j$-th UAV, we use a similar approach to [15] as follows

$\displaystyle\mathcal{L}_{I_{u}}(s_{1})=\mathbb{E}_{I_{u}}\left[\exp\left(-s_{1}\sum_{j=1}^{N_{u}}I_{u,y_{j}}\right)\right]\stackrel{{\scriptstyle(a)}}{{=}}\prod_{j=1}^{N_{u}}\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]=\left(\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]\right)^{N_{u}},$
(35)

where (a) follows from the iid distribution of the fading gains and from their
independence of the interferers distances in the interference expression.
Since the UE associates to a BS, for any of the $N_{u}$ interfering UAVs, it
is either a LOS UAV located further than $E_{g_{l}}(x_{g})$ or a NLOS UAV
further than $E_{g_{n}}(x_{g})$. The probabilities of these two events are
$\frac{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w}{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}$
and
$\frac{\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}$,
respectively, where $f_{W}(w)$ is given in (9). Thus,
$\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]$ can
be calculated as:

$\displaystyle\mathbb{E}_{I_{u,y_{j}}}\left[\exp\left(-s_{1}I_{u,y_{j}}\right)\right]=\frac{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w}{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}\mathbb{E}_{v_{u_{l},y_{j}}}\left[\exp\left(-s_{1}P_{u}\Omega_{u_{l},y_{j}}v_{u_{l},y_{j}}^{-\eta_{l}}\right)\right]$
(36)
$\displaystyle+\frac{\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}{\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}\mathbb{E}_{v_{u_{n},y_{j}}}\left[\exp\left(-s_{1}P_{u}\Omega_{u_{n},y_{j}}v_{u_{n},y_{j}}^{-\eta_{n}}\right)\right].$

where $v_{u_{l},y_{j}}$ and $v_{u_{n},y_{j}}$ are the distances of the UE to
the $j$-th interfering LOS UAV and the $j$-th interfering NLOS UAV. By
omitting $y_{j}$ we obtain

$\displaystyle\mathbb{E}_{v_{u_{l}}}\left[\exp\left(-s_{1}P_{u}\Omega_{u_{l}}v_{u_{l}}^{-\eta_{l}}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{v_{u_{l}}}\left[\left(1+\frac{s_{1}P_{u}v_{u_{l}}^{-\eta_{l}}}{m_{l}}\right)^{-m_{l}}\right]=\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{V_{u_{l}}}\left(v,E_{g_{l}}(x_{g})\right)\left(1+\frac{s_{1}P_{u}{v}^{-\eta_{l}}}{m_{l}}\right)^{-m_{l}}\mathrm{d}v$
(37)

where (a) follows from the moment generating functional (MGF) of the fading
gain $\Omega_{u_{l}}$ that follows a gamma distribution and from the iid
distribution of the interferers distances. All LOS UAVs are further than
$E_{g_{l}}(x_{g})$ from the UE. $f_{V_{u_{l}}}(v,x)$ is the distribution of
the distance from the interfering LOS UAV to the reference UE given in Lemma
$4$ in [15] as
$f_{V_{u_{l}}}(v,x)=\frac{f_{W}(v)\kappa_{u_{l}}(v)}{\int_{x}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w}$.
Similarly, for a NLOS UAV:

$\small\mathbb{E}_{v_{u_{n}}}\left[\exp\left(-s_{1}P_{u}\Omega_{u_{n}}v_{u_{n}}^{-\eta_{n}}\right)\right]=\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{V_{u_{n}}}\left(v,E_{g_{n}}(x_{g})\right)\left(1+\frac{s_{1}P_{u}v^{-\eta_{n}}}{m_{n}}\right)^{-m_{n}}\mathrm{d}v.$
(38)

$f_{V_{u_{n}}}\left(v,E_{g_{n}}(x_{g})\right)$ is the distance distribution
from the NLOS interfering UAV to the reference UE given as
$f_{V_{u_{n}}}(v,x)=\frac{f_{W}(v)\kappa_{u_{n}}(v)}{\int_{x}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w}$.
By plugging (37), (38), $f_{V_{u_{l}}}\left(v,E_{g_{l}}(x_{g})\right)$ and
$f_{V_{u_{n}}}\left(v,E_{g_{n}}(x_{g})\right)$ in (36) and (35), we can get
the expression in (13).

