# Performance Analysis of Charging Infrastructure Sharing in UAV and EV-
involved Networks

Yujie Qin, Mustafa A. Kishk, Member, IEEE, and Mohamed-Slim Alouini, Fellow,
IEEE Yujie Qin and Mohamed-Slim Alouini are with Computer, Electrical and
Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah
University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia
Arabia. Mustafa Kishk was with CEMSE, KAUST during this work. Currently he is
with the Department of Electronic Engineering, National University of Ireland,
Maynooth, W23 F2H6, Ireland. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>slim.alouini@kaust.edu.sa).

###### Abstract

Electric vehicles (EVs) and unmanned aerial vehicles (UAVs) show great
potential in modern transportation and communication networks, respectively.
However, with growing demands for such technologies, the limited energy
infrastructure becomes the bottleneck for their future growth. It might be of
high cost and low energy efficiency for all the operators to each have their
own dedicated energy infrastructure, such as charging stations. In this paper,
we analyze a wireless charging infrastructure sharing strategy in UAV and EV-
involved networks. We consider a scenario where UAVs can charge in EV charging
stations and pay for the sharing fee. On the EVs’ side, sharing infrastructure
can earn extra profit but their service quality, such as waiting time, might
slightly reduce. On the UAVs’ side, if renting EV charging stations can
achieve an acceptable system performance, say high coverage probability, while
considering the cost, they may not need to build their dedicated charging
stations. In this case, we use tools from stochastic geometry to model the
locations and propose an optimization problem that captures the aforementioned
trade-offs between cost or profit and quality of service. Our numerical
results show that sharing infrastructure slightly increases the waiting time
of EVs, say within $5$ min, but dramatically decreases the waiting time of
drones, say more than $50$ min, and deploying more charging stations do
achieve better performances, but all these better performances are expected to
cost more.

###### Index Terms:

Stochastic geometry, Poisson Point Process, Electric Vehicles, Unmanned Aerial
Vehicles, Infrastructure sharing.

##  I Introduction

Electric vehicles (EVs) and unmanned aerial vehicles (UAVs) have great
application potential to achieve green and efficient future transportation
[1]. Compared with traditional vehicles, EVs cause less impact on the
environment and are more flexible since they can recharge at home during the
night. Hence, in the past few decades, EVs and their related infrastructures,
such as charging stations, have been widely studied and developed in real life
[2].

UAVs have recently increased in the market due to their high relocation
flexibility based on dynamic demands[3, 4]. They are expected to play an
essential rule in next-generation wireless networks [5, 6]. UAVs can be very
useful in both rural and urban areas [7], such as improving the quality of
service [8], providing service to remote Internet of Things users [9],
maintaining connectivity in nature disasters [10] and providing additional
capacity [11]. However, UAVs cannot work without frequent charging, which
highly restricts the performance of UAV-enabled wireless networks. Since UAVs
rely on their internal batteries, which are energy limited, their services are
likely to be interrupted if the energy runs low. Meanwhile, compared with EVs,
UAV-assisting networks are still new, and their infrastructures are poor.
While researchers mainly focus on investigating the communication or
application parts of UAV-involved networks, the infrastructures of UAVs, such
as the deployment of charging stations, should catch up [7].

The traditional model of single ownership of all the physical infrastructures
now faces significant challenges: it may be high cost and waste lots of
energy. Take charging stations for instance, the cost of building new charging
stations includes installation, maintenance, electricity grid distribution,
storage [12], resulting in huge pressure on operators. Meanwhile, it is not
energy-efficient for all operators to each have their own grid infrastructure.
In the current market environment, infrastructure sharing idea may be a
sustainable way to follow. Instead of building dedicated charging stations,
operators can consider sharing the charging stations while maintaining an
acceptable service quality.

In this paper, we explore an infrastructure sharing strategy in the EVs and
UAVs-involved network: EVs’ operators are willing to share their charging
stations with UAVs as far as their services are acceptable, and UAVs pay for
the sharing. In other words, UAVs can charge in EV charging stations to avoid
the traffic, hence, to achieve a better system performance. However, the
charging time may be longer in EV charging stations since they are not
designed for UAVs. We would like to emphasize that we are considering wireless
charging in this work which enable using the charging station by two different
technologies such as the UAVs and the EVs. More details are presented later in
this section and Section II-D.

###  I-A Related Work

Literature related to this work can be categorized into (i) design of EV
charging stations, (ii) UAV-assisted networks, and (iii) design of
infrastructure sharing. We provide a brief introduction to each of them in the
following lines.

EV charging stations placement and charging schedule. A comprehensive survey
on EV transportation was provided in [13], which was mainly about key
technologies, such as energy storage, for transportation electrification in
smart grid scenarios. EV charging infrastructure deployment was analyzed in
[14]. Optimization of waiting time for EVs is studied in [15] using a fuzzy
inference system. A Japanese case study about quick charging of EVs based on
waiting time and cost-benefit analyses is investigated in [16]. The authors
studied the infrastructure placement from the perspective of the agent and
used the road network data of the Chicagoland area. Authors in [17] presented
an optimization of minimum plug-in electric vehicles (PEV) infrastructure
deployment for highway corridors. Their models allowed for a straightforward
analysis despite uncertain input data such as the uncertainty of unavailable
information on PEV drivers’ behavior and charging demand data. Besides, much
of existing work on EVs is related to charging scheduling in a vehicle-to-grid
(V2G) system, such as [18, 19]. The authors used queuing networks to model the
dynamics of EVs and investigated the joint scheduling, which allowed the
operators to optimize the total cost of the system [20].

Stochastic geometry-based UAV-enabled network analysis. Stochastic geometry is
a strong mathematical tool that enables large-scale wireless networks and has
been demonstrated that it provides a tight approximation to real networks. In-
depth tutorial and survey about modeling base stations (BSs) and
characterizing interference had been provided in [21, 22]. Authors in [23, 24]
modeled the locations of UAVs and charging stations by two independent Poisson
point processes (PPPs) and modified the definition of coverage probability
based on queuing theory by considering the energy limitation of UAVs. In [25],
the authors proposed a laser-powered UAV system and introduced a new concept
of energy coverage probability, which is a joint probability of harvesting
energy and SNR coverage. Authors in [26] considered renewable energy-powered
UAVs, which can harvest energy from solar or wind resources, and derived the
probability density function (PDF) and cumulative density function (CDF) of
harvest energy and outage probability. Besides harvesting energy, authors in
[27, 11, 28] studied the tethered UAV, which is physically connected to a
ground station. While the tether provides the UAV with a stable power supply
and reliable data rate, it highly restricts the mobility and freedom of UAVs.

Economic analysis of wireless network infrastructure sharing. A brief review
about infrastructure sharing was provided in [29], which captured a conflict
between high demands of infrastructures and high cost. They categorized four
types of infrastructure sharing models, analyzed them from the perspective of
economic dimensions, and provided a practical use case. Authors in [30]
modeled and analyzed an infrastructure sharing system composed of a single
buyer mobile network operator and multiple sellers. Specifically, they modeled
the locations of BSs by PPPs and found the coverage probability of downlink
signal-to-interference-plus-noise (SINR) under the sharing environment. Using
game theory, authors in [31] proposed a system with a switching off decision,
which enabled the operators to switch off some BSs during low traffic time,
such as night. Besides BSs, spectrum licenses can also be shared as mentioned
in [32]. Similarly, authors in [33] studied the spectrum sharing in
Millimeter-wave cellular networks from the perspective of economic.

While most of the existing literature considers the energy resources
separately, none of them analyze the possibility of energy infrastructure
sharing.

###  I-B Contribution

In this paper, our main goal is to explore a charging infrastructure sharing
strategy among the EV and UAV-involved network, based on operators’ decisions,
high profit, or good service quality. We tap a new concept which is charging
infrastructure sharing among UAVs and EVs for the first time. Since it is a
totally new idea, it is difficult to find supporting data, and our system
models and analysis are based on some reasonable assumptions. The main
contributions of this work included the following points.

Modeling of waiting time. While we consider that UAVs can charge in EV
charging stations, the proposed system combines M/G/$c$ and D/D/$c$ queues,
waiting times of both EVs and UAVs become complex to model. We derive some
tight approximate equations about the waiting time of EVs and UAVs in
continuous time cases using the results from renewal processes and under some
reasonable assumptions. We then show the waiting time gap between the
approximated analysis and simulations.

Coverage probability. We consider two association policies between UAVs and
charging stations: (i) biased distance, in which the association cells form a
multiplicatively weighed Cox-Voronoi tessellation, and (ii) independent
thinning, in which the association regions form two independent Cox-Voronoi
tessellations. Our results show that association based on independent thinning
is slightly worse, however, easier than biased distance. Building upon these
two association policies and the waiting time we derived, we formulate a more
accurate expression for coverage probability. It captures the queuing of UAVs
and the influence of limited energy resources.

Economic insights based on charging infrastructure sharing. We consider two
scenarios where the operators care more about their service or profit. The
established optimization problem analyzes the weights of cost, profit, quality
of service from the perspective of EVs and UAVs, respectively. Our results
show that charging infrastructure sharing benefits both UAVs and EVs
operators, especially when the quality of the charging station is high, say
can charge multiple EVs or UAVs. More details shown in Section VII.

##  II System Model

TABLE I: Table of Notations

Notation | Description
---|---
$\Phi_{c,ev}$, $\Phi_{c,d}$; $\Phi_{t}$ | Locations of UAV, EV charging stations; TBSs
$\lambda_{l}$; $\lambda_{p,ev}$, $\lambda_{p,d}$ | Line density, point density
$\lambda_{c,d}$, $\lambda_{c,ev}$ | UAV, EV charging station density
$\lambda_{t}$, $\lambda_{u}$ | TBS density, UAV density
$\mu$, $\sigma$ | Average and deviation of SOC
$P_{\rm cha}$ | EV charging rate
$a,b$ | CV (or MWCV) tessellation fitting parameters
$p_{m}$, $p_{s}$ | Traveling-related, service-related power
$h$, $v$ | UAV altitude, velocity during traveling
$B_{\rm max,ev}$, $B_{\rm max}$ | EV, UAV battery capacity
$T_{\rm ch,d,ev}$, $T_{\rm ch,d,d}$; $T_{\rm tra}$, $T_{\rm ser}$ | Charging time of EVs, UAVs; service, traveling time of UAVs
$R_{s,d}$, $R_{s,ev}$ | Distance to the nearest UAV, EV charging station
$r_{c}$ | Radius of MCP disk
$c_{1},c_{2}$ | N/LoS environment variable
$\rho_{u}$, $\rho_{t}$ | Transmission power of UAVs, TBSs, respectively
$\gamma$, $\sigma^{2}_{n}$ | SINR threshold, noise power
$\alpha_{n},\alpha_{l},\alpha_{t}$ | N/LoS and active charging station path-loss exponent
$m_{n},m_{l}$, $\eta_{n},\eta_{l}$ | N/LoS fading gain, additional loss
$\mathcal{A}_{\rm LoS}(r)$, $\mathcal{A}_{\rm NLoS}(r)$, $\mathcal{A}_{\rm TBS}(r)$ | Probability of the reference user associating with nearest LoS/NLoS UAVs, or TBS, respectively
$\mathcal{A}_{d}(\beta_{d})$, $\mathcal{A}_{ev}(\beta_{d})$ | Probability of the reference UAV associating with EV, UAV charging stations
$N_{d,ev}$, $N_{d,d}$ | Number of drones in EV, UAV charging stations
$T_{\rm w,ev}$; $T_{\rm w,d,d}$, $T_{\rm w,d,ev}$ | Waiting time of EVs, waiting time of UAVs in UAV, EV charging stations, respectively

We consider a network composed of EVs, UAVs, their own charging stations
$\Phi_{c,ev}$ and $\Phi_{c,d}$, and terrestrial base stations (TBSs)
$\Phi_{t}$. The system is shown in Fig. 1 and related notations are explained
in Table I. The locations of EV charging stations and UAV charging stations
are modeled as two independent Poisson line cox processes (PLCPs)
$\Phi_{c,ev}$ and $\Phi_{c,d}$ with the same line density $\lambda_{l}$ and
different point densities $\lambda_{p,ev}$ and $\lambda_{p,d}$. While we
consider that EVs are more widely used than UAVs, $\lambda_{p,ev}$ is higher
than $\lambda_{p,d}$. That is, $\lambda_{c,ev}>\lambda_{c,d}$, where
$\lambda_{c,ev}$ and $\lambda_{c,d}$ are densities of EV and UAV charging
stations, given by $\lambda_{c}=\pi\lambda_{l}\lambda_{p}$.

Figure 1: Illustration of the system network.

