#  Laser-Powered UAVs for Wireless Communication Coverage: A Large-Scale
Deployment Strategy

Mohamed-Amine Lahmeri, Mustafa A. Kishk, and Mohamed-Slim Alouini Mohamed-
Amine Lahmeri was with King Abdullah University of Science and Technology
(KAUST), Computer, Electrical, and Mathematical Science and Engineering
(CEMSE) division, when he carried this work. Currently, he is a Ph.D. student
at the Institute for Digital Communications, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Germany; Mustafa Kishk is with the Department of Electronic
Engineering, National University of Ireland, Maynooth, W23 F2H6, Ireland.
Mohamed-Slim Alouini is with the CEMSE Division, KAUST, Thuwal 23955, Saudi
Arabia (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>slim.alouini@kaust.edu.sa).

###### Abstract

The use of unmanned aerial vehicles (UAVs) is strongly advocated for sixth-
generation (6G) networks, as the 6G standard will not be limited to improving
broadband services, but will also target the extension of the geographical
cellular coverage. In this context, the deployment of UAVs is considered a key
solution for seamless connectivity and reliable coverage. That being said, it
is important to underline that although UAVs are characterized by their high
mobility and their ability to establish line-of-sight (LOS) links, their use
is still impeded by several factors such as weather conditions, their limited
computing power, and, most importantly, their limited energy. In this work, we
are aiming for the novel technology that enables indefinite wireless power
transfer for UAVs using laser beams. We propose a novel UAV deployment
strategy, based on which we analyze the overall performance of the system in
terms of wireless coverage. To this end, we use tractable tools from
stochastic geometry to model the complex communication system. We analyze the
user’s connectivity profile under different laser charging capabilities and in
different type of environments. We show a decrease in the coverage probability
by more than 12% in moderate-to-strong turbulence conditions compared to low
turbulence conditions. We also show how the connection rate to the aerial
network significantly decreases in favor of the terrestrial network for short
laser charging ranges. We conclude that laser-powered drones are considered
interesting alternatives when placed in LOS with users, in low-to-moderate
optical turbulence, and at reasonable ranges from the charging stations.

###### Index Terms:

Stochastic geometry, laser-powered UAVs, wireless power transfer, 6G, coverage
probability, Poisson point process, Matérn cluster process.

##  I Introduction

With the recent upsurge in the demand for fast and reliable connectivity,
researchers have been gravitating incessantly towards integrating aerial and
terrestrial architectures. It is expected that low latency, seamless
connection, and wide coverage areas will be a real paramount to sixth-
generation (6G) networks [1, 2]. In this context, unmanned aerial vehicles
(UAVs) are deemed to be the road to meet the aforementioned requirements of
the future 6G standards in light of their multiple striking features. For
instance, we mention their high flexibility and ability to establish line-of-
sight (LOS) links with users. Without a doubt, drones were used in a plethora
of applications such as in the military, farming, traffic control, and
delivery [3]. Notwithstanding these several use case scenarios, one recent
application attracted the researchers’ attention worldwide. It consists of
using drones as aerial base stations (ABSs) to provide coverage for ground
users (GUEs). The latter scenario is believed to be compelling as it might
provide interesting alternatives for users in an outage, for instance at
gatherings or in the hardly accessible regions. During such scenarios, the
classical cellular network architecture gets overwhelmed with multiple service
requests, and hence UAVs intervene as rescuers. On this basis, literature
related to deploying UAVs as ABS is gradually maturing with a wealth of
publications in this area. On the one hand, several classical techniques have
been used to propose overarching studies for the UAV deployment problem such
as optimization and stochastic geometry [4, 5]. On the other hand, machine
learning was also used to provide efficient solutions enabling a smart UAV
deployment [6, 7]. Although deploying UAVs as ABSs seems to be tremendously
exciting, its practical implementation faces myriad challenges that we cannot
hide. Undoubtedly, the notable vulnerability to bad weather conditions is one
of the downsides related to the use of drones as ABSs. Added to that the fact
that drones must always keep a visual LOS with the controller, according to
most of the latest drone aviation regulations. Furthermore, drones are known
to be unable to process heavy tasks on board as their equipped hardware is
usually limited. This could be explained by the limited payload that can be
carried on commercial drones, which is irrevocably linked to the quality of
hardware that can be installed onboard. That being said, the UAV constrained
energy remains undeniably the ultimate UAV drawback that will be our main
focus in this work.

Broadly speaking, there have been several attempts in the literature to
optimize the drone’s endurance, especially for long-duration missions. To make
it simple, we classify these solutions into (i) software-based solutions where
mathematical solutions and algorithms have been proposed, and (ii) hardware-
based solutions where special devices have been designed to prolong the
battery capacity. Starting with a software-based solution, we mention resource
allocation and scheduling for UAV-based networks where authors in [8] proposed
an energy-efficient resource allocation framework based on multi-agent
reinforcement learning. In the same context, Sun et al. [9] investigated
resource allocation for solar-powered drones by maximizing the system sum
throughput. Along with resource allocation, trajectory optimization also
attracted researchers’ attention. Authors in [10] proposed an energy-efficient
UAV trajectory by maximizing the communication throughput and minimizing the
UAV’s energy consumption. Furthermore, battery optimization is also
investigated in literature [11, 12]. With this in mind, a great source of
concern might arise when considering running such powerful solutions onboard
UAV processors, especially with the present limited UAV resources. As an
alternative to the previously mentioned solutions, a growing body of
literature has examined equipping drones with physical solutions to lengthen
their hovering time. Within this frame, we bring solar-powered drones to the
reader’s attention. As its name suggests, it consists of equipping drones with
adequate solar panels to harvest daylight energy. Although several interesting
research works targeted this area [9, 13, 14], this solution remains limited
since it loses its efficiency at night. Added to that, the considerable
complexity of equipping lightweight and energy-efficient solar panels onboard
small drones. This is why other research publications targeted the use of
tethered drones as an attempt to provide UAVs with indefinite battery life
[15, 16]. This obviously comes with the heavy cost of physically linking
drones with the ground stations. That being said, laser light recently stole
the limelight from tethered drones, as laser beams have been used to provide
indefinite wireless charging for drones.

For many years, laser light was used primarily for applications related to
surgery, industry, and especially communication [17]. However, in recent
decades, several companies have begun to use lasers to enable high-power
beaming. For instance, LaserMotive, also known as PowerLight Technologies,
managed to win the NASA power beaming prize for their laser-powered robot. The
same company succeeded in powering a drone for over 48 hours [18]. More
recently, they held a demonstration of transferring more than 400W at a
distance of 325m, which is indeed an exciting achievement. In addition to
powering drones, high-power laser beaming has also been used to attack and
eliminate drones. Companies such as Lockheed Martin and Boeing have developed
several exciting projects in this area. The proposed system, usually referred
to as laser beam director (LBD), consists of using a dedicated optical system
formed by a set of mirrors to beamform the laser arrays and direct them
towards their target. Although it is beyond the scope of this paper, within
such a complex setup, it is reasonable to worry about safety measures. This is
why several systems are put in place to guarantee safety such as the tracking
systems, cooling systems, and eye-safety systems.

In this paper, we analyze the performance of the deployment of laser-powered
UAVs as ABSs to assist terrestrial base stations (TBSs) in providing coverage
for GUEs. We use ground LBDs to provide energy to the UAVs, so that they can
carry out long-duration missions. As proposed in our previous work [19], we
assume simultaneous energy and information transfer to enable a backhaul link
between the drone and its serving LBD. That being said, we propose keeping a
backup battery to be used in critical cases, when the drone has to leave its
mission area or if by accident a LOS is lost. To this end, we use tools from
stochastic geometry to model the system and provide the expression of some
performance metrics based on which we assess the system performance. In what
follows, we review the most related works to the present paper and highlight
the novelty and contribution of our work.

###  I-A Related works

The use of stochastic geometry in system modeling is now a mature field where
a surfeit of publications already exists. With the recent integration of ABSs
into existing networks, researchers attempted to use tools from stochastic
geometry to provide comprehensive studies and performance analysis for UAV-
assisted networks. In this context, authors in [20] used stochastic geometry
tools to provide performance analysis for a network of UAVs modeled as a
Poisson point process (PPP) for serving one user in an urban environment.
Authors in [21] consider a network of UAVs and LTE base stations (BSs), both
modeled with PPPs, where the BS provides the UAV with a backhaul link.
Furthermore, in [22] a coverage and rate analysis has been proposed for a
network of mixed TBSs and ABSs, modeled with PPPs and serving a typical user.
Moreover, Poisson cluster processes (PCPs) have been used to model the
distribution of users on the ground. In [23], authors used Thomas cluster
process and Matérn cluster process (MCP) to model the distribution of users,
while in [24], users were modeled with MCP where the drones are positioned
above the cluster centers. PPP is known to be tractable and easy to exploit
compared to other point processes, however, it assumes that the user’s
position is completely random. Compared to PPP, MCP is considered more
realistic in scenarios where the distribution of users is described by
gathering groups, called clusters. This being said, MCP requires knowledge of
the statistics of these clusters to adjust the parameters of the point process
such as the cluster radius. Furthermore, authors in [25] used binomial point
processes (BPP) to model the distribution of the drones. In the same context,
authors in [26] model the drones with a BPP, consider the existence of TBSs,
and provide a downlink coverage and rate analysis. None of the previously
mentioned research works have focused on the energy aspect of drones which is
a crucial factor, especially when deploying UAVs as ABSs.

Compared to the use of stochastic geometry in modeling, the use of laser beams
for drones is still under investigation. Authors in [27] studied the downlink
throughput maximization problem for a laser-powered UAV. The considered fixed-
wing drone in this work uses the laser beam to harvest enough energy for
hovering and for communication with the ground terminal. Moreover, authors in
[28] consider the use of laser-powered drones in assisting an Internet-of-
Things (IoT) network. An optimization problem is formulated to maximize the
number of IoT devices connected with the drone. In [29] the problem of joint
power and trajectory optimization was studied for rotary-wing laser-powered
drones. In [30] authors studied a UAV-aided multicasting system for
simultaneous FSO backhaul and power transfer. Under PPP distributed users, the
UAV altitude and the FSO and RF transmissions are optimized. Statistics of the
FSO communication channel between a UAV and a central unit were proposed in
[31]. Channel models were derived taking into account the position and
orientation fluctuation of the drone as well as the non-orthogonality of the
laser beam. In another context, Jaafer et al. provided a battery dynamics
study for a laser-powered drone under different charging techniques [12].
Authors in [32] investigated different charging techniques for drone networks
including laser-powered drones. These research works do not provide any
performance analysis for BS-mounted drones. We provided a stochastic geometry-
based analysis in our previous work [19]. We studied the communication
coverage and energy coverage of a single UAV powered by a network of ground
LBDs, however, we did not consider using multiple drones, multiple users, or
multiple TBSs.