###  -C Proof of Lemma 4

The event $X_{g}>x_{g}$ is equivalent to the event $R_{g}>x_{g}$ given that
the UE associates with a ground BS, where $R_{g}$ is the horizontal distance
separating the nearest BS from the reference UE. The complementary cumulative
distribution function (CCDF) of $X_{g}$ is given as

$\displaystyle\bar{F}_{X_{g}}(x_{g})=\mathbb{P}[R_{g}>x_{g}\mid
s=g]=\frac{\mathbb{P}[R_{g}>x_{g},s=g]}{\mathbb{P}[s=g]}$ (39)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{A_{g}}\int_{x_{g}}^{\infty}f_{R_{g}}(r)\left(\int_{E_{g_{l}}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(r)}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}}\mathrm{d}r$

where (a) follows from the iid distribution of $N_{u}$ UAVs and the exclusion
regions $E_{g_{l}}(\cdot)$ and $E_{g_{n}}(\cdot)$ given in (6) on the
locations of the LOS and NLOS UAVs, respectively.
$f_{R_{g}}(r)=2\pi\lambda_{g}r\exp\left(-\pi\lambda_{g}r^{2}\right)$ is the
PDF of the horizontal distance separating the nearest BS from the reference UE
[38]. The CDF of $x_{g}$ is $F_{X_{g}}(x_{g})=1-\bar{F}_{X_{g}}(x_{g})$ and
the PDF is given as

$\small
f_{X_{g}}(x_{g})=\frac{\mathrm{d}F_{X_{g}(x_{g})}}{\mathrm{d}x_{g}}=\frac{1}{A_{g}}f_{R_{g}}(x_{g})\left(\int_{E_{g_{l}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{l}}(w)\mathrm{d}w+\int_{E_{g_{n}}(x_{g})}^{w_{p}}f_{W}(w)\kappa_{u_{n}}(w)\mathrm{d}w\right)^{N_{u}},\vspace{-0.2cm}$
(40)

where $f_{W}(\cdot)$ and $\kappa_{u_{l}}(\cdot)$ are given in (9) and (2),
respectively. Finally the distance distribution of $x_{g}$ can be obtained as
in (14).

###  -D Proof of Lemma 5

We start by providing the distributions of distances from the reference UAV to
the nearest LOS BS in $\phi_{g_{l}}$ and the nearest NLOS BS in
$\phi_{g_{n}}$. Denoting by $s_{b_{l}}$ the horizontal distance from the
reference UAV to the nearest LOS BS in $\phi_{g_{l}}$, the CCDF of $s_{b_{l}}$
can be calculated as

$\displaystyle\bar{F}_{s_{b_{l}}}(s)$
$\displaystyle=\mathbb{P}(s_{b_{l}}>s)=\mathbb{P}(\mathrm{No\;backhaul-
enabled\;LOS\;BS\;closer\;than\;}s)=e^{-2\pi\lambda_{b}\int_{0}^{s}\kappa_{b_{l}}(r)r\mathrm{d}r}.$
(41)

Therefore, the CDF is
$1-e^{-2\pi\lambda_{b}\int_{0}^{s}\kappa_{b_{l}}(r)r\mathrm{d}r}$ and the PDF
can be found as

$\small
f_{s_{b_{l}}}(s)=2\pi\lambda_{b}s\kappa_{b_{l}}(s)e^{-2\pi\lambda_{b}\int_{0}^{s}\kappa_{b_{l}}(r)r\mathrm{d}r}\vspace{-0.1cm}$
(42)

Similarly, the CDF and the PDF of the horizontal distance $s_{b_{n}}$ from the
UAV to the nearest NLOS BS from $\phi_{g_{n}}$ are given as
$\bar{F}_{s_{b_{n}}}(s)=e^{-2\pi\lambda_{b}\int_{0}^{s}\kappa_{b_{n}}(r)r\mathrm{d}r}$
and
$f_{s_{b_{n}}}(s)=2\pi\lambda_{b}s\kappa_{b_{n}}(s)e^{-2\pi\lambda_{b}\int_{0}^{s}\kappa_{b_{n}}(r)r\mathrm{d}r}$,
where $\kappa_{b_{n}}=1-\kappa_{b_{l}}$ and $\kappa_{b_{l}}$ is given in (3).