Figure 2: Illustration of the system model, two types of UAVs’ association
policies. (a) association based on independent thinning: the association cell
forms two independent CV tessellations. (b) association based on biased
distance: the association cell forms one MWCV tessellation.

The locations of UAVs are modeled as a PPP $\Phi_{u}$ with density
$\lambda_{u}$. Given that UAVs are hovering at a fixed altitude $h$ above user
cluster centers to provide service and they only travel to the charging
station to recharge/swap their battery. To analyze the benefits of
infrastructure sharing, we assume that UAVs can recharge in both types of
charging stations and we analyze two different association policies, as shown
in Fig. 2: (i) independent thinning, in which a fraction of UAVs recharge in
EV charging stations denoted by the offloading ratio $\beta_{o}$, and (ii)
biased distance, in which UAVs associate with the charging station based on
the biased distance, $\min(R_{s,ev},\beta_{d}R_{s,d})$, where $R_{s,ev}$ and
$R_{s,d}$ are the distances between the UAV and its nearest EV/UAV charging
station, respectively, and $\beta_{d}$ is the association weight. In the first
policy, the densities of UAVs recharge in EV charging stations and UAV
charging stations are $\beta_{o}\lambda_{u}$ and $(1-\beta_{o})\lambda_{u}$,
respectively, and the association regions of the UAVs with two types of
charging stations form two independent Cox-Voronoi (CV) tessellations. In the
second policy, the association regions form one multiplicatively weighed Cox-
Voronoi (MWCV) tessellation. Notice that $\beta_{d}=1$ is the special case of
the presented model with equal distance: UAVs go to the nearest charging
stations. Without loss of generality, we perform our analysis in the rest of
the paper at a typical UAV located at the origin and the typical association
(CV or MWCV) region that contains the origin, denoted by association cell.

The arrival process of EVs is considered as a Poisson process with an average
arrival rate $\mu_{e}$ and the EV charging stations charge EVs and UAVs
depending on different serving policies. Moreover, we assume that EV charging
stations can serve $c$ customers simultaneously, which is also known as the
capacity of EV charging stations, and $m$ UAVs can be charged together, due to
the size of EVs being larger than UAVs. It means that while we assume that
UAVs can recharge in EV charging stations, say two or three UAVs can charge
within the same time slot. To be more realistic, our system also includes
TBSs, to serve the users in the clusters, whose locations are modeled by an
independent PPP with density $\lambda_{t}$.

###  II-A Waiting Time

The remaining state of charging (SOC) of the EV battery level is a random
variable modeled by a truncated lognormal distribution with average ($\mu$)
and typical deviation ($\sigma$) [34] as follows

$\displaystyle f_{\rm
SOC}(e)=\frac{1}{e\sigma\sqrt{2\pi}}\frac{\exp(-\frac{(\ln
e-\mu)^{2}}{2\sigma^{2}})}{F_{\rm SOC}(100)-F_{\rm SOC}(0)},$ (1)

where $F_{\rm SOC}(e)=\frac{1}{2}\bigg{[}1+\operatorname{erf}\bigg{(}\frac{\ln
e-\mu}{2\sigma^{2}}\bigg{)}\bigg{]}$. Assume that the EVs are fully charged
when they leave the charging station, and the charging time $T_{\rm ch,ev}$ is

$\displaystyle T_{\rm ch,ev}=\frac{B_{\rm max,ev}}{P_{\rm cha}}(1-\frac{{\rm
SOC}}{100}),$ (2)

where $P_{\rm cha}$ is the charging rate.

As mentioned, UAVs can recharge in both types of charging stations,
$\Phi_{c,ev}$ and $\Phi_{c,d}$ with different charging time $T_{\rm ch,d,ev}$
and $T_{\rm ch,d,d}$, respectively. We compare the system performance under
two serving policies: first in first serve (FIFS) and EV first, as shown in
Fig. 3.

Figure 3: Illustration of the system model, two types of serving policies: (i)
charge depends on arrival time and (ii) EVs have higher priority.

Recall that EV charging stations have capacity $c$ and can charge $m$ UAVs
simultaneously.

######  Definition 1 (FIFS).

In the case of FIFS, $\Phi_{c,ev}$ charge EVs and UAVs based on their arrival
time. In this case, the waiting time of UAVs and EVs are respectively denoted
by $T_{\rm w,ev,fifs}$ and $T_{\rm w,d,fifs}$.

######  Definition 2 (EV First).

In the case of EV first, EVs have higher priority than UAVs: $\Phi_{c,ev}$
charge the EVs first and UAVs can only be charged when no EV waiting. In this
case, the waiting time of UAVs and EVs are respectively denoted by $T_{\rm
w,ev,evfirst}$ and $T_{\rm w,d,evfirst}$.

Assume that the UAV is available when it is hovering above the cluster center
and providing service, and is unavailable when traveling to the nearest
charging station and waiting to recharge. Hence, the availability probability
of a UAV is a time fraction.

######  Definition 3 (Availability Probability).

We define the event $\mathcal{A}$ that indicates the availability of the
typical UAV, which denotes that the UAV is available and can provide service.
Conditioned on $N$ UAVs in the typical association cell, the availability
probability of the UAV is

$\displaystyle\mathbb{P}({\mathcal{A}|N})=\mathbb{E}_{\Phi_{c}}\bigg{[}\frac{T_{\rm
ser}(x)}{T_{\rm ser}(x)+T_{\rm ch}+T_{\rm w,d\mid N}+2T_{\rm
tra}(x)}\bigg{]},$ (3)

where $x$ annotates the location of the typical UAV, and

$\displaystyle T_{\rm tra}(x)$ $\displaystyle=\frac{R_{s,\\{ev,d\\}}(x)}{v},$
(4) $\displaystyle T_{\rm ser}(x)$ $\displaystyle=\frac{B_{\rm
max}-2p_{m}\frac{R_{s,\\{ev,d\\}}(x)}{v}}{p_{s}},$ (5)

in which $B_{\rm max}$, $v$, $p_{m}$ and $p_{s}$ is the maximum battery size,
velocity, traveling- and serving-related power consumption of UAVs, and
waiting time of UAVs depends on $N$, either $T_{\rm w,d,fifs\mid N}$ or
$T_{\rm w,d,evfirst\mid N}$, and $N$ can be either $N_{d,ev}$ or $N_{d,d}$ ,
which are explained in Section III. Hence, the unconditional availability
probability is

$\displaystyle P_{\rm a}$
$\displaystyle=\mathbb{E}_{N}[\mathbb{P}(\mathcal{A}|N)].$ (6)

###  II-B Power Consumption

We consider the UAV’s power consumption composed of two parts: (i) service-
related power $p_{s}$, including hovering and communication-related power,
(ii) traveling-related power $p_{m}$, which denotes the power consumed in
traveling to/from the EV/UAV charging station through the horizontal distance
$R_{s,\\{ev,d\\}}$.

Given in [35], ${p}_{m}$ is a function of the UAV’s velocity $v$ and given by

$\displaystyle p_{m}=P_{0}\left(1+\frac{3v^{2}}{U_{\rm
tip}^{2}}\right)+\frac{P_{i}v_{0}}{v}+\frac{1}{2}d_{0}\rho sAv^{3},$ (7)

where $P_{0}$ and $P_{i}$ present the blade profile power and induced power,
$U_{\rm tip}$ is the tip speed of the rotor blade, $v_{0}$ is the mean rotor
induced velocity in hover, $\rho$ is the air density, $A$ is the rotor disc
area, $d_{0}$ is fuselage drag ratio, and $s$ is rotor solidity. In this case,
the energy consumed during traveling to or from the associated charging
station is

$\displaystyle E_{t}$ $\displaystyle=\frac{R_{s,\\{ev,d\\}}(x)}{v}p_{m}$
$\displaystyle=\frac{R_{s,\\{ev,d\\}}(x)}{v}\left(P_{0}\left(1+\frac{3v^{2}}{U_{\rm
tip}^{2}}\right)+\frac{P_{i}v_{0}}{v}+\frac{1}{2}d_{0}\rho sAv^{3}\right).$
(8)

In the rest of the paper, we use the optimal value of $v$ that minimizes
$E_{t}$.

###  II-C User Association

Without loss of generality, we focus on a typical user randomly selected from
the typical user cluster centered at the origin. From the perspective of the
typical user, we denote the typical UAV as cluster UAV. Assume that the user
associates with the cluster UAV if it is available (hovering and providing
service), if not, associates with a nearby available UAV or TBS, depending on
the average received power. Let $\Phi_{u_{o}}$, $\Phi_{u^{{}^{\prime}}}$ and
$\Phi_{t}$ be the point sets of serving or interference BSs: cluster UAV,
available UAVs and TBSs. Notice that the set $\Phi_{u_{o}}$ is composed of
only one point, which is the location of the cluster UAV in case it is
available, otherwise, $\Phi_{u_{o}}=\emptyset$. When the cluster UAV is
unavailable, the user associates with a UAV in $\Phi_{u^{{}^{\prime}}}$ or the
closest TBS in $\Phi_{t}$. The point process $\Phi_{u^{{}^{\prime}}}$ is
constructed by independently thinning $\Phi_{u}$ with the probability $P_{a}$.
Therefore, the density of $\Phi_{u^{{}^{\prime}}}$ is
$\lambda_{u}^{{}^{\prime}}=P_{a}\lambda_{u}$.

When the typical user associates with a UAV, it can be either line-of-sight
(LoS) or non line-of-sight (NLoS), the received power is

$\displaystyle p_{u}$ $\displaystyle=\left\\{\begin{aligned}
p_{l}=\eta_{l}\rho_{u}G_{l}R_{u}^{-\alpha_{l}},&\quad\text{\rm in case of
LoS},\\\ p_{n}=\eta_{n}\rho_{u}G_{n}R_{u}^{-\alpha_{n}},&\quad\text{\rm in
case of NLoS},\\\ \end{aligned}\right.$

where $\rho_{u}$ is the transmission power of the UAVs, $R_{u}$ denotes the
Euclidean distance between the typical user and the serving UAV, $\alpha_{l}$
and $\alpha_{n}$ present the path-loss exponent, $G_{l}$ and $G_{n}$ are the
fading gains that follow gamma distribution with shape and scale parameters
$(m_{l},\frac{1}{m_{l}})$ and $(m_{n},\frac{1}{m_{n}})$, $\eta_{l}$ and
$\eta_{n}$ denote the mean additional losses for LoS and NLoS transmissions,
respectively. Based on [36], the probability of establishing LoS or NLoS
channels between users and UAVs is

$\displaystyle P_{l}(r)$
$\displaystyle=\frac{1}{1+c_{1}\exp\bigg{(}-c_{2}\bigg{(}\frac{180}{\pi}\arctan\bigg{(}\frac{h}{\sqrt{r^{2}-h^{2}}}\bigg{)}-c_{1}\bigg{)}\bigg{)}},$
(9)

where $r$ is the Euclidean distance, $c_{1}$ and $c_{2}$ are two environment-
related variables (e.g., urban, dense urban, and highrise urban), and $h$ is
the altitude of the UAV. Consequently, the probability of NLoS is
$P_{n}(r)=1-P_{l}(r)$.

When the user associates with TBS, the received power is

$\displaystyle p_{t}$ $\displaystyle=\rho_{t}HR_{t}^{-\alpha_{t}},$

in which $\rho_{t}$ is the transmission power of TBSs, $R_{t}$ denotes the
distance between the user and the nearest TBS, $H$ is the fading gain that
follows exponential distribution with unity mean, and $\alpha_{t}$ presents
the path-loss exponent.

Let $\mathcal{A}_{\rm NLoS}(r)$, $\mathcal{A}_{\rm LoS}(r)$ and
$\mathcal{A}_{\rm TBS}(r)$ be the probabilities that the reference user
associates with the nearest N/LoS UAV or TBS, which is at $r$ away, are
respectively given by

$\displaystyle\mathcal{A}_{\rm\\{NL,L\\}oS}(r)$
$\displaystyle=\mathbb{P}(p_{\\{n,l\\}}(r)>p_{t}),$
$\displaystyle\mathcal{A}_{\rm TBS}(r)$
$\displaystyle=\mathbb{P}(p_{t}(r)>p_{u}).$ (10)

We then define the coverage probability as the probability that the typical
user is successfully served, which is the event that SINR of the related link
is above a predefined threshold.