###  I-B Novelty and Contributions

All of the above-mentioned works either consider applying stochastic geometry
for UAV-assisted networks or studying laser-powered drones from other angles
rather than stochastic geometry, whereas in our work:

  * •

We design a complex system setup, composed of a network of TBSs, multiple
UAVs, multiple GUEs, and a network of LBDs. We use various point processes to
model their spatial distribution.

  * •

We propose and analyze a novel deployment strategy for the laser-powered
drones that enables a trade-off between energy and wireless coverage.

  * •

We derive the theoretical expressions of the coverage probability, the Laplace
transform of interference, and the coverage probability. We validate these
expressions through numerical simulations in different types of environments.

  * •

Finally, we provide high-level insights about the proposed system based on the
derived performance metrics.

Notations: In this paper, we use $\mathbb{E}(\cdot)$ to denote the expectation
operator and $\mathbb{P}(\cdot)$ for the probability of an event.
$\mathds{1}(\cdot)$ denotes the indicator function while $\delta(\cdot)$ is
the Dirac delta function, equal to one at zero and null otherwise. Function
$F_{x}(\cdot)$ denotes the cumulative distribution function (CDF) of a given
random variable $x$. Set $\mathbb{R}_{+}$ denotes non-negative real numbers
and $\mathbb{R}^{2}$ denotes the set of 2-dimensional vectors with real-
entries. For $x\in\mathbb{R}^{2}$, $||x||$ denotes the euclidean norm of $x$.
We denote by $|\cdot|$ the absolute value operator.

##  II System Model

Figure 1: Laser-powered UAV network.

In this work, we consider the scenario of assisting a poor terrestrial TBSs
architecture in providing wireless coverage to GUEs through the use of drones
that are acting as ABSs. The UAVs, while performing their task, benefit from
the energy supply provided by the terrestrial LBDs that are statically placed
on the ground. Figure 1 represents one use case for laser-powered drones used
to provide coverage for GUEs. Such users may be located in small geographic
areas, which motivates modeling them with clustered processes. Figure 1 also
hints at the fact that the deployment of LBDs can be either static or dynamic.
Moreover, the LBDs guarantee energy coverage for the drones as long as the
UAVs are placed within an adequate range $R^{*}$ that is derived later.
Consequently, the drones will not rely on their battery but use them only in
critical situations, allowing them to carry out long-duration missions. We
assume that an LBD can only serve one UAV at a time, so temporal coordination
is beyond the scope of this paper. Therefore, the case where more than one UAV
requires recharging is included in the critical situation we have mentioned
previously. The environments that we target in this study are suburban, urban,
and dense urban, where the distributions of buildings may affect the
performance analysis significantly. In what follows, we present the different
models used in this work.

###  II-A Spatial Distributions

We assume that we do not have any prior knowledge about the optimal positions
of the LBDs and model their distribution with a homogeneous PPP, denoted by
$\Phi_{LBD}$, with corresponding density $\lambda_{LBD}$. Consequently, our
performance results can serve as a lower bound since we do not optimize the
LBD locations. We also consider the existence of a network of TBSs that is
modeled with another homogeneous PPP, denoted by $\Phi_{b}$, with density
$\lambda_{b}$. Taking advantage of the nature of the users’ gathering, that
usually forms random groups creating a congesting in the wireless network
traffic, we model the GUEs distribution with a MCP where the distribution of
the parent nodes, referring to the cluster center, forms a PPP denoted by
$\Phi_{cc}$ with density $\lambda_{cc}$ and maximum cluster radius $r_{max}$.
For the moment, we denote the UAV distribution by $\Phi_{u}$ with density
$\lambda_{u}$, where $x_{i}$ is the position of the $i^{th}$ UAV. We explain,
later in this work, our proposed deployment strategy for the drones.

Figure 2: System model summary.

###  II-B LOS Blockage Probability Model

It is crucial in our work to identify the link type between the transmitter
and the receiver. To this end, we model the probability of LOS link such that
it captures the effect of the distribution of the surrounding buildings within
an urban environment. Various complex models were proposed in the literature
[33], however, in our work, we use the simplified LOS probability, derived in
[26], to model the link UAV-user and TBS-user:

$\displaystyle\mathcal{P}_{\rm
i}(r)=-a\;\exp\left(-b\;\arctan\left(\frac{h_{i}}{r}\right)\right)+c,\forall
i\in\\{Lu,Lb\\},$ (1) $\displaystyle\mathcal{P}_{\rm Nu}(r)=1-\mathcal{P}_{\rm
Lu}(r),$ (2) $\displaystyle\mathcal{P}_{\rm Nb}(r)=1-\mathcal{P}_{\rm Lb}(r),$
(3)

where $\mathcal{P}_{\rm Lu}$, $\mathcal{P}_{\rm Nu}$, $\mathcal{P}_{\rm Lb}$
and $\mathcal{P}_{\rm Nb}$ represent respectively the UAV LOS, UAV non-LOS
(NLOS), TBS LOS, and TBS NLOS probabilities, $r$ is the horizontal distance to
the respective node, $h_{\rm Lu}=h_{\rm Nu}=h$ is the UAV altitude, $h_{\rm
Lb}=h_{\rm Nb}=h_{b}$ is the TBS height. The parameters $a$, $b$, and $c$
depend on the environment nature.

###  II-C PPP Thinning Procedure

In this section, we derive the different distances to the nearest LOS UAV,
NLOS UAV, LOS TBS, and NLOS TBS. To this end, a common technique in dealing
with LOS and NLOS distributions of a given point process is to take advantage
of the thinning operation in dividing the parent PPP into two non-homogeneous
PPPs. The operation consists of deleting each point with a given probability.
In our case, we use $\mathcal{P}_{\rm Lu}(r)$ in dividing $\Phi_{u}$ into a
set of LOS UAVs denoted by $\Phi_{\rm Lu}$ and a set of NLOS UAVs, denoted
$\Phi_{\rm Nu}$. The same is done to $\Phi_{b}$ by dividing it into $\Phi_{\rm
Lb}$ and $\Phi_{\rm Nb}$. The density of each point process is given by:

$\displaystyle\lambda_{i}(r)=\lambda_{\rm u}\mathcal{P}_{\rm i}(r),i\in\\{\rm
Lu,Nu\\},$ (4) $\displaystyle\lambda_{i}(r)=\lambda_{\rm b}\mathcal{P}_{\rm
i}(r),i\in\\{\rm Lb,Nb\\}.$ (5)

###  II-D Channel Modeling

The UAV-to-user channel is modeled by a Nakagami-$m$ multi-path fading denoted
by $G_{\rm Lu}$ and $G_{\rm Nu}$. However, the terrestrial BS-to-user channel
is modeled by a Rayleigh fading denoted by $G_{\rm Lb}$ and $G_{\rm Nb}$. We
use this notation to provide a compact notation for the equations later. The
instantaneous power received from a node placed at $x_{\rm i}$ at the
reference user is given by:

$\displaystyle\mathcal{R}_{\rm j}(r(x_{i}))=\rho_{\rm j}\;\eta_{\rm
j}\;\left(r(x_{i})^{2}+h_{j}^{2}\right)^{-\frac{\alpha_{\rm j}}{2}}G_{\rm
j}\textit{ , }x_{i}\in\Phi_{\rm j},\textit{ and }j\in\mathcal{C},$ (6)

where $\rho_{\rm Lu}=\rho_{\rm Nu}=\rho_{\rm u}$ denotes the UAV transmit
power, $\rho_{\rm Lb}=\rho_{\rm Nb}=\rho_{b}$ denotes the TBS transmit power,
$\eta_{\rm i}$ represents the additional path loss related to a LOS condition
or NLOS conditions, and $r(x_{\rm i})$ is the horizontal distance to the point
located at $x_{\rm i}\in\mathbb{R}^{2}$. Hereinafter, we denote by
$\mathcal{C}$ the set $\\{\rm Lu,Nu,Lb,Nb\\}$. For the ease of notation, we
also define the average power received from a node place at $x_{i}$ at the
reference user as follows:

$\displaystyle\overline{\mathcal{R}}_{\rm j}(r(x_{i}))=\rho_{\rm j}\;\eta_{\rm
j}\;\left(r(x_{i})^{2}+h_{j}^{2}\right)^{-\frac{\alpha_{\rm j}}{2}}\textit{ ,
}x_{i}\in\Phi_{\rm j},\textit{ and }j\in\mathcal{C}.$ (7)

###  II-E Energy Harvesting and Backhaul Link

As proposed in our previous work [19], we consider simultaneous communication
and energy transfer under a power splitting technique at the drone level,
where the LBD at the ground does not only guarantee energy supply but also
provides a backhaul link to the drones. To model the energy transfer link
between the drones and the LBDs, we use the following model based on the Beer-
Lambert equation [34]:

$\displaystyle p_{\rm harv}(R)=h_{t}\frac{\left(1-\delta_{s}\right)\omega
T\chi\;p_{\rm
trans}\exp\left(-\alpha\sqrt{R^{2}+h^{2}}\right)}{\left(D+\sqrt{R^{2}+h^{2}}\Delta\theta\right)^{2}},$
(8)

where $R$ is the horizontal distance to the drone, $h_{t}$ is a log-normal
random variable representing the turbulence effect, $\delta_{s}$ is the power
splitting factor used to separate the energy transfer link and the
communication link. The parameter $w$ denotes the receiver conversion
efficiency, $T$ is the area of the receiver telescope or collection lens, $D$
is the size of the initial laser beam, $\alpha$ is the attenuation coefficient
of the medium, $\chi$ is the combined transmission receiver optical
efficiency, and $\Delta\theta$ is the angular spread of the laser beam. The
angular spread $\Delta\theta$ is equal to $\frac{D_{d}}{f}$, where $D_{d}$ is
the size of the detector and $f$ its focal length.

Regarding the backhaul link, we propose using a basic laser intensity
modulation such as on-off keying for communication. Accordingly, the signal-
to-noise ratio (SNR) level at the UAV is given by [35, 34]:

$\displaystyle{\rm SNR_{\rm UAV}}(R)=\frac{\delta_{s}\;\eta\;h_{t}\;\omega
A\chi\;p_{\rm
trans}\;\exp\left(-\alpha\sqrt{R^{2}+h^{2}}\right)}{2\;\hbar\;\nu\;\Delta
f\left(D+\sqrt{R^{2}+h^{2}}\Delta\theta\right)^{2}},$ (9)

where $\hbar$ is Planck’s constant, $\nu=\frac{c}{\lambda}$ is the photon’s
frequency, $\lambda$ is the wavelength, $c$ is the speed of light, and $\Delta
f$ is the modulation frequency bandwidth.