The reference UAV connects with a backhaul-enabled LOS BS in $\phi_{g_{l}}$ to
get backhaul support if the nearest LOS BS has smaller path-loss than that of
the nearest NLOS BS in $\phi_{g_{n}}$. Thus, the probability $A_{b_{l}}$ that
the reference UAV is associated with a LOS BS can be derived as follows

$\displaystyle
A_{b_{l}}=\mathbb{P}\left[C_{l}z_{b_{l}}^{-\eta_{l}}>C_{n}z_{b_{n}}^{-\eta_{n}}\right]=\int_{0}^{\infty}\mathbb{P}\left[C_{l}(s_{b_{l}}^{2}+\Delta_{h}^{2})^{-\frac{\eta_{l}}{2}}>C_{n}\left(s_{b_{n}}^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{n}}{2}}\right]f_{s_{b_{l}}}(s_{b_{l}})\mathrm{d}s_{b_{l}}$
(43)
$\displaystyle=\int_{0}^{\infty}\mathbb{P}\left[s_{b_{n}}>E_{b_{l}}(s_{b_{l}})\right]f_{s_{b_{l}}}(s_{b_{l}})\mathrm{d}s_{b_{l}}=\int_{0}^{\infty}\bar{F}_{s_{b_{n}}}(E_{b_{l}}(s_{b_{l}}))f_{s_{b_{l}}}(s_{b_{l}})\mathrm{d}s_{b_{l}}$

where $z_{b_{l}}$, $z_{b_{n}}$, $s_{b_{l}}$, $s_{b_{n}}$ are the actual and
horizontal distances from the UAV to the nearest LOS and NLOS BSs and
$E_{b_{l}}(s)=\sqrt{\left(\frac{C_{n}}{C_{l}}\right)^{\frac{2}{\eta_{n}}}({s}^{2}+\Delta_{h}^{2})^{\frac{\eta_{l}}{\eta_{n}}}-\Delta_{h}^{2}}$.
By replacing $s_{b_{l}}$ with $x$, $A_{b_{l}}$ is obtained as in (15).
$A_{b_{n}}$ is obtained with the same procedure.

###  -E Proof of Lemma 6

Denote $X_{b_{l}}$ as the horizontal distance between the reference UAV and
its serving LOS BS. Since the event $X_{b_{l}}>x$ is the event of
$s_{b_{l}}>x$ given that the reference UAV connects to a LOS BS to get
backhaul support, the probability of $X_{b_{l}}>x$ can be given as

$\small\mathbb{P}[X_{b_{l}}>x]=\mathbb{P}[s_{b_{l}}>x\mid
s=b_{l}]=\frac{\mathbb{P}[s_{b_{l}}>x,s=b_{l}]}{\mathbb{P}[s=b_{l}]}$ (44)

where $\mathbb{P}[s=b_{l}]=A_{b_{l}}$ is the probability that the UE
associates to a LOS BS. The joint probability of $s_{b_{l}}>x$ and $s=b_{l}$
is

$\displaystyle\mathbb{P}[s_{b_{l}}>x,s=b_{l}]=\mathbb{P}\left[s_{b_{l}}>x,C_{l}(s_{b_{l}}^{2}+\Delta_{h}^{2})^{-\frac{\eta_{l}}{2}}>C_{n}(s_{b_{n}}^{2}+\Delta_{h}^{2})^{-\frac{\eta_{n}}{2}}\right]$
(45)
$\displaystyle=\int_{x}^{\infty}\mathbb{P}[s_{b_{n}}>E_{b_{l}}(s_{b_{l}})]f_{s_{b_{l}}}(s_{b_{l}})\mathrm{d}s_{b_{l}}=\int_{x}^{\infty}\bar{F}_{s_{b_{n}}}(E_{b_{l}}(s_{b_{l}}))f_{s_{b_{l}}}(s_{b_{l}})\mathrm{d}s_{b_{l}}.$

