# UAV Relay-Assisted Emergency Communications in IoT Networks: Resource
Allocation and Trajectory Optimization

$\text{Dinh-Hieu Tran},\textit{Graduate Student Member, IEEE}$, $\text{Van-
Dinh Nguyen},\textit{Member, IEEE}$, $\text{Symeon Chatzinotas},\textit{Senior
Member, IEEE},$
$\text{Thang X. Vu},\textit{Member, IEEE},$ $\text{and Bj{\"{o}}rn
Ottersten},\textit{Fellow, IEEE}$ This work is supported in part by the
Luxembourg National Research Fund under project FNR CORE ProCAST, grant
C17/IS/11691338 and FNR 5G-Sky, grant C19/IS/13713801.Dinh-Hieu Tran, V.-D.
Nguyen, Symeon Chatzinotas, Thang X. Vu, and Björn Ottersten are with the
Interdisciplinary Centre for Security, Reliability and Trust (SnT), the
University of Luxembourg, Luxembourg. (e-mail: {hieu.tran-dinh, dinh.nguyen,
symeon.chatzinotas, thang.vu, bjorn.ottersten } @uni.lu).Part of this work was
presented in IEEE GLOBECOM 2020 - Workshop on Future of Wireless Access for
Industrial IoT (FutureIIoT), Taipei, Taiwan [1].

###### Abstract

Unmanned aerial vehicle (UAV) communication has emerged as a prominent
technology for emergency communications (e.g., natural disaster) in the
Internet of Things (IoT) networks to enhance the ability of disaster
prediction, damage assessment, and rescue operations promptly. A UAV can be
deployed as a flying base station (BS) to collect data from time-constrained
IoT devices and then transfer it to a ground gateway (GW). In general, the
latency constraint at IoT devices and UAV’s limited storage capacity highly
hinder practical applications of UAV-assisted IoT networks. In this paper,
full-duplex (FD) radio is adopted at the UAV to overcome these challenges. In
addition, half-duplex (HD) scheme for UAV-based relaying is also considered to
provide a comparative study between two modes (viz., FD and HD). Herein, a
device is considered to be successfully served iff its data is collected by
the UAV and conveyed to GW timely during flight time. In this context, we aim
to maximize the number of served IoT devices by jointly optimizing bandwidth,
power allocation, and the UAV trajectory while satisfying each device’s
requirement and the UAV’s limited storage capacity. The formulated
optimization problem is troublesome to solve due to its non-convexity and
combinatorial nature. Towards appealing applications, we first relax binary
variables into continuous ones and transform the original problem into a more
computationally tractable form. By leveraging inner approximation framework,
we derive newly approximated functions for non-convex parts and then develop a
simple yet efficient iterative algorithm for its solutions. Next, we attempt
to maximize the total throughput subject to the number of served IoT devices.
Finally, numerical results show that the proposed algorithms significantly
outperform benchmark approaches in terms of the number of served IoT devices
and system throughput.

###### Index Terms:

Emergency communications, full-duplex, information freshness, Internet-of-
Things, timely data collection, unmanned aerial vehicle (UAV).

##  I Introduction

In 1999, British technology pioneer Kevin Ashton introduced the concept of the
Internet-of-Things (IoT) to describe a system in which all devices equipped
with sensors can connect to each other [2]. IoT has the potential to
significantly enhance the quality of human life such as smart home, health
care, wearable devices, agriculture, smart city, autonomous vehicles, and
smart grid [3, 4]. The number of IoT connections of all types is estimated to
reach close to 25 billion by 2025 [5]. However, the growing demand for
communications is becoming a major challenge for IoT networks due to limited
spectral resources at terrestrial base stations (BSs). Besides, BSs are
deployed at fixed locations and antenna height to serve a fixed geographical
area, and resources cannot be rapidly shifted elsewhere. Especially in
emergency communications, whereas BSs are potentially isolated or damaged
after a natural disaster, or when BSs are unable to serve all users as they
are overloaded during peak hours. This raises a question of how to support the
communication needs of a massive number of IoT devices with restricted
resources without compromising the network performance [6]. Fortunately, due
to the high maneuverability and flexible deployment, unmanned aerial vehicle
(UAV) communications could become a promising technology to overcome the above
mentioned shortcomings [7]. Due to energy constraints, IoT devices are
commonly unable to propagate their signals long distances. On the other hand,
the UAV can fly closer to the devices, harvest the IoT data, and then transmit
it to the BS/control center, which is out of the transmission range of these
devices.

Extensive studies have been carried out to investigate UAV-assisted IoT
communication networks [8, 9, 10, 11, 12, 13]. The work in [8] studied the
joint optimal 3D deployment of UAVs, uplink (UL) power control, and device
association in an IoT network. Specifically, the authors proposed a new
framework for efficiently distributing UAVs to collect information in the UL
from IoT users. In [9], the authors optimized the data gathering efficiency of
a UAV-assisted IoT network, subject to the power budget, energy capacity, and
total transmission time of IoT devices. Herein, a multi-antenna UAV was
operated, which followed a circular trajectory and served IoT devices to
create a virtual multi-input multi-output (MIMO) channel. Reference [10]
presented a robust central system orchestrator (SO) that was designed to
provide value-added IoT services (VAIoTS). Whereas SO keeps the entire details
about UAVs including their current locations, flight missions, total energy
budget, and their onboard IoT devices. To obtain an efficient UAV selection
mechanism corresponding to each task requirement, the authors proposed three
solutions, namely, energy-aware UAV, fair trade-off UAV, and delay-aware UAV
selection. A novel UAV-aided IoT communication network to provide energy-
efficient data gathering and accurate 3D device positioning of IoT devices was
proposed in [11], whereas a UAV was deployed as an aerial anchor node and a
flying data collector. Particularly, UAVs could serve not only as aerial BSs
but also as powerful IoT components that are capable of performing
communications, sensing, and data analysis while hovering in the air [12]. To
extend the coverage for IoT-based emergency communications, Liu et al. [13]
integrated multi-hop device-to-device (D2D) and UAV communication during
natural disasters so that helps out-of-range IoT users can be effectively
connected to UAVs. Note that none of the above-mentioned works in [8, 9, 10,
11, 12, 13] take crucial latency constraint into consideration.

Recently, the delay-sensitive data collection has attracted much attention
from researchers [14, 15, 16, 17]. For example, in the emergency case or
during the natural disaster, the out-of-date gathering data may result in
unreliable controllable decisions, which may ultimately be disastrous [17]. On
the other hand, IoT devices often have limited storage capacity, and thus
their generated data need to be collected timely before it becomes worthless
due to obsolete transmissions or being overwritten by incoming data [18].
Therefore, the UAV must reach the right place at the right time. In [14], the
authors proposed two UAV trajectories, termed Max-AoI-optimal and Ave-AoI-
optimal, to efficiently collect data from ground sensor nodes under the impact
of age of information (AoI) metric. Specifically, the Max-AoI-optimal and Ave-
AoI-optimal trajectory planning minimize the age of the oldest information and
the average AoI of all sensor nodes, respectively. The work in [15] studied
the role of a UAV acting as a relay to minimize the average Peak AoI for a
transmitter-receiver link, which was accomplished via a joint optimization of
the UAV trajectory, energy spending, and service time allocations for packet
transmissions. In [16], the authors designed the UAV trajectory to minimize
expired data packets in UAV-enabled wireless sensor networks (WSNs) and then
applied the reinforcement learning (RL) method for the solution, which
enhances the time-effectiveness and path design performance. The authors in
[17] optimized the UAV trajectory as well as service bandwidth allocation to
maximize the total number of served ground IoT users, in which UAV needs to
collect data from users within their latency constraint. Different from [14,
15, 16, 17], which only studied the aspect of data collection on the UL
channel, the works in [19] and [20] further considered the latency constraint
on the DL channel.

Despite noticeable achievements for data collection in UAV-assisted IoT
networks [8, 9, 10, 11, 12, 14, 15, 16, 17], aforementioned works have not
exploited benefits of FD radios. To efficiently exploit the radio spectrum, FD
transmission was adopted in UAV communications [21, 22, 23, 24]. By applying a
circular trajectory and decode-and-forward (DF) relaying strategy, the work in
[21] maximized instantaneous data rate by a joint design of beam-forming and
power allocation, under individual and sum-power constraint for the source and
relay users. In [22], the authors investigated the spectrum sharing planning
problem for FD UAV relaying systems with underlaid device-to-device (D2D)
communications, which aims to maximize the sum throughput. The work in [23]
maximized the energy efficiency (EE) by jointly optimizing UAV trajectory, as
the transmit and jamming powers of a source and a UAV, respectively. Besides,
a new system model for UAV-enabled FD wireless-powered IoT networks was
proposed in [24], in which three optimization problems, namely, total-time
minimization, sum-throughput maximization, and total energy minimization
problem, were investigated.

Unlike previous studies such as [14, 15, 16, 17, 19, 20] that only investigate
timely data exchange on the UL or DL channel utilizing HD mode, this work
proposes a novel system model in UAV relay-assisted IoT networks that further
explores the impact of requested timeout (RT) constraints for both UL and DL
transmissions. To the best of our knowledge, this is the first work to jointly
optimizes total bandwidth, transmission power, trajectory design, storage
capacity, and latency constraint in UAV relay-assisted IoT networks. To this
end, we formulate two optimization problems and develop efficient iterative
algorithms to obtain a sub-optimal solution. In summary, our contributions are
as follows:

  * •

We propose a novel UAV relay-assisted IoT model that takes into account the
latency requirement for UL and DL channels to improve the freshness of
information. Therein, UAV-enabled FD relaying is exploited as an effective
mean to enhance network performance, i.e., increasing the number of served IoT
devices, throughput, and reducing latency. For instance, the reduced latency
and high throughput owing to FD operation can take the virtual/augmented
reality (VR/AR) experiences or emergency communications to the next level.
Besides, it also helps to overcome UAV’s limited storage capacity. Moreover,
UAV-enabled HD relaying is also investigated to fully capitalize on UAV
benefits for time-sensitive data collection in IoT networks.

  * •

We formulate a generalized optimization problem to maximize the total number
of served IoT devices under the UAV’s maximum speed constraint, total
traveling time constant, maximum transmit power of devices/UAV, limited cache
size of UAV, and latency constraints for both UL and DL. The formulation
belongs to the difficult class of mixed-integer non-convex optimization
problem, which is generally NP-hard. We first relax binary variables into
continuous ones and penalize the objective by introducing a penalty function.
We then develop an iterative computational procedure for its solutions, which
guarantees convergence to at least a local optimal. The key idea behind our
approach is to derive newly approximated functions for non-convex parts by
employing the inner approximation (IA) framework [25].

  * •

Inspired by the practical requirement in human safety measurements, the more
data we have collected, the better our predictions are. This motivates us to
investigate the optimization problem in order to maximize the total collected
throughput subject to a given number of served IoT users.

  * •

The proposed schemes’ effectiveness is revealed via numerical results, which
show significant improvements in both number of served IoT devices and the
total amount of collected throughput compared with the benchmarks. More
specifically, the Benchmark FD and Benchmark HD schemes are respectively
designed similar to the proposed FD-based and HD-based methods but with fixed
resource allocation or fixed trajectory.

  * •

Compared to our conference [1], we have made the following major revisions.
Firstly, the work in [1] only considers the throughput maximization problem
with an assumption of perfect CSI from IoT devices to UAV. Moreover, the
details of mathematical analysis are not provided in [1]. In this manuscript,
we have updated the channel model considering the approximated rate functions
for both uplink and downlink, as given in Lemma 1. Besides, we have provided
the IA framework in Section III and detailed the proof of Proposition 1 in
Appendix D. We have added an efficient method to generate an initial feasible
point to start the IA-based algorithm in Section III-B. Lastly, we have
reproduced all simulation results in Section V due to the change of channel
model. In addition, we have also added Fig. 3 to illustrate the UAV’s
trajectories.

Figure 1: System model: the UAV is deployed as a flying BS to collect the data
from IoT devices and then transmit to GW.  Figure 2:  Illustration of the data
transmission process of 2 IoT devices with $N$ time intervals. The first IoT
device with initial data transmission time at $n_{{\rm start},1}=2$, timeout
at $n_{{\rm end},1}=5$. The second IoT device with initial data transmission
time at $n_{{\rm start},2}=3$, timeout at $n_{{\rm end},2}=6$. The UAV
operates in the FD mode from time slots 5 to 6 since two devices utilize the
same sub-carrier.

The rest of the paper is organized as follows. The system model and problem
formulation are given in Section II. The proposed iterative algorithm for FD
is presented in Section III. Section IV devotes for the HD scheme. Numerical
results are illustrated in Section V, and Section VI concludes the paper.