##  III UAV Deployment strategy

In this section, we present the details of our proposed UAV deployment. We
consider it a large-scale UAV deployment strategy since we use a PPP to model
multiple drones and simulate them in a large area of interest. In what
follows, we derive the critical charging distance $R^{*}$, then we use its
value to define our deployment strategy. Finally, we identify the resulting
UAV distribution after applying the proposed placement strategy.

###  III-A Critical Charging Distance

To guarantee a safe energy level and successful communication with the LBD,
the UAV flies inside a ball centered around the serving LBD and of radius
$R^{*}$. This critical charging distance was the subject of [19], where we
provided its closed-form expression in the non-turbulent 111Refers to the case
where the effect of the optical turbulence on energy harvesting is not taken
into consideration.  regime as follows:

$\displaystyle\medmath{R^{*}=\Bigg{[}\Bigg{\\{}\frac{2}{\alpha}W_{0}\left(\frac{\alpha}{2\Delta\theta}\sqrt{\frac{\left(1-\delta_{s}\right)\omega
T\chi p_{\rm trans}e^{\frac{\alpha D}{\Delta\theta}}}{p_{\rm prop}+p_{\rm
comm}}}\right)-\frac{D}{\Delta\theta}\Bigg{\\}}^{2}-H^{2}\Bigg{]}^{\frac{1}{2}},}$
(10)

where $W_{0}(.)$ is the principal branch of the Lambert W function [36],
$P_{\rm prop}$ is the UAV propulsion power, and $P_{\rm comm}$ is the UAV
communication power. However, the laser intensity is usually affected by
optical turbulence, especially for long range applications.

######  Lemma 1 (Critical charging distance).

In the Case of log-normal turbulence effect, where
$h_{t}\sim\operatorname{lognormal}(-2\sigma,2\sqrt{\sigma})$, the value of the
critical charging distance $R^{*}$, is derived numerically by solving the
below equation:

$P_{\rm
harv}(R^{*})\exp\left(2\sqrt{2\sigma(R^{*})}\operatorname{erf}^{-1}(1-2\mathcal{B})-2\sigma(R^{*})\right)-p_{\rm
prop}-p_{\rm comm}=0,$ (11)

where $R^{*}$ is derived such that the joint SNR and energy coverage
probability is above a certain level denoted by $\mathcal{B}$.

###### Proof:

See Appendix -A. ∎

We bring to the reader’s attention the following points: (i) $\sigma$ is
denoted by $\sigma(R^{*})$ in (11) to emphasize the fact that it is a function
of $R^{*}$ and that a closed-form solution could not be easily derived since a
commonly used expression is
$\sigma(R^{*})=\sqrt{0.3k^{\frac{7}{6}}C^{2}_{n}(h){R^{*}}^{\frac{11}{6}}}$,
where $C^{2}_{n}(h)$ is the refraction index at an altitude h, and k is the
wavenumber. Moreover, $p_{\rm harv}(R^{*})$ is not linear in $R^{*}$, as (8)
shows, which complicates further a closed-form solution. (ii) Typical values
for the minimal threshold $\mathcal{B}$ are [0.9, 0.99], which means that
successful communication energy coverage is guaranteed with probability 90% to
99%. Regarding the outage probability, we rely on the drone’s backup battery
to get back to the LBD’s field of view or to a safe-charging range.

TABLE I: System notations

Notations
---
Notation | Description | Notation | Description
Distribution notations | $\delta_{s}$ | Power splitting factor
LBD | Laser beam director | $\alpha$ | Attenuation coefficient of the medium
Lu | LOS UAV | $\Delta\theta$ | Angular spread of the beam
Nu | NLOS UAV | $w$ | Receiver conversion effiency
Lb | LOS TBS | $T$ | Area of the receiver telescope or collection lens
Nb | NLOS TBS | $k$ | Wavenumber
Distribution-dependent notations | $\chi$ | Combined receiver optical efficiency
$\Phi_{i}$ | Distribution $i$ | $p_{harv}$ | Harvested power at the UAV
$\rho_{i}$ | Transmit power | $p_{\rm prop}$ | Propulsion power for the UAV
$\eta_{i}$ | Additional path loss | $p_{\rm comm}$ | Communication power for the UAV
$\alpha_{i}$ | Path loss | $\sigma(R^{*})$ | Turbulence standard deviation
$\lambda_{i}$ | Density | $D$ | Size of the initial laser beam
$r_{i}$ | Distance to the nearest neighbor from $\Phi_{i}$ | $C^{2}_{n}(h)$ | Refraction index at altitude h
$G_{i}$ | Nakagami-m random multipath fading | Other notations
$m_{i}$ | Nakagami-m fading term | $h$ | UAV altitude
$h_{i}$ | Altitude of distribution $\Phi_{i}$ nodes | $h_{b}$ | TBS altitude
$\mathcal{P}_{i}$ | Probability of LOS | $r_{\rm max}$ | Cluster radius
Laser link notations | $\hbar$ | Planck’s constant
$R^{*}$ | Critical charging distance | $\mathcal{S}$ | Deployment strategy
$\mathcal{B}$ | Joint energy-communication coverage threshold | $\mathcal{C}$ | The set $\\{\rm Lu,Nu,Lb,Nb\\}$

###  III-B UAV Deployment Strategy Description

Figure 3: Deployment strategy overview.

Positioning the UAV above the cluster centers might provide good coverage for
the users, but can not be optimal in terms of energy coverage as it is not
guaranteed that the distance to the serving LBD is always in safe ranges.
Based on the distance $R^{*}$, we propose a deployment strategy, denoted by
$\mathcal{S}$, described by Figure 3 and Algorithm 1, that establishes a
trade-off between energy coverage and the quality of service provided. Below,
we summarize the main points of the UAV placement strategy:

  * •

One UAV is dedicated to each user cluster center.

  * •

Case1. $\mathcal{S}=``r_{c}\leq R^{*}"$: If the distance from the cluster
center to its nearest LBD is below $R^{*}$, the drone decides to hover above
the cluster center to provide maximum coverage for the users inside the
cluster center. This is assumed to provide the lowest average euclidean
distance to the users inside the cluster and thus results in better coverage.

  * •

Case2. $\mathcal{S}=``r_{c}>R^{*}"$: If the distance from the cluster center
to its nearest LBD is above $R^{*}$, The UAV is forced to shift along the line
linking the cluster center and its nearest LBD. This is because the harvested
energy level at the cluster center cannot compensate for the consumed energy.
Thus the UAV is placed at a distance $R^{*}$ on the line linking the drone
from its nearest LBD.

Algorithm 1 UAVs Relocation Algorithm

1:procedure Relocate($\Phi_{LBD},\Phi_{cc}$)

2: $R^{*}\leftarrow\textit{solve(\ref{eq:Rstar})}$

3: $\Phi_{u}\leftarrow\Phi_{cc}$

4: for $(x,y)\in\Phi_{u}$ do

5: for $(x^{\prime},y^{\prime})\in\Phi_{LBD}$ do

6:
$d(x^{\prime},y^{\prime})\leftarrow\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}$

7: end for

8: $r_{c}\leftarrow$min($d$)

9: $(x_{n},y_{n})\leftarrow$argmin($d$) $\triangleright$ $(x_{n},y_{n})$ is
the nearest neighbor of (x,y) from $\Phi_{LBD}$

10: slope $\leftarrow\frac{y-y_{n}}{x-x_{n}}$

11: if $(r_{c}\geq R^{*})$ then$\triangleright$ set (x,y) such that the
distance to the nearest LBD is set to $R^{*}$

12: $x\leftarrow x_{n}+\frac{{\rm sign}(x-x_{n})R^{*}}{\sqrt{1+slope^{2}}}$

13: $y\leftarrow y_{n}+\frac{{\rm
sign}(y-y_{n})R^{*}}{\sqrt{1+slope^{2}}}$$\triangleright$ In case
$r_{c}<R^{*}$ keep (x,y) unchanged.

14: end if

15: end for

16: return $\Phi_{u}$

17:end procedure

###  III-C UAV distribution

In this subsection, we present the resulting UAV distribution after applying
Algorithm 1. Applying random translations to a PPP only affects its density,
and the resulting density is provided in [37]. However, in our case, we cannot
apply the displacement theorem to derive the new density. This is because the
translations we are making are not independent. In this case, and from the
reference user point of view, we consider the existence of a central UAV,
which is the drone associated with the cluster of the reference user and we
approximate the rest of the UAV distribution as a PPP. We show later through
the numerical simulation that this approximation is decent. In what follows,
we provide the ranges of the critical charging distance at which this
approximation is decent. To do so, we analyze the displacement magnitude,
which is the distance crossed by a drone following the placement strategy. Let
$x,y\in\mathbb{R}^{2}$, where $x\in\Phi_{u}$ denotes the location of one UAV
and y denotes its new position after the relocation described in Algorithm 1.
Let $\mathcal{NN}(x)\in\Phi_{LBD}$ be the nearest LBD to $x$. Notice that
$||x-y||$ is a mixed random variable from the following PDF:

$\displaystyle f_{||x-y||}(\alpha)=\left(1-\exp\left(-\pi\lambda_{\rm
LBD}{R^{*}}^{2}\right)\right)\delta(\alpha)+2\lambda_{\rm
LBD}\pi(R^{*}+\alpha)\exp\left(-\pi\lambda_{\rm
LBD}(R^{*}+\alpha)^{2}\right)\mathds{1}(\alpha\geq 0).$ (12)

This is because the CDF of $||x-y||$ is derived as follows:

$\displaystyle F_{||x-y||}(\alpha)$
$\displaystyle=\mathbb{P}\left(||x-y||\leq\alpha\right),$
$\displaystyle=\mathbb{P}\left(||\mathcal{NN}(x)-x||\leq R^{*}+\alpha\right),$
$\displaystyle=1-\exp\left(-\pi\lambda_{\rm LBD}{(R^{*}+\alpha)}^{2}\right).$
(13)

Thus, the displacement magnitude $||x-y||$ is a mixed random variable with a
continuous part:

$\displaystyle C(\alpha)=\begin{dcases}\exp\left(-\pi\lambda_{\rm
LBD}{R^{*}}^{2}\right)-\exp\left(-\pi\lambda_{\rm
LBD}{(R^{*}+\alpha)}^{2}\right)&\alpha\geq 0\\\ 0&else.\end{dcases}$ (14)

Accordingly, the discrete part which discribes the probability of not
displacing the node $x$, is given by:

$\displaystyle D(\alpha)=\begin{dcases}1-\exp\left(-\pi\lambda_{\rm
LBD}{R^{*}}^{2}\right)&\alpha\geq 0\\\ 0&else.\end{dcases}$ (15)

In Figure 4, we plot the theoretical CDF of the displacement norm versus the
empirical CDF for different ranges of the critical charging distance $R^{*}$
and the displacement distance $\alpha$. Based on this figure, we can notice
that for reasonable ranges of the critical charging distance
$[900m\;\;1300m]$, the probability of displacement is low and the nodes are
often not relocated. This is represented by the blue-colored facet of the
cube. Moreover, even if the node is relocated then the displacement distance
is not significant as shown by the green-colored facets of the cube in Fig .4.