######  Definition 4 (Coverage Probability).

The total coverage probability is defined as

$\displaystyle P_{\rm cov}=$ $\displaystyle P_{a}P_{\rm
cov,U_{o}}+(1-P_{a})P_{\rm cov,\hat{U}_{o}}$ $\displaystyle=$ $\displaystyle
P_{a}(P_{\rm cov,U_{o},l}+P_{\rm cov,U_{o},n})$
$\displaystyle+(1-P_{a})(P_{\rm cov,\hat{U}_{o},l}+P_{\rm
cov,\hat{U}_{o},n}+P_{\rm cov,t}),$ (11)

in which,

$\displaystyle P_{\rm cov,{U_{o}}}$ $\displaystyle=\sum_{bs\in\\{\text{\rm
N/LoS}\\}}\mathbb{E}[\mathcal{A}_{bs}(r)\mathbb{P}({\rm
SINR}\geq\gamma|r,bs)],$ $\displaystyle P_{\rm cov,{\hat{U}_{o}}}$
$\displaystyle=\sum_{bs\in\\{\text{\rm
TBS,N/LoS}\\}}\mathbb{E}[\mathcal{A}_{bs}(r)\mathbb{P}({\rm
SINR}\geq\gamma|r,bs)],$

where $P_{\rm cov,{U_{o}}}$ and $P_{\rm cov,{\hat{U}_{o}}}$ are the coverage
probabilities when the cluster UAV is available and unavailable, respectively.
$P_{\rm cov,U_{o},l}$ and $P_{\rm cov,U_{o},n}$ are the coverage probabilities
when associating with the LoS/NLoS cluster UAV. $P_{\rm cov,\hat{U}_{o},l}$,
$P_{\rm cov,\hat{U}_{o},n}$ and $P_{\rm cov,t}$ are the coverage probabilities
when associating with the nearby available LoS/NLoS UAV and the nearest TBS,
respectively.

$\Phi_{u^{{}^{\prime}}}$ is composed of two subsets
$\Phi_{u^{{}^{\prime}}_{l}}$ and $\Phi_{u^{{}^{\prime}}_{n}}$, which denote
the locations of available LoS UAVs and NLoS UAVs, respectively. Conditioning
on the serving BS $b_{s}$, the SINR and the aggregate interference is defined
as

$\displaystyle{\rm SINR}$
$\displaystyle=\frac{\max(p_{u},p_{t})}{I+\sigma^{2}},$ $\displaystyle I$
$\displaystyle=\sum_{N_{i}\in\Phi_{u^{{}^{\prime}}_{n}}/b_{s}}\eta_{n}\rho_{u}G_{n}D_{N_{i}}^{-\alpha_{n}}+\sum_{L_{j}\in\Phi_{u^{{}^{\prime}}_{l}}/b_{s}}\eta_{l}\rho_{u}G_{l}D_{L_{j}}^{-\alpha_{l}}$
$\displaystyle+\sum_{T_{k}\in\Phi_{t}/b_{s}}\rho_{t}HD_{T_{k}}^{-\alpha_{t}},$

in which $D_{N_{i}}$, $D_{L_{j}}$ and $D_{T_{k}}$ are the distances between
the typical user and the interfering NLoS, LoS UAVs, and TBSs, respectively.

###  II-D Infrastructure Sharing

We consider a scenario where EV operators share their infrastructure (EV
charging stations) with UAV operators as far as their own services are
qualified, and UAV operators pay for the corresponding service. Besides, if
the infrastructure shared by EVs cannot maintain an acceptable network
performance, say coverage probability is too low, UAVs need to install more
dedicated charging stations, of which the density is denoted by
$\Delta\lambda_{c,d}$.

The proposed objective functions are the total profit on the sides of UAVs’
and EVs’ operators, respectively. Note that in these two objective functions,
the only constraints are the values of $\beta_{\\{d,o\\}}$, from 0 to the
optimal values, where 0 denotes no sharing and optimal values are obtained
from the maximal coverage probability (More details are provided in Section
VII).

######  Definition 5 (Objective Function).

For EVs’ operator, the formulated function deals with the total extra waiting
time of EVs and the profit paid by UAVs’ operator per charging station per
year:

$\displaystyle C_{e}$ $\displaystyle=w_{\rm wait}\Delta T_{\rm w,ev}\cdot
365\cdot 24\mu_{e}+w_{\rm inf,ev}C_{\rm inf,d}(\beta_{\\{d,o\\}}),$ (12)

where $w_{\rm inf,ev}$, $w_{\rm wait}$ are objective function coefficients
(weights of objectives), $\Delta T_{\rm w,ev}\times 365\times 24\mu_{e}$ is
the total extra waiting time of EVs per charging station and $C_{\rm
inf,d}(\beta_{\\{d,o\\}})$ is the infrastructure sharing fee that UAVs’
operators payed for EVs’ operators. For example, if $w_{\rm wait}$ is larger
than $w_{\rm inf,ev}$, it means that EVs’ operator care more about the quality
of their own services.

For UAVs’ operator, the formulated function deals with the improvement of
network performance (coverage probability), cost of installing new charging
stations and infrastructure sharing fee,

$\displaystyle C_{u}$ $\displaystyle=w_{\rm cov}\frac{P_{\rm
cov}^{{}^{\prime}}(\Delta\lambda_{c,d})}{P_{\rm
cov,in}}+\frac{w_{c}\Delta\lambda_{c,d}}{\lambda_{c,d}}+w_{\rm inf,d}C_{\rm
inf,d}(\beta_{\\{d,o\\}}),$ (13)

where $w_{c}$, $w_{\rm cov}$ and $w_{\rm inf,d}$ are objective function
coefficients, $P_{\rm cov,in}$ and $P_{\rm
cov}^{{}^{\prime}}(\Delta\lambda_{c,d})$ are the initial performance (without
infrastructure sharing), and performance with infrastructure sharing and more
dedicated charging stations.

In the following text, we analyze the system performance from the perspective
of UAVs and EVs. Since UAVs are for communication, we study the coverage
probability, which is defined as the probability of the reference user being
successfully served, e.g., SINR is greater than the predefined threshold. To
do so, in Section III we compute the waiting time of UAVs under the
aforementioned two association policies and charging station serving policies.
We then in Section IV obtain the availability probability of UAVs, which is
needed in coverage probability analysis, say (11), and the final results of
coverage probability is given in Section V. For EVs, we are concerned about
the waiting time for charging. Hence, we in Section III compute the waiting
time of EVs in the case of sharing charging stations.

##  III Waiting Time

This section analyzes the waiting time of both EVs and UAVs and is the most
important technical section of this paper. Notice that deriving the exact
waiting time equations is tricky, our results are based on some reasonable
assumptions and tight approximations.

Given that $T_{\rm ser}$ is much longer than $T_{\rm tra}$, therefore, we
ignore $T_{\rm tra}$ when we analysis the waiting time and assume that all the
drones have the same service time. Let $N_{d,ev}$ and $N_{d,d}$ be the number
of drones recharging in EV charging stations and UAV charging stations,
respectively.

We first derive the waiting time of drones in their dedicated charging
stations.

######  Lemma 1 (Waiting Time of UAVs in UAV Charging Stations).

In the typical association cell of UAV charging station, conditioned on
$N_{d,d}$, the waiting time $T_{w,d,d}$ is given by

$\displaystyle T_{\rm w,d,d|N_{d,d}}=T_{\rm
ch,d,d}\bigg{(}N_{d,d}-\frac{T_{\rm ser}}{T_{\rm ch,d,d}}-1\bigg{)},$ (14)

where $T_{\rm ch,d,d}$ is the charging time of UAVs in their dedicated
charging stations.

We then derive the waiting time of drones in EV charging stations based on the
aforementioned serving priorities (Definition 1 and Definition 2).

Figure 4: Illustration of the proof of waiting time (FIFS and EV first).

######  Lemma 2 (Waiting Time of UAVs in EV Charging Stations (FIFS)).

When the charging policy in the typical cell of EV charging station is FIFS,
conditioned on the number of associated drones $N_{d,ev}$, the waiting time of
UAVs is, in the case of $N_{d,ev}\geq mc(1+\frac{T_{\rm ser}}{T_{\rm
ch,d,ev}})$,

$\displaystyle T_{\rm w,d,ev|N_{d,ev}}=$ $\displaystyle\frac{T_{\rm
ch,d,ev}(\frac{N_{d,ev}}{mc}-1)+\frac{T_{\rm ch,d,ev}+T_{\rm ser}}{c}\mu_{\rm
e}\mathbb{E}[T_{\rm ch,ev}]-T_{\rm ser}}{1-\frac{\mu_{e}\mathbb{E}[T_{\rm
ch,ev}]}{c}}.$ (15)

where $c$ is the capacity of EV charging stations (the number of EVs that can
be served together) and $m$ denotes that $m$ UAVs can charge together within
one charging slot.

###### Proof:

Here we provide the proof for $c=1$, as shown in Fig. 4. From the perspective
of a typical UAV, the waiting time can be computed as follows

$\displaystyle($ $\displaystyle T_{\rm w,d,ev\mid N_{d,ev}}+T_{\rm
ch,d,ev}+T_{\rm ser})\mu_{e}\mathbb{E}[T_{\rm ch,ev}]$
$\displaystyle+(N_{d,ev}-1)T_{\rm ch,d,ev}-T_{\rm ser}=T_{\rm w,d,ev\mid
N_{d,ev}},$ (16)

where the first term $(T_{\rm w,d,ev\mid N_{d,ev}}+T_{\rm ch,d,ev}+T_{\rm
ser})\mu_{e}\mathbb{E}[T_{\rm ch,ev}]$ is the total charging time of EVs
coming after the arrival of the typical UAV, $(N_{d,ev}-1)T_{\rm ch,d,ev}$ is
the total charging time of other UAVs. The proof completes by simplifying the
above equation. ∎

######  Lemma 3 (Waiting time of UAVs in EV Charging Stations (EV First)).

When EVs have higher priority while charging in EV charging stations,
conditioned on the number of associated drones $N_{d,ev}$, the waiting time of
UAVs is, in the case of $N_{d,ev}\geq mc(1+\frac{T_{\rm ser}}{T_{\rm
ch,d,ev}})$,

$\displaystyle T_{\rm w,d,ev|N_{d,ev}}=$ $\displaystyle\frac{T_{\rm
ch,d,ev}(\frac{N_{d,ev}}{mc}-1)+\frac{T_{\rm ch,d,ev}+T_{\rm ser}}{c}\mu_{\rm
e}\mathbb{E}[T_{\rm ch,ev}]-T_{\rm ser}}{1-\frac{\mu_{e}\mathbb{E}[T_{\rm
ch,ev}]}{c}}.$ (17)

###### Proof:

Similar to the Proof of Lemma 2, as shown in Fig. 4. From the perspective of a
typical UAV, the waiting time is

$\displaystyle($ $\displaystyle T_{\rm w,d,ev\mid N_{d,ev}}+T_{\rm
ch,d,ev}+T_{\rm ser})\mu_{e}\mathbb{E}[T_{\rm ch,ev}]$
$\displaystyle+(N_{d,ev}-1)T_{\rm ch,d,ev}-T_{\rm ser}=T_{\rm w,d,ev\mid
N_{d,ev}}.$ (18)

Here we find that the equations are exactly the same which is because of the
scheduling of UAVs: UAVs visit the charging stations after a fixed amount of
time frequently. If the UAV let the EVs charge first, there are no accumulated
EVs waiting when the UAV start charging, however, it needs to wait the later
arrival EVs charging during waiting in the charging stations. ∎

######  Remark 1.

Note that since the arrival processes of UAVs are correlated in time and
scheduled, the equations of waiting time of FIFS and EV first are exactly the
same when $N_{d,ev}\geq mc(1+\frac{T_{\rm ser}}{T_{\rm ch,d,ev}})$. Besides,
the charging time of UAVs in EV charging station is very long and the density
of UAVs is much higher than the density of charging stations, hence,
$N_{d,ev}\geq cm(1+\frac{T_{\rm ser}}{T_{\rm ch,d,ev}})$ can be be satisfied
in the given system. While the waiting time of $N_{d,ev}<mc(1+\frac{T_{\rm
ser}}{T_{\rm ch,d,ev}})$ is given in Appendix -A, it has a low probability.
Therefore, in these two policies the waiting time of UAVs are approximately
the same.

