#  Effects of 3D Position Fluctuations on Air-to-Ground mmWave UAV
Communications

Cunyan Ma, Xiaoya Li, Chen He, Member, IEEE, Jinye Peng, and Z. Jane Wang,
Fellow, IEEE

###### Abstract

Millimeter wave (mmWave)-based unmanned aerial vehicle (UAV) communication is
a promising candidate for future communications due to its flexibility and
sufficient bandwidth. However, random fluctuations in the position of hovering
UAVs will lead to random variations in the blockage and signal-to-noise ratio
(SNR) of the UAV-user link, thus affecting the quality of service (QoS) of the
system. To assess the impact of UAV position fluctuations on the QoS of air-
to-ground mmWave UAV communications, this paper develops a tractable
analytical model that jointly captures the features of three-dimensional (3D)
position fluctuations of hovering UAVs and blockages of mmWave (including
static, dynamic, and self-blockages). With this model, we derive the closed-
form expressions for reliable service probability respective to blockage
probability of UAV-user links, and coverage probability respective to SNR,
respectively. The results indicate that the greater the position fluctuations
of UAVs, the lower the reliable service probability and coverage probability.
The degradation of these two evaluation metrics confirms that the performance
of air-to-ground mmWave UAV systems largely depends on the UAV position
fluctuations, and the stronger the fluctuation, the worse the QoS. Finally,
Monte Carlo simulations demonstrate the above results and show UAVs’ optimal
location to maximize the reliable service and coverage probability,
respectively.

###### Index Terms:

Millimeter wave, unmanned aerial vehicle communications, position
fluctuations, blockages, QoS.

##  I Introduction

The advantage of unmanned aerial vehicle (UAV) makes it more likely to be
employed as the aerial base station (BS) for future communications, which will
significantly satisfy the diversified requirements on data rate, transmission
delay, system capacity, etc.[1, 2]. Compared with traditional BSs, UAVs have
higher flexibility and lower cost. Therefore, UAV communications have
attracted increasing attention and are particularly suitable for on-demand
scenarios, such as hot spots or disaster sites where existing communications
infrastructures are overloaded or damaged [3]. Meanwhile, to achieve ultra-
high data rates, more and more researchers consider millimeter wave (mmWave)
with large available bandwidth as the carrier frequency of UAV
communications[4]. Such communications are hereinafter called as mmWave UAV
communications. mmWave UAV communications have significant advantages in
flexibility, system capacity, etc.

One of the critical challenges of deploying mmWave UAVs is that the position
and orientation of hovering UAVs, compared to ground BSs, can change randomly
due to wind, inaccurate positioning, etc.[5, 6, 7]. Such random variations
will lead to random changes in the quality of the UAV-user links, which will
lead to unreliable communications. Some innovative studies have investigated
the impact of hovering fluctuations of UAVs on mmWave UAV communication
systems. For example, the work in [6] experimentally investigated the effect
of random fluctuations of hovering UAVs on mmWave signals. It demonstrated
that the mismatch of antenna beams caused by the random fluctuations would
deteriorate the performance of UAV-to-ground links. The authors in [7, 8]
investigated the influence of UAV orientation fluctuations on system outage
performance. They showed that the beam misalignment and antenna gain mismatch
caused by UAV orientation fluctuations lead to a decline in the reliability of
the communication system. The authors in [9] analyzed the effect of UAV
attitude angle fluctuations on the antenna orientation, and they proposed a
UAV beam training method to improve the accuracy of antenna angle estimation.
However, [6, 7, 8, 9] focused on investigating the impact of antenna beam
misalignment or poor beam selection caused by hovering UAV fluctuations on
system performances.

In addition to the above impacts on antenna beam alignment, fluctuations,
especially position fluctuations of UAVs may also significantly impact mmWave
blockage characteristics. As we know, mmWave signals are easily blocked by
obstacles, and the blockage state is mainly determined by the relative
positions of transceivers and obstacles [10]. The random position deviation of
hovering UAVs will make the blockage characteristics more randomized, thus
affecting the system’s performance. So investigating the influence of UAV
position fluctuations on link blockage, then evaluating the QoS is necessary.
Specifically, UAV position fluctuation is three-dimensional (3D), and a line-
of-sight (LoS) mmWave link may encounter three types of blockages: static
blockage caused by static obstacles such as buildings, dynamic blockage caused
by moving blockers, and self-blockage caused by the user’s own body[11].
Therefore, it is essential to obtain the relationship between the 3D position
fluctuations of hovering UAVs and the three kinds of blockages of mmWave, as
well as the impact on the system performance.

The effect of blockages on the mmWave system without considering the position
fluctuations of UAVs has been widely studied. For instance, the work in [12,
11, 13, 14, 15, 16] studied the blockage effects of mmWave signals in
terrestrial communications. For mmWave UAV communications, the static blockage
due to buildings has been studied in [17, 18, 19], and the LoS probability is
obtained to guide the deployment of UAVs. In [4, 20, 21], the coverage
performance was analyzed under static blockage. Furthermore, the authors in
[22] contributed an analytical framework to characterize mmWave backhaul links
by considering dynamic blockage, and it is shown that the assistance of UAVs
can significantly improve the performance of mmWave backhaul links. The
authors in [23] presented a method of optimal deployment, which provided the
optimal 3D position and coverage radius of a UAV by considering human
blockage. More recently, in [24], the authors experimentally verified the
impact of human body on air-to-ground links, and the results showed that the
effect depended largely on the hovering UAVs’ positions. However, the above
results are obtained under the supposition that the position of the UAV is
perfectly stable. This assumption is too ideal because, in practice, the
fluctuation of UAVs is unavoidable.

As far as we know, there are few studies on mmWave blockage characteristics
under the effect of UAV position fluctuations. In [25], the authors studied
the blockage characteristics and beam misalignment considering the hovering
fluctuation effect of UAV, and they proposed a UAV deployment and beamforming
optimization method to establish robust transmission. However, their
theoretical model only analyzed the blockage probability considering UAV
position fluctuations. But, more importantly, further system performance
analysis still needs improvement. Especially the system reliability analysis
under the joint consideration of fluctuations and blockages has yet to be
carried out. In addition, in our previous work[26], we developed a theoretical
analysis model to study the impact of UAV position fluctuations on UAV-user
link blockages and the reliability of air-to-ground mmWave UAV communications.
However, the results of [26] are obtained based on a simple model that only
considered the height fluctuations of UAVs.

Although the above work provides valuable insights for mmWave UAV
communications, more research still needs to be done on the effect of UAV 3D
position fluctuations on the system’s QoS. A comprehensive analytical model
that builds the relationship between the hovering UAVs’ position fluctuations
and the mmWave link’s blockages is greatly requested to benefit from mmWave
UAV communications. This is what we are trying to do through this work. The
major contributions of this paper are given as follows:

$\bullet$ We propose a novel model to analyze the effect of UAV 3D position
fluctuations on the QoS of air-to-ground mmWave UAV systems. Different from
existing works, this work establishes the theoretical relationship between the
position fluctuations of UAVs and the blockages of mmWave links (including
static, dynamic, and self-blockages), which enables the evaluation of the
blockage characteristics of UAV-user links under position fluctuations.
Especially, we find that the dynamic blockage is obviously affected by the
random fluctuations of UAV position, and its fluctuation variance is directly
proportional to the variance of UAV position fluctuations.

$\bullet$ We derive the closed-form expressions for reliable service
probability respective to the probability of UAV-user links blockages, and
coverage probability respective to the signal-to-noise ratio (SNR),
respectively. The results indicate that the greater the position fluctuation
strength of the UAV is, the lower the reliable service probability and
coverage probability is. Therefore, UAV position fluctuations considerably
influence these two QoS metrics of the considered system. To our best
knowledge, this is the first work that analytically evaluates the impact of
UAV 3D position fluctuations on the QoS of air-to-ground mmWave UAV
communication systems while considering the static, dynamic, and self-blockage
of mmWave links.

$\bullet$ We provide Monte Carlo simulations to verify the theoretical
results, and the simulation results also show that there exists an optimal
horizontal position and height of UAVs to maximize the reliable service
probability and coverage probability, respectively, which help establish
reliable air-to-ground mmWave UAV communications.

The rest of this paper is organized as follows: In Section II, we present the
system model. In Section III, we analyze the effect of UAV position
fluctuation on link blockages. In Section IV, the closed-form expression of
reliable service probability is derived, while in Section V, the closed-form
expression of coverage probability is derived. Section VI provides the
simulation results, and Section VII concludes this paper. Notations used in
this paper are given in Table I.

##  II System Model

We consider a mmWave UAV communications scenario, as shown in Fig. 1, where
ground users are randomly distributed and served by UAVs as aerial BSs. Since
obstacles easily block UAV-user links, we assume the user can immediately link
to another unblocked link once the current UAV-user link is blocked. Without
losing generality, we randomly select a user as the typical user. The typical
user’s location is denoted as $(x,y,z)\\!=\\!(0,0,h_{R})$, where $h_{R}$ is
the height of the typical user. The specific model settings are given as
follows:

TABLE I: Description of Notation Notation |  Description
---|---
$(x_{i},y_{i},h_{i})$ | Real-time position of the $i$-th UAV under fluctuations.
$(\mu_{x_{i}},\mu_{y_{i}},\mu_{h_{i}})$ | Mean of the real-time position $(x_{i},y_{i},h_{i})$.
$\lambda_{T}$/$\lambda_{B}$/$\lambda_{S}$ | Density of UAVs/human blockers/buildings.
$v$ | Velocity of moving human blockers.
$r_{i}$ | 2D distance from the $i$-th UAV to the user.
$B_{i}^{\rm{d}}$/$B_{i}^{\rm{s}}$ | Indicator for dynamic/static blockage for the $i$-th link.
$B^{\rm{sel}}$ | Indicator for self-blockage for a single UAV-user link.
$C_{i}$ | Indicator for the $i$-th UAV is available.
$M$/$N$ | Number of all/available UAVs.

Figure 1: The considered mmWave UAV communications, where the position of the
hovering UAVs will fluctuate randomly, the QoS provided to the user will be
affected by various blockages and fluctuations.