Figure 4: Displacement magnitude as a function of $R^{*}$ and the displacement
distance $\alpha$, $\lambda_{LBD}=0.52\times 10^{-6}$ unit/m2 and
$\lambda_{u}=3\times 10^{-6}$ unit/m2.

##  IV Performance Analysis

To assess the performance of the system, we start by deriving the important
distance distributions, the necessary association probabilities, the Laplace
transform of interference, and finally the coverage probability.

###  IV-A Important Distance Distributions

####  IV-A1 Distance to the Nearest Neighbor from a PPP

The distance distribution to the nearest point from any PPP is given in [37]
by:

$\displaystyle f_{r_{\rm
i}}(r)=2\pi\;\lambda_{i}(r)\;r\;\exp\left(-2\pi\int_{0}^{r}\lambda_{i}(x)x\;dx\right),$
(16)

where $r_{i}$ is the distance to the nearest neighbor from $\Phi_{i}$, with
corresponding density $\lambda_{i}$, and such that
$i\in\\{Lu,Nu,Lb,Nb,LBD\\}$.

####  IV-A2 Distance Distribution Between the Reference User and its Central
UAV

Following the deployment strategy that we proposed previously in Sec. III-B,
it is clear that a single UAV is associated with each user cluster. From the
reference user point of view, the UAV associated with its cluster center is
considered as the central UAV. Deriving the distance distribution between the
reference user and its central UAV, denoted by $r_{\rm cu}$, is important as
it is used later in deriving the association probabilities and finally the
coverage probability. As indicated in Algorithm 1, this distance depends on
$r_{c}$, which is the distance from the cluster center to the nearest LBD.
Case1. Central UAV is placed above the cluster center ($r_{c}\leq R^{*}$)
In this scenario, as the UAV is placed above the cluster center to maximize
its coverage for GUEs, and since the users are uniformly distributed inside
the cluster of radius $r_{max}$, the distance distribution to the reference
user is given by:

$\displaystyle f_{\rm r_{cu}|r_{c}\leq
R^{*}}(r)=\frac{2r}{r_{max}^{2}}\mathds{1}\left(0\leq r\leq r_{max}\right),$
(17)

where $r_{max}$ is the maximal cluster radius.

Case2. Central UAV is placed at a distance $R^{*}$ from its nearest LBD
($r_{c}>R^{*}$)
In the case where being placed above the hotspot center does not guarantee
enough energy level for the drone, the UAV is obliged to be placed at the
distance $R^{*}$ from its nearest LBD.

######  Proposition 1.

For $\alpha\in\mathbb{R}_{+}$, the PDF of the distance between the UAV and a
reference user in the case where $r_{c}>R^{*}$, denoted by $f_{r_{\rm
cu|r_{c}>R^{*}}}(\alpha)$, is given by:

$\begin{dcases}f_{\rm r_{\rm cu}|r_{c}\leq R^{*}}\left(\alpha-
r_{cc}\right)&|\alpha-r_{\rm cc}|\geq r_{max}\\\ \\\ f_{r_{\rm cu}|r_{c}\leq
R^{*}}\left(\alpha-r_{\rm cc}\right)-\frac{2\left(\alpha-r_{\rm
cc}\right)\operatorname{arcos}\left(f\left(|\alpha-r_{\rm
cc}|,\alpha\right)\right)}{\pi r_{max}^{2}}&|\alpha-r_{\rm cc}|<r_{max}\leq
r_{\rm cc}+\alpha\\\ +\int_{|\alpha-r_{\rm cc}|}^{r_{max}}\frac{2\alpha}{\pi
r_{max}^{2}r_{\rm cc}\sqrt{1-f^{2}(r,\alpha)}}dr&\textit{ and }\alpha\neq
r_{\rm cc}\\\ \\\ \int_{0}^{r_{max}}\frac{2\alpha}{\pi r_{max}^{2}r_{\rm
cc}\sqrt{1-f^{2}(r,\alpha)}}dr&|\alpha-r_{\rm cc}|<r_{max}\leq
r_{cc}+\alpha\\\ &\textit{ and }\alpha=r_{\rm cc}\\\ f_{r_{\rm cu}|r_{c}\leq
R^{*}}(\alpha-r_{\rm cc})-\frac{2(\alpha-r_{\rm
cc})\operatorname{arcos}(f(|\alpha-r_{\rm cc}|,\alpha))}{\pi r_{max}}&|\alpha-
r_{\rm cc}|<r_{\rm cc}+\alpha<r_{max}\\\ +\int_{|\alpha-r_{\rm
cc}|}^{\alpha+r_{\rm cc}}\frac{2\alpha}{\pi r_{max}^{2}r_{\rm
cc}\sqrt{1-f^{2}(r,\alpha)}}dr+\frac{2(\alpha+r_{\rm
cc})\operatorname{arcos}(f(\alpha+r_{\rm cc},\alpha))}{\pi
r_{max}^{2}}&\textit{ and }\alpha\neq r_{\rm cc}\\\ \\\
\int_{0}^{\alpha+r_{\rm cc}}\frac{2\alpha}{\pi r_{max}^{2}r_{\rm
cc}\sqrt{1-f^{2}(r,\alpha)}}dr+\frac{2(\alpha+r_{\rm
cc})\operatorname{arcos}(f(\alpha+r_{\rm cc},\alpha))}{\pi
r_{max}^{2}}&\textit{else,}\end{dcases}$

(18)

where $r_{cc}=r_{c}-R^{*}$, and
$f(r,\alpha)=\frac{(r_{c}-R^{*})^{2}+r^{2}-\alpha^{2}}{2(r_{c}-R^{*})r}.$

###### Proof:

See Appendix -B. ∎

######  Remark 1.

To speed up the integration in the simulation, it is important to identify the
support of the PDF. The support of the PDF $f_{r_{\rm cu}}(\alpha)$ is itself
random and depends on the placement strategy $\mathcal{S}$ and consequently on
the random variable $r_{c}$. However, when conditioning on the placement
strategy, the support of the PDF of $r_{\rm cu}$ is given by
$[(r_{c}-R^{*}-r_{\rm max})^{+},r_{\rm max}+(r_{c}-R^{*})^{+}]$, where $x^{+}$
denotes $\max(0,x)$. This can be justified as follows:

  * •

The first case corresponds to placing the UAV above the cluster center
($r_{c}\leq R^{*})$. Based on the properties of MCP the support of the PDF is
$[0,r_{\rm max}]$.

  * •

The second case corresponds to relocating the UAV according to Algorithm 1 and
placing it inside the reference cluster as shown in Figure 3, which is
mathematically explained by ($0<r_{c}-R^{*}\leq r_{\rm max})$. In this case,
we can show that the support of the PDF function is $[0,r_{\rm
max}+r_{c}-R^{*}]$.

  * •

The third case corresponds to relocating the UAVs but placing it outside the
reference cluster i.e. $(r_{\rm max}<r_{c}-R^{*})$. In this case we can show
that the support of the PDF is $[r_{c}-R^{*}-r_{\rm max},r_{\rm
max}+r_{c}-R^{*}]$.

####  IV-A3 Distance to the Nearest Interferes

The reference user associates towards the nearest point from each distribution
since it provides, by definition, the strongest signal. Since we use multiple
distributions and in order to avoid any confusion, we provide a compact form
for the distance to the nearest interferer from each distribution.

######  Lemma 2 (Distance to the nearest interferer).

Suppose that the reference user is connected to the nearest point from a
distribution $\Phi_{j}$, distant by $r\geq 0$. The distance to the nearest
interferer from the distribution $\Phi_{i}$ is given by:

$\displaystyle
f_{j}^{i}(r)=\sqrt{\max\left(0,\left(\frac{\rho_{i}\eta_{i}}{\rho_{j}\eta_{j}}\right)^{\frac{2}{\alpha_{i}}}\left(h_{j}^{2}+r^{2}\right)^{\frac{\alpha_{j}}{\alpha_{i}}}-h_{i}^{2}\right)}\textit{
, }i,j\in\mathcal{C},$ (19)

where $\rho_{i}$ is the transmit power, $\eta_{i}$ is the additional path
loss, $\alpha_{i}$ is the path loss exponent, and $h_{i}$ is the height of any
node from $\Phi_{i}$. Remark that $f_{i}^{i}(r)=r$, $\forall i\in\mathcal{C}$.

###### Proof:

See Appendix -C. ∎

Following the association policy defined later in Section IV-B1, Lemma 2
defines a shield circle of radius $f_{j}^{i}(r)$ around the reference user
that does not contain any interferer from the distribution $\Phi_{i}$. Thus, a
logical consequence is that $f_{j}^{i}(r)$ an increasing function of the
distance to the serving node $r$ from the distribution $\Phi_{j}$.

###  IV-B Association Probabilities

####  IV-B1 Association Policy

We consider domination by the long-term statistics of the channel. The
reference user connects to the UAV or the TBS providing the strongest average
received power. Consequently, provided that the user is connected to a
distribution $\Phi_{i}$ then the serving entity is the nearest node from
$\Phi_{i}$ to the reference user.

####  IV-B2 Association Probability to the Central UAV

Let $A_{c}(r,Lu)$ denote the association probability to the central UAV given
that it is in LOS condition with the user and distant by r. In contrast,
$A_{c}(r,Nu)$ refers to the case of NLOS condition with the reference user.

######  Lemma 3 (Central association probability).

The expression of $A_{c}(r,i)$, $i$ $\in$ $\\{Lu,Nu\\}$, is given by:

$\displaystyle
A_{c}(r,i)=\prod_{j\in\mathcal{C}}\exp\left(-2\;\pi\;\int_{0}^{f_{i}^{j}(r)}\lambda_{j}(x)\;x\;dx\right),$
(20)

where $f_{i}^{j}(r)$ is given by Lemma 2 and $\lambda_{i}$ is the density of
the point process $\Phi_{i}$.

###### Proof:

See Appendix -D. ∎

####  IV-B3 Association Probability to a Non-Central UAV

Let $A(r,i,\mathcal{S})$ denotes the association probability to the nearest
neighbor, distant by $r\geq 0$, from the distribution $\Phi_{i}$ such that
$\mathcal{S}=``r_{c}\leq R^{*}"$ refers to the case where the UAV is placed
above the cluster center of the reference user and
$\mathcal{S}=``r_{c}>R^{*}"$ refers to the case where the UAV is relocated
according to Algorithm 1.