######  Lemma 4 (Waiting Time of EVs).

The waiting time of EVs are given by,

$\displaystyle T_{\rm w,ev,fifs|N_{d,ev}}$ $\displaystyle=T_{\rm
w,d,ev|N_{d,ev}}+T_{\rm w,ev,noDrone}+T_{\rm d|N_{d,ev}},$ $\displaystyle
T_{\rm w,ev,evfirst|N_{d,ev}}$ $\displaystyle=T_{\rm w,ev,noDrone}+T_{\rm
d|N_{d,ev}},$ (19)

where

$\displaystyle T_{\rm w,ev,noDrone}$
$\displaystyle=\frac{\mu_{e}\mathbb{E}^{2}[T_{\rm
ch,ev}]}{1-\mu_{e}\mathbb{E}^{2}[T_{\rm ch,ev}]}\frac{\mathbb{E}^{2}[T_{\rm
ch,ev}]}{2\mathbb{E}[T_{\rm ch,ev}^{2}]},$ $\displaystyle T_{\rm d|N_{d,ev}}$
$\displaystyle=\min\bigg{(}1,\frac{N_{d,ev}T_{\rm ch,d,ev}}{mc(T_{\rm
ch,d,ev}+T_{\rm ser})}\bigg{)}$ $\displaystyle\int_{0}^{T_{\rm
ch,d,ev}}\frac{c}{T_{\rm ch,d,ev}}(1-\frac{x}{T_{\rm ch,d,ev}})^{c-1}{\rm
d}x.$ (20)

###### Proof:

In the case of EV first, EVs don’t need to wait for UAVs charging unless they
are already charging. Hence, the waiting time of an EV is the sum of charging
time of EVs come before it $T_{\rm w,ev,noDrone}$ and the remaining charging
time of UAVs $T_{\rm d|N_{d,ev}}$.

In the case of FIFS, assume that an EV arrives after the typical UAV, hence
the waiting time of the EV is the sum of the waiting time of the UAV and the
charging time of EVs before it and the remaining charging time of UAVs. ∎

######  Remark 2.

Based on Remark 1 and Lemma 4, we note that the waiting time of UAVs are
approximately the same in these two serving policies while waiting time of EVs
in the case FIFS is much longer than that of EV first. Besides, the waiting
time of EVs increases slightly with the increase of $N_{ev}$ and then stays
constant, in the case of EV first. The same results shown in Section VII.
Therefore, in the following analysis, we focus on analyzing the performance of
infrastructure sharing based on the serving policy: EV first.

##  IV Availability Probability

This section derives the availability probability of UAVs based on two UAVs’
association policies: (i) biased distance and (ii) independent thinning. Based
on the association policies, we use the optimal $\beta_{d}$ and $\beta_{o}$
that maximize the coverage probability.

###  IV-A Biased Distance

In this section, we derive the $P_{a}$ under the biased distance policy: the
typical UAV associates with the charging station based on
$\min(R_{s,ev},\beta_{d}R_{s,d})$. To do so, we first formulate the distance
distribution.

######  Lemma 5 (Distance Distribution).

The cumulative distribution function of the distances between the typical UAV
and the nearest EV/UAV charging station denoted by $F_{c,ev}(r)$ and
$F_{c,d}(r)$, respectively, are given by

$\displaystyle F_{c,ev}$
$\displaystyle=1-\exp\bigg{(}-2\pi\lambda_{l}\int_{0}^{r}1-\exp(-2\lambda_{p,ev}\sqrt{r^{2}-\rho^{2}}){\rm
d}\rho\bigg{)},$ (21) $\displaystyle F_{c,d}$
$\displaystyle=1-\exp\bigg{(}-2\pi\lambda_{l}\int_{0}^{r}1-\exp(-2\lambda_{p,d}\sqrt{r^{2}-\rho^{2}}){\rm
d}\rho\bigg{)},$ (22)

taking the derivative, their PDF are, respectively, given by

$\displaystyle
f_{c,ev}(r)=2\pi\lambda_{l}\int_{0}^{r}\frac{2\lambda_{p,ev}r}{\sqrt{r^{2}-\rho^{2}}}\exp(-2\lambda_{p,ev}\sqrt{r^{2}-\rho^{2}}){\rm
d}\rho$
$\displaystyle\exp\bigg{(}-2\pi\lambda_{l}\int_{0}^{r}1-\exp(-2\lambda_{p,ev}\sqrt{r^{2}-\rho^{2}}){\rm
d}\rho\bigg{)},$ (23) $\displaystyle
f_{c,d}(r)=2\pi\lambda_{l}\int_{0}^{r}\frac{2\lambda_{p,d}r}{\sqrt{r^{2}-\rho^{2}}}\exp(-2\lambda_{p,d}\sqrt{r^{2}-\rho^{2}}){\rm
d}\rho$
$\displaystyle\exp\bigg{(}-2\pi\lambda_{l}\int_{0}^{r}1-\exp(-2\lambda_{p,d}\sqrt{r^{2}-\rho^{2}}){\rm
d}\rho\bigg{)}.$ (24)

The probability that the cluster UAV associates with the typical charging
station is a function of distance and association weights. In the following
lemma, we identify the conditional and unconditional association probability.

######  Lemma 6 (UAV’s Association Probability).

Given that the nearest serving charging station is at $r$ away, the
association probability is given by,

$\displaystyle\mathcal{A}_{ev\mid r}(r,\beta_{d})$
$\displaystyle=\mathbb{P}(R_{s,ev}<\beta_{d}R_{s,d})$
$\displaystyle=\mathbb{P}(\frac{r}{\beta_{d}}<R_{s,d})=\bar{F}_{c,d}(\frac{r}{\beta_{d}}),$
(25) $\displaystyle\mathcal{A}_{d\mid r}(r,\beta_{d})$
$\displaystyle=\mathbb{P}(R_{s,ev}>\beta_{d}R_{s,d})$
$\displaystyle=\mathbb{P}(R_{s,ev}>\beta_{d}r)=\bar{F}_{c,ev}(\beta_{d}r),$
(26)

taking the expectation over the distance, the probabilities of associating
with EV/UAV charging stations are

$\displaystyle\mathcal{A}_{ev}(\beta_{d})$
$\displaystyle=\int_{0}^{\infty}\bar{F}_{c,d}(\frac{r}{\beta_{d}})f_{c,ev}(r){\rm
d}r,$ $\displaystyle\mathcal{A}_{d}(\beta_{d})$
$\displaystyle=\int_{0}^{\infty}\bar{F}_{c,ev}(r\beta_{d})f_{c,d}(r){\rm d}r.$
(27)

Note that for a PPP, the probability of the number of points filling in a
certain cell is proportional to the area of this cell. Hence in the following
lemma, we provide the area approximation. Here we adopt both the fitting and
approximation formulas of cell area based on Poisson Voronoi (PV)
tessellations, however, it shows that they are both tight for CV and MWCV
tessellations, in the case that line density is large while point density is
low.

######  Lemma 7 (Area Approximation).

Using the approximation mentioned in [37], the association area of EV/UAV
charging stations can be approximated as

$\displaystyle f_{C}(c)$
$\displaystyle=\frac{b^{a}}{\Gamma(a)}(\frac{\lambda}{\mathcal{A}(\beta_{d})})(\frac{\lambda}{\mathcal{A}(\beta_{d})}c)^{a-1}\exp(-b\frac{\lambda}{\mathcal{A}(\beta_{d})}c),$
(28)

where $\lambda$ and $\mathcal{A}(\beta_{d})$ are the density of charging
stations ($\lambda_{c,ev}$ or $\lambda_{c,d}$) and UAV’s association
probability defined in Lemma 6 ($\mathcal{A}_{c,ev}(\beta_{d})$ or
$\mathcal{A}_{c,d}(\beta_{d})$). As a random UAV is more likely to lie in a
larger cell, which is know as biased cell, and the PDF of the biased cell is

$\displaystyle f_{C^{{}^{\prime}}}(c)$
$\displaystyle=\frac{b^{a}}{\Gamma(a)}(\frac{\lambda}{\mathcal{A}(\beta_{d})})(\frac{\lambda}{\mathcal{A}(\beta_{d})}c)^{a}\exp(-b\frac{\lambda}{\mathcal{A}(\beta_{d})}c).$
(29)

Knowing the area of the biased cell, apart from the typical UAV, the PMF of
the number of other UAVs are given in the following lemma.

######  Lemma 8 (Average Number of UAVs).

The PMF of the other UAVs associated with the typical EV/UAV charging stations
are given by

$\displaystyle\mathbb{P}(N_{d,ev}=n)=$
$\displaystyle\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}(\frac{\lambda_{c,ev}}{\mathcal{A}_{ev}(\beta_{d})})^{a+1}$
$\displaystyle\times\frac{\lambda_{\mathrm{u}}^{n}}{\left(b\frac{\lambda_{c,ev}}{\mathcal{A}_{ev}(\beta_{d})}+\lambda_{\mathrm{u}}\right)^{a+n+1}},$
$\displaystyle\mathbb{P}(N_{d,d}=n)=$
$\displaystyle\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}(\frac{\lambda_{c,d}}{\mathcal{A}_{d}(\beta_{d})})^{a+1}$
$\displaystyle\times\frac{\lambda_{\mathrm{u}}^{n}}{\left(b\frac{\lambda_{c,d}}{\mathcal{A}_{d}(\beta_{d})}+\lambda_{\mathrm{u}}\right)^{a+n+1}}.$
(30)

###### Proof:

The number of UAVs per CV cell is a Poisson random variable, with parameter
cell area, given by

$\displaystyle\mathbb{P}(N=n)=\mathbb{E}_{C^{\prime}}\left[\mathbb{P}\left(N=n\mid
C^{\prime}\right)\right]=\int_{0}^{\infty}\mathbb{P}(N=n)f_{C^{\prime}}(c)\mathrm{d}c$
$\displaystyle=\int_{0}^{\infty}\frac{\left(\lambda_{\mathrm{u}}c\right)^{n}e^{-\lambda_{\mathrm{u}}c}}{n!}\frac{b^{a}}{\Gamma(a)}(\frac{\lambda}{\mathcal{A}(\beta_{d})})(\frac{\lambda}{\mathcal{A}(\beta_{d})}c)^{a}\exp(-b\frac{\lambda}{\mathcal{A}(\beta_{d})}c){\rm
d}c$
$\displaystyle=\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}(\frac{\lambda}{\mathcal{A}(\beta_{d})})^{a+1}\frac{\lambda_{\mathrm{u}}^{n}}{\left(b\frac{\lambda}{\mathcal{A}(\beta_{d})}+\lambda_{\mathrm{u}}\right)^{a+n+1}}.$
(31)

∎

Notice that conditioned on associating with the tagged charging station, the
distance distribution is different from the first contact distance in PLCP,
since it is also influenced by nearby charging stations and association
weight.

######  Lemma 9 (Conditional Distance Distribution).

Let $Y_{\\{ev,d\\}}$ be the distance between the typical UAV and its serving
EV/UAV charging station. Conditioned on association, the PDF of
$Y_{\\{ev,d\\}}$ is

$\displaystyle f_{Y_{ev}}(y)$
$\displaystyle=\frac{\bar{F}_{c,d}(\frac{y}{\beta_{d}})f_{c,ev}(y)}{\mathcal{A}_{ev}(\beta_{d})},$
(32) $\displaystyle f_{Y_{d}}(y)$
$\displaystyle=\frac{\bar{F}_{c,ev}(y\beta_{d})f_{c,d}(y)}{\mathcal{A}_{d}(\beta_{d})}.$
(33)

###### Proof:

$Y_{ev}$ has the same distance distribution as $R_{s,ev}$ conditioned on the
typical UAV being associated with the EV charging station,

$\displaystyle\mathbb{P}(Y_{ev}>y)$
$\displaystyle=\frac{\mathbb{P}(R_{s,ev}>y\mid\text{associate with EV charging
station})}{\mathbb{P}(\text{associate with EV charging station})}$
$\displaystyle=\frac{\int_{y}^{\infty}\bar{F}_{c,d}(\frac{r}{\beta_{d}})f_{ev}(r){\rm
d}r}{\mathcal{A}_{ev}(\beta_{d})},$ (34)

proof completes by taking the derivative. ∎

Now we proceed to present the conditional and unconditional availability
probability of UAVs in the case of biased distance association policy.