###  II-A UAV Models

This subsection models the position and position fluctuations of UAVs. Since
hovering UAVs will fluctuate randomly, the real-time position of each UAV
fluctuates randomly around its central position [5, 25]. We denote
$(\mu_{x_{i}},\mu_{y_{i}},\mu_{h_{i}})$ and $({x_{i}},{y_{i}},{h_{i}})$ as the
central position and the real-time position for the $i$-th UAV, respectively.
The central positions on the two-dimensional (2D) plane is modeled as a
homogeneous Poisson point process (PPP) with density $\lambda_{T}$ [11]. We
assume that the maximum service radius of the UAV is $R$. Therefore, the
number of all potential serving UAVs $M$ for the typical user is Poisson
distributed i.e.,
$\mathbb{P}_{M}(m)=\frac{{[\lambda_{T}\pi{R}^{2}]}^{m}}{m!}\exp\left({\\!-\\!\lambda_{T}\pi{R}^{2}}\right)$.
Then, due to the real-time position of the hovering UAV fluctuates randomly,
and numerous literature has modeled UAV position fluctuations as Gaussian
distributions [27, 7, 5, 28], the exact position of the $i$-th UAV can be
modeled as $P_{i}\sim N(\mu_{P_{i}},\sigma^{2}),P\in\\{x,y,h\\}$, where
$\sigma^{2}$ is the variance of the UAV position fluctuations. It is clear
that the greater the $\sigma$ is, the greater the fluctuation is. Therefore,
we regard $\sigma$ as a measure of UAV position fluctuation strength.

###  II-B Blockage Models

####  II-B1 Dynamic Blockage Model

Dynamic blockage due to moving humans has been thoroughly studied in mmWave
systems with traditional BSs [12, 11, 13]. Following [11], we assume that
humans are located in the region according to a homogeneous PPP of density
$\lambda_{B}$, and they move randomly at velocity $v$. We define effective
blockage length to reflect whether blockage occurs when a human blocker passes
through a UAV-user link. The effective blockage length $r_{i}^{\mathrm{eff}}$
for the $i$-th UAV-user link is
$r_{i}^{\mathrm{eff}}\\!=\\!\frac{h_{B}-h_{R}}{h_{i}-h_{R}}r_{i}$, where
$h_{B}$ is the height of the human blocker,
$r_{i}\\!=\\!\sqrt{x_{i}^{2}+y_{i}^{2}}$. Then, the dynamic blockage of the
$i$-th link is modeled as an exponential on-off process with blocking and non-
blocking rates of $\eta_{i}$ and $\omega$, respectively [11], where
$\eta_{i}=\frac{2}{\pi}\lambda_{B}vr_{i}^{\mathrm{eff}}$ and $\omega$ is
simplified to a constant. Moreover, we know that the $i$-th link is
dynamically blocked with probability (w.p.)
$\phi_{i}(h_{i},r_{i})=\frac{\eta_{i}}{\eta_{i}+\omega}$. We define a random
variable $B_{i}^{\rm{d}}\\!=\\!\\{1,0\\}$ to indicate whether dynamic blockage
occurs on the $i$-th UAV-user link, and $B_{i}^{\rm{d}}$ can be represented as

$\displaystyle B_{i}^{\rm{d}}\\!=\\!\left\\{\begin{array}[]{l l
l}1&{\text{w.p.}}&\phi_{i}(h_{i},r_{i})=\frac{\rho{r_{i}}}{{\rho
r_{i}+\omega(h_{i}-h_{R})}},\\\
0&{\text{w.p.}}&\tilde{\phi_{i}}(h_{i},r_{i})=1-\phi_{i}(h_{i},r_{i}),\\\
\end{array}\right.$ (3)

where $\rho={2}{\lambda_{B}}v({h_{B}-h_{R}})/{\pi}$ is used for convenience.

####  II-B2 Static Blockage Model

Static blockage caused by static obstacles (e.g., high-rise buildings) has
been well studied in [18, 4, 16] based on random shape theory. We consider the
blockage model given in [16] to model the static blockage, and the $i$-th link
is blocked with probability $\psi_{i}(h_{i},r_{i})$, where
$\psi_{i}(h_{i},r_{i})$ will be given later. Similar to dynamic blockage, we
define $B_{i}^{\rm{s}}\\!=\\!\\{1,0\\}$ to describe whether static blockage
occurs on the $i$-th link, and $B_{i}^{\rm{s}}$ can be represented as

$\displaystyle B_{i}^{\rm{s}}\\!=\\!\left\\{\begin{array}[]{l l
l}1&{\text{w.p.}}&\psi_{i}(h_{i},r_{i})=1-{\exp{\left(-(\epsilon
r_{i}+\epsilon_{0})\right)}},\\\
0&{\text{w.p.}}&\tilde{\psi_{i}}(h_{i},r_{i})=1-\psi_{i}(h_{i},r_{i}),\\\
\end{array}\right.$ (6)

where $\epsilon_{0}=\lambda_{S}\mathbb{E}(l)\mathbb{E}(w)$ and
$\epsilon=\frac{2}{\pi}\lambda_{S}(\mathbb{E}(l)+\mathbb{E}(w))$,
$\lambda_{S}$, $\mathbb{E}(l)$, and $\mathbb{E}(w)$ are the density, expected
length, and width of the buildings, respectively.

####  II-B3 Self-Blockage Model

Except for dynamic and static blockages, a fraction of links will also be
blocked by the user’s own body. Following [11], we define a sector of angle
$\theta$ behind the user as the self-blockage zone. The user himself will
block all UAV-user links in this zone. Therefore, the self-blockage
probability of a UAV-user link is the probability that the UAV is located in
the self-blockage zone. We define an indicative random variable
$B^{\rm{sel}}\\!=\\!\\{1,0\\}$ to describe whether self-blockage occurs on a
single UAV-user link, and $B^{\rm{sel}}$ can be represented as

$\displaystyle B^{\rm{sel}}\\!=\\!\left\\{\begin{array}[]{l l
l}1&{\text{w.p.}}&\frac{\theta}{2\pi},\\\
0&{\text{w.p.}}&1-\frac{\theta}{2\pi}.\\\ \end{array}\right.$ (9)

##  III Effect of Fluctuations on Blockages

This section focuses on obtaining the relationship between UAV position
fluctuations and UAV-user link blockages. Specifically, according to (3) and
(6), we observe that the probability of dynamic and static blockages of UAV-
user link is related to the position of the UAV (i.e., $h_{i}$ and $r_{i}$).
Due to random fluctuations in the UAV’s position, $\phi_{i}(h_{i},r_{i})$ and
$\psi_{i}(h_{i},r_{i})$ are randomly changing and only reflect an
instantaneous moment’s blockage characteristic. On the one hand, the random
variation in the blockage state is detrimental to users, as it will lead to
unreliable communications. On the other hand, analyzing the performance of a
system at a specific moment is difficult and of limited value. Hence, to
evaluate the average performance of the system, it is important to obtain the
distribution of $\phi_{i}(h_{i},r_{i})$ and $\psi_{i}(h_{i},r_{i})$ under UAV
position fluctuations. Accordingly, in the following, we derive the relevant
probability density function (PDF).

###  III-A Effect of Fluctuations on Dynamic Blockage

######  Theorem 1.

The PDF of $\phi_{i}(h_{i},r_{i})$ can be well approximately modeled as

$\displaystyle
f_{\phi_{i}}(\phi_{i})\approx\frac{1}{\sqrt{2\pi}\sigma_{\phi_{i}}}{\exp\left({-{\frac{{(\phi_{i}-\mu_{\phi_{i}})}^{2}}{2{\sigma_{\phi_{i}}^{2}}}}}\right)},$
(10)

where the mean $\mu_{\phi_{i}}$ and variance $\sigma^{2}_{\phi_{i}}$ are given
in (38) and (39), respectively. Note that the notation ${\phi_{i}}$ is a
compressed version of $\phi_{i}(h_{i},r_{i})$, which we use for convenience.

###### Proof.

The proof is given in Appendix A. ∎

Remark 1: From Theorem 1, (38) and (39), we can observe that the mean
$\mu_{\phi_{i}}$ of $\phi_{i}(h_{i},r_{i})$ is independent of the variance
$\sigma^{2}$ of the UAV position fluctuations. However, the variance
$\sigma^{2}_{\phi_{i}}$ of $\phi_{i}(h_{i},r_{i})$ is an increasing function
of $\sigma^{2}$. Therefore, the higher the fluctuation strength of the UAV,
the larger the deviation of $\phi_{i}(h_{i},r_{i})$ from its mean
$\mu_{\phi_{i}}$.

Then, the relationship between $\mu_{\phi_{i}}$, $\sigma_{\phi_{i}}$ and
average 2D distance $\mu_{r_{i}}$ is given as follows:

######  Corollary 1.

$\mu_{\phi_{i}}$ and $\sigma_{\phi_{i}}$ are increasing and decreasing
functions of $\mu_{r_{i}}$, respectively, $i\\!=\\!1,\cdots,m$. Therefore, the
maximum values of $\mu_{\phi_{i}}$ can be expressed as

$\displaystyle\mu^{\max}_{\phi_{i}}=\frac{\rho{\mu^{\max}_{r_{i}}}}{{\rho{\mu^{\max}_{r_{i}}}+\omega(\mu_{h_{i}}\\!-\\!h_{R})}}$
(11)

where $\mu^{\max}_{r_{i}}$ denotes the maximum value of $\mu_{r_{i}}$,
$i=1,\cdots,m$. Similarly, we can obtain the maximum value of
$\sigma_{\phi_{i}}$ by replacing $\mu_{r_{i}}$ in (39) with
$\mu^{\min}_{r_{i}}$.

###### Proof.

The proof is given in Appendix B. ∎

Remark 2: According to Corollary 1, we can conclude that if the UAV is
deployed closer to the user, i.e., $\mu_{r_{i}}$ is smaller, the mean
$\mu_{\phi_{i}}$ of $\phi_{i}(h_{i},r_{i})$ will be smaller. However, in this
case, the variance $\sigma_{\phi_{i}}^{2}$ of $\phi_{i}(h_{i},r_{i})$ will
become larger. Therefore, there may be an appropriate $\mu_{r_{i}}$ to obtain
a trade-off between $\mu_{r_{i}}$ and $\sigma_{\phi_{i}}^{2}$ so that the
system can achieve a better QoS.

###  III-B Effect of Fluctuations on Static Blockage

######  Corollary 2.

The PDF of $\psi_{i}(h_{i},r_{i})$ can be well approximately modeled as

$\displaystyle
f_{\psi_{i}}(\psi_{i})\approx\frac{1}{\sqrt{2\pi}{(1\\!-\\!\psi_{i})}\sigma_{\hat{r_{i}}}}{\exp\left(-{{\frac{({\ln\left({1-\psi_{i}}\right)-\mu_{\hat{r_{i}}})}^{2}}{2{\sigma_{\hat{r_{i}}}^{2}}}}}\right)},$
(12)

where $\sigma_{\hat{r_{i}}}$ and $\mu_{\hat{r_{i}}}$ can be seen in the proof,
and ${\psi_{i}}$ is a compressed version of $\psi_{i}(h_{i},r_{i})$.