######  Lemma 4 (Non-central association probability).

For $i\in\mathcal{C}$, the expression of $A(r,i,\mathcal{S})$ is given by :

$\displaystyle
A(r,i,\mathcal{S})=\overline{A}(r,i,\mathcal{S})\times\prod_{j\in\mathcal{C}\setminus
i}\exp\left(-2\;\pi\;\int_{0}^{f_{i}^{j}(r)}\lambda_{j}(x)\;x\;dx\right),$
(21)

where $\overline{A}(r,i,\mathcal{S})$ is the probability that the serving
entity from $\Phi_{i}$ is providing stronger signal than the central UAV. The
expression of $\overline{A}(r,i,\mathcal{S})$ is given by :

$\displaystyle\begin{split}&\overline{A}(r,i,\mathcal{S})=\int_{f_{i}^{\rm
Lu}(r)}^{\infty}f_{r_{\rm cu}|\mathcal{S}}(x)\;\mathcal{P}_{\rm
Lu}(x)\;dx+\int_{f_{i}^{\rm Nu}(r)}^{\infty}f_{\rm r_{\rm
cu}|\mathcal{S}}(x)\;\mathcal{P}_{\rm Nu}(x)\;dx.\end{split}$ (22)

###### Proof:

See Appendix -E. ∎

###  IV-C Laplace Transform of Interference

In this subsection, the Laplace transform of interference is derived. To this
end, several scenarios are possible depending on what node the reference user
is connected to.

####  IV-C1 Reference user connected to the central UAV

Let $\mathcal{L}_{I}^{c}(s,i,r)$ denotes the Laplace transform of the
interference experienced at the reference user when it is connected to its
central UAV distant by $r$ and conditioned on the placement strategy
$\mathcal{S}$.

######  Lemma 5 (Central Laplace interference).

The Laplace transform of interference in the case where the UAV is connected
to the central UAV, distant by $r\geq 0$, conditioned on the channel type
$i\in\\{Lu,Nu\\}$, is given by:

$\displaystyle\mathcal{L}_{I}^{c}(s,i,r)=\prod_{j\in\mathcal{C}}\exp\left(-2\pi\int_{f_{i}^{j}(r)}^{\infty}\left(1-\left(\frac{m_{j}}{m_{j}+s\rho_{j}\eta_{j}(x^{2}+h_{j}^{2})^{-\frac{\alpha_{j}}{2}}}\right)^{m_{j}}\right)\lambda_{j}(x)xdx\right).$
(23)

The Laplace transform of the interference is used later to derive the coverage
probability. It is a decreasing function of the distance to the serving node
$r$, which means that the interference $I$ is increasing with $r$, by
definition of the Laplace transform, i.e.
$\mathcal{L}_{I}^{c}(s,i,r)=\mathbb{E}(\exp\left(-sI\right))$ . This is a
logical consequence since when the connected node is far then a large shield
circle radius $f_{j}^{i}(r)$ is imposed around the reference user.

###### Proof:

See Appendix -F. ∎

####  IV-C2 Reference user not-connected to the central UAV

Suppose that the user is connected to the nearest point from the distribution
$\Phi_{i}$, distant by $r_{i}$, and different from the central UAV. In the
following lemma, we derive the Laplace transform of the interference generated
by the central UAV.

######  Lemma 6 (Non-central Laplace interference).

The Laplace interference generated by the central UAV conditioned on the
distance to the serving entity $r_{i}$ and the placement strategy
$\mathcal{S}$ is given by:

$\displaystyle\mathcal{L}_{I}(s,i,r,\mathcal{S})=\overline{\mathcal{L}_{I}}(s,i,r,\mathcal{S})\times\prod_{j\in\mathcal{C}\setminus
i}\exp\left(-2\pi\int_{f_{i}^{j}(r)}^{\infty}1-\left(\frac{m_{j}}{m_{j}+s\rho_{j}\eta_{j}(x^{2}+h_{j}^{2})^{-\frac{\alpha_{j}}{2}}}\right)^{m_{j}}\lambda_{j}(x)xdx\right),$
(24)

where $\overline{\mathcal{L}_{I}}(s,r,i,\mathcal{S})$ is the Laplace transform
of interference generated by the central UAV conditioned on the distance to
the serving entity $r$ and the placement strategy $\mathcal{S}$:

$\displaystyle\overline{\mathcal{L}_{I}}(s,r,i,\mathcal{S})=\sum_{j\in\\{Lu,Nu\\}}\int_{f_{i}^{j}(r)}^{\infty}\left(\frac{m_{j}}{m_{j}+s\rho_{j}\eta_{j}(z^{2}+h^{2})^{-\frac{\alpha_{j}}{2}}}\right)^{m_{j}}\frac{\mathcal{P}_{j}(z)f_{r_{\rm
cu}|\mathcal{S}}(z)}{\int_{f_{i}^{j}(r_{i})}^{\infty}f_{r_{\rm
cu}|\mathcal{S}}(x)dx}dz.$ (25)

###### Proof:

See Appendix -G. ∎

###  IV-D Coverage probability

In this section, we derive the analytical expression of the coverage
probability $C(\gamma)$, defined as the probability that the received signal
at the user is below a certain threshold $\gamma$.

######  Proposition 2.

The coverage probability $C(\gamma)$ is given by:

$\displaystyle\begin{split}C(\gamma)=\int_{0}^{\infty}\Big{(}\sum\limits_{i\in\mathcal{C}}\mathds{E}_{\mathcal{S}}\\{L(r,i,\mathcal{S})A(r,i,\mathcal{S})\\}f_{R_{i}}(r)+\sum_{i\in\\{\rm
Lu,Nu\\}}L_{c}(r,i)A_{c}(r,i)\mathcal{P}_{i}(r)\mathds{E}_{\mathcal{S}}\\{f_{r_{cu|S}}(r)\\}\Big{)}dr,\end{split}$
(26)

where the expectation over the placement strategy $\mathcal{S}$ are given by:

$\displaystyle\mathds{E}_{\mathcal{S}}\\{L(r,i,\mathcal{S})A(r,i,\mathcal{S})\\}=L(r,i,r_{c}\leq
R^{*})A(r,i,r_{c}\leq
R^{*})\left(1-\exp\left(-\lambda_{LBD}\pi{R^{*}}^{2}\right)\right)$ (27)
$\displaystyle+\int_{R^{*}}^{\infty}L(r,i,r_{c}>R^{*})A(r,i,r_{c}>R^{*})f_{\rm
R_{c}}(r_{c})dr_{c},$
$\displaystyle\mathds{E}_{\mathcal{S}}\\{f_{r_{cu|S}}(r)\\}=\int_{R^{*}}^{\infty}f_{r_{cu}|rc>R^{*}}(r)f_{R_{c}}(r_{c})dr_{c}+\left(1-\exp\left(-\lambda_{LBD}\pi{R^{*}}^{2}\right)\right)f_{r_{cu}|rc\leq
R^{*}}(r).$ (28)

Moreover, $L(r,i,\mathcal{S})$ is given by:

$\displaystyle
L(r,i,\mathcal{S})=\sum_{k=0}^{m_{i}-1}(-1)^{k}\frac{s^{k}}{k!}\frac{d^{k}}{ds^{k}}\mathcal{L}_{I+\sigma^{2}}(s_{r},i,r,\mathcal{S}),$
(29)

whereas $L_{c}(r,i)$ is given by:

$\displaystyle
L_{c}(r,i)=\sum_{k=0}^{m_{i}-1}(-1)^{k}\frac{s^{k}}{k!}\frac{d^{k}}{ds^{k}}\mathcal{L}_{I+\sigma^{2}}^{c}(s_{r},i,r),$
(30)

and finally $s_{r}$ is given by:

$\displaystyle s_{r}=\frac{\gamma
m_{i}(h_{i}^{2}+r^{2})^{\alpha_{i}/2}}{\rho_{i}\eta_{i}}.$ (31)

###### Proof:

See Appendix -H. ∎

##  V Numerical Results

TABLE II: System parameters [19, 26]

Parameter | Value | Parameter | Value
---|---|---|---
Spatial distribution | $\Delta f$ | 1 Ghz
Simulation area | 900 km2 | $k$ |  $5.92\times 10^{6}$ m-1
$\lambda_{\rm LBD}$ | $0.52\times 10^{-6}$ | $\alpha$ |  $10^{-6}$ m
$\lambda_{\rm cc}$ | $3\times 10^{-6}$ | $\delta_{s}$ | $10^{-5}$
$\lambda_{\rm TBS}$ | $2\times 10^{-6}$ | $p_{\rm trans}$ | 2 kW
Environmental parameters | $p_{\rm prop}$ | 100 W
$\mathcal{B}$ | 90% | $p_{comm}$ | 2 W
$a$ | 1 | Other parameters
$c$ | 1 | $\eta_{\rm Lu}=\eta_{\rm Lb}$ | 0.9
$b_{\rm SubUrban}$ | 6.581 | $\eta_{\rm Nu}=\eta_{\rm Nb}$ | 0.7
$b_{\rm Urban}$ | 0.151 | $\rho_{\rm Lu}=\rho_{\rm Nu}$ | 1 W
$b_{\rm denseUrban}$ | 0.101 | $\rho_{\rm Lb}=\rho_{\rm Nb}$ | 30 W
Laser-link parameters | $\alpha_{\rm Lu}=\alpha_{\rm Lb}$ | 2
$D$ | 0.1 m | $\alpha_{\rm Nu}=\alpha_{\rm Nb}$ | 3
$\Delta\theta$ | $3.4\times 10^{-5}$ | $h_{\rm Lb}=h_{\rm Nb}$ | 25 m
$wT\chi$ | 0.004 m2 | $h_{\rm Lu}=h_{\rm Nu}$ | 100 m
$C_{n}^{2}(n)$ | $0.5\times 10^{-14}$ | $m_{\rm Lu}=m_{\rm Lb}$ | 1
$\hbar$ |  $6.63\times 10^{-34}$ m2 kg s-1 | $m_{\rm Nu}=m_{\rm Nb}$ | 3

In this section, we validate the theoretical results we have proposed in this
paper through Monte Carlo simulations. We analyze various figures to reveal
some relevant trends related to the setup we have suggested. Finally, we
provide high-level insights about the system we proposed for future practical
deployment of laser-powered drones. In all of the figures, we use markers for
the simulation results whereas solid lines are used to represent the analytic
expressions.

Figure 5: Conditional association probability to the central UAV as a function
of the serving distance.

In Figure 5, we plot the association probability conditioned on the distance
to the reference user in different environments. The figure shows that the
probability to connect to the central UAV in dense urban environments is
significantly higher than in suburban environments. This means that in dense
urban environments, since links between drones and the reference user are
often NLOS, as long as a LOS drone is placed close to the reference user there
is a high chance to be the serving drone, this is not the case for suburban
environments since links are probably LOS and the probability of having
another LOS drone providing a stronger signal is not negligible. The same
figure brings to light the effect of the link type on the association
probability since the association to an NLOS central drone, plotted with
dashed lines, is lower than the association probability to a central drone in
LOS condition with the reference user, showing how much building blockage
could affect the system setup.