######  Theorem 1 (Availability Probability).

Availability probability under the presented UAVs’ association policy is given
by

$\displaystyle P_{\rm
a,bias}=\sum_{n=0}^{\infty}\bigg{(}\int_{0}^{\frac{vB_{\rm
max}}{2p_{m}}}g_{ev}(y\mid n)\bar{F}_{c,d}(\frac{y}{\beta_{d}})f_{ev}(y){\rm
d}y$
$\displaystyle\quad\times\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}(\frac{\lambda_{c,ev}}{\mathcal{A}_{ev}(\beta_{d})})^{a+1}\frac{\lambda_{\mathrm{u}}^{n}}{\left(b\frac{\lambda_{c,ev}}{\mathcal{A}_{ev}(\beta_{d})}+\lambda_{\mathrm{u}}\right)^{a+n+1}}\bigg{)}$
$\displaystyle+\sum_{n=0}^{\infty}\bigg{(}\int_{0}^{\frac{vB_{\rm
max}}{2p_{m}}}g_{d}(y\mid n)\bar{F}_{c,ev}(y\beta_{d})f_{d}(y){\rm d}y$
$\displaystyle\quad\times\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}(\frac{\lambda_{c,d}}{\mathcal{A}_{d}(\beta_{d})})^{a+1}\frac{\lambda_{\mathrm{u}}^{n}}{\left(b\frac{\lambda_{c,d}}{\mathcal{A}_{d}(\beta_{d})}+\lambda_{\mathrm{u}}\right)^{a+n+1}}\bigg{)},$
(35)

where $g_{ev}(y\mid n)=\frac{vB_{\rm max}-2yp_{m}}{vB_{\rm
max}-2y(p_{m}-p_{s})+vp_{s}(T_{\rm ch,d,ev}+T_{\rm w,d,ev\mid N_{d,ev}})}$ and
$g_{d}(y\mid n)=\frac{vB_{\rm max}-2yp_{m}}{vB_{\rm
max}-2y(p_{m}-p_{s})+vp_{s}(T_{\rm ch,d,d}+T_{w,d,d\mid N_{d}})}$.

###### Proof:

See Appendix -B. ∎

###  IV-B Independent Thinning

In this section, we derive the $P_{a}$ under the independent thinning policy:
the typical UAV associates with the charging station based on the independent
thinning.

In this policy, we consider UAVs are divided into two parts: associating with
$\Phi_{c,ev}$ and associating with $\Phi_{c,d}$ with probability $\beta_{o}$
and $1-\beta_{o}$, respectively. Following the same steps in (31) and
substituting the UAVs’ density $\beta_{o}\lambda_{u}$ and
$(1-\beta_{o})\lambda_{u}$ and association probabilities here equal to 1, we
derive the availability probability $P_{a,th}$.

######  Theorem 2 (Availability Probability).

Availability probability under the independent thinning association policy is

$\displaystyle P_{\rm a,th}$ $\displaystyle=\beta_{o}\sum_{n=0}^{\frac{vB_{\rm
max}}{2p_{m}}}\bigg{(}\int_{0}^{\infty}g_{ev}(y\mid n)f_{c,ev}(y){\rm d}y$
$\displaystyle\quad\times\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}\frac{\bigg{(}\frac{\beta_{o}\lambda_{\mathrm{u}}}{\lambda_{c,ev}}\bigg{)}^{n}}{\left(b+\frac{\beta_{o}\lambda_{\mathrm{u}}}{\lambda_{c,ev}}\right)^{a+n+1}}\bigg{)}$
$\displaystyle+(1-\beta_{o})\sum_{n=0}^{\frac{vB_{\rm
max}}{2p_{m}}}\bigg{(}\int_{0}^{\infty}g_{d}(y\mid n)f_{c,d}(y){\rm d}y$
$\displaystyle\quad\times\frac{\Gamma(a+n+1)}{\Gamma(a)}\frac{b^{a}}{n!}\frac{\bigg{(}\frac{(1-\beta_{o})\lambda_{\mathrm{u}}}{\lambda_{c,d}}\bigg{)}^{n}}{\left(b+\frac{(1-\beta_{o})\lambda_{\mathrm{u}}}{\lambda_{c,d}}\right)^{a+n+1}}\bigg{)}.$
(36)

###### Proof:

Similar to Proof of Theorem 1, thus omitted here. ∎

Note that in these two association policies, the expectation of distance under
the biased distance policy is shorter than the other one. Hence, its
performance is expected to be slightly better than the other one. More details
are shown in Section VII.

In the following part of the paper, we assume that the optimal value of
$\beta_{o}$ and $\beta_{d}$ are used to maximize the coverage probability
given in the next part, and to simplify the notation we use $P_{a}$ as the
optimal value for both $P_{\rm a,bias}$ and $P_{\rm a,th}$.

##  V Coverage Probability

In this section, our goal is to analyze the coverage probability. To do so,
the distance between the BSs and users are required and given in the following
lemma. Recall that $\Phi_{u^{{}^{\prime}}}$ is the set formed by available
UAVs from the original point process $\Phi_{u}$, with thinning probability
$P_{a}$, the density of available UAVs is
$\lambda_{u}^{{}^{\prime}}=P_{a}\lambda_{u}$.

######  Lemma 10 (Distance Distribution).

The probability density function of the distances between the typical user and
the cluster UAV, the nearest available NL/LoS UAV, and the nearest TBS,
denoted by $f_{R_{u_{o}}}(r)$, $f_{R_{u^{{}^{\prime}}_{n}}}(r)$,
$f_{R_{u^{{}^{\prime}}_{l}}}(r)$ and $f_{R_{t}}(r)$, are respectively given by

$\displaystyle f_{R_{u_{o}}}(r)=\frac{2r}{r_{c}^{2}},\quad h\leq
r\leq\sqrt{r_{c}^{2}+h^{2}},$ (37) $\displaystyle
f_{R_{u^{{}^{\prime}}_{n}}}(r)=2\pi\lambda_{u}^{{}^{\prime}}P_{n}(r)r$
$\displaystyle\quad\times\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{r^{2}-h^{2}}}zP_{n}(\sqrt{z^{2}+h^{2}}){d}z\bigg{)},$
(38) $\displaystyle
f_{R_{u^{{}^{\prime}}_{l}}}(r)=2\pi\lambda_{u}^{{}^{\prime}}P_{l}(r)r$
$\displaystyle\quad\times\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{r^{2}-h^{2}}}zP_{l}(\sqrt{z^{2}+h^{2}}){d}z\bigg{)},$
(39) $\displaystyle f_{R_{t}}(r)=2\pi r\lambda_{t}\exp(-\pi\lambda_{t}r^{2}),$
(40)

where $P_{n}(r)$ and $P_{l}(r)$ are defined in (9).

Recall that we assume that the typical user associates with the cluster UAV if
it is available, otherwise, it associates with a nearby available UAV or TBS
based on the average received power. The following lemma gives the association
probability of the typical user.

######  Lemma 11 (Association Probability).

Given the serving BS located at $r$ away, the association probabilities of the
typical user are, respectively, given by

$\displaystyle\mathcal{A}_{\rm LoS}(r)$
$\displaystyle=\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{d_{n}^{2}(r)-h^{2}}}zP_{n}(\sqrt{z^{2}+h^{2}}){\rm
d}z\bigg{)}$
$\displaystyle\quad\times\exp\bigg{(}-2\pi\lambda_{t}d_{lt}^{2}(r)\bigg{)},$
(41) $\displaystyle\mathcal{A}_{\rm NLoS}(r)$
$\displaystyle=\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{d_{l}^{2}(r)-h^{2}}}zP_{l}(\sqrt{z^{2}+h^{2}}){\rm
d}z\bigg{)}$
$\displaystyle\quad\times\exp\bigg{(}-2\pi\lambda_{t}d_{nt}^{2}(r)\bigg{)},$
(42) $\displaystyle\mathcal{A}_{\rm TBS}(r)$
$\displaystyle=\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{d_{tl}^{2}(r)-h^{2}}}zP_{l}(\sqrt{z^{2}+h^{2}}){\rm
d}z\bigg{)}$
$\displaystyle\times\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{d_{tn}^{2}(r)-h^{2}}}zP_{n}(\sqrt{z^{2}+h^{2}}){\rm
d}z\bigg{)},$ (43)

where
$d_{lt}(r)=(\frac{\rho_{t}}{\rho_{u}\eta_{l}})^{\frac{1}{\alpha_{t}}}r^{\frac{\alpha_{l}}{\alpha_{t}}}$,
$d_{n}(r)=\max\bigg{(}(\frac{\eta_{n}}{\eta_{l}})^{\frac{1}{\alpha_{n}}}r^{\frac{\alpha_{l}}{\alpha_{n}}},h\bigg{)}$,
$d_{l}(r)=\max\bigg{(}h,(\frac{\eta_{l}}{\eta_{n}})^{\frac{1}{\alpha_{l}}}r^{\frac{\alpha_{n}}{\alpha_{l}}}\bigg{)}$,
$d_{nt}(r)=(\frac{\rho_{t}}{\rho_{u}\eta_{n}})^{\frac{1}{\alpha_{t}}}r^{\frac{\alpha_{n}}{\alpha_{t}}}$,
$d_{tl}(r)=\max\bigg{(}h,(\frac{\rho_{u}\eta_{l}}{\rho_{t}})^{\frac{1}{\alpha_{l}}}r^{\frac{\alpha_{t}}{\alpha_{l}}}\bigg{)}$
and
$d_{tn}(r)=\max\bigg{(}h,(\frac{\rho_{u}\eta_{n}}{\rho_{t}})^{\frac{1}{\alpha_{n}}}r^{\frac{\alpha_{t}}{\alpha_{n}}}\bigg{)}$.

###### Proof:

See Appendix -C. ∎

Laplace transform of the aggregate interference is the final requirement to
the coverage probability.

######  Lemma 12 (Laplace Transform).

Given the serving BS $b_{s}$, the Laplace transform of the interference is
given by

$\displaystyle\mathcal{L}_{I}(s,r)$ $\displaystyle=$
$\displaystyle\exp\biggl{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{a(\|x\|)}^{\infty}\bigg{[}1-\bigg{(}\frac{m_{n}}{m_{n}+s\eta_{n}\rho_{u}(z^{2}+h^{2})^{-\frac{\alpha_{n}}{2}}}\bigg{)}^{m_{n}}\bigg{]}$
$\displaystyle\quad\times zP_{n}(\sqrt{z^{2}+h^{2}}){d}z\biggl{)}$
$\displaystyle\times\exp\biggl{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{b(\|x\|)}^{\infty}\bigg{[}1-\bigg{(}\frac{m_{l}}{m_{l}+s\eta_{l}\rho_{u}(z^{2}+h^{2})^{-\frac{\alpha_{l}}{2}}}\bigg{)}^{m_{l}}\bigg{]}$
$\displaystyle\quad\times zP_{l}(\sqrt{z^{2}+h^{2}}){\rm d}z\biggl{)}$
$\displaystyle\times\exp\biggl{(}-2\pi\lambda_{t}\int_{c(\|x\|)}^{\infty}\bigg{[}1-(\frac{1}{1+s\rho_{t}z^{-\alpha_{t}}})\bigg{]}z{\rm
d}z\biggl{)},$ (44)

in which,

$\displaystyle a(\|x\|)=\left\\{\begin{aligned} 0,&\quad\text{\rm if}\quad
bs\in\Phi_{u_{o}},\\\ \sqrt{d_{n}^{2}(\|x\|)-h^{2}},&\quad\text{\rm if}\quad
bs\in\Phi_{u^{{}^{\prime}}_{l}},\\\ \sqrt{\|x\|^{2}-h^{2}},&\quad\text{\rm
if}\quad bs\in\Phi_{u_{n}^{{}^{\prime}}},\\\
\sqrt{d_{tn}^{2}(\|x\|)-h^{2}},&\quad\text{\rm if}\quad bs\in\Phi_{t},\\\
\end{aligned}\right.$ $\displaystyle b(\|x\|)=\left\\{\begin{aligned}
0,&\quad\text{\rm if}\quad bs\in\Phi_{u_{o}},\\\
\sqrt{\|x\|^{2}-h^{2}},&\quad\text{\rm if}\quad
bs\in\Phi_{u^{{}^{\prime}}_{l}},\\\
\sqrt{d_{l}^{2}(\|x\|)-h^{2}},&\quad\text{\rm if}\quad
bs\in\Phi_{u_{n}^{{}^{\prime}}},\\\
\sqrt{d_{tl}^{2}(\|x\|)-h^{2}},&\quad\text{\rm if}\quad bs\in\Phi_{t},\\\
\end{aligned}\right.$ $\displaystyle c(\|x\|)=\left\\{\begin{aligned}
0,&\quad\text{\rm if}\quad bs\in\Phi_{u_{o}},\\\ d_{lt}(\|x\|),&\quad\text{\rm
if}\quad bs\in\Phi_{u^{{}^{\prime}}_{l}},\\\ d_{nt}(\|x\|),&\quad\text{\rm
if}\quad bs\in\Phi_{u_{n}^{{}^{\prime}}},\\\ \|x\|,&\quad\text{\rm if}\quad
bs\in\Phi_{t},\\\ \end{aligned}\right.$

###### Proof:

The Laplace transform is computed using moment generating function (MGF) of
Gamma distribution, probability generation functional (PGFL) of inhomogeneous
PPP. For more details, please refer to [24]. ∎

Using the results derived thus far, the total coverage probability can be
obtained as given in the following theorem.