###### Proof.

We first derive the PDF of
$\tilde{\psi_{i}}(h_{i},r_{i})={\exp{\left(-(\epsilon
r_{i}+\epsilon_{0})\right)}}$. According to (37), $r_{i}$ can be approximately
modeled as a normal distribution with mean $\mu_{r_{i}}$ and variance
$\sigma^{2}$. As a result, we have $-(\epsilon r_{i}+\epsilon_{0})$ is
normally distributed with mean
$\mu_{\hat{r_{i}}}\\!=\\!-(\epsilon\mu_{r_{i}}+\epsilon_{0})$ and variance
${\sigma_{\hat{r_{i}}}^{2}}\\!=\\!\epsilon^{2}\sigma^{2}$. Therefore,
$\tilde{\psi_{i}}(h_{i},r_{i})$ is log-normally distributed [29], and its PDF
is given by
$f_{{\tilde{\psi_{i}}}}(\tilde{\psi_{i}})\approx\frac{1}{\sqrt{2\pi}{\tilde{\psi_{i}}}\sigma_{\hat{r_{i}}}}{\exp\left({-{\frac{({\ln\left({\tilde{\psi_{i}}}\right)-\mu_{\hat{r_{i}}})}^{2}}{2{\sigma_{\hat{r_{i}}}^{2}}}}}\right)}$.
We further denote the cumulative distribution function (CDF) of
$\tilde{\psi_{i}}(h_{i},r_{i})$ and ${\psi_{i}(h_{i},r_{i})}$ as
$F_{\tilde{\psi_{i}}}(\tilde{\psi_{i}})$ and $F_{\psi_{i}}({\psi_{i}})$,
respectively. It is easy to get
$F_{\psi_{i}}({\psi_{i}})\\!=\\!\mathbb{P}(1\\!-\\!{\exp{\left(-(\epsilon
r_{i}+\epsilon_{0})\right)}}\leq{\psi_{i})}=1-F_{\tilde{\psi_{i}}}(1-{\psi_{i}})$,
and $f_{{{\psi_{i}}}}({\psi_{i}})=f_{{\tilde{\psi_{i}}}}(1-{\psi_{i}})$ by
taking the derivative of $F_{{{\psi_{i}}}}({\psi_{i}})$. Finally, the PDF of
${\psi_{i}(h_{i},r_{i})}$ is obtained as shown in (12).∎

Using Corollary 2, we can obtain the mean and variance of
$\tilde{\psi_{i}}(h_{i},r_{i})$ separately as

$\displaystyle\mu_{\tilde{\psi_{i}}}$
$\displaystyle={\exp{\left(\mu_{\hat{r_{i}}}+{\sigma_{\hat{r_{i}}}^{2}}/2\right)}},$
(13) $\displaystyle\sigma_{\tilde{\psi_{i}}}^{2}$
$\displaystyle={\exp{\left(2(\mu_{\hat{r_{i}}}+\sigma_{\hat{r_{i}}}^{2})\right)}}-{\exp{\left(2(\mu_{\hat{r_{i}}}+\sigma_{\hat{r_{i}}}^{2}/2)\right)}}$
(14)

Remark 3: From (13) and (14), we can infer that the mean and variance of
$\psi_{i}(h_{i},r_{i})$ may also be affected by the position fluctuation of
UAV. However, in our system assumption, we can obtain:
$\mu_{\tilde{\psi_{i}}}\\!\approx\\!\exp{\left(-(\epsilon\mu_{r_{i}}\\!+\\!\epsilon_{0}\right))}$
and $\sigma_{\tilde{\psi_{i}}}^{2}\approx 0$, where the detailed proof can be
seen in Appendix C. The results indicate that the $\psi_{i}(h_{i},r_{i})$
hardly changes with the position fluctuation of the UAV, which matches the
intuition that the dimension of the position fluctuation of the UAV is very
small compared to the size of buildings. Therefore, the impact of UAV position
fluctuations on the static blockage probability of the link is negligible, and
only when the UAV is located at the edge of the building will position
fluctuations of the UAV affect the link’ static blockage state.

##  IV Effect of Fluctuations on Reliable Service

This section analyzes the effect of UAV position fluctuations on the reliable
service probability. Since static and self-blockage will lead to permanent
blockage, a UAV is available if UAV-user link is not blocked by static
blockage and self-blockage. We are more concerned that at least one UAV is
available. Due to blockages affecting the system reliability, we propose a new
QoS metric in terms of links blockages, called reliable service probability,
given by Definition 1.

######  Definition 1.

A user is said to be in reliable service if at least one UAV is available with
a blockage probability not higher than a predefined threshold. The reliable
service probability is denoted as $\mathbb{P}_{\mathrm{rel}}$ and given by

$\displaystyle\mathbb{P}_{\mathrm{rel}}\triangleq
1\\!-\\!\sum_{n=0}^{\infty}\mathbb{P}_{N}(n)\prod_{i=0}^{n}\mathbb{P}\left(\phi_{i}(h_{i},r_{i})\\!>\\!p_{\mathrm{th}}\right),$
(15)

where $\mathbb{P}_{N}(n)$ denotes the probability that $n$ UAVs are available,
$\mathbb{P}_{N}(n)\prod_{i=0}^{n}\mathbb{P}\left(\phi_{i}(h_{i},r_{i})\\!>\\!p_{\mathrm{th}}\right)$
denotes the probability that $n$ UAVs are available and the dynamic blockage
probability of each link is greater than the threshold $p_{\mathrm{th}}$.
Therefore, the second term of (15) represents the probability that the user is
not in a reliable service. Note that $n=0$ will result in $i=0$, which means
there are no UAVs available. So we let
$\mathbb{P}\left(\phi_{i}(h_{i},r_{i})\\!>\\!p_{\mathrm{th}}\right)|_{i=0}=1$.
To calculate $\mathbb{P}_{\mathrm{rel}}$, we first give the distribution of
UAVs availability in the following.

###  IV-A Distribution of UAVs Availability

We use an indicative random variable $C_{i}$ to indicate that the $i$-th UAV
is available, $\mathbb{P}(C_{i})$ denotes the probability of $C_{i}$, which is
also the probability that the $i$-th link is not blocked by static and self-
blockage. Then, the distribution of the number of available UAVs is obtained
as:

######  Lemma 1.

The distribution of the number of available UAVs $N$ is Poisson distributed
with parameter $\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}$, i.e.,

$\displaystyle\mathbb{P}_{N}(n)=\frac{{[\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}]}^{n}}{n!}\exp\left({-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}}\right),$
(16)

where
$\mathbb{P}(C_{i})\approx\left(1\\!-\\!\frac{\theta}{2\pi}\right)\frac{2\exp{(\\!-\epsilon_{0})}}{R^{2}\epsilon^{2}}\left(1\\!-\\!(1\\!+\\!R\epsilon)\exp{(\\!-R\epsilon)}\right)$.

###### Proof.

The proof is given in Appendix C. ∎

Then, we derive the closed-form expression of reliable service probability for
single and multiple UAV cases according to Definition 1. It indicates that the
greater the fluctuation strength $\sigma$ of the UAV, the lower the
$\mathbb{P}_{\mathrm{rel}}$, and the worse the QoS. The specific details are
given as follows.

###  IV-B Reliable Service Probability for Single UAV Case

In this case, we assume that the typical user can only link to one UAV, and
denote the UAV as the $i$-th UAV. Let
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ represent the reliable service
probability for the single UAV case, it indicates the probability that the
$i$-th UAV is available and the dynamic blockage probability of the link is
lower than the threshold $p_{\mathrm{th}}$. Then,
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ can be obtained as shown in Theorem
2.

######  Theorem 2.

The reliable service probability for the single UAV case is given by

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}\approx\mathbb{P}(C_{i})\left(\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\right),$
(17)

where $\mathbb{P}(C_{i})$ is given in (47), $\mu_{\phi_{i}}$ and
$\sigma_{\phi_{i}}$ are given in (38) and (39), respectively.

###### Proof.

Since in this case we only consider one UAV is available, (15) can be
rewritten as

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}=1\\!-\\!\sum_{n=0}^{1}\mathbb{P}_{N}(n)\prod_{i=0}^{n}\mathbb{P}\left(\phi_{i}(h_{i},r_{i})\\!>\\!p_{\mathrm{th}}\right),$
(18)

where $\mathbb{P}_{N}(1)$ is indeed the probability that the $i$-th UAV is
available, i.e., $\mathbb{P}_{N}(1)=\mathbb{P}(C_{i})$, and
$\mathbb{P}_{N}(0)=1-\mathbb{P}(C_{i})$. Substituting $\mathbb{P}_{N}(0)$ and
$\mathbb{P}_{N}(1)$ into (18), then using Theorem 1, we can get

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$
$\displaystyle=\mathbb{P}(C_{i})-\mathbb{P}(C_{i})\mathbb{P}\left(\phi_{i}(h_{i},r_{i})\\!>\\!p_{\mathrm{th}}\right)$
$\displaystyle=\mathbb{P}(C_{i})\int_{0}^{p_{\mathrm{th}}}f_{\phi_{i}}(\phi_{i})d{\phi_{i}}$
$\displaystyle\approx\mathbb{P}(C_{i})\left(\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\right),$
(19)

where the last step is obtained by integrating $f_{\phi_{i}}(\phi_{i})$
according to [15, Eq. (3)]. ∎

According to Theorem 2, we can obtain the effect of $\sigma$ on the QoS. In
(17), $\mathbb{P}(C_{i})$ and $\mu_{\phi_{i}}$ are not affected by $\sigma$
according to (47) and (38), respectively. So only $\sigma_{\phi_{i}}$ is
related to $\sigma$, and $\sigma_{\phi_{i}}$ is an increasing function of
$\sigma$, which can be observed from (39), we first analyze the effect of
$\sigma_{\phi_{i}}$ on $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. Since error
function erf$(\cdot)$ is a monotonically function and erf$(0)\\!=\\!0$,
$\sigma_{\phi_{i}}$ will show two different effects on
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. For
$p_{\mathrm{th}}>\mu_{\phi_{i}}$, erf$(\cdot)>0$ and
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ decreases as $\sigma_{\phi_{i}}$
increases. For $p_{\mathrm{th}}<\mu_{\phi_{i}}$, erf$(\cdot)\\!<\\!0$ and
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ increases as $\sigma_{\phi_{i}}$
increases. However, in the latter situation, the value of
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is lower, even less than 0.5, which
indicates that the QoS is evil, further analyze the effect of $\sigma$ on the
QoS is almost meaningless. Hence, throughout this paper, we pay more attention
to the former case ($p_{\mathrm{th}}>\mu_{\phi_{i}}$). We can conclude that
the greater the $\sigma$ is, the larger the $\sigma_{\phi_{i}}$ is, the lower
the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is, and the worse the QoS is.
It is worth noting that from the derivation of (IV-B), it can be found that
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is similar to the CDF of
$\phi_{i}(h_{i},r_{i})$, which is normally distributed according to Theorem 1.
Therefore, when $\sigma_{\phi_{i}}$ changes, the variation of
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is similar to that of the CDF of
$\phi_{i}(h_{i},r_{i})$. The above analysis is also consistent with this
phenomenon. Moreover, for an open area scenario (park/stadium/square)[11], we
can rewrite $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ as follows.