Plugging (45) in (44) gives
$\mathbb{P}[X_{b_{l}}>x]=\frac{1}{A_{b_{l}}}\int_{x}^{\infty}\bar{F}_{s_{b_{n}}}(E_{b_{l}}(s_{b_{l}}))f_{s_{b_{l}}}(s_{b_{l}})\mathrm{d}s_{b_{l}}.$
The CDF of $X_{b_{l}}$ is $F_{X_{b_{l}}}(x)=1-\mathbb{P}[X_{b_{l}}>x]$ and the
PDF is given as

$\small
f_{X_{b_{l}}}(x)=\frac{\mathrm{d}F_{X_{b_{l}}}(x)}{\mathrm{d}x}=\frac{1}{A_{b_{l}}}\bar{F}_{s_{b_{n}}}(E_{b_{l}}(x))f_{s_{b_{l}}}(x)$
(46)

By plugging $\bar{F}_{s_{b_{n}}}(\cdot)$ and $f_{s_{b_{l}}}(\cdot)$ in (46),
the PDF of the horizontal distance to the serving LOS BS is given as in (16)
in Lemma 6. Following the same procedure, the PDF of the horizontal distance
between the UAV and its serving NLOS BS is determined and presented in (16).

###  -F Proof of Theorem 1

Given that the reference UAV is connected to a BS in $\phi_{g_{l}}$, and that
the desired link has a length of $x_{b_{l}}=x$, by Slivnyak’s Theorem, the
conditional backhaul probability is given as

$\small
S_{l}(\tau_{b})=\mathbb{P}[\mathrm{SINR}>\tau_{b}]=\int_{0}^{\infty}\mathbb{P}\left[\Omega_{b_{l},0}>\frac{\tau_{b}(\sigma_{b}^{2}+I_{b})}{P_{b}G_{b,0}C_{l}(x^{2}+\Delta_{h}^{2})^{-\frac{\eta_{l}}{2}}}\right]f_{X_{b_{l}}}(x)\mathrm{d}x$
(47)

where $I_{b}=I_{b_{l}}+I_{b_{n}}$ is the total interference from the LOS and
the NLOS backhaul-enabled BSs, $\sigma_{b}^{2}$ is the noise power and
$G_{b,0}$ is the maximum antennas gain. Noting that $\Omega_{b_{l},0}$ is a
normalized gamma random variable with parameter $m_{l}$, we have the following
approximation

$\displaystyle\mathbb{P}\left[\Omega_{b_{l},0}>\frac{\tau_{b}(\sigma_{b}^{2}+I_{b})}{P_{b}G_{b,0}C_{l}(x^{2}+\Delta_{h}^{2})^{-\frac{\eta_{l}}{2}}}\right]=1-\mathbb{P}\left[\Omega_{b_{l},0}<\frac{\tau_{b}\left(\sigma_{b}^{2}+I_{b}\right)}{P_{b}G_{b,0}C_{l}\left(x^{2}+\Delta
h^{2}\right)^{-\frac{\eta_{l}}{2}}}\right]$ (48)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\approx}}1-\mathbb{E}_{\phi_{g}}\left[\left(1-e^{-\frac{\gamma_{l}\tau_{b}(\sigma_{b}^{2}+I_{b})}{P_{b}G_{b,0}C_{l}(x^{2}+\Delta_{h}^{2})^{-\frac{\eta_{l}}{2}}}}\right)^{m_{l}}\right]\stackrel{{\scriptstyle(b)}}{{\approx}}\sum_{q=1}^{m_{l}}(-1)^{q+1}{m_{l}\choose
q}\mathbb{E}_{\phi_{g}}\left[e^{-\frac{q\gamma_{l}\tau_{b}(\sigma_{b}^{2}+I_{b})(x^{2}+\Delta_{h}^{2})^{\frac{\eta_{l}}{2}}}{P_{b}G_{b,0}C_{l}}}\right]$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\sum_{q=1}^{m_{l}}(-1)^{q+1}{m_{l}\choose
q}e^{-q\mu_{l}\tau_{b}\sigma_{b}^{2}}\mathcal{L}_{I_{b_{l}}}\left(q\mu_{l}\tau_{b}\right)\mathcal{L}_{I_{b_{n}}}\left(q\mu_{l}\tau_{b}\right)$