######  Theorem 3 (Coverage Probability).

When the typical user is associated with the cluster UAV, conditioned on LoS
or NLoS, the coverage probability is

$\displaystyle P_{\rm cov,U_{o,l}}$
$\displaystyle=\int_{h}^{\sqrt{h^{2}+r_{c}^{2}}}\sum_{k=0}^{m_{l}-1}\frac{(-m_{l}g_{l}(r))^{k}}{k!}$
$\displaystyle\times\frac{\partial^{k}}{\partial
s^{k}}\bigg{[}\mathcal{L}_{\sigma^{2}+I}(s,r)\bigg{]}_{s=m_{l}g_{l}(r)}P_{l}(r)\frac{2r}{r_{c}^{2}}{\rm
d}r,$ (45) $\displaystyle P_{\rm cov,U_{o,n}}$
$\displaystyle=\int_{h}^{\sqrt{h^{2}+r_{c}^{2}}}\sum_{k=0}^{m_{n}-1}\frac{(-m_{n}g_{n}(r))^{k}}{k!}$
$\displaystyle\times\frac{\partial^{k}}{\partial
s^{k}}\bigg{[}\mathcal{L}_{\sigma^{2}+I}(s,r)\bigg{]}_{s=m_{n}g_{n}(r)}P_{n}(r)\frac{2r}{r_{c}^{2}}{\rm
d}r.$ (46)

where $g_{l}(r)=\gamma(\eta_{l}\rho_{u})^{-1}r^{\alpha_{l}}$ and
$g_{n}(r)=\gamma(\eta_{n}\rho_{u})^{-1}r^{\alpha_{n}}$.

When the cluster UAV is unavailable, the coverage probability when associating
with the nearest LoS/NLoS available UAV $P_{\rm cov,\hat{U}_{o},l}$ and
$P_{\rm cov,\hat{U}_{o},n}$ can be given by

$\displaystyle P_{\rm cov,\hat{U}_{o},l}$
$\displaystyle=\int_{h}^{\infty}\mathcal{A}_{\rm
LoS}(r)\sum_{k=0}^{m_{l}-1}\frac{(-m_{l}g_{l}(r))^{k}}{k!}$
$\displaystyle\times\frac{\partial^{k}}{\partial
s^{k}}\mathcal{L}_{\sigma^{2}+I}(s,r)|_{s=m_{l}g_{l}(r)}\bigg{]}f_{R_{u^{{}^{\prime}},l}}(r){\rm
d}r,$ (47) $\displaystyle P_{\rm cov,\hat{U}_{o},n}$
$\displaystyle=\int_{h}^{\infty}\mathcal{A}_{\rm
NLoS}(r)\sum_{k=0}^{m_{n}-1}\frac{(-m_{n}g_{n}(r))^{k}}{k!}$
$\displaystyle\times\frac{\partial^{k}}{\partial
s^{k}}\mathcal{L}_{\sigma^{2}+I}(s,r)|_{s=m_{n}g_{n}(r)}\bigg{]}f_{R_{u^{{}^{\prime}},n}}(r){\rm
d}r,$ (48)

where $f_{R_{u^{{}^{\prime}},l}}(r)$ is given in (39).

When the typical UAV is unavailable, the coverage probability of associating
with the nearest TBS $P_{\rm cov,t}$ can be written as

$\displaystyle P_{\rm cov,t}$
$\displaystyle=\int_{0}^{+\infty}\mathcal{L}_{\sigma^{2}+I}(s,r)|_{s=\theta\rho_{t}^{-1}r^{\alpha_{t}}}\mathcal{A}_{\rm
TBS}(r)f_{R_{t}}(r){\rm d}r.$ (49)

###### Proof:

The coverage probability is derived by the fact that (i) the uniform
distribution of the users in the disk with radius $r_{c}$ and
$g_{l}(r)=\frac{\gamma\sigma^{2}}{\eta_{l}r^{-\alpha_{l}}\rho_{u}}$, (ii) the
definition: $\bar{F}_{G}(g)=\frac{\Gamma_{u}(m,g)}{\Gamma(m)}$, where
$\Gamma_{u}(m,g)=\int^{\infty}_{mg}t^{m-1}e^{-t}dt$ is the upper incomplete
Gamma function, and (iii) the definition
$\frac{\Gamma_{u}(m,g)}{\Gamma(m)}=\exp(-g)\sum^{m-1}_{k=0}\frac{g^{k}}{k!}$.
∎

TABLE II: Table of Parameters

Parameter | Symbol | Simulation Value
---|---|---
Line density, point density |  $\lambda_{l}$; $\lambda_{p,ev}$, $\lambda_{p,d}$ |  $24/\pi\times 10^{-3}$; $2.1\times 10^{-5}$, $1.05\times 10^{-5}$
Charging station density |  $\lambda_{c,d}$, $\lambda_{c,ev}$ |  $0.5$/km2, $0.25$/km2
TBS density, UAV density |  $\lambda_{t}$, $\lambda_{u}$ |  $1$/km2, $4$/km2
Average and deviation of SOC |  $\mu$, $\sigma$ | 3 to 0.6
EV charging rate | $P_{\rm cha}$ | 120 kw
CV (or MWCV) tessellation fitting parameters | $a,b$ | 3.5
Traveling-related power | $p_{m}$ | 161.8 W
Service-related power | $p_{s}$ | 177.5 W
UAV altitude | $h$ | 100 m
UAV velocity | $v$ | 18.46 m/s
Battery capacity | $B_{\rm max}$ | 177.6 W$\cdot$H
Charging time |  $T_{\rm ch,d,ev}$, $T_{\rm ch,d,d}$ | 30, 5 min
Radius of MCP disk | $r_{c}$ | 100 m
N/LoS environment variable | $c_{1},c_{2}$ | 25.27,0.2
Transmission power |  $\rho_{u}$, $\rho_{t}$ | 0.2 W, 10 W
SINR threshold | $\gamma$ | 0 dB
Noise power | $\sigma^{2}_{n}$ |  $10^{-9}$ W
N/LoS and active charging station path-loss exponent | $\alpha_{n},\alpha_{l},\alpha_{t}$ | $4,2.1,4$
N/LoS fading gain | $m_{n},m_{l}$ | $1,3$
N/LoS additional loss | $\eta_{n},\eta_{l}$ |  $-20,0$ dB
Cost | $c_{\rm vol},c_{\rm main}$ |  $0.2$ USD/kwh, $1131$ USD/year
Weight |  $w_{\rm wait},w_{\rm inf,ev}$, $w_{\rm cov},w_{c}$ | (-1/3,2/3) or (-2/3,1/3), (8,1) or (6,1)

It can be seen that the above coverage probability equations require
evaluating higher order of derivatives of the Laplace transform. Using the
upper bound of the CDF of the Gamma distribution [38], the above equations can
be approximated in the following lemma.

######  Lemma 13 (Approximated Coverage Probability).

Following the steps in [39] and [40], the approximate coverage probabilities
are given by,

$\displaystyle P_{\rm
cov,U_{o,\\{l,n\\}}}=\sum_{k=1}^{m_{\\{l,n\\}}}\binom{m_{\\{l,n\\}}}{k}(-1)^{k+1}\int_{h}^{\sqrt{h^{2}+r_{c}^{2}}}$
$\displaystyle\quad\mathcal{L}_{\sigma^{2}+I}(k\beta_{2}(m_{\\{l,n\\}})m_{\\{l,n\\}}g_{\\{l,n\\}}(r),r)P_{\\{l,n\\}}(r)\frac{2r}{r_{c}^{2}}{\rm
d}r,$ (50) $\displaystyle P_{\rm
cov,\hat{U}_{o,\\{l,n\\}}}=\sum_{k=1}^{m_{\\{l,n\\}}}\binom{m_{\\{l,n\\}}}{k}(-1)^{k+1}\int_{h}^{\infty}\mathcal{A}_{\rm\\{L,NL\\}oS}(r)$
$\displaystyle\times
f_{R_{u^{{}^{\prime}}_{\\{l,n\\}}}}(r)\mathcal{L}_{\sigma^{2}+I}(k\beta_{2}(m_{\\{l,n\\}})m_{\\{l,n\\}}g_{\\{l,n\\}}(r),r){\rm
d}r,$ (51)

in which $\beta_{2}=(m_{\\{l,n\\}}!)^{(}-\frac{1}{m_{\\{l,n\\}}})$.

###### Proof:

The approximation is derive by using the definition of CCDF of Gamma function,
upper imcomplete Gamma function and Binomial theorem. ∎

##  VI Infrastructure Sharing

Figure 5: Analysis and simulation results of waiting time of UAVs under
different average interarrival time of EVs in the case of sharing and no
sharing. $c$ and $m$ are charging slots and the number of UAVs that can be
charged together in EV charging stations. The dash lines are the waiting time
of UAVs in their own charging stations without infrastructure sharing, and the
sold lines are with the infrastructure sharing: increasing $\beta_{\\{d,o\\}}$
(the direction of the arrows) until the optimal value (maximize the coverage
probability). Figure 6: Analysis and simulation results of waiting time of EVs
under different average interarrival time in the case of sharing and no
sharing. $c$ and $m$ are charging slots and the number of UAVs that can be
charged together in EV charging stations. The dash curves are the worst case
of EVs, help to offload a large scale of UAVs to maximize the coverage
probability and the solid lines are for the waiting time of EVs without
infrastructure sharing. Figure 7: Analysis and simulation results of coverage
probability of UAVs under different average interarrival time in the case of
sharing and no sharing. $c$ and $m$ are charging slots and the number of UAVs
that can be charged together in EV charging stations. The minima and maxima
curves are the coverage probability of UAVs without infrastructure sharing and
upper bound of given system (no waiting time and traveling time).

In this section, we relate the system performance and the cost. While we show
that infrastructure sharing can improve the coverage probability of UAVs,
however, the operators also care about the fee they paid for the EV
infrastructure sharing and the cost of extra dedicated charging stations. The
same for EV operators, while they earn extra money from sharing, their own
service quality decreases.

As mentioned in Definition 5, UAVs’ operators pay for the EV infrastructure,
we simply consider this fee is composed of voltage regulation and maintenance
cost,

$\displaystyle C_{\rm inf,d}=C_{\rm vol}+C_{\rm main}$
$\displaystyle=\frac{365\times 24\times
60\times\mathbb{E}[N_{d,ev}]}{\mathbb{E}[T_{\rm ser}+T_{\rm ch}+T_{\rm
tra}+T_{\rm w}]}\cdot B_{\rm max}\cdot c_{\rm vol}+c\cdot c_{\rm main},$ (52)

where $\frac{24\times 60}{\mathbb{E}[T_{\rm ser}+T_{\rm ch}+T_{\rm tra}+T_{\rm
w}]}\times B_{\rm max}$ is the total energy UAV needed in a day, $c_{\rm vol}$
is the price EV operators are paid and $c_{\rm main}$ is the maintenance cost
per charger [34].

EVs providers solve their optimization problem first: how many UAVs they can
help to offload from UAV charging stations to maximize the profit from UAVs,
given that the extra waiting time of EVs are tolerable. Then, UAVs operators
solve their problem: based on the charging infrastructure EVs shared, how to
minimize the cost while ensure an acceptable system performance, pay more for
building their charging stations to achieve a better performance, or pay less
for less dedicated charging stations and a lower system performance.

Observing that the objective functions (12) and (13) are functions of the
electricity price, which is different at each hour of the day and varies from
the region, and weights of objectives, it is difficult to obtain an exact
value and say it provides the best performance. However, providing a general
model to analyze this trade-off and considering the particularity of each
realistic scenario is the most suitable solution to follow. Besides, the
proposed system model and analysis can be easily extended to difference
scenarios. For instance, given the electricity price is different of the day,
EVs’ operators can change offloading UAV densities and UAVs’ operators pay
different amounts of fees for infrastructure sharing according to the
electricity price, and then build their dedicated charging stations by jointly
considering multiple offloading ratios. We do admit that such optimization
highly depends on operators’ decision and is parameter-based, our goal here is
to capture the constraint of cost on system performance: deploying more
charging stations do achieve better performances, but all these better
performances are expected to cost more.

##  VII Numerical Results

In this section we validate our analytical results with simulations and
evaluate the improvement of system performance using infrastructure sharing.
Unless stated otherwise, we use the simulation parameters as listed herein
Table II.

For the simulation of the considered setup, we apply Monte-Carlo simulations
with a large number of iterations to ensure accuracy. In each iteration, we
first generate $10^{4}$ exponential distributed random variables to simulate
the arrival processes of EVs. To compute the number of UAVs associate with
EV/UAV charging stations, we generate three independent PPPs and compute the
association cell. The average waiting time of EVs and UAVs are obtained based
on the two association policies and serving policies. We then generate another
independent PPP realizations to model the locations of user cluster centers
and TBSs and generate the locations of the reference user. Conditioned on the
typical UAVs located at the origin, we derive the coverage probability. In all
the figures we plot, markers are the simulation results and curves are the
analysis results.