######  Corollary 3.

For an open area communication scenario,
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ can be rewritten as follows:

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}\approx\left(1-\frac{\theta}{2\pi}\right)\left(\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\right)$
(20)

###### Proof.

For the open area scenario, such as a public park, buildings play a small
role. Therefore, we can assume that $\lambda_{S}\\!\approx\\!0$, and
$\psi_{i}(h_{i},r_{i})\approx\\!0$ by using (6). Then, the conditional
probability that the $i$-th UAV is available is
$\mathbb{P}(C_{i}|h_{i},r_{i})=1-\frac{\theta}{2\pi}$. Since $\theta$ is
independent of $h_{i}$ and $r_{i}$, we have
$\mathbb{P}(C_{i})=\mathbb{P}(C_{i}|h_{i},r_{i})$. Substituting the
$\mathbb{P}(C_{i})$ into (17), we can obtain
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ as shown in (20).∎

###  IV-C Reliable Service Probability for Multiple UAVs Case

In this case, we assume that the typical user has more than one UAVs that can
be connected, and it will immediately switch to any other available UAV when
the currently serving link is blocked. We define
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ as the reliable service probability
for this case, it represents the probability that at least one UAV is
available with a dynamic blockage probability lower than the predefined
threshold $p_{\mathrm{th}}$. Then, we obtain the approximate
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ as shown in Theorem 3.

######  Theorem 3.

The reliable service probability for the multiple UAVs case is given by

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}\approx$ $\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{\infty}\mathbb{P}_{N}(n)\prod_{i=1}^{n}\left(\frac{1}{2}\\!-\\!\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\\!\right),$
(21)

where $\mu_{\phi_{i}}$ and $\sigma_{\phi_{i}}$ are given in (38) and (39),
respectively, $i=1,\cdots,n$.

###### Proof.

In this case, $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is expressed as shown
in (15). Therefore, we can get

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}=$ $\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{\infty}\mathbb{P}_{N}(n)\prod_{i=1}^{n}\mathbb{P}\left(\phi_{i}(h_{i},r_{i})\\!>\\!p_{\mathrm{th}}\right)$
$\displaystyle\approx$ $\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{\infty}\mathbb{P}_{N}(n)\\!\prod_{i=1}^{n}\left(\frac{1}{2}\\!-\\!\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\right),$
(22)

where
$\mathbb{P}\left(\phi_{i}(h_{i},r_{i})>p_{\mathrm{th}}\right)=1-\int_{0}^{p_{\mathrm{th}}}f_{\phi_{i}}(\phi_{i})d{\phi_{i}}\approx\frac{1}{2}-\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)$
according to (IV-B).∎

Although Theorem 3 gives an approximate expression of
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, the expression is not tractable
due to $n\in[0,\infty]$. Fortunately, since it is unrealistic to have an
infinite number of UAVs in practice, we can assume the maximum number of UAVs
is $K$. Then, $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ can be rewritten as:

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}\\!\approx$
$\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{K}\mathbb{P}_{N}(n)\prod_{i=1}^{n}\left(\frac{1}{2}\\!-\\!\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\right),$
(23)

which is solvable. Furthermore, we can get the lower bound of
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ as shown in Corollary 4.

######  Corollary 4.

A lower bound of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is given by

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}\geq
1\\!-\\!\exp\left({\\!-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}\left(\frac{1}{2}\\!+\\!\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi}}{\sqrt{2}\sigma_{\phi}}\right)\right)}\right),$
(24)

where $\mu_{\phi}=\max(\mu_{\phi_{i}})$ and
$\sigma_{\phi}=\max(\sigma_{\phi_{i}})$, $i=1,\cdots,n$.

###### Proof.

Since erf$(\cdot)$ is an increasing function and we focus on
$p_{\mathrm{th}}>\mu_{\phi_{i}}$,
$\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)$
decreases with the increase of $\mu_{\phi_{i}}$ and $\sigma_{\phi_{i}}$,
respectively. Hence, $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ in (21) can
achieve a lower bound when
$\mu_{\phi_{1}}=\cdots=\mu_{\phi_{n}}=\max(\mu_{\phi_{i}})$ and
$\sigma_{\phi_{1}}=\cdots=\sigma_{\phi_{n}}=\max(\sigma_{\phi_{i}})$,
$i=1,\cdots,n$, i.e.,

$\displaystyle\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ $\displaystyle\geq
1-\sum_{n=0}^{\infty}\mathbb{P}_{N}(n)\left(\frac{1}{2}-\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}-\mu_{\phi}}{\sqrt{2}\sigma_{\phi}}\right)\right)^{n},$
$\displaystyle=1\\!-\\!\exp\left({\\!-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}\left(\frac{1}{2}\\!+\\!\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi}}{\sqrt{2}\sigma_{\phi}}\right)\right)}\right),$
(25)

where the last step is obtained by substituting (16) into (21). ∎

According to Theorem 3 and Corollary 4, we can conclude that the larger the
$\sigma$ is, the smaller the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is,
and the lower the QoS is, which is consistent with the single UAV case.
Similarly, when $\mu_{\phi}=\min(\mu_{\phi_{i}})$ and
$\sigma_{\phi}=\min(\sigma_{\phi_{i}})$, $i=1,\cdots,n$, we can obtain an
upper bound of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, which is similar to
(IV-C) and is omitted for brevity. For an open area scenario,
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ can be obtained by setting
$\mathbb{P}(C_{i})\\!=\\!1\\!-\\!\frac{\theta}{2\pi}$, the reason is
consistent with the proof of Corollary 3.

Finally, a graphical example of the impact of $\sigma$ on the blockage and the
QoS is shown in Fig. 2. This example depicts two snapshots of the links’
blockage status. As we can see, the blockage status and reliable service in
the two snapshots is different even though the blockers are the same, which is
due to the position fluctuations of the UAV. Moreover, as discussed in Theorem
2 and Corollary 4, the function of reliable service probability has almost
similar variation characteristics to the CDF of normal distribution.
Therefore, when $p_{\mathrm{th}}>\mu_{\phi_{i}}$, the higher the UAV’s
fluctuation, the smaller the probability of reliable service. In conclusion,
the larger the $\sigma$ is, the lower the reliable service probability is, and
the worse the QoS is.

Figure 2:  An example of the effect of hovering UAVs positions fluctuations on
UAV-user links’ blockage status and reliable service. The central positions of
UAVs in (a) and (b) are the same. The snapshots in (a) and (b) show the
possible positions of UAVs and blockage status under weak and strong
fluctuations. As can be seen, the greater the variance of position
fluctuation, the more the UAV deviates from its central position. So the
greater the dynamic blockage probability deviates from its mean value.
Therefore, when $p_{\mathrm{th}}>\mu_{\phi_{i}}$, the larger the fluctuation
of UAV positions, the smaller the reliable service probability.

##  V Effect of Fluctuations on Coverage

Section IV evaluated the effect of $\sigma$ on reliable service probability,
which is defined respective to the blockage probability. This may not capture
the full picture of the considered system. Hence, this section refers to
coverage probability in terms of SNR as another QoS measurement and evaluates
the impact of $\sigma$ on this measurement. The coverage probability is
defined as follows:

######  Definition 2.

A user is said to be in coverage if at least one UAV is available with a SNR
not smaller than a predefined threshold. The coverage probability is denoted
as $\mathbb{P}_{\mathrm{cov}}$ and given by

$\displaystyle\mathbb{P}_{\mathrm{cov}}\triangleq
1\\!-\\!\sum_{n=0}^{\infty}\mathbb{P}_{N}(n)\prod_{i=0}^{n}\mathbb{P}\left(\gamma_{i}<\gamma_{0}\right),$
(26)

where $\gamma_{i}$ denotes the SNR at the user from the $i$-th UAV,
$\mathbb{P}_{N}(n)\prod_{i=0}^{n}\mathbb{P}\left(\gamma_{i}<\gamma_{0}\right)$
denotes the probability that $n$ UAVs are available and the SNR of each link
is smaller than the threshold $\gamma_{0}$. Therefore, the second term of (26)
denotes the probability that the user is not in coverage, and we consider that
$\mathbb{P}\left(\gamma_{i}<\gamma_{0}\right)=1$ when $i=0$ since in this case
there are no available UAVs.

Next, we first develop a channel model for the considered system. Then, the
SNR at the user is expressed, and the coverage probability for single and
multiple UAV cases are derived.

###  V-A Channel Model

Since UAV-user link is only dynamically blocked when the UAV is available, the
LoS probability of the $i$-th link can be expressed as follows:

$\mathbb{P}^{\rm{LoS}}_{i}=\tilde{\phi_{i}}(h_{i},r_{i})=\frac{\omega(h_{i}-h_{R})}{{\rho
r_{i}+\omega(h_{i}-h_{R})}},$ (27)

and we assume that the channel between UAVs and the target user is based on
the dominant LoS. Therefore, the channel gain $g_{i}$ of the $i$-th UAV-user
link is given by [30]:

$\displaystyle g_{i}=\mathbb{P}^{\rm{LoS}}_{i}\beta_{0}d_{i}^{-\alpha},$ (28)

where $\beta_{0}$ is the path loss (PL) at unit distance, $\alpha$ is
parameter of the PL,
$d_{i}\\!=\\!\sqrt{r_{i}^{2}\\!+\\!(h_{i}\\!-\\!h_{R})^{2}}$ is the 3D
distance from the $i$-th UAV to the typical user. Then, the SNR at the user is
given by

$\gamma_{i}=\frac{P_{t}g_{i}}{N_{0}},$ (29)

where $P_{t}$ is the transmit power of the UAV and $N_{0}$ is the noise power.
It is clear that $\gamma_{i}$ is a function of random variables $r_{i}$ and
$h_{i}$, which means that the position fluctuations of UAVs will affect the
$\gamma_{i}$. Therefore, the coverage probability given in Definition 2 can
capture these effects.

###  V-B Coverage Probability for Single UAV Case

Similar to $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$, we define
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ as the coverage probability for the
single UAV case, it represents the probability that the $i$-th UAV is
available and the SNR of the $i$-th link is not smaller than the threshold
$\gamma_{0}$. Therefore, using Definition 2 and similar to (18) and (IV-B), we
first can get

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$
$\displaystyle=1\\!-\\!\sum_{n=0}^{1}\mathbb{P}_{N}(n)\prod_{i=0}^{n}\mathbb{P}\left(\gamma_{i}<\gamma_{0}\right)$
$\displaystyle=\mathbb{P}(C_{i})\mathbb{P}\left(\gamma_{i}\geq\gamma_{0}\right).$
(30)

Then, $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is approximately obtained as
shown in Theorem 4.