_Notation_ : Scalars and vectors are denoted by lower-case letters and
boldface lower-case letters, respectively. For a set $\mathcal{{K}}$,
$|\mathcal{{K}}|$ denotes its cardinality. For a vector $v$,
$\left\|v\right\|_{1}$ and $\left\|v\right\|$ denote its $\ell_{1}$ and
Euclidean ($\ell_{2}$) norm, respectively. $\mathbb{R}$ represents for the
real matrix. $\mathbb{R}^{+}$ denotes the non-negative real numbers, i.e.,
$\mathbb{R}^{+}=\\{x\in\mathbb{R}|x\geq 0\\}$. $x\sim{\cal{CN}}(0,\sigma^{2})$
represents circularly symmetric complex Gaussian random variable with zero
mean and variance $\sigma^{2}$. Finally, $\triangledown f$ is the first
derivative of a function $f$. $\mathbb{E}[x]$ denotes the expected value of
$x$.

##  II System Model and Problem Formulation

We consider a UAV-aided cooperative wireless IoT network, where a UAV is
deployed to assist the existing terrestrial communication infrastructure in
the case of adverse conditions or natural calamities, as shown in Fig. 1. In
emergency communications, the ground base station (GBS) is either partially or
completely damaged after a natural disaster or in the case that the GBS is
overloaded during the peak hours due to its incapability of handling all the
devices at the same time (e.g., a sporting event) [26]. The latter case has
been recognized as one of the key scenarios that need to be effectively solved
by fifth-generation (5G) wireless communication [27, 26]. Concretely, a UAV
helps to relay data from a set of $K$ IoT devices (or GUs), denoted by
$\mathcal{K}\triangleq\\{1,\ldots,K\\}$, to a GW. Each IoT device is equipped
with a single antenna and works in HD mode. Due to the SWAP (size, weight, and
power) limitations, the UAV, acting as an on-demand relay, is equipped with
one FD antenna, which can simultaneously be used for data transmission and
reception. Specifically, the UAV can operate in FD or HD mode depending on the
system designer. It hovers over the considered area to effectively gather data
from IoT devices and then transmit it to the GW using UL and DL
communications, respectively. Due to limited energy budget, we restrain the
total serving time of UAV as $T$ [28]. We assume that each device is active at
different time instances $t$, where $0\leq t\leq T$. The location of device
$k$ is denoted as ${{\bf w}_{k}}\in{\mathbb{R}^{2\times 1}},k\in\mathcal{K}$.
We assume that the locations of IoT devices together with their data sizes,
the initial data transmission time (i.e., $n_{{\rm start},k}$ with $k\in{\cal
K}$), and latency requirement (i.e., $n_{{\rm end},k}$ with $k\in{\cal K}$)
are known to the UAV through the control center.111The control center can take
care of the corresponding computations and inform the UAV through dedicated
signaling, without affecting the performance of the considered framework.
Denote $n_{{\rm start},k}$ and $n_{{\rm end},k}$ by the initial data
transmission time and timeout constraint of the device $k$, respectively, for
$k\in\cal{K}$. It is assumed that the UAV collects data from device $k$ within
$n_{{\rm end},k}$ units of time. For simplicity, we assume that the UAV flies
at a constant altitude of $H$ (m), e.g., imposed by the regulatory authority
for safety considerations. The location of UAV projected on the ground at time
t is denoted as ${\bf q}(t)\in{\mathbb{R}^{2\times 1}},$ with $0\leq t\leq T$
[29].

###  II-A UAV-to-Ground and Ground-to-UAV Channel Model

For ease of exposition, the time horizon $T$ is discretized into $N$ equally
spaced time intervals, i.e., $T=N\delta_{t}$ with $\delta_{t}$ being the
primary slot length. Moreover, let ${\cal N}=\\{1,\dots,N\\}$ denote a set of
all time slots. Note that the UAV location can be assumed to be approximately
unchanged during each time slot compared to the distance from the UAV to IoT
devices since $\delta_{t}$ is chosen sufficiently small [30]. Then, the UAV
trajectory ${\bf q}(t)$ during time horizon $T$ can be represented as
$\left({{\bf q}[n]}\right)_{n=1}^{N},$ where ${\bf q}[n]$ denotes the UAV’s
horizontal location at $n$-th time interval. Let $V_{\rm max}$ denote the
maximum velocity of the UAV, then the UAV’s speed constraint can be presented
as

$\displaystyle\left\|{{\bf q}[n]-{\bf q}[n-1]}\right\|\leq\delta_{d}=V_{\rm
max}{\delta_{t}},n=2,...,N.$ (1)

For notation convenience, let us denote the $k$-th IoT device and UAV by $k$
and $\rm U$, respectively. Henceforth, $1k$ and $2k$ represent for the UL
(i.e., $k\to\rm U$) and DL (i.e., ${\rm U}\to\rm GW$), respectively. Then, the
time-dependence distance from $k\to\rm U$ or ${\rm U}\to\rm GW$ (i.e., $1k$ or
$2k$), is given by

$\displaystyle{d_{ik}}[n]=\sqrt{{H^{2}}+{\left\|{{\bf q}[n]-{\bf
w}}\right\|}^{2}},i\in\\{1,2\\},\;\forall n,\;k,$ (2)

where ${\bf w}\in\\{{\bf w}_{k},{\bf w}_{0}\\}$, with ${\bf w}_{0}$ denoting
the location of GW.

In realistic scenarios, the devices are located in different environments,
e.g., rural, urban, suburban, etc. Thus, a generalized channel model
consisting of both line-of-sight (LOS) and non-line-of-sight (NLOS) channel
elements is considered. In this work, we consider a practical channel model
that takes into account both large-scale and small-scale fading channels [31].
Specifically, the channel coefficient at the $n$-th time slot, denoted by
$h_{ik}[n]$, can be written as [32, 17]

$\displaystyle h_{ik}[n]=\sqrt{\omega_{ik}[n]}{\tilde{h}}_{ik}[n],$ (3)

where $\omega_{ik}[n]$ represents for the large-scale fading effects and
${\tilde{h}}_{ik}[n]$ accounts for Rician small-scale fading coefficient.
Specifically, $\omega_{ik}[n]$ can be modeled as

$\displaystyle\omega_{ik}[n]=\omega_{0}d_{ik}^{-\alpha}[n],$ (4)

where $\omega_{0}$ is the average channel power gain at the reference distance
$d=$ 1 m, and $\alpha\geq 2$ is the path loss exponent for the Rician fading
channel [17]. The small scale fading ${\tilde{h}}_{ik}[n]$ with an expected
value $\mathbb{E}\left[|{\tilde{h}}_{ik}[n]|^{2}\right]=1$, is given by

$\displaystyle{\tilde{h}}_{ik}[n]=\sqrt{\frac{G}{1+G}}\bar{h}_{ik}[n]+\sqrt{\frac{1}{1+G}}\hat{h}_{ik}[n],$
(5)

where $G$ is the Rician factor; $\bar{h}_{ik}[n]$ and
$\hat{h}_{ik}[n]\sim\mathcal{CN}(0,1)$ denote the deterministic LoS and the
NLoS component (Rayleigh fading) during time slot $n$, respectively.

Due to the UL and DL channels’ coexistence using the same frequency at $n$-th
time slot, the self-interference (SI) may occur at the UAV. Without loss of
generality, once the UAV finishes data collection from device $k$, then the
transmission from UAV to GW can be conducted.222In this work, we adopt a
(decode-and-forward) DF relaying technique [33]; thus, the UAV needs to
complete receiving all the data from device $k$ before relaying to GW to
guarantee the data encoding properly. Moreover, a sufficiently large time
period is assumed to carry out the data transfer as well as the decoding
process at the UAV.

Let us denote by $x_{1k}[n]$ and $x_{2k}[n]$ the data symbols with unit power
(i.e., $\mathbb{E}\left[|x_{1k}[n]|^{2}\right]=1$ and
$\mathbb{E}\left[|x_{2k}[n]|^{2}\right]=1$) from $k\to\rm U$ and $\rm U\to\rm
GW$ at time slot $n$, respectively. As a result, the received signals of
device $k$ at the UAV and GW are respectively given by

$\displaystyle y_{1k}[n]$ $\displaystyle=\sqrt{p_{1k}[n]}h_{1k}[n]x_{1k}[n]$
$\displaystyle+\sqrt{\rho^{\rm RSI}}g_{\rm U}[n]\sum\limits_{k^{\ast}\in{\cal
K}\setminus k}\sqrt{p_{2k^{\ast}}[n]}x_{2k^{\ast}}[n]+n_{0},$ (6)
$\displaystyle y_{2k}[n]$
$\displaystyle=\sqrt{p_{2k}[n]}h_{2k}[n]x_{2k}[n]+n_{0},$ (7)

where RSI represents for residual self-interference term, $\sqrt{\rho^{\rm
RSI}}g_{\rm U}[n]\sum\limits_{k^{\ast}\in{\cal K}\setminus
k}\sqrt{p_{2k^{\ast}}[n]}x_{2k^{\ast}}[n]$ is the RSI power after all
interference cancellations [34, 35, 36, 37], $\rho^{\rm RSI}\in[0,1)$ is the
degree of RSI, $n_{0}\sim{\cal{CN}}(0,\sigma^{2})$ denotes the additive white
Gaussian noise (AWGN); $p_{1k}[n]$ and $p_{2k}[n]$ are the transmit power of
the device $k$ and UAV on the UL and DL to transmit the device $k$’s data at
time slot $n$, respectively; $g_{\rm U}[n]$ denotes the fading loop channel at
the UAV, which interferes UL reception due to concurrent downlink transmission
[38, 39].

To deal with the issues involved in limited resources and the UAV’s self-
interference, we consider the resources allocation (i.e., bandwidth and
transmit power) for bold the UL and DL. Thus, the achievable rate (bits/s) of
links from $k\to{\rm U}$ or ${\rm U}\to{\rm GW}$ to transmit the data of
device $k$ at time slot $n$ are respectively given as

$\displaystyle r_{ik}[n]$
$\displaystyle=a_{ik}[n]B\log_{2}\left(1+\Gamma_{ik}\right),i\in\\{1,2\\},$
(8)

where
$\Gamma_{1k}\triangleq\frac{p_{1k}[n]|{\tilde{h}}_{1k}[n]|^{2}\omega_{0}}{{\left({H^{2}}+{{\left\|{{\bf
q}[n]-{\bf w}_{k}}\right\|}^{2}}\right)^{\alpha/2}}\big{(}\phi^{\rm
RSI}\sum\limits_{k^{\ast}\in{\cal K}\setminus
k}p_{2k^{\ast}}[n]+\sigma^{2}\big{)}}$,
$\Gamma_{2k}\triangleq\frac{p_{2k}[n]{|{\tilde{h}}_{2k}[n]|^{2}\omega_{0}}}{\left({H^{2}}+{{\left\|{{\bf
q}[n]-{\bf w}_{0}}\right\|}^{2}}\right)^{\alpha/2}\sigma^{2}}$, $\phi^{\rm
RSI}\triangleq\rho^{\rm RSI}|g_{\rm U}[n]|^{2}$; $B$ denotes the total
bandwidth in hertz (Hz) of the system; $a_{1k}[n]B$ and $a_{2k}[n]B$ are the
bandwidth allocated for the UL and DL to transmit data of $k$-th device during
time slot $n$, respectively. Herein, $a_{1k}[n]$ and $a_{2k}[n]$ represent for
the spectrum allocation for devices and the UAV, respectively. Note that
instantaneous CSI elements (i.e., $|{\tilde{h}}_{1k}[n]|^{2}$ and
$|{\tilde{h}}_{2k}[n]|^{2}$) are difficult to obtain in advance. Moreover,
$|{\tilde{h}}_{1k}[n]|^{2}$ and $|{\tilde{h}}_{2k}[n]|^{2}$ are random
variables, thus instantaneous rates (i.e., $r_{1k}[n]$ and $r_{2k}[n]$) are
also random variables. Therefore, the expected values of received rates at the
UAV/GW are expressed as [40, 41]

$\displaystyle\mathbb{E}\big{[}r_{ik}[n]\big{]}$
$\displaystyle=a_{ik}[n]B\mathbb{E}[\log_{2}\big{(}1+\Gamma_{ik}\big{)}],i\in\\{1,2\\},k\in{\cal
K}.$ (9)

Due to the troublesome of deriving the probability density function, it raises
a difficulty in obtaining the closed-form expression of
$\mathbb{E}\big{[}r_{ik}[n]\big{]}$. Thus, we provide lower-bound functions of
$\mathbb{E}\big{[}r_{ik}[n]\big{]}$ as follows:

###### Lemma 1

The lower bounds of $\mathbb{E}\big{[}r_{1k}[n]\big{]}$ and
$\mathbb{E}\big{[}r_{2k}[n]\big{]}$ are respectively given as

$\displaystyle\bar{r}_{1k}[n]$
$\displaystyle=a_{1k}[n]B\log_{2}\Bigg{(}1+\frac{e^{-E}p_{1k}[n]\omega_{0}}{{({H^{2}}+{{\left\|{{\bf
q}[n]-{\bf w}_{k}}\right\|}^{2}})^{\alpha/2}}\nu_{1k}[n]}\Bigg{)},$ (10)
$\displaystyle\bar{r}_{2k}[n]$
$\displaystyle=a_{2k}[n]B\log_{2}\left(1+\frac{e^{-E}p_{2k}[n]{\omega_{0}}}{({H^{2}}+{{\left\|{{\bf
q}[n]-{\bf w}_{0}}\right\|}^{2}})^{\alpha/2}\sigma^{2}}\right),$ (11)

where $\nu_{1k}[n]\triangleq\big{(}\phi^{\rm RSI}\sum\limits_{k^{\ast}\in{\cal
K}\setminus k}p_{2k^{\ast}}[n]+\sigma^{2}\big{)}$.