Figure 6: Coverage Probability as a function of the SINR threshold.

Figure 6 shows the coverage probability as a function of the signal-to-
interference-plus-noise ratio (SINR) threshold $\gamma$ in different type of
environments. It shows that coverage in dense urban environments is higher
than coverage in urban and suburban environments. The degradation of wireless
coverage is a logical consequence of the significant interference generated by
all the nodes in the system, especially when most of them are in LOS condition
with the user. In other words, it is a confirmation of Figure 8, but from a
different angle.

Figure 7: Joint energy and communication coverage as function of the distance
to the drone.

Figure 7 shows the joint energy and communication coverage probability as a
function of the distance to the drone R under different turbulence conditions.
The figure is informative regarding the charging requirement for the drones as
the coverage probability decreases when increasing the critical charging
distance, in other words in strong turbulent environments. Moreover, we can
state that reasonable ranges for the critical charging distance are [1000
1300] meters for energy and communication coverage threshold
$\mathcal{B}=90\%$ . The critical charging distance is also related to the
energy coverage and communication threshold $\mathcal{B}$, since it decreases
when targeting highly reliable charging and backhaul link for the drones when
increasing $\mathcal{B}$.

(a) Environment effect on the connectivity profile.

(b) Effect of the critical charging distance $R^{*}$.

Figure 8: Average user connectivity profile.

Figure 8 shows the connectivity diagram of the user in different types of
environments and in different laser charging coverage. The purpose of this
figure is to answer the question to what extent the UAV network could support
the terrestrial architecture. We distinguish 3 different connection modes: (1)
connection to the central UAV, (2) connection to any other UAV, and (3)
connection to a TBS. As we consider LOS and NLOS links, the number of
connection modes increases to 6 accordingly. The trends in Figure 8(a) show a
considerable tendency to connect to terrestrial architecture in dense urban
environments. This is due to the fact that LOS links to UAVs are largely
affected by building blockage. In addition, the transmit power of UAVs (1W) is
remarkably lower than the transmit power of TBSs (30W), which explains why
links to UAVs are very sensitive to LOS probability. That said, in urban and
suburban environments, the drone network successfully assists the TBS by
providing stronger signals to users. We note that the connection rate to
airborne base stations is 7.4% for in dense urban environments, 66.4% in urban
environments and increases to 73.5% in suburban environments. In a nutshell,
this confirms that UAVs must be deployed in LOS conditions to be effective in
wireless coverage scenarios, i.e. in urban to suburban environments. Figure
8(b) captures the effect of the maximal laser charging radius on the user
connectivity. We can easily notice that under poor laser charging conditions,
i.e under low critical charging distance $R^{*}=800m$, the association rate to
the central UAV is 33.8%. However, under better laser critical charging
distance, $R^{*}=1300m$, the association probability to the central drone
increases to 47.16%. This is the consequence of expanding the drones’
flexibility, allowing them to be positioned close to user clusters. At the
same time, the association rate to the TBSs decreases by 11% when increasing
the critical charging distance, as the performance of the drones is severely
affected.

Figure 9: UAV altitude effect on the coverage probability.

In Figure 9, we plot the coverage probability as a function of drone altitude
for different building densities. This figure shows how drone altitudes can
affect the users’ quality of service in certain scenarios. With the current
system configuration, we can see that the optimal drone placement is at 50 m
altitude in a suburban environment and at 80 m altitude in an urban
environment. The same figure confirms the results observed in Figure 8 where
the coverage probability is not affected by the altitudes of the UAVs in dense
urban environments because the probability of association to the aerial
network is low. This problem can be solved by optimizing the altitude of each
drone individually, which is not feasible in our system model.

Figure 10: Optical turbulence effect on the coverage probability for different
energy coverage ($\mathcal{B}$) levels.

Figure 10 presents the turbulence effect on the coverage probability for
different levels of the drone energy reliability, denoted by $\mathcal{B}$.
This figure is equivalent to plotting against the critical charging distance
$R^{*}$ since there is a direct relation between $R^{*}$ and $\mathcal{B}$.
The figure shows a degradation in the coverage probability for moderate-to-
strong optical turbulence and relatively constant energy coverage probability
for weak optical turbulence ($C_{n}^{2}=10^{-15}$). We can also remark that
coverage probability at the user is affected by the energy coverage
reliability level. For instance, the coverage probability is better for low
joint energy and communication coverage levels. We can also notice that the
gap induced by the drone energy coverage increases in moderate to strong
optical turbulence. The aforementioned observations could be explained as
follows; (1) when increasing the energy coverage threshold $\mathcal{B}$ the
critical charging distance $R^{*}$ decreases to meet this requirement.
Consequently, this fact induces shifting the UAV more close to their nearest
LBDs instead of being positioned above the cluster center, thus wireless
coverage is sacrificed for better energy coverage. (2) The same phenomenon
happens when increasing the refractive index, i.e increasing the optical
turbulence effect, as the critical charging distance decreases in such
environments. In a nutshell, a trade-off between energy and wireless coverage
is visible in all the figures we have presented as a logical consequence of
the placement strategy we have proposed. The best-case scenario corresponds to
the placement of the drone above the cluster centers while maintaining energy
harvesting, which yields the best average performance. However, in the worst
case, drones are attracted to the energy sources, and therefore users tend to
connect to the available terrestrial network architecture.

##  VI Conclusion

In this paper, we have presented a comprehensive coverage analysis for
assisting a terrestrial network architecture through a laser-powered drone
network. We used stochastic geometry to model the system and proposed a novel
deployment strategy for the laser-powered UAVs that ensures concomitantly
energy and wireless coverage for GUEs. As proposed in our previous work, we
assume simultaneous communication and energy harvesting at the drone, enabling
a reliable backhaul link. Within this framework, we shed light on an
interesting trade-off between energy reliability and coverage efficiency. We
have also showed how optical turbulence badly affect the drone’s wireless
charging and consequently results in a degradation of the user’s coverage
probability. In the light of these analyses, we believe that laser-powered
UAVs strongly compete with tethered, solar-powered, and untethered UAVs and
could offer interesting alternatives for future wireless communication
networks.

###  -A Proof of Lemma 1

To start with, we have previously demonstrated in [19] that the joint energy
and communication coverage probability is dominated by the energy coverage
probability, and thus to derive the critical charging distance, we rely on
lower bounding the energy coverage probability defined in the same work. Let
us denote the lower bound on the energy coverage probability by
$\mathcal{B}\in[0\;1]$:

$\displaystyle P_{\rm energy}$ $\displaystyle=\mathbb{P}\left(h_{t}p_{\rm
harv}(r)\geq p_{prop}+p_{comm}\right)\geq\mathcal{B}.$
$\displaystyle\Rightarrow$ $\displaystyle\ \mathbb{P}(h_{t}\geq\frac{p_{\rm
prop}+p_{\rm comm}}{p_{\rm harv}(r)})\geq\mathcal{B}.$
$\displaystyle\Rightarrow$ $\displaystyle\ 1-F_{h_{t}}(\frac{p_{\rm
prop}+p_{\rm comm}}{p_{\rm harv}(r)})\geq\mathcal{B}.$
$\displaystyle\overset{(a)}{\Rightarrow}$ $\displaystyle\
\frac{1}{2}-\frac{1}{2}\operatorname{erf}\left(\frac{\ln(\frac{p_{\rm
prop}+p_{\rm comm}}{p_{\rm
harv}(r)})+2\sigma}{\sqrt{2}\sigma}\right)\geq\mathcal{B}.$
$\displaystyle\overset{(b)}{\Rightarrow}$ $\displaystyle\
\frac{\ln(\frac{p_{\rm prop}+p_{\rm comm}}{p_{\rm
harv}(r)})+2\sigma}{\sqrt{2}\sigma}\leq\operatorname{erf}^{-1}\left(1-2\mathcal{B}\right).$
(32)

where (a) is due to the fact that the turbulence effect is following a
lognormal distribution and (b) is due to the fact that
$\operatorname{erf}^{-1}$ is an increasing function. Based on the inequality
(32), we conclude that $R^{*}$ should satisfy (11) provided in Lemma 1.

###  -B Proof of Proposition 1

To derive the PDF of the distance between the reference user and the central
UAV, denoted by $r_{cu}$, in the case of $\mathcal{S}=``r_{c}>R^{*}"$, we
start by deriving $F_{r_{cu}|r_{c}>R^{*}}$. Let $\alpha\in\mathbb{R}_{+}$, we
can write the following:

$\displaystyle F_{r_{cu}|r_{c}>R^{*}}(\alpha)$
$\displaystyle=\mathbb{P}(r_{\rm cu}\leq\alpha|r_{c}>R^{*})=\mathbb{P}(r_{\rm
cu}^{2}\leq\alpha^{2}|r_{c}>R^{*}).$ (33)

We exploit the cosine rule applied to the colored triangle shown is Figure 2.
Notice that $r$, $d$, and $\theta$ are random variables, but, since we
conditioned on $r_{c}$ then $r_{cc}$ is also fixed due the fact that
$r_{cc}=r_{c}-R^{*}$.