######  Theorem 4.

The coverage probability for the single UAV case is given by

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$
$\displaystyle\approx\mathbb{P}(C_{i})\left(1\\!-\\!Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\tau_{i}}{\sigma}\right)\right),$
(31)

where $\mathbb{P}(C_{i})$ is given in (47), $Q_{1}(a,b)$ is the Marcum
Q-function and is expressed in (52),
$\mu_{d_{i}}=\sqrt{\mu_{r_{i}}^{2}+(\mu_{h_{i}}-h_{R})^{2}}$, and
$\tau_{i}={\sqrt[\alpha]{\frac{P_{t}\beta_{0}\omega(\mu_{h_{i}}\\!-\\!h_{R})}{N_{0}\gamma_{0}({\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}\\!-\\!h_{R})})}}}$
is used for convenience.

###### Proof.

The proof is given in Appendix D. ∎

Since Marcum Q-function is a standard function that is easy to compute[7],
using Theorem 4, we can quickly evaluate the QoS of the considered system
without resorting to time-consuming simulation, especially the impact of UAV
position fluctuations on the coverage probability. In addition, when an open
area communication scenario is considered, the coverage probability can be
obtained by setting $\mathbb{P}(C_{i})\\!=\\!1\\!-\\!\frac{\theta}{2\pi}$, the
reason is consistent with the proof of Corollary 3.

###  V-C Coverage Probability for Multiple UAVs Case

Similar to $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, we define
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ as the coverage probability for the
multiple UAVs case, it represents the probability that at least one UAV is
available with a SNR not smaller than the threshold $\gamma_{0}$. Then, we can
obtain $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ as follows:

######  Theorem 5.

The coverage probability for the multiple UAVs case is given by

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}\approx$ $\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{\infty}\mathbb{P}_{N}(n)\prod_{i=1}^{n}Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\tau_{i}}{\sigma}\right),$
(32)

where
$\tau_{i}={\sqrt[\alpha]{\frac{P_{t}\beta_{0}\omega(\mu_{h_{i}}\\!-\\!h_{R})}{N_{0}\gamma_{0}({\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}\\!-\\!h_{R})})}}}$,
$i=1,\cdots,n$.

###### Proof.

In this case, $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ is expressed as shown
in (26). Therefore, we can get

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}=$ $\displaystyle
1-\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{\infty}\mathbb{P}_{N}(n)\prod_{i=1}^{n}\mathbb{P}\left(\gamma_{i}<\gamma_{0}\right)$
$\displaystyle\approx$ $\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{\infty}\mathbb{P}_{N}(n)\prod_{i=1}^{n}Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\tau_{i}}{\sigma}\right),$
(33)

where
$\mathbb{P}\left(\gamma_{i}<\gamma_{0}\right)\\!=\\!1\\!-\mathbb{P}\left(\gamma_{i}\geq\gamma_{0}\right)\approx
Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\tau_{i}}{\sigma}\right)$ with the
aid of (V-B) and (31).∎

Similar to (23), given the limited number of UAVs in practice,
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ can be rewritten as

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}\approx$ $\displaystyle
1\\!-\\!\mathbb{P}_{N}(0)\\!-\\!\sum_{n=1}^{K}\mathbb{P}_{N}(n)\prod_{i=1}^{n}Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\tau_{i}}{\sigma}\right).$
(34)

Furthermore, a lower bound of $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ is
obtained as shown in Corollary 5.

######  Corollary 5.

A lower bound of $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ is given by

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}\geq
1\\!-\\!\exp\left({\\!-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}\left(1-Q_{1}\left(\frac{\mu_{d}}{\sigma},\frac{\tau}{\sigma}\right)\right)}\right),$
(35)

where $\mu_{d}=\max(\mu_{d_{i}})$ and $\tau=\min(\tau_{i})$, $i=1,\cdots,n$.

###### Proof.

According to the characteristics of Marcum Q-function, we know that
$Q_{1}(a,b)$ is an increasing function of $a$ and a decreasing function of
$b$. Therefore, $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ in (32) can achieve
the lower bound when
$\mu_{d_{1}}\\!=\\!\cdots\\!=\\!\mu_{d_{n}}=\max(\mu_{d_{i}})$ and
$\tau_{1}\\!=\\!\cdots\\!=\\!\tau_{n}=\min(\tau_{i})$, $i=1,\cdots,n$, i.e.,

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ $\displaystyle\geq
1-\sum_{n=0}^{\infty}\mathbb{P}_{N}(n)Q_{1}\left(\frac{\mu_{d}}{\sigma},\frac{\tau}{\sigma}\right)^{n}$
$\displaystyle=1\\!-\\!\exp\left({\\!-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}\left(1-Q_{1}\left(\frac{\mu_{d}}{\sigma},\frac{\tau}{\sigma}\right)\right)}\right),$
(36)

where the last step is obtained by substituting (16) into (32). ∎

##  VI Simulation Results

This section performs extensive simulations to confirm the theoretical
results. For the simulation, we consider that the blockers are uniformly
distributed within a circular area centered at the typical user with a radius
of $R=100$ m. The movement of the human blockers is generated by a random way-
point mobility model[31], where human blockers will randomly select a
direction and walk in that direction at the speed of 1 m$/$s, the time of
moving in that direction is uniformly distributed for [0, 60] seconds. The
detailed simulation parameters are given in Table II.

TABLE II: Simulation Parameter Values Parameters |  Values
---|---
Height of the user/blockers, $h_{R}$/$h_{B}$ | 1.4/1.8 m [11]
Density of moving blockers, $\lambda_{B}$ | 0.01, 0.02 bl$/$m2
Density of static buildings, $\lambda_{S}$ | 100 sbl$/$km2[11]
Non-blocking rate, $\omega$ |  $2$ bl$/$s [11]
Fluctuation strength of UAVs, $\sigma$ | 0 to 0.2 m[28]
Transmit power of UAV, $P_{t}$ |  $20$ dBm
Noise power, $N_{0}$ |  $-110$ dBm [32]
SNR threshold, $\gamma_{0}$ |  $3$ dB[22]
Size of static buildings, $\mathbb{E}(l)\times\mathbb{E}(w)$ |  $10$ m$\times 10$ m[16]
Path loss parameters, $\alpha$, $\beta_{0}$ | 2, $7\\!\times\\!10^{\\!-5}$ [32]

###  VI-A Analysis of Reliable Service Probability

####  VI-A1 Single UAV case

Fig. 3 visualizes the impact of $\sigma$ on
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. First, as expected, the higher the
$\sigma$ is, the lower the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is.
Therefore, we can get the higher the degree of UAV position fluctuations, the
worse the system reliability. Then, under the same $\sigma$, the higher the
density of blockers is, the smaller the
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is. This is reasonable because the
fewer blockers on the UAV-user link, the smaller the blockage probability and
the higher the reliability of the link. Therefore, the QoS in the scenario
with static buildings is worse than in the open scenario without. However, in
the low $\sigma$ region ($\sigma<0.05$), even for different $\lambda_{B}$, the
value of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is the same and hardly
changes with $\sigma$. This happens because under our simulation parameter
settings, in a low $\sigma$ region,
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}\approx\mathbb{P}(C_{i})$ and is
independent of $\lambda_{B}$ and $\sigma$. The details are described as: Using
(39), we can get $\sigma_{\phi_{i}}^{2}$ is close to 0 when $\sigma$ is small.
For example, when $\sigma\\!=\\!0.05$, $\lambda_{B}\\!=\\!0.02$ and other
parameters given in Table II,
$\sigma_{\phi_{i}}^{2}\\!\approx\\!3.4\\!\times\\!10^{-11}$, so the
fluctuation of $\phi_{i}(h_{i},r_{i})$ is extremely small and
$\phi_{i}(h_{i},r_{i})$ is closely around its mean value $\mu_{\phi_{i}}$.
Therefore, the probability that $\phi_{i}(h_{i},r_{i})$ is less than the
threshold $p_{\mathrm{th}}$ is close to 1 since
$p_{\mathrm{th}}\\!>\\!\mu_{\phi_{i}}$, i.e.,
$\mathbb{P}(\phi_{i}(h_{i},r_{i})\\!\leq
p_{\mathrm{th}})\\!\approx\\!\frac{1}{2}\\!+\\!\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\\!\approx\\!1$,
and the value of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ mainly depends on
$\mathbb{P}(C_{i})$.

Figure 3: Reliable service probability
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ of the single UAV case versus UAV
position fluctuation strength $\sigma$ for different blocker densities, where
$\mu_{h_{i}}\\!=\\!25$ m, $\mu_{r_{i}}\\!=\\!10$ m, $\theta\\!=\\!\pi/3$, and
$p_{\mathrm{th}}=0.001$. As we can see, the larger the $\sigma$ is, the
smaller the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is.
Figure 4: Available probability $\mathbb{P}(C_{i})$ versus UAV position
fluctuation strength $\sigma$. As we can see, the value of $\mathbb{P}(C_{i})$
in (C) and (47) is almost the same. This not only indicates that $\sigma$ has
little impact on $\mathbb{P}(C_{i})$, but also proves that the impact of
$\sigma$ on the static blockage is negligible. Moreover, the curves with
$\theta=\pi/3$ are actually the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$
(when $\sigma<0.05$) in Fig. 3.
Figure 5:  Reliable service probability
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ of the single UAV case versus
average 2D distance $\mu_{r_{i}}$ from the UAV to the user, where
$\sigma\\!=\\!0.2$ m and other parameters are the same as in Fig. 3. As we can
see, there exists an optimal $\mu_{r_{i}}$ to maximize
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$.

Fig. 4 depicts the curves of $\mathbb{P}(C_{i})$ given in (C) and (47),
respectively. First, it is clear that the impact of $\sigma$ on
$\mathbb{P}(C_{i})$ is negligible. Then, the curves with $\theta=\pi/3$ are
actually the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ (when
$\sigma\\!<\\!0.05$) in Fig. 3. Therefore, at a low $\sigma$ region,
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}\approx\mathbb{P}(C_{i})$ and
$\sigma$ has little impact on it.

Fig. 5 shows how the average 2D distance $\mu_{r_{i}}$ from the UAV to the
typical user impacts $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. As we can
see, there exists an optimal $\mu_{r_{i}}$ (about 16 m in our simulation
scenario) for maximizing $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. According
to Corollary 1 and Theorem 2, on the one hand, we know that the greater the
$\mu_{r_{i}}$ is, the smaller the $\sigma_{\phi_{i}}$ is, so the larger the
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is. On the other hand, however,
$\mu_{\phi_{i}}$ is an increasing function of $\mu_{r_{i}}$, and the greater
the $\mu_{\phi_{i}}$ is, the smaller the
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is. Therefore, these two factors
lead to the existence of an optimum $\mu_{r_{i}}$ for maximizing
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$.