Figure 8: Coverage probability under different values of $\beta_{\\{d,o\\}}$.
The dash curves are for $m=2$ and the solid lines are for $m=1$. Along with
the arrow direction, $c$ increases. Figure 9: Results of objective functions.
We consider two cases: operators care more about performance ($w_{\rm
wait}>w_{\rm inf,ev}$ ,$w_{\rm cov}>w_{\rm c},w_{\rm inf,d}$) or more about
profit, and corresponding decisions denoted by
$\beta_{d,p},\Delta\lambda_{d,p}$ and $\beta_{d,c},\Delta\lambda_{d,c}$,
respectively. All the performance-based decisions are plot in blue and sold
lines with circle markers, and all the cost-based decisions are plot in red
and dash lines with upward-pointing triangle markers.

In Fig. 5 and 6, we plot the waiting time of EVs and UAVs with/without
infrastructure sharing: we increase $\beta_{\\{d,o\\}}$ from $0$, which means
no infrastructure sharing, till the optimal values, which maximize the
coverage probability, from the perspective of UAVs. Here we only plot the
worst curve of the waiting time of EVs, under optimal values of
$\beta_{\\{d,o\\}}$ (See Fig. 8 for details) since the waiting time does not
change a lot. Interestingly, while the waiting time of UAVs drops
dramatically, the waiting time of EVs does not increase a lot, as mentioned in
Remark 2, which is because that $T_{w,ev}$ is only a function of $N_{d,ev}$
when $N_{d,ev}$ is a small value, while it stays constant when $N_{d,ev}$ is
large, especially when the quality of EV charging station is high (large
capacity and can charge multiple UAVs).

Fig. 7 shows the benefit of infrastructure sharing for UAVs, as it improves
the coverage probability and approaches the system upper bound where $T_{w}$
and $T_{\rm ch}$ is 0. However, the coverage probability in the case of
infrastructure sharing cannot reach the system upper bound, owing to a longer
charging time in EV charging stations. That is, even though the waiting time
is approximately 0 and traveling time is negligible, gap of availability
probability (which is a time fraction denoted that UAV is available and can
provide service) exists between using infrastructure sharing and deploying
more dedicated charging stations. Hence it requires a trade-off between better
performance and huge cost.

Fig. 8 shows the optimal values for both $\beta_{d}$ and $\beta_{o}$. As we
offloading UAVs to EV charging stations, the waiting time in UAV charging
stations decreases, hence the coverage probability increases. However, if we
offload a large fraction of UAVs to EV charging stations, UAVs start to have a
traffic and long waiting time in EV charging stations, then the coverage
performance decreases. In both Fig. 7 and Fig. 8, we show that the system
performance based on biased distance association policy is slightly better
than independent thinning association policy, which is because of the
traveling distance in the first association policy is shorter than than the
second one.

Fig. 9 shows the results of optimal infrastructure sharing strategy. While we
find that the cost of installing extra dedicated charging stations is much
higher than paying for the infrastructure sharing, we assume that UAVs’
operators install extra charging stations (the density is denoted by
$\Delta\lambda_{d}$) to improve its system performance based on the decision
of $\beta_{d}$ (here we use $\beta_{d}$ to denote both $\beta_{d}$ and
$\beta_{o}$, for simplification). With that being said, since we find
infrastructure sharing fee is much lower than installing new charging
stations, UAVs’ operators deploy their dedicated charging stations based on
the sharing strategy of EVs’ operators. Besides, the sharing fee paid for EVs’
operators are not function of $\Delta\lambda_{d}$, it is a constant added up
to the total cost. Hence, the optimization problem of EV, (12), is actually
about a trade-off between performance and cost based on infrastructure
sharing. If the operators care more about their services, less sharing
($\beta_{d,p}<\beta_{d,c}$) and more dedicated charging stations
($\Delta\lambda_{d,c}<\Delta\lambda_{d,p}$) deployed for EVs and UAVs
respectively (blue and sold lines with circle markers).

As shown, when the quality of EV charging stations is poor (low capacity and
cannot charge multiple UAVs), infrastructure sharing is not a good strategy
since the waiting time of EVs increases dramatically. Hence, UAV operators
need to build more dedicated charging stations to achieve an acceptable
performance. When the quality of EV charging stations is high ($c=3$, $r=2$),
EV charging stations can help to offload a larger scale of UAVs while still
maintain the performance compared with low quality case.

##  VIII Conclusion

In this paper, we presented an optimization problem to analyze the possibility
of sharing charging infrastructure in EV and UAV-involved networks. We first
approximated waiting time for both EVs and UAVs in continuous time, and
derived the availability probability of UAVs in a more accurate method,
compared with existing literature. We then solve our optimization problem from
the perspective of EVs’ and UAVs’ operators and based on weights of service
quality and profit or cost. Our results show some interesting system insights:
if the charging stations are high quality, charging infrastructure sharing can
benefit both UAVs’ and EVs’ operators.

###  -A Waiting Time when $N_{d,ev}<mc(1+\frac{T_{\rm ser}}{T_{\rm ch,d,ev}})$

In this part, we provide the waiting time equations for
$N_{d,ev}<mc(1+\frac{T_{\rm ser}}{T_{\rm ch,d,ev}})$. As mentioned in Remark
1, the probability of $N_{d,ev}<mc(1+\frac{T_{\rm ser}}{T_{\rm ch,d,ev}})$
happened is low and it is difficult to compute. Hence, we also present an
approximation here. The following equations derived by conditioned on the
number of EVs arrival during the serving time interval of the typical UAV, and
then take the expectation.

In the case of $N_{d,ev}<mc(1+\frac{T_{\rm ser}}{T_{\rm ch,d,ev}})$, if
$n\geq\frac{c(T_{\rm ser}-T_{\rm ch,d,ev})-T_{\rm
ch,d,ev}\frac{N_{d,ev}}{m}}{\mathbb{E}[T_{\rm ch,ev}]}$,

$\displaystyle T_{\rm w,d,ev|N_{d,ev}}$
$\displaystyle=\mathbb{E}_{n}\bigg{[}\frac{n\mathbb{E}[T_{\rm
ch,ev}]}{c}+\bigg{(}\frac{\mathbb{E}[T_{\rm ch,ev}]\mu_{e}}{c}-1\bigg{)}$
$\displaystyle\times\bigg{(}T_{\rm ser}-(\frac{N_{d,ev}}{mc}-1)T_{\rm
ch,d,ev}\bigg{)}\bigg{]},$

else,

$\displaystyle T_{\rm
w,d,ev|N_{d,ev}}=\mathbb{E}_{n}\bigg{[}\sum_{x=0}^{\infty}(T_{\rm
gap|n}\mu_{e})^{x}\frac{\exp(-T_{\rm gap|n}\mu_{e})}{x!}T_{w|x}\bigg{]},$

where $n\sim Exp\bigg{(}T_{\rm w,d,ev|N_{d,ev}}+T_{\rm
ch,d,ev}\frac{N_{d,ev}}{c}\bigg{)}$ and

$\displaystyle T_{\rm gap|n}=T_{\rm ser}-(\frac{N_{d,ev}}{mc}-1)T_{\rm
ch,d}-\frac{\mathbb{E}[nT_{\rm ch,ev}]}{c},$ $\displaystyle
T_{w|x}=\int_{0}^{T_{\rm gap|n}}\int_{t_{1}}^{T_{\rm
gap|n}}\cdots\int_{t_{x}}^{T_{\rm
gap|n}}\frac{1}{c}f_{t_{1},t_{2}\cdots,t_{x}}(t_{1},t_{2},\cdots)$
$\displaystyle\quad\max\bigg{(}\max(\max(t_{1}^{{}^{\prime}}+\mathbb{E}[T_{\rm
ch,ev}],t_{2})+\cdots,t_{x})+\mathbb{E}[T_{\rm ch,ev}]$ $\displaystyle\quad-
T_{\rm gap|n},0\bigg{)}{\rm d}t_{x}\cdots{\rm d}t_{1},$ (53)

when $x=1,2,3$ the above equation $T_{w|x}$ is,

$\displaystyle\int_{0}^{T_{\rm gap|n}}\frac{1}{c}\frac{1}{T_{\rm
gap|n}}\max(0,t_{1}^{{}^{\prime}}+\mathbb{E}[T_{\rm ch,ev}]-T_{\rm gap|n}){\rm
d}t_{1},$ (54) $\displaystyle\int_{0}^{T_{\rm gap|n}}\int_{t_{1}}^{T_{\rm
gap|n}}\frac{1}{c}\frac{2}{T_{\rm
gap|n}^{2}}\max(\max(t_{1}^{{}^{\prime}}+\mathbb{E}[T_{\rm ch,ev}],t_{2})$
$\displaystyle\qquad\qquad\qquad+\mathbb{E}[T_{\rm ch,ev}]-T_{\rm
gap|n},0){\rm d}t_{2}{\rm d}t_{1},$ (55) $\displaystyle\int_{0}^{T_{\rm
gap|n}}\int_{t_{1}}^{T_{\rm gap|n}}\int_{t_{2}}^{T_{\rm
gap|n}}\frac{1}{c}\frac{6}{T_{\rm
gap|n}^{3}}\max(\max(\max(t_{1}^{{}^{\prime}}$
$\displaystyle+\mathbb{E}[T_{\rm ch,ev}],t_{2})+\mathbb{E}[T_{\rm
ch,ev}],t_{3})+\mathbb{E}[T_{\rm ch,ev}]-T_{\rm gap|n},0){\rm d}t_{3}{\rm
d}t_{2}{\rm d}t_{1},$ (56)

for $x>3$, using the following lower bound approximation,

$\displaystyle\int_{0}^{T_{\rm gap|n}}\frac{x}{T_{\rm gap|n}}$
$\displaystyle\frac{1}{c}(1-\frac{t_{1}}{T_{\rm gap|n}})^{x-1}$
$\displaystyle\max(0,t_{1}^{{}^{\prime}}+\frac{x}{c}\mathbb{E}[T_{\rm
ch,ev}]-T_{\rm gap|n}){\rm d}t_{1},$ (57)

in which,

$\displaystyle t_{1}^{{}^{\prime}}$
$\displaystyle=(t_{1}<T)(T+t_{1})+(t_{1}>T)t_{1},$ $\displaystyle T$
$\displaystyle=\frac{n\mu_{e}\mathbb{E}^{2}[T_{\rm ch,ev}]}{c^{2}}.$ (58)

###  -B Proof of Theorem 1

Conditioned on $N$ UAVs fill in the typical association cell, availability
probability is given by,

$\displaystyle\mathbb{P}(\rm\mathcal{A}\mid N)$
$\displaystyle=\mathcal{A}_{ev}(\beta)\mathbb{E}_{Y_{ev}}\bigg{[}\frac{T_{\rm
ser}(y)}{T_{\rm ser}(y)+T_{\rm ch,d,ev}+T_{\rm w,d,ev\mid N_{d,ev}}+2T_{\rm
tra}(y)}\bigg{]}$
$\displaystyle+\mathcal{A}_{d}(\beta)\mathbb{E}_{Y_{d}}\bigg{[}\frac{T_{\rm
ser}(y)}{T_{\rm se}(y)+T_{\rm ch,d,d}+T_{\rm w,d,d\mid N_{d,d}}+2T_{\rm
tra}(y)}\bigg{]}$
$\displaystyle=\mathcal{A}_{d}(\beta)\mathbb{E}_{Y_{d}}\bigg{[}\frac{vB_{\rm
max}-2Y_{d}p_{m}}{vB_{\rm max}-2Y_{d}(p_{m}-p_{s})+vp_{s}(T_{\rm
ch,d,d}+T_{\rm w,d,d\mid N_{d,d}})}\bigg{]}$
$\displaystyle+\mathcal{A}_{ev}(\beta)$
$\displaystyle\quad\mathbb{E}_{Y_{ev}}\bigg{[}\frac{vB_{\rm
max}-2Y_{ev}p_{m}}{vB_{\rm max}-2Y_{ev}(p_{m}-p_{s})+vp_{s}(T_{\rm
ch,d,ev}+T_{\rm w,d,ev\mid N_{d,ev}})}\bigg{]}$
$\displaystyle=\int_{0}^{\frac{vB_{\rm max}}{2p_{m}}}\bigg{[}\frac{vB_{\rm
max}-2yp_{m}}{vB_{\rm max}-2y(p_{m}-p_{s})+vp_{s}(T_{\rm ch,d,ev}+T_{\rm
w,d\mid N_{d,ev}})}\bigg{]}$
$\displaystyle\quad\times\bar{F}_{c,d}(\frac{y}{\beta})f_{c,ev}(y){\rm d}y$
$\displaystyle+\int_{0}^{\frac{vB_{\rm max}}{2p_{m}}}\bigg{[}\frac{vB_{\rm
max}-2yp_{m}}{vB_{\rm max}-2y(p_{m}-p_{s})+vp_{s}(T_{\rm ch,d,d}+T_{\rm
w,d,d\mid N_{d,d}})}\bigg{]}$
$\displaystyle\quad\times\bar{F}_{c,ev}(y\beta)f_{c,d}(y){\rm d}y,$ (59)

then the unconditional availability probability is given by

$\displaystyle P_{\rm a,bias}=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{P}(\rm\mathcal{A}\mid
N)\mathbb{P}(N=n),$

proof completes by taking the expectation over $N_{d,ev}$ and $N_{d,d}$,
respectively.