###### Proof:

See Appendix A. ∎

Further, $a_{1k}[n]$ and $a_{2k}[n]$ are approximately continuous between 0
and 1 [17]. Thus, the bandwidth allocation should satisfy:

$\displaystyle\sum\limits_{k\in{\cal K}}a_{ik}[n]\leq 1,\forall
n,\;i\in\\{1,2\\},$ (12) $\displaystyle 0\leq a_{ik}[n]\leq 1,\forall k,n.$
(13)

Based on (10) and (11), the throughput (in bits) received on the UL or DL to
transmit device $k$’s data during time slot $n$, can be written as

$\displaystyle C_{ik}[n]=\delta_{t}R_{ik}[n],\;{\rm where}\;i\in\\{1,2\\},$
(14)

where

$\displaystyle R_{ik}[n]$
$\displaystyle=\left\\{\begin{array}[]{l}\bar{r}_{ik}[n],\hfill\text{if}\;n\in{\cal
T}_{ik},\\\ 0,\hfill\text{otherwise},\end{array}\right.$ (17)

with ${\cal T}_{1k}\triangleq\\{n_{{\rm start},k},\ldots,n_{{\rm end},k}\\}$,
${\cal T}_{2k}\triangleq\\{n_{{\rm end},k}+1,\ldots,N\\}$; (17) means that the
UAV only can collect the data from device $k$ (or transmit data to the GW)
during time period ${\cal T}_{1k}$ (or ${\cal T}_{2k}$); Otherwise, the data
transmission rate is treated as zero. Specifically, the UAV only transmits
device $k$’s data to GW iff it finishes the data collection process for that
device. Moreover, the total throughput over $N$ time slots received on the UL
and DL are denoted as $C_{1k}=\sum\limits_{n\in{\cal
T}_{1k}}\delta_{t}R_{1k}[n]$ and $C_{2k}=\sum\limits_{n\in{\cal
T}_{2k}}\delta_{t}R_{2k}[n]$, respectively.

To assist in the mathematical problem formulation, we introduce a new binary
variable $\lambda_{k}$ as

$\displaystyle{\lambda_{k}}=\left\\{\begin{array}[]{l}1,\hfill{\rm{Device}}\;k\;{\rm{is\;successfully\;served\;by\;the\;UAV}},\\\
0,\hfill{\rm{otherwise.}}\end{array}\right.$ (20)

###### Definition 1

The value of $\lambda_{k}$ should be equal to one iff the device $k$’s data is
collected by the UAV while additionally guaranteeing its successful reception
at the GW.

Let $S_{k}$ denote the data size (in bits) needed to transmit from device $k$
to GW. Then, we have the RT constraint for transmitting the device $k$’s data
on the UL and DL are expressed as, respectively

$\displaystyle\lambda_{k}\frac{S_{k}}{R_{1k}}\leq(n_{{\rm end},k}-n_{{\rm
start},k}+1)\delta_{t},\forall k,$ (21)
$\displaystyle\lambda_{k}\frac{S_{k}}{R_{2k}}\leq(N-n_{{\rm
end},k})\delta_{t},\forall k,$ (22)

where $R_{ik}=\sum\limits_{n\in{\cal T}_{ik}}R_{ik}[n]$, with $i\in\\{1,2\\}$;
(21) means that device $k$ must transmit information to the UAV before timeout
constraint, i.e., $n_{{\rm end},k}$; (22) implies that the data transmission
process to transmit the device $k$’s data from $\rm U\to\rm GW$ is performed
during the serving time of the UAV.333We consider the system model in which
the UAV does not transmit the data during taking off and landing [42]. Thus,
the data transmission process only occurs when the UAV is flying in the sky.

###  II-B Caching Model

The UAV has a cache with a storage capacity of $C$. Due to the limited cache
size of the UAV, it can utilize FD mode to release the storage and improve the
network throughput. Considering a storage limitation, the total number of
files cached at the UAV should not exceed its available storage capacity,
i.e.,

$\displaystyle\sum\limits_{k\in{\cal
K}}\Bigg{(}\sum\limits_{l=1}^{n}C_{1k}[l]-\sum\limits_{l=1}^{n-1}C_{2k}[l]\Bigg{)}\leq
C,$ (23)

where
$\sum\limits_{l=1}^{n}C_{1k}[l]\triangleq\lambda_{k}S_{k}-\sum\limits_{l=n+1}^{N}C_{1k}[l]$.

Note that, in order to spend a part of storage capacity for future use, i.e.,
a free cache size to store new data streams, the amount of data stored at the
UAV is calculated as the size of files collected from all devices till $n$-th
time slot minus files transmitted to GW till $(n-1)$-th time slot as in (23).

###  II-C Problem Formulation

In this section, we aim to maximize the total number of served IoT devices by
jointly optimizing the UAV trajectory ${{\bf q}}[n]$, the allocation of
resources (i.e., bandwidth and transmit power assigned for UL and DL), and
taking into account the storage limitation, the locations, initial
transmission time, and the timeout constraint of all IoT devices.

Let us define ${\bf q}\triangleq\\{{\bf q}[n],\forall n\\}$, ${\bf
a}\triangleq\\{a_{1k}[n],a_{2k}[n],k\in{\cal K},n\in{\cal N}\\},$ ${\bf
p}\triangleq\\{p_{1k}[n],p_{2k}[n],k\in{\cal K},n\in{\cal N}\\},$
${\boldsymbol{\lambda}}\triangleq\\{{\lambda_{k}},k\in{\cal K}\\}$. Based on
the above discussions, the problem for maximizing number of served IoT devices
can be mathematically formulated as follows:

$\displaystyle{\cal P}^{\rm FD}:\ $ $\displaystyle\max_{{\bf q},{\bf a},{\bf
p},{\boldsymbol{\lambda}}}~{}~{}\left\|{\boldsymbol{\lambda}}\right\|_{1}$
(24a) $\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle{\lambda_{k}\in\\{0,1\\},\forall k,}$
$\displaystyle\delta_{t}\;{{\rm
min}(R_{1k},R_{2k})\geq\lambda_{k}S_{k},\forall k,}$
$\displaystyle\lambda_{k}\frac{S_{k}}{R_{1k}}\leq(n_{{\rm end},k}-n_{{\rm
start},k}+1)\delta_{t},\forall k,$
$\displaystyle\lambda_{k}\frac{S_{k}}{R_{2k}}\leq(N-n_{{\rm
end},k})\delta_{t},\forall k,$ $\displaystyle\sum\limits_{k\in{\cal
K}}\left(\lambda_{k}S_{k}-\sum\limits_{l=n+1}^{N}\delta_{t}R_{1k}[l]-\sum\limits_{l=1}^{n-1}\delta_{t}R_{2k}[l]\right)\leq
C,$ $\displaystyle\hfill\forall n,$ (35a) $\displaystyle\sum\limits_{k\in{\cal
K}}a_{ik}[n]\leq 1,\forall n,$ $\displaystyle 0\leq a_{ik}[n]\leq 1,\forall
k,n,$ $\displaystyle{\bf q}[1]={\bf q}_{\rm I},{\bf q}[N]={\bf q}_{\rm F},$
$\displaystyle\left\|{{\bf q}[n]-{\bf
q}[n-1]}\right\|\leq\delta_{d},n=2,\dots,N,$ $\displaystyle 0\leq
p_{1k}[n]\leq P_{k}^{\rm max}[n],\forall k,n,$ $\displaystyle
0\leq\sum\limits_{k\in{\cal K}}p_{2k}[n]\leq P_{\rm U}^{\rm max}[n],\forall
n,$

where constraint (II-C) means that each IoT device needs to upload an amount
of data $S_{k}$. In constraint (35a), ${\bf q}_{\rm I}$ and ${\bf q}_{\rm
F}\in\mathbb{R}^{2\times 1}$ denote the beginning and ending locations of UAV
projected onto horizontal plane, respectively; (35a) signifies the maximum
speed constraint of the UAV; constraints (35a) and (35a) imply maximum
transmit power constraints.

The problem ${\cal P}^{\rm FD}$ is a mixed integer non-linear program (MINLP),
which is generally NP-hard. Moreover, the binary constraint (II-C) and non-
convex constraints (II-C) to (35a) cause intractability. Therefore, it is
cumbersome to find an efficient solution of ${\cal P}^{\rm FD}$ directly.
However, a suitable solution (local or global optimal) may be obtained by
employing adequate relaxations to ${\cal P}^{\rm FD}$. In this regard, we
provide a transformation mechanism for ${\cal P}^{\rm FD}$, followed by its
corresponding solution in the succeeding section.

##  III Proposed Iterative Algorithm for solving ${\cal P}^{\rm FD}$

This section provides an iterative algorithm based on the IA method to solve
the design problem. The principle of IA framework [25, 43] can be detailed as
follows. Let us consider the following non-convex problem

$\displaystyle\min_{x\in\mathbb{R}^{n}}~{}~{}f(x)$ (36a)
$\displaystyle\mathtt{s.t.}~{}~{}$ $\displaystyle h_{m}(x)\leq
0,m=\\{1,\dots,M\\},$ (37a)

where $f(x)$ and $h_{m}(x)$ are non-convex and continuous differentiable
functions over $\mathbb{R}^{n}$. The key idea of IA is to replace non-convex
functions by its approximated convex ones. Let us denote $\bar{f}^{(j)}(x)$
and $\bar{h}_{m}^{(j)}(x),\forall m$ are new convex functions, satisfying the
properties listed in [43], i.e.,

$\displaystyle f(x)$
$\displaystyle\leq\bar{f}^{(j)}(x)\;\text{and}\;h_{m}(x)\leq\bar{h}_{m}^{(j)}(x),$
(38) $\displaystyle f(x^{(j)})$
$\displaystyle=\bar{f}^{(j)}(x^{(j)})\;\text{and}\;h_{m}(x^{(j)})=\bar{h}_{m}^{(j)}(x^{(j)}),$
(39) $\displaystyle\frac{\partial f(x)}{\partial x}\bigg{|}_{x=x^{(j)}}$
$\displaystyle=\frac{\partial\bar{f}^{(j)}(x)}{\partial
x}\bigg{|}_{x=x^{(j)}}\;$ $\displaystyle\text{and}\;\frac{\partial
h_{m}(x)}{\partial
x}\bigg{|}_{x=x^{(j)}}=\frac{\partial\bar{h}_{m}^{(j)}(x)}{\partial
x}\bigg{|}_{x=x^{(j)}},$ (40)

where $x^{(j)}\in{\mathcal{F}}\triangleq\\{x|{\rm s.t.}\;\eqref{IP:b}\\}$ is a
feasible point of problem (36a) at iteration $j$. In some special cases, the
approximated functions $(\bar{f}^{(j)}(x),\bar{h}_{m}^{(j)}(x))$ can be easily
obtained by adopting first-order Taylor approximation of $(f(x),h_{m}(x))$ at
feasible point $x^{(j)}$. Consequently, we solve the approximate convex
program at iteration $j$ of an iterative algorithm, which is given by

$\displaystyle\min_{x\in\mathbb{R}^{n}}~{}~{}\bar{f}^{(j)}(x)$ (41a)
$\displaystyle\mathtt{s.t.}~{}~{}$ $\displaystyle\bar{h}^{(j)}_{m}(x)\leq
0,m=\\{1,\dots,M\\}.$ (42a)

A general iterative algorithm to solve (41a) is presented as follows: $i$)
Generate the initial feasible point $x^{(0)}\in{\mathcal{F}}$; $ii$) At
iteration $j$, the optimal solution $x^{\star}$ is obtained by solving (41a);
$iii$) Update $x^{(j+1)}\triangleq x^{\star}$ and $j=j+1$; $iv$) Repeats steps
$(ii)-(iii)$ until convergence. The detailed proof of convergence can be found
in [25, 43].