Figure 6: Reliable service probability
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ of the multiple UAVs case versus
UAV position fluctuation strength $\sigma$ for different blocker densities,
where $\lambda_{T}\\!=\\!100/$km2, $\mu_{h_{i}}\\!=\\!25$ m,
$\min(\mu_{r_{i}})\\!=\\!10$ m, $\max(\mu_{r_{i}})\\!=\\!15$ m,
$i=1,\cdots,n$, $\theta=\pi/3$, and $p_{\mathrm{th}}\\!=\\!0.001$. As we can
see, the larger the $\sigma$ is, the smaller the
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is.
Figure 7: Reliable service probability
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ of the multiple UAVs case versus
minimum average 2D distance $\min(\mu_{r_{i}})$, $i\\!=\\!1,\cdots,n$ from the
UAV to the user with different blocker densities, where $\sigma=0.2$ m and
other parameters are the same as in Fig. 6. As we can see, there exists an
optimal $\min(\mu_{r_{i}})$ for maximizing
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, and a lower bound for
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ in (24) is given.

####  VI-A2 Multiple UAVs case

Fig. 6 and Fig. 7 plot reliable service probability
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ according to (23), where $K=6$. As
we can see, the larger the $\sigma$ is, the smaller the
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is. Therefore, the influence of UAV
position fluctuations on the QoS of this case is the same as that of the
single UAV case. However, comparing these two cases, it is obvious that
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}\\!>\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$.
This mainly benefits from the multi-connectivity strategy, i.e., when the
current link is blocked, the user can quickly switch to other available UAVs
to reduce the blockage. Therefore, using multiple UAVs can alleviate the
impact of blockage on the QoS. Then, similar to the single UAV case, when
$\sigma$ is small ($\sigma\\!<\\!0.05$), we can infer
$\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\\!\approx\\!1$,
so
$\frac{1}{2}-\frac{1}{2}\mathrm{erf}\left(\frac{p_{\mathrm{th}}\\!-\\!\mu_{\phi_{i}}}{\sqrt{2}\sigma_{\phi_{i}}}\right)\\!\approx\\!0$,
and
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}=1\\!-\\!\exp\left({\\!-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}}\right)$.
This indicates that in the considered scenario,
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is independent of $\lambda_{B}$ and
$\sigma$ at a low $\sigma$ region. Noted that in this case, we assume that all
UAVs are distributed on a ring with an inner diameter of $\min(\mu_{r_{i}})$
an outer diameter of $\min(\mu_{r_{i}})\\!+\\!5$ m. Fig. 7 studies the impact
of $\min(\mu_{r_{i}})$ on $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, it is
clear that an optimal $\min(\mu_{r_{i}})$ exists for maximizing
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$. The reason is a combination of two
factors, as described in the analysis of Fig. 5. In addition, Fig. 7 compares
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ and a lower bound of
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ in (24) for
$\lambda_{B}\\!=\\!0.01$ and $\lambda_{S}\\!=\\!0$. It can be observed that
the value of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is slightly greater
than the lower bound.

###  VI-B Analysis of Coverage Probability

####  VI-B1 Single UAV case

Fig. 8 studies the effect of $\sigma$ on the coverage probability
${P}_{\mathrm{cov}}^{\mathrm{sig}}$ for the single UAV case. First, we observe
that ${P}_{\mathrm{cov}}^{\mathrm{sig}}$ decreases as $\sigma$ increases. This
happens because strong fluctuations in the UAV position will result in strong
fluctuations in the SNR $\gamma_{i}$, which increases the probability that
$\gamma_{i}$ will fall below the given threshold $\gamma_{0}$. Therefore,
${P}_{\mathrm{cov}}^{\mathrm{sig}}$ decreases with the increase of $\sigma$.
Then, it is clear that in the low $\sigma$ region ($\sigma<0.05$), even for
different $\lambda_{B}$, the value of
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is the same and hardly changes with
$\sigma$. This happens because in a low $\sigma$ region, the effect of
$\sigma$ on the $\gamma_{i}$ is negligible. Therefore, the probability that
$\gamma_{i}$ is less than $\gamma_{0}$ is close to 0, as $\gamma_{i}$ is
slightly greater than $\gamma_{0}$ in our simulation settings, i.e.,
$\mathbb{P}\left(\gamma_{i}\geq\gamma_{0}\right)\approx 1$. Therefore,
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}\approx\mathbb{P}(C_{i})$ and is
independent of $\lambda_{B}$ and $\sigma$ by using (V-B) and (47).

Figure 8:  Coverage probability $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ of
the single UAV case versus UAV position fluctuation strength $\sigma$ for
different blocker densities, where $\mu_{h_{i}}=25$ m, $\mu_{r_{i}}=50$ m, and
$\theta=\pi/3$. As we can see, the larger the $\sigma$ is, the smaller the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is.
Figure 9:  Coverage probability $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ of
the single UAV case versus average 2D distance $\mu_{r_{i}}$ from the UAV to
the user, where $\sigma=0.2$ m and other parameters are the same as in Fig. 8.
As we can see, the larger the $\mu_{r_{i}}$ is, the smaller the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is.

Fig. 9 investigates the impact of average 2D distance $\mu_{r_{i}}$ from the
UAV to the user on $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$. It is clear
that the larger the $\mu_{r_{i}}$ is, the smaller the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is. The reason is that increasing
$\mu_{r_{i}}$ will increase the distance between transceivers and the blockage
probability. Therefore, the SNR received by the user is reduced, and
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is decreased according to (V-B).
The result is also consistent with Theorem 4. Specifically,
$\mu_{d_{i}}=\sqrt{\mu_{r_{i}}^{2}+(\mu_{h_{i}}-h_{R})^{2}}$ is an increasing
function of $\mu_{r_{i}}$, and
$Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\tau_{i}}{\sigma}\right)$
increase with the increase of $\mu_{d_{i}}$ according to the characteristics
of Marcum Q-function. Therefore, $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is
a decreasing function of $\mu_{r_{i}}$. Additionally, combined with the
analysis of reliable service probability in Fig. 5, one thing we found very
interesting is that when $\mu_{r_{i}}$ is at a relatively low region, the user
has a higher probability of getting reliable service and coverage. After all,
selecting an appropriate $\mu_{r_{i}}$ can improve the service quality.

Fig. 10 shows $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ as a function of
average UAV height $\mu_{h_{i}}$. As we expected, there is an optimal
$\mu_{h_{i}}$ that maximizes $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$. Two
factors cause the result: On the one hand, when the UAV is at a low height,
the LoS probability will be small. The resulting small channel gain $g_{i}$
increases the probability of SNR falling below the given threshold, thereby
reducing coverage. On the other hand, for the UAV with high height, even
though the link is in LoS, the increasing distance between the user and the
UAV will also reduce the SNR of the link, thus leading to a low probability of
coverage. Therefore, these two contributors eventually lead to the existence
of an optimum $\mu_{h_{i}}$ for maximum
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$.

Figure 10:  Coverage probability $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ of
the single UAV case versus mean height of UAV $\mu_{h_{i}}$ for different
blocker densities, where $\sigma=0.2$ m, $\theta=\pi/3$, and $\mu_{r_{i}}=55$
m. As we can see, there exists an optimal $\mu_{h_{i}}$ that maximize
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$.
Figure 11: Coverage probability $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ of
the multiple UAVs case versus UAV position fluctuation strength $\sigma$ with
different blocker densities, where $\lambda_{T}\\!=\\!100/$km2,
$\mu_{h_{i}}=25$ m, $\min(\mu_{r_{i}})\\!=\\!45$ m,
$\max(\mu_{r_{i}})\\!=\\!50$ m, $i=1,\cdots,n$, and $\theta=\pi/3$. As we can
see, the larger the $\sigma$ is, the smaller the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ is.

####  VI-B2 Multiple UAVs case

Fig. 11 plots the effect of $\sigma$ on coverage probability
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$. Similar to the single UAV case in
Fig. 8, we can see that in a low $\sigma$ region ($\sigma\\!<\\!0.05$), the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ for different $\lambda_{B}$ is the
same and hardly changes with $\sigma$. However, when $\sigma$ is larger, the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ decreases with the increase of
$\sigma$. Meanwhile, when comparing Fig. 11 and Fig. 8, it is obvious that
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}>\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$,
the reason is that using multiple UAVs can reduce the blockage of the link,
thus improving the SNR and the coverage performance. Note that in this case,
we assume that all UAVs are distributed on a ring with an inner diameter of
$\min(\mu_{r_{i}})$ an outer diameter of $\min(\mu_{r_{i}})+5$ m.

Figure 12: Coverage probability $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ of
the multiple UAVs case versus minimum average 2D distance
$\min(\mu_{r_{i}})$,$i=1,\cdots,n$ from the UAV to the user, where
$\sigma=0.2$ m and other parameters are the same as in Fig. 11. As we can see,
the larger the $\min(\mu_{r_{i}})$ is, the smaller the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ is, and a lower bound for
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ in (35) is given.
Figure 13:  Coverage probability $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ of
the multiple UAVs case versus mean height of UAVs $\mu_{h_{i}}$ for different
blocker densities, where $\sigma\\!=\\!0.2$ m, $\theta\\!=\\!\pi/3$, and
$\min(\mu_{r_{i}})\\!=\\!55$ m. As we can see, there exists an optimal
$\mu_{h_{i}}$ that maximize $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$.

Fig. 12 shows the impact of $\min(\mu_{r_{i}})$ on
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$. It is clear that the
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ decrease with the increase of
$\min(\mu_{r_{i}})$, the reason is similar to the analysis of Fig. 9 of the
single UAV case, and is also consistent with the general conclusion that the
service quality of edge users is generally the worst. In addition, Fig. 12
compares the $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ and a low bound of
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ in (35) for $\lambda_{B}=0.01$ and
$\lambda_{S}=0$. We observe that the value of
$\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ in (34) is slightly greater than
the lower bound of $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$, which is
reasonable. Finally, Fig. 13 proves that there is an optimal $\mu_{h_{i}}$ for
maximum $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$. The reason is a
combination of two factors, as described in the analysis of Fig. 10.

##  VII Conclusion

In this paper, we studied the critical issues affecting the QoS of air-to-
ground mmWave UAV communication systems, especially those arising from the
random position fluctuations of hovering UAVs. Accordingly, we considered the
reliable service probability respective to links blockages, and the coverage
probability respective to SNR as the system’s key QoS measures. Then, we
derived the closed-form expressions to evaluate the impact of UAV position
fluctuations on these two QoS metrics. Our results indicated that the larger
the position fluctuations of the UAV, the smaller the reliable service
probability, and the smaller the coverage probability. Therefore, unlike
ground mmWave communications, the QoS of the air-to-ground mmWave UAV
communication system largely depends on the position fluctuations of the UAV.
Specifically, the fluctuation will cause random changes in the blockage
characteristic and the SNR of the link, thus resulting in unreliable
communications. Fortunately, we theoretically analyzed the effect of UAV
position fluctuations on link blockages and SNR, and the relevant results made
it possible to quickly evaluate the QoS of the system under different levels
of UAV position fluctuations. In addition, according to the simulation
results, we find the optimal horizontal position and height of the UAV to
maximize the reliable service probability and the coverage probability,
respectively, which helps establish reliable mmWave UAV communications.