where $\gamma_{l}=m_{l}(m_{l}!)^{-\frac{1}{m_{l}}}$, $(a)$ follows from the
upper bound of the CDF of a gamma random variable deduced from Theorem $1$ of
[39], $(b)$ follows from the binomial theorem and the assumption that $m_{l}$
is an integer. (c) follows from denoting
$\mu_{l}=\frac{\gamma_{l}(x^{2}+\Delta_{h}^{2})^{\frac{\eta_{l}}{2}}}{P_{b}G_{b,0}C_{l}}$,
and from denoting the Laplace functionals of the interference of the LOS and
NLOS BSs as $\mathcal{L}_{I_{b_{l}}}(s)=\mathbb{E}[e^{-sI_{b_{l}}}]$ and
$\mathcal{L}_{I_{b_{n}}}(s)=\mathbb{E}[e^{-sI_{b_{n}}}]$, respectively, and
the fact that $\phi_{g_{l}}$ and $\phi_{g_{n}}$ are independent.

Given that the backhaul link is LOS of length $x_{b_{l}}$, based on the
association rule, all the LOS interfering BSs are further than $x_{b_{l}}$,
and all NLOS BSs are further than $E_{b_{l}}(x_{b_{l}})$. The Laplace
transform $\mathcal{L}_{I_{b_{l}}}(t)$ is derived as

$\displaystyle\mathcal{L}_{I_{b_{l}}}(t)=\mathbb{E}\left[e^{-tI_{b_{l}}}\right]=\mathbb{E}\left[e^{-t\sum_{x_{i}\in\phi_{g_{l}}\backslash
x_{b_{l}}}P_{b}\Omega_{b_{l},x_{i}}G_{b,I}C_{l}\left(s_{b_{l},x_{i}}^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{l}}{2}}}\right]$
(49)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp\left[-2\pi\int_{x_{b_{l}}}^{\infty}\left(1-\mathbb{E}_{\Omega_{b_{l}},G_{b}}\left[e^{-tP_{b}\Omega_{l}G_{b}C_{l}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{l}}{2}}}\right]\right)\right.\lambda_{b}\kappa_{b_{l}}(r)r\mathrm{d}r\Bigg{]}.$

where (a) follows from the PGFL of a PPP [18] and from omitting the indices
$x_{i}$ and $I$. The expectation term in (49) is given as

$\displaystyle\mathbb{E}_{\Omega_{b_{l}},G_{b}}\left[e^{-tP_{b}\Omega_{b_{l}}G_{b}C_{l}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{l}}{2}}}\right]$
(50)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\sum_{k=1}^{4}p_{k}\mathbb{E}_{\Omega_{b_{l}}}\left[e^{-tP_{b}\Omega_{b_{l}}G_{k}C_{l}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{l}}{2}}}\right]\stackrel{{\scriptstyle(b)}}{{=}}\sum_{k=1}^{4}p_{k}\left(1+\frac{tP_{b}G_{k}C_{l}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{l}}{2}}}{m_{l}}\right)^{-m_{l}}$

where (a) follows from the discrete directivity gain in the interference
channels, and (b) follows from computing the Laplace transform of
$\Omega_{b_{l}}$ which follows a gamma distribution. Similarly, for the NLOS
interfering links the Laplace transform $\mathcal{L}_{I_{b_{n}}}(t)$ is given
as

$\small\mathcal{L}_{I_{b_{n}}}(t)=\exp\left(-2\pi\int_{E_{b_{l}}(x_{b_{l}})}^{\infty}\left(1-\mathbb{E}_{\Omega_{b_{n}},G_{b}}\left[e^{-tP_{b}\Omega_{b_{n}}G_{b}C_{n}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{n}}{2}}}\right]\right)\right.\lambda_{b}\kappa_{b_{n}}(r)r\mathrm{d}r\Bigg{)}\vspace{-0.5cm}$
(51)

where

$\small\mathbb{E}_{\Omega_{b_{n}},G_{b}}\left[e^{-tP_{b}\Omega_{b_{n}}G_{b}C_{n}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{n}}{2}}}\right]=\sum_{k=1}^{4}p_{k}\left(1+\frac{tP_{b}G_{k}C_{n}\left(r^{2}+\Delta_{h}^{2}\right)^{-\frac{\eta_{n}}{2}}}{m_{n}}\right)^{-m_{n}}\vspace{-0.3cm}$
(52)