###  III-A Tractable Formulation for (24a)

In this section, we aim to make problem $({\cal P}^{\rm FD})$ more tractable
by relaxing the binary variables of (II-C) into continuous values, i.e.,
$0\leq\lambda_{k}\leq 1$. To obtain near-exact binary solutions at optimum, we
introduce the penalty function to penalize uncertainties of the binary nature.
It is straightforward to see that
${\lambda_{k}}\in\left\\{{0,1}\right\\}\Leftrightarrow\big{(}0\leq{\lambda_{k}}\leq
1\;\&\;{\lambda_{k}}-({{\lambda_{k}}})^{2}\leq 0\big{)}.$ We see that the
convex function
$\mathbb{P}(\boldsymbol{\lambda})\triangleq\sum\limits_{k\in{\cal
K}}\lambda_{k}(\lambda_{k}-1)$ with $0\leq\lambda_{k}\leq 1,\forall k$ is
always non-positive and can be used to measure the degree of satisfaction of
(II-C). Similar to [44, 45], instead of handling the non-convex constraint
${\lambda_{k}}-({{\lambda_{k}}})^{2}\leq 0$, we maximize the penalty function
$\mathbb{P}(\boldsymbol{\lambda})$ to achieve its satisfaction by
incorporating it in the objective function (see, e.g., [46, Chapter 16]).
Hence, the parameterized relaxed problem with penalty parameter
$\mu\in\mathbb{R}^{+}$ is expressed as

$\displaystyle\vspace{-0.01cm}{{\cal P}^{\rm FD}_{\rm relaxed}}:$
$\displaystyle\max_{{\bf q},{\bf a},{\bf
p},{\boldsymbol{\lambda}}}~{}~{}\left\|{\boldsymbol{\lambda}}\right\|_{1}+\mu\mathbb{P}(\boldsymbol{\lambda})$
(43a) $\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$ $\displaystyle
0\leq\lambda_{k}\leq 1,\forall k,$ (45a)
$\displaystyle{\eqref{eq:P1:c}-\eqref{eq:P1:m}}.$

###### Remark 1

Note that in the parameterized relaxed problem ${\cal P}^{\rm FD}_{\rm
relaxed}$ (43a), the binary variables in the original problem (24a) are
relaxed to continuous ones between 0 and 1. Therefore, if $\lambda_{k},\forall
k$ are all binary at optimal, then the relaxation is tight and the obtained
solution is also a feasible solution of problem (24a). Theoretically,
${\mathbb{P}}(\boldsymbol{\lambda})$ should be zero at convergence to
guarantee the same objective value with (24a) under the sufficiently large
value of $\mu$. Nevertheless, there exists a numerical tolerance in
computation and it can be accepted if
$\mathbb{P}(\boldsymbol{\lambda})<\epsilon$, where $\epsilon$ is a very small
chosen value corresponding to a large value of $\mu$ [47, 44, 48].

However, a direct application of IA method to solve ${\cal P}_{\rm
relaxed}^{\rm FD}$ is inapplicable due to non-concavity of the objective
function and non-convexity of constraints in (II-C)-(35a) as well as strong
coupling among optimization variables. In what follows, we transform (43a)
into an equivalent non-convex problem where the IA method can be applied. In
this context, we introduce slack variables $z_{1k}[n]$, $z_{2k}[n]$, and
$t_{1k}[n]$ such that $\bigl{(}{H^{2}}+{\left\|{{\bf q}[n]-{\bf
w}_{k}}\right\|}^{2}\bigr{)}\leq(z_{1k}[n])^{2/\alpha}$,
$\bigl{(}{H^{2}}+{\left\|{{\bf q}[n]-{\bf
w}_{0}}\right\|}^{2}\bigr{)}\leq(z_{2k}[n])^{2/\alpha}$, and $\phi^{\rm
RSI}\sum\limits_{k^{\ast}\in{\cal K}\setminus
k}p_{2k^{\ast}}[n]+\sigma^{2}\leq t_{1k}[n]$, respectively, where $\alpha\geq
2$ for Rician fading channel [17, 49, 50], by which (10) and (11) can be
rewritten as

$\displaystyle\bar{r}_{1k}[n]\geq r_{1k}^{\rm lb}[n]\triangleq
a_{1k}[n]B\log_{2}\Bigl{(}1+\frac{e^{-E}p_{1k}[n]\omega_{0}}{z_{1k}[n]t_{1k}[n]}\Bigr{)},$
(46) $\displaystyle\bar{r}_{2k}[n]\geq r_{2k}^{\rm lb}[n]\triangleq
a_{2k}[n]B\log_{2}\Bigl{(}1+\frac{e^{-E}p_{2k}[n]\omega_{0}}{z_{2k}[n]\sigma^{2}}\Bigr{)}.$
(47)

By substituting (46) and (47) into (14) and (17), we respectively obtain
$C^{\rm lb}_{ik}[n]$ and $R^{\rm lb}_{ik}[n]$, with $i\in\\{1,2\\}$. Moreover,
we have $R^{\rm lb}_{ik}=\sum\limits_{n\in{\cal T}_{ik}}R^{\rm lb}_{ik}[n]$
and $C_{ik}^{\rm lb}=\sum\limits_{n\in{\cal T}_{ik}}\delta_{t}R^{\rm
lb}_{ik}[n]$. Let us denote ${\bf z}=\\{z_{1k}[n],z_{2k}[n],n\in{\cal
N},k\in{\cal K}\\},$ ${\bf t}=\\{t_{1k}[n],k\in{\cal K},n\in{\cal N}\\}$.
Then, the problem ${\cal P}_{\rm relaxed}^{\rm FD}$ can be reformulated as

$\displaystyle{\cal P}_{\rm\rm relaxed-1}^{\rm FD}:\ $
$\displaystyle\max_{{\bf q},{\bf a},{\bf p},{\boldsymbol{\lambda}},{\bf
z},{\bf
t}}~{}~{}\left\|{\boldsymbol{\lambda}}\right\|_{1}+\mu\mathbb{P}(\boldsymbol{\lambda})$
(48a) $\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\eqref{eq:P11:b},\eqref{eq:P1:h}-\eqref{eq:P1:m},$ (56a)
$\displaystyle{H^{2}}+{{\left\|{{\bf q}[n]-{\bf
w}_{k}}\right\|}^{2}}\leq\left(z_{1k}[n]\right)^{2/\alpha},\forall k,n,$
$\displaystyle{H^{2}}+{{\left\|{{\bf q}[n]-{\bf
w}_{0}}\right\|}^{2}}\leq\left(z_{2k}[n]\right)^{2/\alpha},\forall n,$
$\displaystyle\phi^{\rm RSI}\sum\limits_{k^{\ast}\in{\cal K}\setminus
k}p_{2k^{\ast}}[n]+\sigma^{2}\leq t_{1k}[n],\forall k,n,$
$\displaystyle\lambda_{k}\frac{S_{k}}{R_{1k}^{\rm lb}}\leq(n_{{\rm
end},k}-n_{{\rm start},k}+1)\delta_{t},\forall k,$
$\displaystyle\lambda_{k}\frac{S_{k}}{R_{2k}^{\rm lb}}\leq(N-n_{{\rm
end},k})\delta_{t},\forall k,$ $\displaystyle\delta_{t}\;{{\rm
min}(R_{1k}^{\rm lb},R_{2k}^{\rm lb})\geq\lambda_{k}S_{k},\forall k,}$
$\displaystyle\sum\limits_{k\in{\cal K}}\delta_{t}R_{2k}^{\rm
lb}\geq\sum\limits_{k\in{\cal K}}{\lambda_{k}S_{k}},\forall k\in{\cal{K}},$
$\displaystyle\sum\limits_{k\in{\cal
K}}\Big{(}\lambda_{k}S_{k}-\sum\limits_{l=n+1}^{N}\delta_{t}R_{1k}[l]-\sum\limits_{l=1}^{n-1}\delta_{t}R_{2k}[l]\Big{)}$
$\displaystyle\leq C,\forall k,n.$

It is noteworthy that ${\cal P}_{\rm\rm relaxed-1}^{\rm FD}$ is a much simpler
form in comparison to ${\cal P}^{\rm FD}$, but the possibility of a direct
solution still seems unviable. This is due to the fact that joint computation
of the optimization parameters (related to (56a)-(56a)) leads to non-convexity
of the problem. However, it is still possible to solve the problem in an
iterative manner. In the following, we discuss the above-mentioned approach in
details.

###  III-B Proposed IA-based Algorithm

We are now in position to convexify (48a) by applying the IA method [25] under
which the non-convex parts are completely exposed.

Approximation of the objective function: The objective (48a) is a convex
function in $\boldsymbol{\lambda}$, which is useful to apply the IA method. In
particular, the convex function $\mathbb{P}(\boldsymbol{\lambda})$ is
iteratively replaced by the linear function
$\hat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$:

$\displaystyle\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$
$\displaystyle\triangleq\mathbb{P}(\boldsymbol{\lambda}^{(j)})+\triangledown\mathbb{P}(\boldsymbol{\lambda}^{(j)})\big{(}\boldsymbol{\lambda}-\boldsymbol{\lambda}^{(j)}\big{)}$
$\displaystyle=\sum\limits_{k\in{\cal
K}}\Big{(}\lambda_{k}(2\lambda_{k}^{(j)}-1)-(\lambda_{k}^{(j)})^{2}\Big{)},$
(57)

where
$\mathbb{P}(\boldsymbol{\lambda}^{(j)})=\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda}^{(j)})$.
As a result, the objective function in problem ${\cal P}_{\rm\rm
relaxed-1}^{\rm FD}$ can be replaced by
$\left\|{\boldsymbol{\lambda}}\right\|_{1}+\mu\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$.

Approximation of $r_{1k}^{\rm lb}[n]$ and $r_{2k}^{\rm lb}[n]$: Before
proceeding further, we can express $r_{ik}^{\rm lb}[n]$, $i\in\\{1,2\\}$ as

$\displaystyle r_{ik}^{\rm lb}[n]=a_{ik}[n]\Phi_{ik}[n],$ (58)

where

$\displaystyle\Phi_{1k}[n]$ $\displaystyle\triangleq
B\log_{2}\left(1+\frac{e^{-E}p_{1k}[n]\omega_{0}}{z_{1k}[n]t_{1k}[n]}\right),$
(59) $\displaystyle\vspace{-0.01cm}\Phi_{2k}[n]$ $\displaystyle\triangleq
B\log_{2}\left(1+\frac{e^{-E}p_{2k}[n]\omega_{0}}{z_{2k}[n]\sigma^{2}}\right).$
(60)

To approximate (59) and (60), we first introduce the following lemmas:

###### Lemma 2

Consider a concave function $h(x,y)\triangleq\sqrt{xy},\;x>0,\;y>0$. Its
convex upper bound at given points $x^{(j)}$ and $y^{(j)}$ can be given by
[34, Appendix B], [43]:

$\displaystyle
h(x,y)\leq\frac{\sqrt{x^{(j)}}}{2\sqrt{y^{(j)}}}y+\frac{\sqrt{y^{(j)}}}{2\sqrt{x^{(j)}}}x.$
(61)

###### Lemma 3

Consider a function $h_{1}(x,y,z)\triangleq\ln\left(1+\frac{x}{yz}\right)$ and
$h_{2}(x,z)\triangleq\ln\left(1+\frac{x}{z}\right),x>0,\;y>0,\;z>0$. The
concave lower bound of $h_{1}(x,y,z)$ and $h_{2}(x,z)$ at given point
$x^{(j)}$, $y^{(j)}$, and $z^{(j)}$ are expressed as

$\displaystyle h_{1}(x,y,z)$
$\displaystyle\geq\ln\left(1+\frac{x^{(j)}}{y^{(j)}z^{(j)}}\right)-\frac{x^{(j)}}{y^{(j)}z^{(j)}}$
$\displaystyle+2\frac{\sqrt{x^{(j)}}\sqrt{x}}{y^{(j)}z^{(j)}}-\frac{x^{(j)}\left(x+\frac{y^{(j)}}{2z^{(j)}}z^{2}+\frac{z^{(j)}}{2y^{(j)}}y^{2}\right)}{y^{(j)}z^{(j)}\left(x^{(j)}+y^{(j)}z^{(j)}\right)},$
(62) $\displaystyle h_{2}(x,z)$
$\displaystyle\geq\ln\left(1+\frac{x^{(j)}}{z^{(j)}}\right)-\frac{x^{(j)}}{z^{(j)}}+2\frac{\sqrt{x^{(j)}}\sqrt{x}}{z^{(j)}}$
$\displaystyle-\frac{x^{(j)}\left(x+z\right)}{z^{(j)}\left(x^{(j)}+z^{(j)}\right)}.$
(63)