$\displaystyle=\mathbb{E}_{\rm r}\left(\mathbb{P}\left(r_{\rm
cc}^{2}+r^{2}-2rr_{\rm
cc}cos(\theta)\leq\alpha^{2}\;\Big{|}\;r_{c}>R^{*},r\right)\right),$
$\displaystyle=\mathbb{E}_{r}\left(\mathbb{P}\left(\cos(\theta)\geq\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm
cc}}\;\Big{|}\;r_{c}>R^{*},r\right)\right),$ $\displaystyle=\mathbb{E}_{\rm
r}\left(\mathbb{P}\left(\cos(\theta)\geq\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm cc}}\big{|}r_{c}>R^{*},r\right)\right).$
(34)

Now using the fact that $\arccos$ is a decreasing function defined on
$[-1\;\;1]$ and that $\theta$ is uniformly distributed across $[0\;\;2\pi]$,
we can further develop as follows:

$\displaystyle=\mathbb{E}_{\rm
r}\Bigg{(}\frac{\operatorname{arcos}\left(\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm
cc}}\right)}{\pi}\mathds{1}\Big{\\{}\Big{|}\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm cc}}\Big{|}\leq
1\Big{\\}}+\mathds{1}\Big{\\{}\frac{r_{\rm cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm
cc}}<-1\Big{\\}}$ $\displaystyle+0\times\mathds{1}\Big{\\{}\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm cc}}>1\Big{\\}}\Bigg{)},$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{\rm
r}\left(\frac{\operatorname{arcos}\left(\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm
cc}}\right)}{\pi}\mathds{1}\Big{\\{}|\alpha-r_{\rm cc}|\leq r\leq r_{\rm
cc}+\alpha\Big{\\}}\right)+\mathbb{E}_{r}\Bigg{(}\mathds{1}\Big{\\{}r<\alpha-
r_{\rm cc}\Big{\\}}\Bigg{)}.$ (35)

At this stage, we average over the random variable $r$ taking into account
that r is derived form the distance of one point and the cluster center from a
MCP. Thus we can write the following:

$\displaystyle=\int_{[0,r_{max}]\cap[|\alpha-r_{\rm cc}|,r_{\rm
cc}+\alpha]}\frac{2r}{\pi r_{max}^{2}}\operatorname{arcos}\left(\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2r_{\rm cc}r}\right)dr+\frac{{(\alpha-r_{\rm
cc})}^{2}}{r_{max}^{2}}\mathds{1}\Big{\\{}0\leq\alpha-r_{\rm cc}\leq
r_{max}\Big{\\}}.$ (36)

In what follows, we explain how (a) is derived. Notice that $r$, $r_{cc}$, and
$\alpha$ are positive:

$\displaystyle\mathds{1}\Big{\\{}\Big{|}\frac{r_{\rm
cc}^{2}+r^{2}-\alpha^{2}}{2rr_{\rm cc}}\Big{|}\leq
1\Big{\\}}=\mathds{1}\Big{\\{}r_{\rm cc}^{2}+r^{2}-\alpha^{2}\leq 2rr_{\rm
cc}\Big{\\}}\times\mathds{1}\Big{\\{}r_{\rm
cc}^{2}+r^{2}-\alpha^{2}\geq-2rr_{\rm cc}\Big{\\}},$
$\displaystyle=\mathds{1}\Big{\\{}(r_{\rm
cc}-r)^{2}\leq\alpha^{2}\Big{\\}}\mathds{1}\Big{\\{}(r_{\rm
cc}+r)^{2}\geq\alpha^{2}\Big{\\}},$ $\displaystyle=\mathds{1}\Big{\\{}|r_{\rm
cc}-r|\leq\alpha\Big{\\}}\mathds{1}\Big{\\{}|r_{\rm
cc}+r|\geq\alpha\Big{\\}},$ $\displaystyle=\mathds{1}\Big{\\{}r_{\rm
cc}-\alpha\leq r\leq r_{\rm cc}+\alpha\Big{\\}}\mathds{1}\Big{\\{}r\geq\alpha-
r_{\rm cc}\Big{\\}},$ $\displaystyle=\mathds{1}\Big{\\{}|\alpha-r_{\rm
cc}|\leq r\leq r_{\rm cc}+\alpha\Big{\\}}.$ (37)

Now that we derived the CDF of $r_{cu}$, we obtain the PDF expression by
deriving the CDF with respect to $\alpha$, to this end, we simply apply the
Leibniz integral rule to the CDF to get the expression described in
Proposition 1. Different cases should be distinguished depending on the result
of $[0,r_{max}]\cap[|\alpha-r_{\rm cc}|,r_{\rm cc}+\alpha]$.

###  -C Proof of Lemma 2

Let $r_{i}$ be the distance from the reference user to its nearest neighbor
from a point process $\Phi_{i}$. Suppose that the reference user is connected
to the nearest neighbor from the point process $\Phi_{j}$, distant by $r_{j}$.
According to the association policy that implies connection to the strongest
signal, the nearest interferer from $\Phi_{i}$ to the reference user must
satisfy the following equation:

$\displaystyle\rho_{j}\;\eta_{j}\;(h_{j}^{2}+r_{j}^{2})^{-\frac{\alpha_{j}}{2}}\geq\rho_{i}\;\eta_{i}\;(h_{i}^{2}+r_{i}^{2})^{-\frac{\alpha_{i}}{2}},$
$\displaystyle\left(\frac{\rho_{i}\;\eta_{i}\;}{\rho_{j}\;\eta_{j}}\right)^{\frac{2}{\alpha_{i}}}\;(h_{j}^{2}+r_{j}^{2})^{\frac{\alpha_{j}}{\alpha_{i}}}-h_{i}^{2}\leq
r_{i}^{2},$
$\displaystyle\sqrt{\max\left(0,\left(\frac{\rho_{i}\;\eta_{i}\;}{\rho_{j}\;\eta_{j}}\right)^{\frac{2}{\alpha_{i}}}\;(h_{j}^{2}+r_{j}^{2})^{\frac{\alpha_{j}}{\alpha_{i}}}-h_{i}^{2}\right)}\leq
r_{i}.$ (38)

Hence, conditioned on the distance to the serving node $r$, the nearest
interferer distance is provided by:

$\displaystyle\begin{split}f_{j}^{i}(r)=\sqrt{\max\left(0,(\frac{\rho_{i}\eta_{i}}{\rho_{j}\eta_{j}})^{\frac{2}{\alpha_{i}}}(h_{j}^{2}+r^{2})^{\frac{\alpha_{j}}{\alpha_{i}}}-h_{i}^{2}\right)},i,j\in\\{\rm
Lu,Nu,Lb,Nb\\}.\end{split}$ (39)

###  -D Proof of Lemma 3

Suppose that the central UAV is located at a distance $r\geq 0$ with channel
type $i$, where $i$ could be either LOS $(i=Lu)$, or NLOS $(i=Nu)$. Then,
based on the association policy defined in Sec. IV-B1, the association
probability to the central UAV is given by:

$\displaystyle
A_{c}(r,i)=\mathbb{P}\left(\overline{\mathcal{R}}_{i}(r)\geq\max\limits_{j\in\mathcal{C}}\left(\overline{\mathcal{R}}_{j}(r_{j})\right)\right),$
$\displaystyle=\prod_{j\in\mathcal{C}}\mathbb{P}\Bigg{(}\overline{\mathcal{R}}_{i}(r)\geq\overline{\mathcal{R}}_{j}(r_{j})\Bigg{)},$
$\displaystyle=\prod_{j\in\mathcal{C}}\mathbb{P}\Bigg{(}r_{j}\geq
f_{i}^{j}(r)\Bigg{)}.$ (40)

Since $r_{j}$ is the nearest neighbor distance from the PPP $\Phi_{j}$ with
density $\lambda_{j}$, we can write the following:

$\displaystyle
A_{c}(r,i)=\prod_{j\in\mathcal{C}}\exp\left(-2\;\pi\;\int_{0}^{f_{i}^{j}(r)}\lambda_{j}(x)\;x\;dx\right).$
(41)

###  -E Proof of Lemma 4

Let $A(r,i,\mathcal{S})$ denotes the association probability to the nearest
neighbor, distant by $r\geq 0$, from the distribution $\Phi_{i}$, and
$\mathcal{S}$ ={ “$r_{c}\leq R^{*}$”,“$r_{c}>R^{*}$” } denotes the deployment
strategy specified in Algorithm 1. Accordingly, for $i\in\\{Lu,Nu,Lb,Nb\\}$,
we can write the following:

$\displaystyle
A(r,i,\mathcal{S})=\overline{A}(r,i,\mathcal{S})\mathbb{P}\Bigg{(}\overline{\mathcal{R}}_{i}(r)\geq\max\limits_{j\in\mathcal{C}\setminus
i}\Big{(}\overline{\mathcal{R}}_{j}(r_{j})\Big{)}\Bigg{)},$
$\displaystyle=\overline{A}(r,i,\mathcal{S})\prod_{j\in\mathcal{C}\setminus
i}\mathbb{P}\Bigg{(}\overline{\mathcal{R}}_{i}(r)\geq\overline{\mathcal{R}}_{j}(r_{j})\Bigg{)},$
$\displaystyle=\overline{A}(r,i,\mathcal{S})\prod_{j\in\mathcal{C}\setminus
i}\mathbb{P}\Bigg{(}r_{j}\geq f_{i}^{j}(r)\Bigg{)},$ $\displaystyle=$
$\displaystyle\overline{A}(r,i,\mathcal{S})\prod_{j\in\mathcal{C}\setminus
i}\exp\left(-2\pi\int_{0}^{f_{i}^{j}(r)}\lambda_{j}(x)x\;dx\right),$ (42)

where $\overline{A}(r,i,\mathcal{S})$ is the probability that the signal from
the serving entity from $\Phi_{i}$ is stronger than the signal from the
central UAV, given that the serving point is at a distance r from the
reference user and conditioned on the placement of the central UAV. Let
$r_{cu}$ be the distance from the reference user to the central UAV. The
expression $\overline{A}(r,i,\mathcal{S})$ is given by:

$\displaystyle\overline{A}(r,i,\mathcal{S})=\mathbb{P}\Bigg{(}\overline{\mathcal{R}}_{\rm
Lu}(r_{\rm cu})\leq\overline{\mathcal{R}}_{\rm i}(r_{\rm
i}),LOS\Bigg{)}+\mathbb{P}\Bigg{(}\ \overline{\mathcal{R}}_{\rm Nu}(r_{\rm
cu})\leq\overline{\mathcal{R}}_{\rm i}(r_{\rm i}),NLOS\Bigg{)},$
$\displaystyle=\mathbb{P}\Bigg{(}r_{\rm cu}\geq f_{i}^{\rm
Lu}(r),LOS\Bigg{)}+\mathbb{P}\Bigg{(}r_{\rm cu}\geq f_{i}^{\rm
Nu}(r),NLOS\Bigg{)},$ $\displaystyle=\int_{f_{i}^{\rm
Lu}(r)}^{\infty}f_{r_{\rm cu}|\mathcal{S}}(r)\;\mathcal{P}_{\rm
Lu}(r)\;dr+\int_{f_{i}^{\rm Nu}(r)}^{\infty}f_{r_{\rm
cu}|\mathcal{S}}(r)\;\mathcal{P}_{\rm Nu}(r)\;dr.$ (43)

###  -F Proof of Lemma 5

Suppose that the reference user is connected to its central UAV. Let $I_{i}$,
$i\in\mathcal{C}$ denotes the interference induced by the point process
$\Phi_{i}$. Let $I_{c}$ denotes the interference generated by the central UAV.
Since the UAV is connected the central UAV placed at $x_{c}$, $I_{c}=0$, and
hence, $\mathcal{L}_{I_{c}}(s)=1$. Let $\mathcal{L}^{c}_{I}(s,i,r)$ denotes
the Laplace interference given that the reference user is connected to the
central UAV with channel type $i\in\\{Lu,Nu\\}$, and $I$ denotes the total
interference:

$\displaystyle I$ $\displaystyle=\sum_{j\in\mathcal{C}}I_{j},$ $\displaystyle
I$ $\displaystyle=\sum_{j\in\mathcal{C}}\sum_{x_{i}\in\Phi_{j}\setminus
x_{c}}\mathcal{R}_{\rm j}(r(\ x_{i})).$ (44)

Hence, the Laplace transform of interference is given by:

$\displaystyle\mathcal{L}_{I}^{c}(s,i,r)=\mathbb{E}(\exp\left(-sI\right)),$
$\displaystyle=\prod_{j\in\mathcal{C}}\mathbb{E}\left(\exp\left(-s\sum_{x_{i}\in\Phi_{j}\setminus
x_{c}}\mathcal{R}_{j}(r(x_{i}))\right)\right),$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\prod_{j\in\mathcal{C}}\exp\left(-2\pi\int_{f_{i}^{j}(r)}^{\infty}\left(1-\mathbb{E}_{G_{j}}\left\\{\exp\left(-s\mathcal{R}_{j}(x)\right)\right\\}\right)\lambda_{j}(x)xdx\right),$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\prod_{j\in\mathcal{C}}\exp\left(-2\pi\int_{f_{i}^{j}(r)}^{\infty}\left(1-\left(\frac{m_{j}}{m_{j}+s\overline{\mathcal{R}}_{\rm
j}(x)}\right)^{m_{j}}\right)\lambda_{j}(x)xdx\right),$ (45)

where (a) is performed by applying the probability generating functional of a
process [37] and (b) is due to the fact that the $G_{j}$ is gamma distributed
with channel-dependent fading parameters $m_{j}$.