Finally, by plugging (48), (49) and (51) in (47) and by replacing $\mu_{l}$ by
$\frac{\gamma_{l}\left(x^{2}+\Delta_{h}^{2}\right)^{\frac{\eta_{l}}{2}}}{P_{b}G_{b,0}C_{l}}$
and $x_{b_{l}}$ by $x$, we get $S_{l}(\tau_{b})$ as in (18) given that the UAV
is served by a LOS BS. The same procedure is followed to obtain
$S_{n}(\tau_{b})$. Here, all NLOS interferers are farther than $x_{b_{n}}$ and
all LOS interferers are farther than $E_{b_{n}}(x_{b_{n}})$. The detailed
proof is omitted here and $S_{n}(\tau_{b})$ is given in (18). Finally, by the
law of total probability, the backhaul probability can be derived as in (17).

###  -G Proof of Lemma 7

The conditional coverage probability $P_{cov,u_{l}}$ is defined as the
coverage probability given that the UE associates to a LOS UAV at distance
$y_{u_{l}}$ and is given as

$\displaystyle
P_{cov,u_{l}}=\mathbb{P}\left[\mathrm{SIR}\geq\tau_{a},\mathrm{SINR}\geq\tau_{b}\right|s=u_{l}]\stackrel{{\scriptstyle(a)}}{{\approx}}\mathbb{E}_{Y_{u_{l}}}\left[\mathbb{E}_{I_{agg,u_{l}}}\left[\mathbb{P}\left(\Omega_{u_{l},0}\geq\frac{\tau_{a}I_{agg,u_{l}}}{P_{u}y_{u_{l}}^{-\eta_{l}}}\right)\right]\right]\times
S(\tau_{b})$ (53)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathbb{E}_{Y_{u_{l}}}\left[\mathbb{E}_{I_{agg,u_{l}}}\left[\sum_{k=0}^{m_{l}-1}\frac{(\hat{I}_{u_{l}}+I_{g_{l}})^{k}}{k!}\left(\frac{m_{l}\tau_{a}y_{u_{l}}^{\eta_{l}}}{P_{u}}\right)^{k}\exp\left(-\left(\frac{m_{l}\tau_{a}y_{u_{l}}^{\eta_{l}}}{P_{u}}\right)\left(\hat{I}_{u_{l}}+I_{g_{l}}\right)\right)\right]\right]\times
S(\tau_{b})$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}S(\tau_{b})\int_{h_{u}}^{w_{p}}\sum_{k=0}^{m_{l}-1}\frac{(-s_{2_{l}})^{k}}{k!}\left[\frac{\partial^{k}}{\partial
s_{2_{l}}^{k}}\mathcal{L}_{I_{g}}(s_{2_{l}})\mathcal{L}_{\hat{I}_{u_{l}}}(s_{2_{l}})\right]f_{Y_{u_{l}}}(y_{u_{l}})\mathrm{d}y_{u_{l}},$

where $s_{2_{l}}=\frac{m_{l}\tau_{a}y_{u_{l}}^{\eta_{l}}}{P_{u}}$ and
$S(\tau_{b})$ is the backhaul probability given in (17), (a) follows from the
independence assumption, (b) from the CCDF of the UE-UAV channel power gain
$\Omega_{u_{l},0}$ and from the expression of the interference
$I_{agg,u_{l}}=\hat{I}_{u_{l}}+I_{g_{l}}$. Finally, (c) follows from the
independence of $\hat{I}_{u_{l}}$ and $I_{g_{l}}$, where $\hat{I}_{u_{l}}$ and
$I_{g_{l}}$ are the interference expressions from all the UAVs except the
serving UAV and from all the BSs. $P_{cov,u_{n}}$ can be derived following a
similar approach.

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