###### Proof:

See Appendix B. ∎

Based on Lemmas 2 and 3, $\Phi_{1k}[n]$ and $\Phi_{1k}[n]$ are lower bounded
by

$\displaystyle\Phi_{1k}[n]$ $\displaystyle\geq\bar{\Phi}_{1k}[n]\triangleq
B\big{(}\Xi_{1}+\Xi_{2}-\Xi_{3}\big{)},$ (64)
$\displaystyle\vspace{-0.01cm}\Phi_{2k}[n]$
$\displaystyle\geq\bar{\Phi}_{2k}[n]\triangleq
B\big{(}\Xi_{4}+\Xi_{5}-\Xi_{6}\big{)},$ (65)

where $\Xi_{1},\Xi_{2},\Xi_{3},\Xi_{4},\Xi_{5}$, and $\Xi_{6}$ are defined in
Appendix C. By introducing slack variable $\Phi_{ik}^{\rm lb}[n]$,
$i\in\\{1,2\\}$, with

$\displaystyle\bar{\Phi}_{ik}[n]$ $\displaystyle\geq\Phi_{ik}^{\rm lb}[n],$
(66)

we rewrite $r_{ik}^{\rm lb}[n]$ as

$\displaystyle r_{ik}^{\rm lb}[n]\geq\bar{r}_{ik}^{\rm lb}[n]\triangleq
a_{ik}[n]\Phi_{ik}^{\rm lb}[n].$ (67)

To tackle non-convex function $a_{ik}[n]\Phi_{ik}^{\rm lb}[n]$ we replace
$a_{ik}[n]\Phi_{ik}^{\rm lb}[n]$ by equivalent Difference of Convex (DC)
function $0.25\big{[}(a_{ik}[n]+\Phi_{ik}^{\rm
lb}[n])^{2}-(a_{ik}[n]-\Phi_{ik}^{\rm lb}[n])^{2}\big{]}$. Then, we apply the
first-order Taylor approximation to approximate the convex function
$(a_{ik}[n]+\Phi_{ik}[n])^{2}$ at the $(j+1)$-th iteration:

$\displaystyle a_{ik}[n]\Phi_{ik}^{\rm lb}[n]$ $\displaystyle\geq$
$\displaystyle\frac{\big{(}a_{ik}^{(j)}[n]+\Phi_{ik}^{{\rm
lb},(j)}[n]\big{)}^{2}}{4}+\frac{\big{(}a_{ik}^{(j)}[n]+\Phi_{ik}^{{\rm
lb},(j)}[n]\big{)}}{2}$ (68) $\displaystyle\times$
$\displaystyle\Big{(}a_{ik}[n]-a_{ik}^{(j)}[n]+\Phi_{ik}^{\rm
lb}[n]-\Phi_{ik}^{{\rm lb},(j)}[n]\Big{)}$ $\displaystyle-$
$\displaystyle\frac{\big{(}a_{ik}[n]-\Phi_{ik}^{\rm
lb}[n]\big{)}^{2}}{4}\triangleq\tilde{r}_{ik}^{\rm lb}[n].$

To convexify (56a)-(56a), we introduce the slack variables
$\widehat{r}_{ik}^{\rm lb}[n]$, with $i\in\\{1,2\\}$, to equivalently express
(68) as

$\displaystyle\tilde{r}_{ik}^{\rm lb}[n]\geq\widehat{r}_{ik}^{\rm
lb}[n],i\in\\{1,2\\}.$ (69)

As a result, substituting $\widehat{r}_{ik}^{\rm lb}[n]$ into (14), (17), we
obtain $\widehat{R}_{ik}^{\rm
lb}[n]\triangleq\left\\{\begin{array}[]{l}\widehat{r}^{\rm
lb}_{ik}[n],\hfill\text{if}\;n\in{\cal T}_{ik},\\\
0,\hfill\text{otherwise},\end{array}\right.$, $\widehat{C}_{ik}^{\rm
lb}[n]\triangleq\delta_{t}\widehat{R}^{\rm lb}_{ik}[n],\;{\rm
where}\;i\in\\{1,2\\}$. Moreover, we have $\widehat{R}_{ik}^{\rm
lb}=\sum\limits_{n\in{\cal T}_{ik}}\widehat{R}_{ik}^{\rm lb}[n]$,
$\widehat{C}_{ik}^{\rm lb}=\sum\limits_{n\in{\cal
T}_{ik}}\widehat{C}_{ik}^{\rm lb}[n]$. Let us define
${\bf\Phi}\triangleq\\{\Phi_{1k}^{\rm lb}[n],\Phi_{2k}^{\rm lb}[n],\forall
k,n\\}$ and ${\bf r}\triangleq\\{\widehat{r}_{1k}^{\rm lb}[n],$
$\widehat{r}_{2k}^{\rm lb}[n],\forall k,n\\}$.

Bearing all the above developments in mind, we solve the following approximate
convex program at the $(j+1)$-th iteration:

$\displaystyle\vspace{-0.01cm}{\cal P}^{\rm FD}_{\rm convex}:\ $
$\displaystyle\max_{\boldsymbol{\Psi}}~{}~{}\sum\limits_{k\in{\cal
K}}\lambda_{k}+\mu\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$ (70a)
$\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\eqref{eq:P1:h}-\eqref{eq:P1:m},\eqref{eq:P11:b},\eqref{eq:P12:c},\eqref{eq:P12:d},\eqref{eq:44},$
(76a) $\displaystyle\lambda_{k}\frac{S_{k}}{\widehat{R}_{1k}^{\rm
lb}}\leq(n_{{\rm end},k}-n_{{\rm start},k}+1)\delta_{t},\forall k,$
$\displaystyle\lambda_{k}\frac{S_{k}}{\widehat{R}_{2k}^{\rm lb}}\leq(N-n_{{\rm
end},k})\delta_{t},\forall k,$ $\displaystyle\delta_{t}\;{{\rm
min}\big{(}{\widehat{R}_{1k}^{\rm lb}},{\widehat{R}_{2k}^{\rm
lb}}\big{)}\geq\lambda_{k}S_{k},\forall k,}$
$\displaystyle\sum\limits_{k=1}^{K}\delta_{t}{\widehat{R}_{2k}^{\rm
lb}}\geq\sum\limits_{k=1}^{K}{\lambda_{k}S_{k}},$
$\displaystyle\sum\limits_{k\in{\cal
K}}\Big{(}\lambda_{k}S_{k}-\sum\limits_{l=n+1}^{N}\delta_{t}{\widehat{R}_{1k}^{\rm
lb}}[l]-\sum\limits_{l=1}^{n-1}\delta_{t}{\widehat{R}_{2k}^{\rm
lb}}[l]\Big{)}$ $\displaystyle\leq C,\forall k,n,$

where $\boldsymbol{\Psi}\triangleq\\{{{\bf q}},{\bf a},{\bf
p},{\boldsymbol{\lambda}},{\bf z},{\bf t},{\bf\Phi},{\bf r}\\}$ and
$\boldsymbol{\Psi}^{(j)}\triangleq\\{{\bf q}^{(j)},{\bf a}^{(j)},{\bf
p}^{(j)},{\boldsymbol{\lambda}}^{(j)},{\bf z}^{(j)},$ ${\bf
t}^{(j)},{\bf\Phi}^{(j)}\\}$ as the feasible point for (70a) at iteration $j$.
The convex program (70a) can be solved by using standard convex optimization
solvers [51]. To ensure the feasibility of (70a) at the first iteration, an
appropriate starting point $\boldsymbol{\Psi}^{(0)}$ is necessary. This
selection should be made such that the feasibility of (76a) is always
guaranteed while additionally satisfying other constraints. Therefore, we
successively solve the following simplified version of (70a):

$\displaystyle{\cal P}_{\rm feasible}^{\rm FD}:\ $
$\displaystyle\max_{\boldsymbol{\Psi},\\{\tau_{k}\\}_{k=1}^{K}}~{}~{}\vspace{-0.01cm}\min_{\forall
k}{\tau_{k}}$ (77a) $\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\delta_{t}\;{{\rm min}\big{(}{\widehat{R}_{1k}^{\rm
lb}},{\widehat{R}_{2k}^{\rm lb}}\big{)}-\lambda_{k}S_{k}\geq\tau_{k},\forall
k,}$ (79a)
$\displaystyle\eqref{eq:P13:b}-\eqref{eq:P13:d},\eqref{eq:P13:f},\eqref{eq:P13:g},\vspace{-0.01cm}$

where $\tau_{k}$ is the slack variable. The initial feasible point
$\Psi^{(0)}$ is obtained until problem (77a) is successfully solved and
$\tau_{k}\geq 0,\forall k$. Then, the sub-optimal solution is obtained by
successively solving (70a) and updating the involved variables until
satisfying the convergence condition (discussed below in detail). Finally, a
pseudo-code for solving (24a) is summarized in Algorithm 1.

0: Set $j:=0$ and solve (77a) to generate an initial feasible point
$\boldsymbol{\Psi}^{(0)}$.

1: repeat

2: Solve (70a) to obtain the optimal solution
$\boldsymbol{\Psi}^{\star}\triangleq\left({\bf q}^{\star},{\bf a}^{\star},{\bf
p}^{\star},{\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{\star},{\boldsymbol{t}}^{\star},{\boldsymbol{\Phi}}^{\star},{\boldsymbol{r}}^{\star}\right)$.

3: Update ${\bf q}^{(j+1)}:={\bf q}^{\star},{\bf a}^{(j+1)}:={\bf
a}^{\star},{\bf p}^{(j+1)}:={\bf
p}^{\star},{\boldsymbol{\lambda}}^{(j+1)}:={\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{(j+1)}:={\boldsymbol{z}}^{\star},{\boldsymbol{t}}^{(j+1)}:={\boldsymbol{t}}^{\star}$,
${\boldsymbol{\Phi}}^{(j+1)}:={\boldsymbol{\Phi}}^{\star}$.

4: Set $j:=j+1.$

5: until Convergence

Algorithm 1 Proposed IA Based Design to Solve (24a)

###  III-C Convergence and Complexity Analysis

####  III-C1 Convergence Analysis

Algorithm 1 is mainly based on inner approximation, where its convergence is
proved in [25, 43]. To be self-contained, we introduce the following
proposition.

###### Proposition 1

The proposed Algorithm 1 yields a sequence of improved solutions converging to
at least a local optimum of the relaxed problem ${\cal P}^{\rm FD}_{\rm
relaxed}$.

###### Proof:

See Appendix D. ∎

####  III-C2 Complexity Analysis

We now provide the worst-case complexity analysis for each iteration in
Algorithm 1. Since problem (70a) is convex, several solvers employing the
interior point method can be applied to solve efficiently [51]. More
specifically, the convex problem (70a) involves $N(7+8K)+4K$ linear and
quadratic constraints, and $5N(1+3K)+K$ scalar real variables. As a result,
the per-iteration computational complexity required to solve (70a) is
$\mathcal{O}(N(7+8K)+4K)^{0.5}(5N(1+3K)+K)^{3}$ [52, Chapter 6]. It results in
the overall complexity of
$\mathcal{O}\Big{(}N_{i}(N(7+8K)+4K)^{0.5}(5N(1+3K)+K)^{3}\Big{)}$, where
$N_{i}$ is the number of iterations to reach a local optimal solution.

###  III-D Throughput Maximization

In an emergency case or during a natural disaster, data need to be collected
timely to assess the current situation in a given area. The more collected
information we have, the better our predictions are. This motivates us to
present a new problem that maximizes the total amount of collected data with a
given number of served IoT devices subjected to certain quality-of-service
(QoS) constraints:

$\displaystyle{\cal P}_{\rm rate}^{\rm FD}:\ $ $\displaystyle\max_{{\bf
q},{\bf a},{\bf p},{\boldsymbol{\lambda}}}~{}~{}\sum_{k\in{\cal
K}}\delta_{t}{\rm min}(R_{1k},R_{2k})$ (80a)
$\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\left\|{\boldsymbol{\lambda}}\right\|_{1}\geq\lambda_{\rm
thresh},$ (82a) $\displaystyle\eqref{eq:P1:b}-\eqref{eq:P1:m},$

where constraint (82a) means that the total number of served IoT devices must
be larger than or equal to a predefined threshold value, i.e., $\lambda_{\rm
thresh}$.