###  -C Proof of Lemma 11

When the cluster UAV is not available, users associate with a nearby available
UAV or the nearest TBSs, which provides the strongest average received power.
The probability of associating with a nearby LoS UAV is,

$\displaystyle\mathcal{A}_{\rm LoS}(r)=\mathcal{A}_{\rm LoS-
NLoS}(r)\mathcal{A}_{\rm LoS-TBS}(r)$
$\displaystyle=\mathbb{P}(\eta_{l}\rho_{u}r^{-\alpha_{l}}>\eta_{n}\rho_{u}R_{u_{n}^{{}^{\prime}}}^{-\alpha_{n}})\mathbb{P}(\rho_{u}\eta_{l}r^{-\alpha_{l}}>\rho_{t}R_{t}^{-\alpha_{t}})$
$\displaystyle=\mathbb{P}\bigg{(}R_{u_{n}^{{}^{\prime}}}>(\frac{\eta_{n}}{\eta_{l}})^{\frac{1}{\alpha_{n}}}r^{\frac{\alpha_{l}}{\alpha_{n}}}\bigg{)}\mathbb{P}\bigg{(}R_{t}>(\frac{\rho_{t}}{\rho_{u}\eta_{l}})^{\frac{1}{\alpha_{t}}}r^{\frac{\alpha_{l}}{\alpha_{t}}}\bigg{)}$
$\displaystyle=\exp\bigg{(}-2\pi\lambda_{u}^{{}^{\prime}}\int_{0}^{\sqrt{d_{n}^{2}(r)-h^{2}}}zP_{n}(\sqrt{z^{2}+h^{2}}){\rm
d}z\bigg{)}$
$\displaystyle\quad\times\exp\bigg{(}-2\pi\lambda_{t}d_{lt}^{2}(r)\bigg{)},$
(60)

where $d_{lt}(r)$ and $d_{n}(r)$ are defined in Lemma 11. $\mathcal{A}_{\rm
NLoS}(r)$ and $\mathcal{A}_{\rm TBS}(r)$ follow the same way, therefore
omitted here.

## References

  * [1] L. Gan, U. Topcu, and S. H. Low, “Optimal decentralized protocol for electric vehicle charging,” _IEEE Transactions on Power Systems_ , vol. 28, no. 2, pp. 940–951, 2012.
  * [2] S. Liu, X. Xia, Y. Cao, Q. Ni, X. Zhang, and L. Xu, “Reservation-based EV charging recommendation concerning charging urgency policy,” _Sustainable Cities and Society_ , vol. 74, p. 103150, 2021.
  * [3] S. Sekander, H. Tabassum, and E. Hossain, “Multi-tier drone architecture for 5G/B5G cellular networks: Challenges, trends, and prospects,” _IEEE Communications Magazine_ , vol. 56, no. 3, pp. 96–103, 2018.
  * [4] M. Mozaffari, W. Saad, M. Bennis, Y.-H. Nam, and M. Debbah, “A tutorial on UAVs for wireless networks: Applications, challenges, and open problems,” _IEEE communications surveys & tutorials_, vol. 21, no. 3, pp. 2334–2360, 2019.
  * [5] B. Li, Z. Fei, and Y. Zhang, “UAV communications for 5G and beyond: Recent advances and future trends,” _IEEE Internet of Things Journal_ , vol. 6, no. 2, pp. 2241–2263, 2018.
  * [6] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: Opportunities and challenges,” _IEEE Communications Magazine_ , vol. 54, no. 5, pp. 36–42, 2016.
  * [7] Y. Qin, M. A. Kishk, and M.-S. Alouini, “Drone charging stations deployment in rural areas for better wireless coverage: Challenges and solutions,” _IEEE Internet of Things Magazine_ , vol. 5, no. 1, pp. 148–153, 2022.
  * [8] H. Liao, Z. Zhou, W. Kong, Y. Chen, X. Wang, Z. Wang, and S. Al Otaibi, “Learning-based intent-aware task offloading for air-ground integrated vehicular edge computing,” _IEEE Transactions on Intelligent Transportation Systems_ , 2020.
  * [9] N. Cheng, F. Lyu, W. Quan, C. Zhou, H. He, W. Shi, and X. Shen, “Space/aerial-assisted computing offloading for IoT applications: A learning-based approach,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 5, pp. 1117–1129, 2019.
  * [10] M. Matracia, M. A. Kishk, and M.-S. Alouini, “On the topological aspects of UAV-assisted post-disaster wireless communication networks,” _IEEE Communications Magazine_ , vol. 59, no. 11, pp. 59–64, 2021.
  * [11] M. Kishk, A. Bader, and M.-S. Alouini, “Aerial base station deployment in 6G cellular networks using tethered drones: The mobility and endurance tradeoff,” _IEEE Vehicular Technology Magazine_ , vol. 15, no. 4, pp. 103–111, 2020.
  * [12] Q. Cui, Y. Weng, and C.-W. Tan, “Electric vehicle charging station placement method for urban areas,” _IEEE Transactions on Smart Grid_ , vol. 10, no. 6, pp. 6552–6565, 2019.
  * [13] N. Shaukat, B. Khan, S. Ali, C. Mehmood, J. Khan, U. Farid, M. Majid, S. Anwar, M. Jawad, and Z. Ullah, “A survey on electric vehicle transportation within smart grid system,” _Renewable and Sustainable Energy Reviews_ , vol. 81, pp. 1329–1349, 2018. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1364032117307190
  * [14] T. Sweda and D. Klabjan, “An agent-based decision support system for electric vehicle charging infrastructure deployment,” in _2011 IEEE Vehicle Power and Propulsion Conference_. IEEE, 2011, pp. 1–5.
  * [15] S. Hussain, Y.-S. Kim, S. Thakur, and J. G. Breslin, “Optimization of waiting time for electric vehicles using a fuzzy inference system,” _IEEE Transactions on Intelligent Transportation Systems_ , pp. 1–12, 2022.
  * [16] T. Oda, M. Aziz, T. Mitani, Y. Watanabe, and T. Kashiwagi, “Mitigation of congestion related to quick charging of electric vehicles based on waiting time and cost–benefit analyses: A japanese case study,” _Sustainable cities and society_ , vol. 36, pp. 99–106, 2018.
  * [17] N. Sathaye and S. Kelley, “An approach for the optimal planning of electric vehicle infrastructure for highway corridors,” _Transportation Research Part E: Logistics and Transportation Review_ , vol. 59, pp. 15–33, 2013.
  * [18] S. Chen and L. Tong, “iEMS for large scale charging of electric vehicles: Architecture and optimal online scheduling,” in _2012 IEEE Third International Conference on Smart Grid Communications (SmartGridComm)_ , 2012, pp. 629–634.
  * [19] J. J. Yu, V. O. Li, and A. Y. Lam, “Optimal V2G scheduling of electric vehicles and unit commitment using chemical reaction optimization,” in _2013 IEEE Congress on Evolutionary Computation_ , 2013, pp. 392–399.
  * [20] A. Y. Lam, K.-C. Leung, and V. O. Li, “Capacity management of vehicle-to-grid system for power regulation services,” in _2012 IEEE Third International Conference on Smart Grid Communications (SmartGridComm)_ , 2012, pp. 442–447.
  * [21] H. ElSawy, A. Sultan-Salem, M.-S. Alouini, and M. Z. Win, “Modeling and analysis of cellular networks using stochastic geometry: A tutorial,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 1, pp. 167–203, 2017\.
  * [22] H. ElSawy, E. Hossain, and M. Haenggi, “Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey,” _IEEE Communications Surveys Tutorials_ , vol. 15, no. 3, pp. 996–1019, 2013.
  * [23] Y. Qin, M. A. Kishk, and M.-S. Alouini, “Performance evaluation of UAV-enabled cellular networks with battery-limited drones,” _IEEE Communications Letters_ , vol. 24, no. 12, pp. 2664–2668, 2020.
  * [24] ——, “On the influence of charging stations spatial distribution on aerial wireless networks,” _IEEE Transactions on Green Communications and Networking_ , vol. 5, no. 3, pp. 1395–1409, 2021.
  * [25] M.-A. Lahmeri, M. A. Kishk, and M.-S. Alouini, “Stochastic geometry-based analysis of airborne base stations with laser-powered UAVs,” _IEEE Communications Letters_ , vol. 24, no. 1, pp. 173–177, 2020.
  * [26] S. Sekander, H. Tabassum, and E. Hossain, “Statistical performance modeling of solar and wind-powered UAV communications,” _IEEE Transactions on Mobile Computing_ , 2020.
  * [27] M. A. Kishk, A. Bader, and M.-S. Alouini, “On the 3-D placement of airborne base stations using tethered UAVs,” _IEEE Transactions on Communications_ , vol. 68, no. 8, pp. 5202–5215, 2020.
  * [28] Z. Lou, A. Elzanaty, and M.-S. Alouini, “Green tethered UAVs for EMF-aware cellular networks,” _IEEE Transactions on Green Communications and Networking_ , vol. 5, no. 4, pp. 1697–1711, 2021.
  * [29] D.-E. Meddour, T. Rasheed, and Y. Gourhant, “On the role of infrastructure sharing for mobile network operators in emerging markets,” _Computer networks_ , vol. 55, no. 7, pp. 1576–1591, 2011.
  * [30] T. Sanguanpuak, S. Guruacharya, E. Hossain, N. Rajatheva, and M. Latva-aho, “Infrastructure sharing for mobile network operators: Analysis of trade-offs and market,” _IEEE Transactions on Mobile Computing_ , vol. 17, no. 12, pp. 2804–2817, 2018.
  * [31] A. Bousia, E. Kartsakli, A. Antonopoulos, L. Alonso, and C. Verikoukis, “Game-theoretic infrastructure sharing in multioperator cellular networks,” _IEEE Transactions on Vehicular Technology_ , vol. 65, no. 5, pp. 3326–3341, 2015.
  * [32] R. Jurdi, A. K. Gupta, J. G. Andrews, and R. W. Heath, “Modeling infrastructure sharing in mmWave networks with shared spectrum licenses,” _IEEE Transactions on Cognitive Communications and Networking_ , vol. 4, no. 2, pp. 328–343, 2018.
  * [33] F. Fund, S. Shahsavari, S. S. Panwar, E. Erkip, and S. Rangan, “Spectrum and infrastructure sharing in millimeter wave cellular networks: An economic perspective,” _arXiv preprint arXiv:1605.04602_ , 2016.
  * [34] J. Domnguez-Navarro, R. Dufo-Lpez, J. Yusta-Loyo, J. Artal-Sevil, and J. Bernal-Agustín, “Design of an electric vehicle fast-charging station with integration of renewable energy and storage systems,” _International Journal of Electrical Power & Energy Systems_, vol. 105, pp. 46–58, 2019.
  * [35] Y. Zeng, J. Xu, and R. Zhang, “Energy minimization for wireless communication with rotary-wing UAV,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 4, pp. 2329–2345, 2019.
  * [36] A. Al-Hourani, S. Kandeepan, and S. Lardner, “Optimal LAP altitude for maximum coverage,” _IEEE Wireless Communications Letters_ , vol. 3, no. 6, pp. 569–572, 2014.
  * [37] S. Singh, H. S. Dhillon, and J. G. Andrews, “Offloading in heterogeneous networks: Modeling, analysis, and design insights,” _IEEE transactions on wireless communications_ , vol. 12, no. 5, pp. 2484–2497, 2013.
  * [38] H. Alzer, “On some inequalities for the incomplete Gamma function,” _Mathematics of Computation_ , vol. 66, no. 218, pp. 771–778, 1997.
  * [39] M. Alzenad and H. Yanikomeroglu, “Coverage and rate analysis for vertical heterogeneous networks (VHetNets),” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 12, pp. 5643–5657, 2019.
  * [40] T. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 2, pp. 1100–1114, 2014.