###  -G Proof of Lemma 6

Suppose that the UAV is not connected to the central UAV, placed at $x_{c}$,
with channel type $j$, instead it is connected to the nearest point from the
distribution $\Phi_{i}$, distant by $r_{i}$, and placed at $x_{i}$. Suppose
also that the placement strategy $\mathcal{S}$ is used. The generated
interference expression is given by:

$\displaystyle I$ $\displaystyle=I_{c}+\sum_{j\in\mathcal{C}\setminus
i}I_{j},$ (46)

where $I_{c}$ is the interference generated by the central UAV, and $I_{j}$ is
the interference generated by the other point porcesses except the point
process $\Phi_{i}$ since $I_{i}=0$. Hence, based on the proof in Appendix. -E,
we can write the Laplace transform of interference as follows:

$\displaystyle\mathcal{L}_{I}(s,r,i,\mathcal{S})=\overline{\mathcal{L}_{I}}(s,r,i,\mathcal{S})\times$
$\displaystyle\prod_{j\in\mathcal{C}\setminus
i}\exp\left(-2\pi\int_{f_{i}^{j}(r)}^{\infty}1-\Big{(}\frac{m_{j}}{m_{j}+s\overline{\mathcal{R}}_{\rm
j}(x)}\Big{)}^{m_{j}}\lambda_{j}(x)xdx\right),$ (48)

where the Laplace transform of the interference generated by the central UAV,
$\overline{\mathcal{L}_{I}}(s,r,i,\mathcal{S})$, is given by:

$\displaystyle\overline{\mathcal{L}_{I}}(s,r,i,\mathcal{S})=\mathbb{E}(exp(-s\;I_{c})),$
$\displaystyle=\mathbb{E}_{r(x_{c})}\Big{(}\mathbb{E}_{j,G_{j}}(exp(-s\;\mathcal{R}_{\rm
j}(r(x_{\rm c})))\Big{)},$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\sum_{j\in\\{Lu,Nu\\}}\int\Big{(}\frac{m_{j}}{m_{j}+s\overline{\mathcal{R}}_{\rm
j}(z)}\Big{)}^{m_{j}}\mathbb{P}\left(r(x_{c})=z,j|\mathcal{S},\overline{\mathcal{R}}_{\rm
j}(r(x_{\rm c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm i}))\right)dz,$
$\displaystyle=\sum_{j\in\\{Lu,Nu\\}}\int\Big{(}\frac{m_{j}}{m_{j}+s\overline{\mathcal{R}}_{\rm
j}(z)}\Big{)}^{m_{j}}\frac{\mathbb{P}(\overline{\mathcal{R}}_{\rm j}(r(x_{\rm
c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i})),r(x_{c})=z,j|\mathcal{S})}{\mathbb{P}(\overline{\mathcal{R}}_{\rm
j}(r(x_{\rm c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i}))|\mathcal{S})}dz,$
$\displaystyle=\sum_{j\in\\{Lu,Nu\\}}\int\Big{(}\frac{m_{j}}{m_{j}+s\overline{\mathcal{R}}_{\rm
j}(z)}\Big{)}^{m_{j}}\frac{\mathbb{P}(\overline{\mathcal{R}}_{\rm j}(r(x_{\rm
c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i})),r(x_{c})=z,j|\mathcal{S})}{\mathbb{P}(\overline{\mathcal{R}}_{\rm
j}(r(x_{\rm c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i}))|\mathcal{S})}dz,$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\sum_{j\in\\{Lu,Nu\\}}\int_{f_{i}^{j}(r)}^{\infty}\Big{(}\frac{m_{j}}{m_{j}+s\overline{\mathcal{R}}_{\rm
j}(z)}\Big{)}^{m_{j}}\frac{\mathcal{P}_{\rm j}(z)f_{r_{\rm
cu}|\mathcal{S}}(z)}{\int_{f_{i}^{j}(r)}^{\infty}f_{r_{\rm
cu}|\mathcal{S}}(x)dx}dz.$ (49)

(a) is due to the moment generating function of Gamma distribution. To explain
(b), we can write:

$\displaystyle\mathbb{P}\Big{(}\overline{\mathcal{R}}_{\rm j}(r(x_{\rm
c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i}))|\mathcal{S}\Big{)}=\int_{f_{i}^{j}(r)}^{\infty}f_{r_{\rm
cu}|\mathcal{S}}(x)dx,$
$\displaystyle\mathbb{P}\Big{(}\overline{\mathcal{R}}_{\rm j}(r(x_{\rm
c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i})),r(x_{c})=r,j|\mathcal{S}\Big{)}=\mathbb{P}\Big{(}\overline{\mathcal{R}}_{\rm
j}(r(x_{\rm c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i}))|r(x_{c})=r,j,\mathcal{S}\Big{)}\mathbb{P}\Big{(}r(x_{c})=r,j|\mathcal{S}\Big{)},$
$\displaystyle=\mathbb{P}\Big{(}\overline{\mathcal{R}}_{\rm j}(r(x_{\rm
c}))\leq\overline{\mathcal{R}}_{\rm i}(r(x_{\rm
i}))|r(x_{c})=r,j,\mathcal{S}\Big{)}\mathbb{P}\Big{(}j|r(x_{c})=r,\mathcal{S}\Big{)}\times\mathbb{P}\Big{(}r(x_{c})=r|\mathcal{S}\Big{)},$
$\displaystyle=\mathds{1}\big{(}r\geq
f_{i}^{j}(r)\big{)}\mathcal{P}_{j}(r)f_{r_{\rm cu}|\mathcal{S}}(r).$ (50)

###  -H Proof of Proposition 2

$\displaystyle C(\gamma)=\mathbb{P}(\frac{S}{I+\sigma^{2}}\geq\gamma)$
$\displaystyle=\mathbb{E}_{\mathcal{S},r,i|r,I|r,i,\mathcal{S}}\Big{\\{}\mathbb{P}(\frac{\rho_{i}\eta_{i}G_{i}(h_{i}^{2}+r^{2})^{-\frac{\alpha_{i}}{2}}}{I+\sigma^{2}}\geq\gamma)\Big{\\}},$
$\displaystyle=\mathbb{E}_{\mathcal{S},r,i|r,I|r,i,\mathcal{S}}\Big{\\{}\mathbb{P}(G_{i}\geq\frac{\gamma(I+\sigma^{2})(h_{i}^{2}+r^{2})^{\frac{\alpha_{i}}{2}}}{\rho_{i}\eta_{i}})\Big{\\}},$
$\displaystyle\overset{(a)}{=}\mathbb{E}_{\mathcal{S},r,i|r,I|r,i,\mathcal{S}}\Big{\\{}\frac{\Gamma(m_{i},m_{i}(I+\sigma^{2})X)}{\Gamma(m_{i})}\Bigg{|}_{X=\frac{\gamma(h_{i}^{2}+r^{2})^{\frac{\alpha_{i}}{2}}}{\rho_{i}\eta_{i}}}\Big{\\}},$
$\displaystyle\overset{(b)}{=}\mathbb{E}_{\mathcal{S},r,i|r,I|r,i,\mathcal{S}}\Big{\\{}\exp\left(-m_{i}(I+\sigma^{2})X\right)\sum_{k=0}^{m_{i}-1}\frac{m_{i}^{k}(I+\sigma^{2})^{k}X^{k}}{k!}\Big{\\}},$
$\displaystyle=\mathbb{E}_{\mathcal{S},r,i|r}\Bigg{\\{}\mathbb{E}_{I|r,i,\mathcal{S}}\Big{(}\exp\left(-m_{i}X(I+\sigma^{2})\right)(I+\sigma^{2})^{k}\Big{)}\sum_{k=0}^{m_{i}-1}\frac{m_{i}^{k}X^{k}}{k!}\Bigg{\\}},$
$\displaystyle=\mathbb{E}_{\mathcal{S},r,i|r}\Bigg{\\{}\sum_{k=0}^{m_{i}-1}(-1)^{k}\frac{{s_{r}}^{k}}{k!}\frac{d^{k}}{{ds_{r}}^{k}}\mathcal{L}_{I+\sigma^{2}}(s_{r},r,i,\mathcal{S})\Bigg{\\}},$
$\displaystyle\overset{(c)}{=}\mathbb{E}_{\mathcal{S}}\Bigg{\\{}\int_{0}^{\infty}\sum_{i\in\mathcal{A}}\sum_{k=0}^{m_{i}-1}(-1)^{k}\frac{{s_{r}}^{k}}{k!}\frac{d^{k}}{{ds_{r}}^{k}}\mathcal{L}_{I+\sigma^{2}}(s_{r},r,i,\mathcal{S})A_{i}(r,i,\mathcal{S})f_{R_{i}}(r)$
$\displaystyle+\sum_{i\in\mathcal{B}}\sum_{k=0}^{m_{i}-1}(-1)^{k}\frac{{s_{r}}^{k}}{k!}\frac{d^{k}}{{ds_{r}}^{k}}\mathcal{L}^{c}_{I+\sigma^{2}}(s_{r},r,i)A_{i}^{c}(r,i)\mathcal{P}_{i}(r)f_{r_{cu|\mathcal{S}}}(r)dr\Bigg{\\}}.$
(51)

In (a), since $G_{i}$ follows a Gamma distribution, we use the CDF of a Gamma
distribution, (b) comes form the power series expansion of the lower
incomplete gamma function, and in (c), we average over the distance to the
serving node and the channel link type (LOS,NLOS).

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