##  Appendix A Proof of Theorem 1

We denote the PDF of $\phi_{i}(h_{i},r_{i})$ as $f_{\phi_{i}}(\phi_{i})$, and
according to (3), $f_{\phi_{i}}(\phi_{i})$ can first be expressed by taking
the derivative of the corresponding CDF
$\mathbb{P}\left(\frac{\rho{r_{i}}}{{\rho
r_{i}+\omega(h_{i}-h_{R})}}\leq\phi_{i}\right)$, i.e.,
$f_{\phi_{i}}(\phi_{i})=\frac{\partial\mathbb{P}\left(\frac{\rho{r_{i}}}{{\rho
r_{i}+\omega(h_{i}-h_{R})}}\leq\phi_{i}\right)}{\partial\phi_{i}}$. Therefore,
it is important to analyze the CDF of $\frac{\rho{r_{i}}}{{\rho
r_{i}+\omega(h_{i}-h_{R})}}$ in order to obtain $f_{\phi_{i}}(\phi_{i})$.
Since $x_{i}$ and $y_{i}$ are Gaussian distributed,
$r_{i}\\!=\\!\sqrt{x_{i}^{2}+y_{i}^{2}}$ follows a Rician distribution [28],
and the PDF can be denoted as
$f_{r_{i}}({r_{i}})=\frac{r_{i}}{\sigma^{2}}{\exp{\left(-{\frac{{r_{i}^{2}}+\mu_{r_{i}}^{2}}{2{\sigma}^{2}}}\right)}}I_{0}\left(\frac{r_{i}\mu_{r_{i}}}{\sigma^{2}}\right)$,
where $\mu_{r_{i}}\\!=\\!\sqrt{\mu^{2}_{x_{i}}\\!+\\!\mu^{2}_{y_{i}}}$,
$I_{0}(\cdot)$ denotes the modified Bessel function of the first kind of zero
order. In general, the 2D distance from the UAV to the user is far greater
than the fluctuation strength of the UAV, i.e., $\mu_{r_{i}}\\!\gg\\!\sigma$.
Therefore, $r_{i}$ can be well approximated by a Gaussian random variable
according to [33, 34], and then, the PDF of $r_{i}$ can be rewritten as
follows:

$\displaystyle
f_{r_{i}}({r_{i}})\approx\frac{1}{\sqrt{2\pi}\sigma}{\exp\left({-{\frac{{(r_{i}-\mu_{r_{i}})}^{2}}{2{\sigma}^{2}}}}\right)}.$
(37)

Then, we can get $\rho{r_{i}}\\!\sim\\!N(\rho\mu_{r_{i}},\rho^{2}\sigma^{2})$
and
$\rho{r_{i}}\\!+\\!\omega(h_{i}-h_{R})\\!\sim\\!N\left(\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}-h_{R}),(\rho^{2}+\omega^{2})\sigma^{2}\right)$.
Finally, using the conclusions in [35, 36], $\frac{\rho{r_{i}}}{{\rho
r_{i}+\omega(h_{i}-h_{R})}}$ can be well modeled as a Gaussian distribution
with mean $\mu_{\phi_{i}}$ and variance $\sigma_{\phi_{i}}^{2}$. Among them,
$\mu_{\phi_{i}}$ and $\sigma_{\phi_{i}}^{2}$ are given as follows:

$\displaystyle\mu_{\phi_{i}}$
$\displaystyle=\frac{\rho{\mu_{r_{i}}}}{{\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})}},$
(38) $\displaystyle\sigma_{\phi_{i}}^{2}\\!$
$\displaystyle=\\!\frac{\rho^{2}\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{2}}+\frac{(\rho\mu_{r_{i}})^{2}(\rho^{2}+\omega^{2})\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{4}}.$
(39)

Therefore, $f_{\phi_{i}}(\phi_{i})$ can finally be obtained as shown in (10).

##  Appendix B Proof of Corollary 1

First, we take the derivative of (38) with respect to $\mu_{r_{i}}$ and have

$\displaystyle\frac{\partial\mu_{\phi_{i}}}{\partial\mu_{r_{i}}}=\frac{\rho\omega(\mu_{h_{i}}-h_{R})}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}-h_{R})\right)^{2}}>0.$
(40)

The reason why (40) is greater than 0 is that in the considered communication
system, generally, the height of the UAV is higher than that of the user,
i.e., $\mu_{h_{i}}>h_{R}$. Therefore, $\mu_{\phi_{i}}$ is an increasing
function of $\mu_{r_{i}}$. Next, to obtain the relationship between
$\mu_{r_{i}}$ and $\sigma_{\phi_{i}}$, we denote the first and second terms of
$\sigma^{2}_{\phi_{i}}$ in (39) as $\sigma^{2}_{\phi_{i,1}}$ and
$\sigma^{2}_{\phi_{i,2}}$ respectively. Then, we can get

$\displaystyle\frac{\partial\sigma^{2}_{\phi_{i,1}}}{\partial\mu_{r_{i}}}\\!=\\!-\frac{2\rho^{3}\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}-h_{R})\right)^{3}}<0,$
(41)

which indicates that $\sigma^{2}_{\phi_{i,1}}$ decreases with the increase of
$\mu_{r_{i}}$. Similarly, we have

$\displaystyle\frac{\partial\sigma^{2}_{\phi_{i,2}}}{\partial\mu_{r_{i}}}\\!=\\!\frac{2\mu_{r_{i}}(\rho^{2}\\!+\\!\omega^{2})\rho^{2}\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}-h_{R})\right)^{5}}{\left(\rho\mu_{r_{i}}\\!-\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)}<0,$
(42)

which means that $\sigma^{2}_{\phi_{i,2}}$ is a decreasing function of
$\mu_{r_{i}}$. Then, $\sigma^{2}_{\phi_{i}}$ also is a decreasing function of
$\mu_{r_{i}}$. Therefore, we can conclude that $\sigma_{\phi_{i}}$ is a
decreasing function of $\mu_{r_{i}}$. The reason why (42) is less than 0 is as
follows: In our system, $v$ is the speed of the moving humans, $h_{B}$ and
$h_{R}$ are the height of the moving humans and user, respectively. Generally
speaking, these parameters are fixed values in the considered communications
scenario, and $v({h_{B}-h_{R}})=0.4$ by using Table II. As such,
$\rho={2{\lambda_{B}}v({h_{B}-h_{R}})}/{\pi}$ is only a fraction of
$\lambda_{B}$, and even for $\lambda_{B}$ as high as $0.1$ bl/m2[11], the
value of $\rho$ is only 0.025. Therefore, $\rho\mu_{r_{i}}\leq\rho R=2.5$.
However, $\mu_{h_{i}}$ is usually several times or even more than ten times of
$h_{R}$ and $\omega=2$. All in all,
$\rho\mu_{r_{i}}\\!-\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})<0$.

##  Appendix C Proof of Lemma 1

We assume that the blockage of each link is independent. Therefore, given the
total number of UAVs $M$, the distribution of the number of available UAVs $N$
can be expressed as [11]:

$\displaystyle\mathbb{P}_{N|M}(n|m)=\binom{m}{n}\mathbb{P}(C_{i})^{n}\left(1-\mathbb{P}(C_{i})\right)^{m-n}.$
(43)

Since $M$ is Poisson distributed, $\mathbb{P}_{N}(n)$ can be obtained as
$\mathbb{P}_{N}(n)=\sum_{m=0}^{\infty}\mathbb{P}_{N|M}(n|m)\mathbb{P}_{M}(m)$
[11, Lemma 2], where $\mathbb{P}_{M}(m)$ is given in Subsection II-A.
Therefore, we can finally get

$\displaystyle\mathbb{P}_{N}(n)$
$\displaystyle=\frac{{[\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}]}^{n}}{n!}\exp\left({-\mathbb{P}(C_{i})\lambda_{T}\pi{R}^{2}}\right),$
(44)

where the available probability $\mathbb{P}(C_{i})$ is calculated as follows:
Using (6) and (9), the conditional probability that the $i$-th UAV is
available is
$\mathbb{P}(C_{i}|h_{i},r_{i})\\!=\\!\left(1-\frac{\theta}{2\pi}\right)\exp{\left(-(\epsilon
r_{i}+\epsilon_{0})\right)}$, and it is obviously that
$\mathbb{P}(C_{i}|h_{i},r_{i})=\mathbb{P}(C_{i}|r_{i})$. Therefore, to obtain
the marginal probability $\mathbb{P}(C_{i})$, we first need to take the
average of $\mathbb{P}(C_{i}|r_{i})$ over the distribution of $r_{i}$, and
according to (37), $r_{i}$ follows a Gaussian distribution with mean value
$\mu_{r_{i}}$. However, $\mu_{r_{i}}$ is still a random variable since the
center positions of the UAVs are modeled as a PPP. As a result, we can get

$\displaystyle\mathbb{P}(C_{i}|\mu_{r_{i}})$
$\displaystyle\\!=\\!\int_{r_{i}}\mathbb{P}(C_{i}|h_{i},r_{i})f_{r_{i}}({r_{i}})d{r_{i}}$
$\displaystyle\\!\approx\\!\left(\\!1\\!-\\!\frac{\theta}{2\pi}\\!\right)\exp{\left(\\!-\epsilon\mu_{r_{i}}\\!-\\!\epsilon_{0}\\!+\\!\frac{\epsilon^{2}\sigma^{2}}{2}\right)}\mathrm{erf}\\!\left(\frac{3\\!+\\!\epsilon\sigma}{\sqrt{2}}\\!\right),$
(45)

the PDF of $\mu_{r_{i}}$ is
$f_{\mu_{r_{i}}}(\mu_{r_{i}})=\frac{2\mu_{r_{i}}}{{R}^{2}};0<\mu_{r_{i}}\leq
R,\forall i=1,\cdots,m.$ [11]. Therefore, we can get

$\displaystyle\mathbb{P}(C_{i})$
$\displaystyle=\int_{\mu_{r_{i}}}\mathbb{P}(C_{i}|\mu_{r_{i}})f_{\mu_{r_{i}}}(\mu_{r_{i}})d{\mu_{r_{i}}}$
$\displaystyle\approx\left(1\\!-\\!\frac{\theta}{2\pi}\right)\mathrm{erf}\left(\frac{3\\!+\\!\epsilon\sigma}{\sqrt{2}}\right)\frac{2\exp{(\\!-\epsilon_{0}\\!+\\!\frac{\epsilon^{2}\sigma^{2}}{2})}}{R^{2}\epsilon^{2}}\left(1\\!-\\!(1\\!+\\!R\epsilon)\exp{(\\!-R\epsilon)}\right).$
(46)