Similar to ${\cal P}^{\rm FD}$, ${\cal P}_{\rm rate}^{\rm FD}$ is also a mixed
integer non-convex problem, which is NP-hard. Fortunately, by reusing the
developments presented in Section III-B, (80a) is rewritten as

$\displaystyle{\cal P}_{\rm rate-convex}^{\rm FD}:\ $
$\displaystyle\max_{\Psi}~{}~{}\sum_{k\in{\cal K}}\delta_{t}{\rm
min}(\widehat{R}_{1k}^{\rm lb},\widehat{R}_{2k}^{\rm
lb})+\mu\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$
$\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\eqref{eq:P13:b}-\eqref{eq:P13:g},$ (84a)

where $\widehat{R}_{ik}^{\rm lb}$ are obtained as in Section III-B.

0: Set $j:=0$ and generate an initial feasible point
$\boldsymbol{\Psi}^{(0)}$.

1: repeat

2: Solve (III-D) to obtain the optimal solution
$\boldsymbol{\Psi}^{\star}\triangleq\left({\bf q}^{\star},{\bf a}^{\star},{\bf
p}^{\star},{\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{\star},{\boldsymbol{t}}^{\star},{\boldsymbol{\Phi}}^{\star},{\boldsymbol{r}}^{\star}\right)$.

3: Update ${\bf q}^{(j+1)}:={\bf q}^{\star},{\bf a}^{(j+1)}:={\bf
a}^{\star},{\bf p}^{(j+1)}:={\bf
p}^{\star},{\boldsymbol{\lambda}}^{(j+1)}:={\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{(j+1)}:={\boldsymbol{z}}^{\star},{\boldsymbol{t}}^{(j+1)}:={\boldsymbol{t}}^{\star},$
${\boldsymbol{\Phi}}^{(j+1)}:={\boldsymbol{\Phi}}^{\star}$.

4: Set $j:=j+1.$

5: until Convergence

Algorithm 2 Proposed IA-based Iterative Algorithm to Solve (80a)

Consequently, the solution of problem ${\cal P}_{\rm rate}^{\rm FD}$ can be
found by successively solving a simpler convex problem in (III-D), as
summarized in Algorithm 2.

##  IV Half Duplex Mode Scheme

###  IV-A Maximizing The Number of Served IoT Devices

In order to stress the benefits of our proposed method using FD mode, we now
describe the problem again by considering HD mode at the UAV. First, (II-A)
and (7) can be rewritten as

$\displaystyle
y_{ik}^{\rm{HD}}[n]=\sqrt{p_{ik}[n]}h_{ik}[n]x_{ik}[n]+n_{0},\;i\in\\{1,2\\}.$
(85)

In (85), the UAV only transmits data to GW when it finishes collecting data
from all GUs in HD mode. Consequently, the RSI is disappeared compared to that
of (II-A). Thus, the achievable rate (bits/s) of link from $k\to{\rm U}$ or
${\rm U}\to{\rm GW}$ to transmit the data of device $k$ at time slot $n$ is
given as

$\displaystyle r_{ik}^{\rm{HD}}[n]$
$\displaystyle=a_{ik}[n]B\log_{2}\Bigg{(}1+\frac{p_{ik}[n]|{\tilde{h}}_{1k}[n]|^{2}\omega_{0}}{\big{(}{H^{2}}+{{\left\|{{\bf
q}[n]-{\bf w}}\right\|}^{2}}\big{)}^{\alpha/2}\sigma^{2}}\Bigg{)},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad i\in\\{1,2\\},$
(86)

where ${\bf w}$ is ${\bf w}_{k}$ and ${\bf w}_{0}$ corresponding to $i$ equals
1 and 2, respectively.

Similar to (11), the approximated result of $r_{ik}^{\rm{HD}}[n]$ can be
expressed as

$\displaystyle\bar{r}_{ik}^{\rm{HD}}[n]=a_{ik}[n]B\log_{2}\Bigg{(}1+\frac{e^{-E}p_{ik}[n]\omega_{0}}{\big{(}{H^{2}}+{{\left\|{{\bf
q}[n]-{\bf w}}\right\|}^{2}}\big{)}^{\alpha/2}\sigma^{2}}\Bigg{)}.$ (87)

By substituting (87) into the equations (14) and (17), we obtain
$C_{1k}^{\rm{HD}}[n]$, $C_{2k}^{\rm{HD}}[n]=C_{2k}[n]$, and
$R_{1k}^{\rm{HD}}[n]$, respectively. Then, we reformulate the problem of
maximizing the total number of served IoT devices as follows:

$\displaystyle\vspace{-0.01cm}{\cal P}^{\rm HD}:\ $ $\displaystyle\max_{{\bf
q},{\bf a},{\bf
p},{\boldsymbol{\lambda}}}~{}~{}\left\|{\boldsymbol{\lambda}}\right\|_{1}$
(88a) $\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\eqref{eq:P1:b},\eqref{eq:P1:f},\eqref{eq:P1:h}-\eqref{eq:P1:m},$
(92a) $\displaystyle\delta_{t}\;{{\rm min}(R_{1k}^{\rm{HD}},R_{2k}^{\rm
HD})\geq\lambda_{k}S_{k},\forall k,}$
$\displaystyle\lambda_{k}\frac{S_{k}}{R_{1k}^{\rm HD}}\leq(n_{{\rm
end},k}-n_{{\rm start},k}+1)\delta_{t},\forall k,$
$\displaystyle\sum\limits_{k\in{\cal
K}}\Big{(}\lambda_{k}S_{k}-\sum\limits_{l=n+1}^{N}\delta_{t}R_{1k}^{\rm
HD}[l]-\sum\limits_{l=1}^{n-1}\delta_{t}R_{2k}^{\rm HD}[l]\Big{)}$
$\displaystyle\leq C,\forall n.$

The problem ${\cal P}^{\rm HD}$ is a mixed integer non-convex due to the
binary constraint (II-C) and non-convex constraints (II-C), (92a), (92a), and
(92a). In order to seek a suitable solution, we first relax binary constraint
(II-C) as in (56a). Then, by introducing $z_{1k}^{\rm{HD}}[n]$ and
$z_{2k}^{\rm{HD}}[n]$ such that $\left({H^{2}}+{\left\|{{\bf q}[n]-{\bf
w}_{k}}\right\|}^{2}\right)\leq(z_{1k}^{\rm{HD}}[n])^{2/\alpha}$ and
$\left({H^{2}}+{\left\|{{\bf q}[n]-{\bf
w}_{0}}\right\|}^{2}\right)\leq(z_{2k}^{\rm{HD}}[n])^{2/\alpha}$, (87) can be
expressed as

$\displaystyle\bar{r}_{ik}^{\rm{HD}}[n]$
$\displaystyle=a_{ik}[n]B\log_{2}\left(1+\frac{e^{-E}p_{ik}[n]\omega_{0}}{z_{ik}^{\rm{HD}}[n]\sigma^{2}}\right),\;{\rm
with}\;i\in\\{1,2\\}.$ (93)

Given that the $\bar{r}_{ik}^{\rm{HD}}[n]$ is the same as $\bar{r}_{2k}^{\rm
lb}[n]$ in (11), we apply IA method for $\bar{r}_{2k}^{\rm lb}[n]$ in Section
III to $\bar{r}_{ik}^{\rm{HD}}[n]$. As a result, $r_{ik}^{\rm{HD}}[n]$ can be
rewritten as

$\displaystyle\bar{r}_{ik}^{\rm{HD}}[n]=a_{ik}[n]\Phi_{ik}^{\rm{HD}}[n],$ (94)

where

$\displaystyle\Phi_{ik}^{\rm{HD}}[n]$
$\displaystyle=B\log_{2}\left(1+\frac{e^{-E}p_{ik}[n]\omega_{0}}{z_{ik}^{\rm{HD}}[n]\sigma^{2}}\right).$
(95)

Similar to (66), $\Phi_{ik}[n]$ is lower bounded by

$\displaystyle\Phi_{ik}^{\rm{HD}}[n]\geq\bar{\Phi}_{ik}^{\rm{HD}}[n],$ (96)

where $\bar{\Phi}_{1k}^{\rm{HD}}[n]$ and $\bar{\Phi}_{2k}^{\rm{HD}}[n]$ can be
calculated as $\bar{\Phi}_{2k}[n]$, shown in Appendix B.

As in (67), it follows that

$\displaystyle r_{ik}^{\rm HD}[n]\geq r_{ik}^{\rm
HD,lb}[n]=a_{ik}[n]\Phi_{ik}^{\rm HD,lb}[n],$ (97)

where $\Phi_{ik}^{\rm HD,lb}[n]$ is a slack variable which is a lower bound of
$\bar{\Phi}_{ik}^{\rm{HD}}[n]$. Then, by applying the first order Taylor
approximation for $a_{ik}[n]\Phi_{ik}^{\rm HD,lb}[n]$, it yields:

$\displaystyle r_{ik}^{\rm HD,lb}[n]\geq\bar{r}_{ik}^{\rm HD,lb}[n],$ (98)

where $\bar{r}_{ik}^{\rm HD,lb}[n]$ and $\bar{r}_{ik}^{\rm HD,lb}[n]$ can be
represented as in (68).

In turn, by introducing a slack variable $\widehat{r}_{ik}^{\rm HD,lb}[n]$,
constraint (98) is innerly approximated by the following convex constraints:

$\displaystyle\bar{r}_{ik}^{\rm HD,lb}[n]\geq\widehat{r}_{ik}^{\rm HD,lb}[n].$
(99)

By substituting $\widehat{r}_{ik}^{\rm HD,lb}[n]$ into (17), we obtain
$\widehat{R}_{ik}^{\rm HD,lb}[n]$. Moreover, we have $\widehat{R}_{ik}^{\rm
HD,lb}=\sum\limits_{n\in{\cal T}_{ik}}\widehat{R}_{ik}^{\rm HD,lb}[n]$. In
Algorithm 3, we propose an iterative algorithm to solve the problem (88a). At
the $(j+1)$-th iteration, it solves the following convex program:

$\displaystyle{\cal P}_{\rm convex}^{\rm HD}:\ $
$\displaystyle\max_{\Psi}~{}~{}\sum\limits_{k\in{\cal
K}}\lambda_{k}+\mu\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$ (100a)
$\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\eqref{eq:P1:h}-\eqref{eq:P1:m},\eqref{eq:P11:b},\eqref{eq:70},$
(106a) $\displaystyle\delta_{t}\;{{\rm
min}(\widehat{R}_{1k}^{\rm{HD,lb}},\widehat{R}^{\rm{HD,lb}}_{2k})\geq\lambda_{k}S_{k},\forall
k,}$
$\displaystyle\lambda_{k}\frac{S_{k}}{\widehat{R}_{1k}^{\rm{HD,lb}}}\leq(n_{{\rm
end},k}-n_{{\rm start},k}+1)\delta_{t},\forall k,$
$\displaystyle\sum\limits_{k\in{\cal
K}}\Big{(}\lambda_{k}S_{k}-\sum\limits_{l=n+1}^{N}\delta_{t}\widehat{R}_{1k}^{\rm{HD,lb}}[l]-\sum\limits_{l=1}^{n-1}\delta_{t}\widehat{R}_{2k}^{\rm{HD,lb}}[l]\Big{)}$
$\displaystyle\leq C,\forall n,$
$\displaystyle\lambda_{k}\frac{S_{k}}{\widehat{R}_{2k}^{\rm{HD,lb}}}\leq(N-n_{{\rm
end},k})\delta_{t},\forall k,$ $\displaystyle\left({H^{2}}+{\left\|{{\bf
q}[n]-{\bf w}_{k}}\right\|}^{2}\right)\leq(z_{1k}^{\rm{HD}}[n])^{2/\alpha},$
$\displaystyle\left({H^{2}}+{\left\|{{\bf q}[n]-{\bf
w}_{0}}\right\|}^{2}\right)\leq(z_{2k}^{\rm{HD}}[n])^{2/\alpha}.$

Similar to (43a), we adopt a penalty function in objective to guarantee an
exact binary value of $\lambda_{k}$, $\forall k\in\mathcal{K}$. The initial
feasible point to solve (100a) can be obtained similar to (77a).

0: Set $j:=0$ and generate an initial feasible point
${\boldsymbol{\Psi}}^{(0)}$.

1: repeat

2: Solve (100a) to obtain the optimal solution
${\boldsymbol{\Psi}}^{\star}\triangleq\left({\bf q}^{\star},{\bf
a}^{\star},{\bf
p}^{\star},{\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{\star},{\boldsymbol{\Phi}}^{\star},{\boldsymbol{r}}^{\star}\right)$.

3: Update ${\bf q}^{(j+1)}:={\bf q}^{\star},{\bf a}^{(j+1)}:={\bf
a}^{\star},{\bf p}^{(j+1)}:={\bf
p}^{\star},{\boldsymbol{\lambda}}^{(j+1)}:={\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{(j+1)}:={\boldsymbol{z}}^{\star},$
${\boldsymbol{\Phi}}^{(j+1)}:={\boldsymbol{\Phi}}^{\star}$.

4: until Convergence

Algorithm 3 Proposed IA-based Iterative Algorithm to Solve (88a)

0: Set $j:=0$ and generate an initial feasible point
${\boldsymbol{\Psi}}^{(0)}$.

1: repeat

2: Solve (109a) to obtain the optimal solution
${\boldsymbol{\Psi}}^{\star}\triangleq\left({\bf q}^{\star},{\bf
a}^{\star},{\bf
p}^{\star},{\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{\star},{\boldsymbol{\Phi}}^{\star},{\boldsymbol{r}}^{\star}\right)$.

3: Update ${\bf q}^{(j+1)}:={\bf q}^{\star},{\bf a}^{(j+1)}:={\bf
a}^{\star},{\bf p}^{(j+1)}:={\bf
p}^{\star},{\boldsymbol{\lambda}}^{(j+1)}:={\boldsymbol{\lambda}}^{\star},{\boldsymbol{z}}^{(j+1)}:={\boldsymbol{z}}^{\star}$,
${\boldsymbol{\Phi}}^{(j+1)}:={\boldsymbol{\Phi}}^{\star}$.

4: Set $j:=j+1.$

5: until Convergence

Algorithm 4 Proposed IA-based Iterative Algorithm to Solve (IV-B)

####  IV-A1 Complexity Analysis

: The convex problem (100a) involves $N(7+8K)+4K$ linear and quadratic
constraints, and $3N(1+4K)+K$ scalar real variables. As a result, the per-
iteration complexity required to solve (100a) is
$(N(7+8K)+4K)^{0.5}(3N(1+4K)+K)^{3}$. It results in the overall complexity is
$\mathcal{O}\Big{(}N_{i}(N(7+8K)+4K)^{0.5}(3N(1+4K)+K)^{3}\Big{)}$, with
$N_{i}$ is the number of iterations to reach a local solution.

###  IV-B Throughput Maximization

In this section, we reuse all the slack variables as introduced in Sections
III-D and IV-A. First, the throughput maximization problem for HD mode can be
presented as:

$\displaystyle{\cal P}_{\rm rate}^{\rm HD}:\ $ $\displaystyle\max_{{\bf
q},{\bf a},{\bf p},{\boldsymbol{\lambda}}}~{}~{}\sum_{k\in{\cal
K}}\delta_{t}{\rm min}(R_{1k}^{\rm HD},R_{2k}^{\rm HD})$
$\displaystyle\mathtt{s.t.}~{}~{}\vspace{-0.01cm}$
$\displaystyle\eqref{eq:P3:b},\eqref{eq:P2:b}-\eqref{eq:P2:e}.$ (108a)

By following the same steps presented in Section III-D, we obtain the
following convex optimization problem:

$\displaystyle{\cal P}_{\rm rate-convex}^{\rm HD}:\ $
$\displaystyle\max_{\boldsymbol{\Psi}}~{}~{}\sum_{k\in{\cal K}}\delta_{t}{\rm
min}(\widehat{R}_{1k}^{\rm{HD,lb}},\widehat{R}_{2k}^{\rm{HD,lb}})+\mu\widehat{\mathbb{P}}^{(j)}(\boldsymbol{\lambda})$
(109a) $\displaystyle\mathtt{s.t.}~{}~{}$
$\displaystyle\eqref{eq:P3:b},\eqref{eq:P21:b}-\eqref{eq:P21:g},$ (110a)

where ${\widehat{R}_{ik}^{\rm{HD,lb}}}$ can be obtained as in Section IV-A.
Due to the convexity of problem ${\cal P}_{\rm rate}^{\rm HD}$, the solution
of problem ${\cal P}_{\rm rate}^{\rm HD}$ can be iteratively obtained as in
Algorithm 4.

##  V Numerical Results

In this section, we present numerical results to evaluate the proposed joint
bandwidth allocation and transmit power for the devices/UAV as well as the UAV
trajectory design in UAV-assisted IoT networks. We consider a system with $K$
IoT devices that are randomly distributed in a horizontal plane, i.e, ${\rm
area}=x^{2}$ $(m^{2})$, with $x=$ 500 m. We assume that the GW, the initial
location, and end location of the UAV are located at (0, 500 m), ${\bf q}_{\rm
I}$ = [500 m, 200 m], and ${\bf q}_{\rm F}$ = [300 m, 0], respectively. The
UAV flight altitude is invariant at $H=100$ m [29]. The total bandwidth is
$B=20$ MHz. Thus, the total AWGN power is
$\sigma^{2}=-174+10\log_{10}(B)=-100.9897$ dBm. The transmit power budget of
the UAV and IoT devices is respectively set as $P_{\rm U}^{\rm max}=$ 18 dBm
and $P_{k}^{\rm max}=$ 10 dBm. Other parameters are set as follows: maximum
speed $V_{\max}=50$ m/s, path loss exponent $\alpha=2.4$, $\omega_{0}=$ -30
dB, $S_{k}\in$ [10, 70] Mbits, one time slot duration $\delta_{t}=0.5$ s, the
maximum collection time deadline for each device $k$ $n_{{\rm end},k}$ is
uniformly distributed between $n_{{\rm end},k}^{\min}$ and $n_{{\rm
end},k}^{\max}$. The RSI suppression $\rho^{\rm RSI}$ is set to -80 dB [53,
54]. To show the superiority of our designs, we compare the proposed methods
with benchmark schemes. Herein, the benchmark FD 2 (BFD2) and benchmark HD 2
(BHD2) are respectively implemented similar to Algorithms 1 and 2 with fixed
resource allocation, i.e., $a_{1k}[n]=a_{1k}[n]=\frac{1}{K}$,
$p_{1k}[n]=P_{k}^{\max}[n]$, $p_{2k}[n]=\frac{P_{U}^{\max}}{K}$. The benchmark
FD 1 (BFD1) and benchmark HD 1 (BHD1) are implemented with a fixed trajectory,
i.e., linear from initial to final locations.

(a) FD mode

(b) HD mode

Figure 3: Geometry distribution of GUs and the UAV trajectory. Figure 4:
Percentage of served IoT devices vs. cache size in FD mode with different
value of $r_{\rm thresh}$. Figure 5: Percentage of served IoT devices vs.
cache size in HD mode with different value of $P_{k}^{\max}[n]$.

(a) Full-duplex mode

(b) Half-duplex mode

Figure 6: Percentage of served IoT devices vs. cache size with different range
of $\eta_{\rm end,k}$. Figure 7: Percentage of served IoT devices vs. cache
sizes with different $S_{k}$. Figure 8: Percentage of served IoT devices vs.
network size (maximum IoT devices located in the network area).

###  V-A Maximizing the Number of Served IoT Devices

Fig. 3 plots the UAV’s designed trajectory corresponding to FD and HD mode,
with $N=$ 70 times slots, $\eta_{{\rm start},k}\in[2,15]$, $\eta_{{\rm
end},k}\in[25,50]$, ${\rm area}=$ 700 m $\times$ 700 m, $C=1000$, and $S_{k}$
values are ranging from 10 to 55 Mbits, $P_{\rm U}^{\rm max}=$ 19 dBm and
$P_{k}^{\rm max}=$ 10 dBm. In additions, the GW, initial location, and end
location of the UAV are respectively set as $(0,700\;\rm m)$, ${{\bf q}_{\rm
I}}=(700\;\rm m,400\;\rm m)$, ${{\bf q}_{\rm I}}=(300\;\rm m,0)$. First, we
observe that the proposed FD method significantly improves the number of
served IoT devices than the HD method, i.e., 20 and 13 served GUs in FD and HD
mode, respectively. Besides, the UAV can fly closer to GW and GUs in FD than
in HD mode. It is because the UAV transfers device $k$’s data to GW right
after it finishes gathering data of that IoT device in FD-based scheme. While
in HD mode, the UAV only operates in the downlink transmission when it
completes the data acquisition for all users on the uplink to prevent RSI at
the UAV. Consequently, the UAV in the FD scheme has more time to fly closer to
GW and GUs. Thus, it obtains a higher probability of satisfying the GUs’ RT.
The UAV in the HD mode can collect information and fulfill the latency
constraint for each IoT device, but it has less time to move forward GUs/GW to
collect/offload the generated data. Thus, the performance in the HD-based
method is degraded.

In Fig. 4, we investigate the performance of FD-based schemes with different
QoS requirements. Specifically, the QoS is defined as the minimum rate
threshold at the UAV/GW to successfully decode the signal, i.e., $r_{1k,{\rm
thresh}}[n]$ and $r_{2k,{\rm thresh}}[n]$. For simplicity, we assume that
$r_{1k,{\rm thresh}}[n]=r_{2k,{\rm thresh}}[n]=r_{\rm thresh}$. It can be seen
that the more the minimum rate threshold is required, the fewer users the
system can serve. This is because the UAV tends to come closer or spend more
time around an IoT device to gain a higher rate requirement. As a result, the
UAV has less chance of serving more devices due to limited flight time and
latency constraints per IoT user. Another observation is that for larger cache
sizes, the number of served users increases. It is due to the fact that the
UAV has more capacity to store incoming data. Thus, the UAV can serve more
users before offloading information to GW. Similar to Fig. 3, our proposed FD
algorithm achieves a much better percentage of served IoT devices compared to
BFD1 and BFD2 schemes, respectively. Particularly, the performance of the BFD2
outperforms BFD1 with a small QoS requirement, i.e., $r_{\rm thresh}$ = 0.5.
However, the BFD2’s performance is inferior to that of BFD1 method with a
large QoS value, i.e., $r_{\rm thresh}$ = 1.2. This is due to the fixed
resource allocation per each time slot $n$ in these algorithms. This
additionally leads to fluctuations in data transmission rate values with low
variance during time slot $n$, i.e., $r_{1k}[n]$ and $r_{2k}[n]$. Thus, when
the $r_{\rm thresh}$ value is still lower than the average rate of the BFD2,
the performance is not significantly affected. Nevertheless, if $r_{\rm
thresh}$ is large enough, the performance of BFD2 will drastically be
influenced.

Fig. 5 depicts the percentage of served IoT devices versus cache size with
different value of $P_{k}^{\max}[n]$. The parameters are set up similarly as
shown in Fig. 4, e.g., $r_{\rm thresh}=0.5$. First, we observe that HD-based
schemes’ performance is interior to that of FD counterparts. In particular, at
$P_{\rm U}^{\max}=20$ dBm and $C=800$, the HD method only serves up to 85 $\%$
number of users, while the FD scheme can serve all IoT devices with $P_{\rm
U}^{\max}=18$ dBm and $C=800$, as shown in Fig. 4. This also confirms the
advantages of the FD system. Second, it can be easily seen that the HD scheme
outperforms benchmark ones, i.e., BHD1 and BHD2. Specifically, at $P_{\rm
U}^{\max}=20$ dBm and $C=500$, the HD algorithm can serve 85$\%$ of GUs, and
the BHD1 achieves less than 15$\%$ OP. In comparison, the BHD2 scheme imposes
a 35$\%$ percentage of served IoT devices. In Figs. 4 and 5, the proposed FD
and HD algorithms provide significantly better performance than those
benchmarks, which shows the superiority of these designed schemes compared to
other ones.