For further insight, we further approximate $\mathbb{P}(C_{i})$ as follows: We
know that $\mathrm{erf}\left(\frac{3}{\sqrt{2}}\right)\approx 1$, and
$3+\epsilon\sigma\geq 3$ due to
$\epsilon\\!=\\!\frac{2}{\pi}\lambda_{S}(\mathbb{E}(l)\\!+\\!\mathbb{E}(w))\\!>\\!0$.
Therefore,
$\mathrm{erf}\left(\frac{3+\epsilon\sigma}{\sqrt{2}}\right)\\!\approx\\!1$
according to the characteristics of error function. Then, since the size of
the buildings are $\mathbb{E}(l)\times\mathbb{E}(w)$, the maximum value of
density of the buildings is
$\lambda_{S}^{\max}\\!=\\!\frac{1}{{\mathbb{E}(l)}\mathbb{E}(w)}$, and the
maximum value of $\epsilon$ is given by
$\epsilon_{\max}=\frac{2}{\pi}\lambda_{S}^{\max}(\mathbb{E}(l)+\mathbb{E}(w))=\frac{2}{\pi}\left(\frac{1}{\mathbb{E}(w)}+\frac{1}{\mathbb{E}(l)}\right)<\frac{4}{\pi}\approx
1$, where the inequality is obtained based on the fact that the value of
$\mathbb{E}(w)$ and $\mathbb{E}(l)$ are larger than 1 in practice, and the
result indicates that $\epsilon\\!<\\!1$, i.e., $\epsilon\\!>\\!\epsilon^{2}$.
We have $\epsilon_{0}\\!=\\!\lambda_{S}\mathbb{E}(l)\mathbb{E}(w)\\!>\\!0$,
$\mu_{r_{i}}\\!\gg\\!\sigma$ and $\sigma<1$ in practice. Hence,
$\epsilon\mu_{r_{i}}\\!+\\!\epsilon_{0}\\!\gg\\!\epsilon^{2}\sigma^{2}$, and
the value of
$-(\epsilon\mu_{r_{i}}\\!+\\!\epsilon_{0})\\!+\\!\frac{\epsilon^{2}\sigma^{2}}{2}$
mainly depends on $-(\epsilon\mu_{r_{i}}\\!+\\!\epsilon_{0})$, i.e.,
$\exp{\left(-\epsilon\mu_{r_{i}}\\!-\\!\epsilon_{0}\\!+\\!\frac{\epsilon^{2}\sigma^{2}}{2}\right)}\\!\approx\\!\exp{\left(-(\epsilon\mu_{r_{i}}\\!+\\!\epsilon_{0}\right))}$,
which verifies our insights in Remark 3. Therefore we can assume that
$\mathbb{P}(C_{i}|\mu_{r_{i}})\approx\left(1\\!-\\!\frac{\theta}{2\pi}\right)\exp{\left(\\!-\\!(\epsilon\mu_{r_{i}}\\!+\\!\epsilon_{0})\right)}$,
and then,

$\displaystyle\mathbb{P}(C_{i})$
$\displaystyle\\!\approx\\!\left(1\\!-\\!\frac{\theta}{2\pi}\right)\frac{2\exp{(\\!-\epsilon_{0})}}{R^{2}\epsilon^{2}}\left(1\\!-\\!(1\\!+\\!R\epsilon)\exp{(\\!-R\epsilon)}\right),$
(47)

Fig. 4 further illustrates the accuracy of this approximation.

##  Appendix D Proof of Theorem 4

According to (V-B), we can get
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}=\mathbb{P}(C_{i})\mathbb{P}\left(\gamma_{i}\geq\gamma_{0}\right)$,
where $\mathbb{P}(C_{i})$ is given in (47). Hence, we only need to calculate
$\mathbb{P}\left(\gamma_{i}\geq\gamma_{0}\right)$. With the aid of (27), (28)
and (29), we can get

$\displaystyle\mathbb{P}\left(\gamma_{i}\geq\gamma_{0}\right)=\mathbb{P}\left({\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}}{d_{i}^{-\alpha}}\geq\frac{N_{0}}{P_{t}}\gamma_{0}\right)$
(48)

which is related to the CDF of
${\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}}{d_{i}^{-\alpha}}$. Therefore, to obtain
$\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$, it is necessary to calculate the
CDF mentioned above by considering random variables $h_{i}$, $r_{i}$, and
$d_{i}$. To this end, we define an indicator random variable $Z$ as follows:

$\displaystyle Z={\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}}{d_{i}^{-\alpha}},$ (49)

and analyze the CDF of $Z$. First, we have
$h_{i}\\!\sim\\!N(\mu_{h_{i}},\sigma^{2})$ and
$r_{i}\\!\sim\\!N(\mu_{r_{i}},\sigma^{2})$. Therefore,
$d_{i}\\!=\\!\sqrt{r_{i}^{2}\\!+\\!(h_{i}\\!-\\!h_{R})^{2}}$ is Rician
distributed, and the PDF is
$f_{d_{i}}({d_{i}})\\!=\\!\frac{d_{i}}{\sigma^{2}}{\exp{\left(\\!-{\frac{{d_{i}^{2}}\\!+\\!\mu_{d_{i}}^{2}}{2{\sigma}^{2}}}\right)}}I_{0}\left(\frac{d_{i}\mu_{d_{i}}}{\sigma^{2}}\right)$,
where
$\mu_{d_{i}}\\!=\\!\sqrt{\mu_{r_{i}}^{2}+(\mu_{h_{i}}\\!-\\!h_{R})^{2}}$.
Then, using (27) and (3), we have
$\mathbb{P}^{\rm{LoS}}_{i}\\!=1-{\phi_{i}}(h_{i},r_{i})$, and
$\phi_{i}(h_{i},r_{i})$ follows a normal distribution as shown in Theorem 1.
Therefore, $\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}$ also follows a normal
distribution with mean $\beta_{0}(1-\mu_{\phi_{i}})$ and variance
$\beta_{0}^{2}\sigma_{\phi_{i}}^{2}$ where $\mu_{\phi_{i}}$ and
$\sigma_{\phi_{i}}^{2}$ are given in (38) and (39), respectively. Next, we
will prove that the randomness of $Z$ mainly depends on $d_{i}$. In this case,
the task of calculating the CDF of $Z$ can be converted to calculating the CDF
of $d_{i}$. We first prove that the random variable
$\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}$ can be approximated by its mean
$\beta_{0}(1-\mu_{\phi_{i}})$. Using the proof of Corollary 1, we know that
$\rho\leq 0.025$, $\rho\mu_{r_{i}}\leq 2.5$, and $\rho$ is several orders of
magnitude smaller than that of $\omega(\mu_{h_{i}}\\!-\\!h_{R})$ for our
system setup. Therefore, we first can get
$\rho^{2}\sigma^{2}\ll\left(\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{2}$ since
$\sigma<1$ in practice. Then, we have
$\frac{\rho^{2}\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{2}}\\!\ll\\!\frac{\omega(\mu_{h_{i}}\\!-\\!h_{R})(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R}))}{{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{2}}}$.
Moreover, we can get $(\rho\mu_{r_{i}})^{2}(\rho^{2}+\omega^{2})\sigma^{2}\leq
2.5^{2}(0.025^{2}+4)0.04\approx 1$ with the aid of Table II. Therefore,
$\frac{(\rho\mu_{r_{i}})^{2}(\rho^{2}+\omega^{2})\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{4}}\ll\frac{\omega(\mu_{h_{i}}\\!-\\!h_{R})(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R}))^{3}}{{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{4}}}$
due to $\mu_{h_{i}}$ is usually several times or even more than ten times of
$h_{R}$ and $\omega=2$. In addition, we have
$\beta_{0}=7\\!\times\\!10^{\\!-5}$[32]. Based on the above discussion, we can
conclude that
$\beta_{0}^{2}\left(\frac{\rho^{2}\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{2}}+\frac{(\rho\mu_{r_{i}})^{2}(\rho^{2}+\omega^{2})\sigma^{2}}{\left(\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})\right)^{4}}\right)\ll\frac{\beta_{0}\omega(\mu_{h_{i}}\\!-\\!h_{R})}{{\rho\mu_{r_{i}}\\!+\\!\omega(\mu_{h_{i}}\\!-\\!h_{R})}}$,
i.e., $\beta_{0}^{2}\sigma_{\phi_{i}}^{2}\ll\beta_{0}(1-\mu_{\phi_{i}})$.
Therefore, $\beta_{0}^{2}\sigma_{\phi_{i}}^{2}$ has little impact on
$\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}$. As a result, we can assume that
$\beta_{0}\mathbb{P}^{\rm{LoS}}_{i}\approx\beta_{0}(1-\mu_{\phi_{i}})$, and
then, substituting (38) into (49), $Z$ can be rewritten as

$\displaystyle
Z\approx\frac{\beta_{0}\omega(\mu_{h_{i}}-h_{R})}{d_{i}^{\alpha}\left(\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}-h_{R})\right)},$
(50)

and the objective of calculating the CDF of $Z$ can be simplified as
calculating the CDF of $d_{i}$. Let us denote the CDF of $d_{i}$ as
$F_{d_{i}}(d)$. Since $d_{i}$ is Rician distributed, $F_{d_{i}}(d)$ can be
expressed in terms of the Marcum Q-function as follows [37, 38]:

$\displaystyle
F_{d_{i}}(d)=1-Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{d}{\sigma}\right)$
(51)

where $Q_{1}(a,b)$ is the Marcum Q-function and is defined as

$\displaystyle
Q_{1}(a,b)=\int_{b}^{\infty}x\exp{\left(-\frac{a^{2}+x^{2}}{2}\right)}I_{0}(ax)d{x}.$
(52)

Finally, with the aid of (V-B), (48), (49), (50) and (51), we have

$\displaystyle\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$
$\displaystyle\approx\mathbb{P}(C_{i})\mathbb{P}\left(\frac{\beta_{0}\omega(\mu_{h_{i}}-h_{R})}{d_{i}^{\alpha}\left(\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}-h_{R})\right)}\\!\geq\\!\frac{N_{0}}{P_{t}}\gamma_{0}\right)$
$\displaystyle=\mathbb{P}(C_{i})\mathbb{P}\left(d_{i}\leq\sqrt[\alpha]{\frac{P_{t}\beta_{0}\omega(\mu_{h_{i}}\\!-\\!h_{R})}{N_{0}\gamma_{0}({\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}\\!-\\!h_{R})})}}\right)$
$\displaystyle=\mathbb{P}(C_{i})\left(1\\!-\\!Q_{1}\left(\frac{\mu_{d_{i}}}{\sigma},\frac{\sqrt[\alpha]{\frac{P_{t}\beta_{0}\omega(\mu_{h_{i}}\\!-\\!h_{R})}{N_{0}\gamma_{0}({\rho\mu_{r_{i}}+\omega(\mu_{h_{i}}\\!-\\!h_{R})})}}}{\sigma}\right)\right),$
(53)

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