Fig. 6 shows the impact of different value of $\eta_{\rm end,k}$ on our
system, with $N=$ 80, $K$ = 20, area = 500 m $\times$ 500 m, $P_{\rm U}^{\rm
max}=$ 18 dBm, $P_{k}^{\rm max}=$ 15 dBm, $\eta_{{\rm start},k}\in[2,20]$, and
$S_{k}$ value is ranging from 10 to 55 Mbits. It is observed that the
percentage of served users increases corresponding to $\eta_{{\rm end},k}\in$
$[65,70],$ $[60,65],$ $[55,60]$, $[50,55]$, $[45,50],$ $[40,45],$
respectively. It can be explained by constraint (II-C), which describes the
condition of the user being successfully served. Since the total throughput
collected is proportional to the time duration allocated to the UL/DL. When
the given time for UL from an IoT device to a UAV is large enough, the number
of served IoT users depends significantly on the time allocation for DL from
UAV to GW. Furthermore, the time period for DL is calculated as $N-\eta_{{\rm
end},k}^{\rm min}$ and $N-\eta_{{\rm end},k}^{\rm max}$ for the FD and HD
schemes, respectively. We see that the period of time allocated for DL in the
FD algorithm is higher than that in the HD algorithm, such that the
performance of the FD scheme outperforms the HD one. Specifically, the total
number of served IoT users obtained from the HD scheme equals that of the FD
method when the value of $N-\eta_{{\rm end},k}^{\rm max}$ is large enough. For
instance, in Figs. 6(a) and 6(b), both proposed methods can serve the maximum
number of IoT devices when $\eta_{{\rm end},k}\in[40,45]$ and $C\geq 600$. In
this scenario, the UAV should work in HD mode for simplicity of operation in
realistic implementation.

Figure 9: Percentage of served IoT devices vs. $P_{\rm U}^{\max}$ with
different data size. Figure 10: Total achievable throughput vs. different
network sizes. Figure 11: Maximum system throughput vs. different bandwidth.

In Fig. 7, we investigate the effect of data size on system performance, where
$K=$ 20, $B=5$ Mhz, $\eta_{{\rm start},k}\in[2,20]$, $n_{{\rm
end},k}^{\min}=30$ time slots, and $n_{{\rm end},k}^{\max}=55$ time slots,
with $N$ = 70 time slots. As inferred from the results, the FD algorithm
significantly improves the percentage of the served IoT devices compared to
the HD algorithm for all values of cache size. Specifically, at $S_{k}\in$
[10, 30] Mbits and $C=$ 400 Mbits, the FD scheme can serve 85% of IoT users on
the network while HD imposes 40% of IoT users served. Furthermore, performance
is degraded by increasing packet size $S_{k}$ due to limited available
resources for IoT devices or the UAV, i.e., $P_{\rm U}^{\rm max}$, $P_{k}^{\rm
max}$, $V_{\rm max}$, and $B$. Besides, when the $S_{k}$ value is small,
corresponding to low data rate IoT devices, i.e., $S_{k}\in$ [1, 10] Mbits,
the number of IoT users successfully served by proposed methods converge to a
saturation value. Therefore, the UAV can operate in HD mode instead of FD one.

Fig. 8 illustrates the percentage of served IoT devices versus network size
(maximum number of IoT devices located in the network area) with different
data sizes, where $B=10$ MHz, $N=70$, $P_{\rm U}^{\rm max}=$ 18 dBm,
$P_{k}^{\rm max}=$ 15 dBm, $n_{\rm start,k}\in[2,15]$ seconds, $n_{\rm
end,k}^{\min}=25$ time slots, $n_{\rm end,k}^{\max}=55$ time slots, and $C=$
1000 Mbits. Similar to Figs. 3-7, the percentage of IoT devices served by the
FD method is better than the HD one. In addition, the percentage of served
users is reduced by increasing the number of IoT users in the same network
area. It is due to limited resources (i.e., bandwidth and transmit power
allocated for UL and DL) and $V_{\rm max}$ when more IoT devices are
considered. Besides, the percentage of served users will enlarge by decreasing
the packet sizes $S_{k}$. This is expected because the UAV needs to spend more
time and resources to compensate for higher $S_{k}$ increase.

Fig. 9 presents the results corresponding to the percentage of served GUs
versus $P_{\rm U}^{\rm max}$ with different $n_{{\rm end},k}$ values. As
shown, the number of served users is enhanced by increasing the power budget,
i.e., $P_{\rm U}^{\rm max}$. Furthermore, FD scheme provides better results
than HD scheme when $P_{\rm U}^{\rm max}$ is relatively small, e.g., $P_{\rm
U}^{\rm max}<22$ dBm with $n_{{\rm end},k}\in$ [40, 50] seconds. Nevertheless,
the HD method can obtain the same number of served users as the FD method when
the $P_{\rm U}^{\rm max}$ value is large, e.g., $P_{\rm U}^{\rm max}\geq 22$
dBm with $n_{{\rm end},k}\in$ [40, 50] seconds. This is because the FD mode
suffers from RSI, which significantly increases the noise power in the UAV
compared to the HD mode. In addition, RSI is linearly proportional to $P_{\rm
U}^{\rm max}$ as in (8). Therefore, when $P_{\rm U}^{\rm max}$ is large, the
UAV should operate in HD mode since the FD mode requires more energy, which
may exceed the system energy budget. It is due to the fact that in FD mode,
the UAV starts to transmit data to GW earlier than in HD mode, which is
highlighted in Fig. 5. This results in higher energy consumption in the UAV
when it manoeuvers in FD mode.

###  V-B Throughput Maximization:

In the following, we present the corresponding results for the total
throughput maximization problem described in Sections III-D and IV-B. In Fig.
10, the total achieved throughput is given as a function of network sizes,
i.e., area is ranging from 500 m $\times$ 500 m to 900 m $\times$ 900 m, with
$K=20$, $S_{k}$ is ranging from 20 to 70 Mbits, $B=10$ Mbits, $N=$ 70 time
slots, $n_{{\rm start},k}\in[2,20]$ seconds, and $n_{{\rm end},k}\in[30,45]$
seconds. Specifically, the achieved throughput is defined as the total
throughput that the UAV transfers from GUs to GW. Herein, we only take into
account the throughput of successfully served GUs. We found that the proposed
algorithms (i.e, FD and HD) significantly improve throughput performance
compared to references (i.e., BFD1, BFD2, BHD1, BHD2) for all values of
network sizes, i.e., $x$ (meters). Specifically, at $x$ = 700 m, FD algorithm
can obtain 788 Mbits and BFD1 algorithm achieves less than 131 Mbits. Whereas
BFD2, HD, BHD1, and BHD2 impose 230, 537, 372, and 140 Mbits, respectively. In
particular, an interesting result is that HD is even better than BFD2, which
underlines the superiority of the proposed algorithms over the references.
That is due to the benefits of optimizing resource allocation.

In Fig. 11, we investigate the effect of system bandwidth on maximum
throughput, with $K=20$, $\rm area=$ 700 m $\times$ 700 m, $S_{k}$ ranging
from 10 to 70 Mbits, $P_{\rm U}^{\rm max}=$ 18 dBm and $P_{k}^{\rm max}=$ 10
dBm, $N=$ 70 time slots, $n_{{\rm start},k}\in[2,20]$ seconds, and $n_{{\rm
start},k}\in[45,55]$ seconds. Maximum throughput is defined as the total
throughput that the UAV can convey to the GW regardless of whether or not each
GU is successfully served. It has been observed that all schemes achieve
better performance with an increase in total bandwidth. This is because the
higher the bandwidth allocation, the greater the transmission can be achieved.
Fig. 11 shows that FD schemes’ performance is significantly better than the HD
ones, since the UAV has more time to transfer collected data to GW in FD-based
methods compared to HD-based ones. Therefore, they can be considered suitable
for practical high throughput applications.

##  VI Conclusion and Future Directions

We investigated the resource allocation and trajectory design for UAV-assisted
FD IoT networks with the emergency communication system, taking into account
latency requirements of IoT devices and the limited storage capacity of the
UAV. In this context, we formulated a novel problem to maximize the total
number of served IoT devices via a joint optimization of the UAV trajectory,
allocated bandwidth, as well as the transmission power of IoT devices and UAV
while satisfying the requested timeout constraints and storage capacity. Due
to non-convexity of the formulated problem, we first transformed the original
problem into a tractable form, which is then solved using an iterative
algorithm with a polynomial computational complexity per iteration. Besides,
pertaining to the realistic requirements for improving the estimation accuracy
in a natural disaster or emergency scenario, we proposed an additional
optimization problem in order to maximize the total collected data while
satisfying the threshold of a minimum number of served IoT devices. We
illustrated via numerical results that the proposed designs outperform the
benchmark schemes in terms of both the total number of served IoT devices and
the amount of collected data. Notably, in the scenarios such as when IoT
devices’ RT is not stringent, in the case of small data size, or required
$P_{\rm U}^{\rm max}$ is large, the UAV should operate in the HD mode for a
simple implementation.

The outcome of this work will motivate future works in UAV-aided wireless
systems. One possible problem is to extend this work to a multi-antenna UAV
system, which imposes higher complexity but might further improve the network
performance. Another promising problem is to consider low complexity yet
efficient machine learning approaches to provide a reliable prediction of the
LoS probability for any pair of UAV and GU locations, hence leading to enhance
performance assurance.

## Appendix A: Proof of Lemma 1

Proof for (10) and (11): We consider a function
$f(z)=\mathbb{E}_{Z}[\log_{2}(1+e^{\ln z})]$, $z>0$. By adopting Jensen’s
inequality for convex function $\log_{2}(1+e^{\ln z})$, it yields

$\displaystyle f(z)\geq\log_{2}\big{(}1+e^{\mathbb{E}_{Z}[\ln z]}\big{)}.$
(A.1)

Let us denote $Z\triangleq\Gamma_{\rm
1k}=\frac{p_{1k}[n]|{\tilde{h}}_{1k}[n]|^{2}\omega_{0}}{{\big{(}{H^{2}}+{{\big{\|}{{\bf
q}[n]-{\bf w}_{k}}\big{\|}}^{2}}\big{)}^{\alpha/2}}\big{(}\phi^{\rm
RSI}\sum\limits_{k^{\ast}\in{\cal K}\setminus
k}p_{2k^{\ast}}[n]+\sigma^{2}\big{)}}$. Thus, this is an exponentially
distributed random variable with parameter
$\lambda_{Z}\triangleq(\mathbb{E}[Z])^{-1}=\frac{\zeta_{1k}}{p_{1k}[n]\omega_{0}}$
with $\zeta_{1k}\triangleq{\big{(}{H^{2}}+{{\big{\|}{{\bf q}[n]-{\bf
w}_{k}}\big{\|}}^{2}}\big{)}^{\alpha/2}}$ $\big{(}\phi^{\rm
RSI}\sum\limits_{k^{\ast}\in{\cal K}\setminus
k}p_{2k^{\ast}}[n]+\sigma^{2}\big{)}$. By applying [55, Eq. 4.331.1],
$\mathbb{E}_{Z}[\ln z]$ can be calculated as

$\displaystyle\mathbb{E}_{Z}[\ln z]$
$\displaystyle=\int_{0}^{+\infty}\lambda_{Z}e^{-z\lambda_{Z}}\ln{z}dz=-\big{(}\ln(\lambda_{Z})+E\big{)},$
$\displaystyle=\ln\frac{p_{1k}[n]\omega_{0}}{\zeta_{1k}}-E,$ (A.2)

where $E$ is the Euler-Mascheroni constant, i.e., $E=0.5772156649$ as in [55,
Eq. 8.367.1].

By substituting (Appendix A: Proof of Lemma 1) into (A.1), we obtain (10).
Similar to (Appendix A: Proof of Lemma 1), we also easily achieve (11) by
adopting $Z\triangleq\Gamma_{\rm 2k}$.

## Appendix B: Proof of Lemma 3

As in [36, Eq. (20)], we have

$\displaystyle h_{1}(x,y,z)\geq$
$\displaystyle\ln\left(1+\frac{x^{(j)}}{y^{(j)}z^{(j)}}\right)-\frac{x^{(j)}}{y^{(j)}z^{(j)}}+2\frac{\sqrt{x^{(j)}}\sqrt{x}}{y^{(j)}z^{(j)}}$
$\displaystyle-\frac{x^{(j)}\left(x+yz\right)}{y^{(j)}z^{(j)}\left(x^{(j)}+y^{(j)}z^{(j)}\right)},$
(B.1) $\displaystyle h_{2}(x,z)\geq$
$\displaystyle\ln\left(1+\frac{x^{(j)}}{z^{(j)}}\right)-\frac{x^{(j)}}{z^{(j)}}+2\frac{\sqrt{x^{(j)}}\sqrt{x}}{z^{(j)}}$
$\displaystyle-\frac{x^{(j)}\left(x+z\right)}{z^{(j)}\left(x^{(j)}+z^{(j)}\right)}.$
(B.2)

By applying (61), the upper bound of $yz$ in (Appendix B: Proof of Lemma 3) is
given by

$\displaystyle
yz\leq\frac{y^{(j)}}{2z^{(j)}}z^{2}+\frac{z^{(j)}}{2y^{(j)}}y^{2},$ (B.3)

with $x>0,$ $y>0,$ $z>0,$ $x^{(j)}>0,$ $y^{(j)}>0,$ $z^{(j)}>0$.

Then, substituting (B.3) into (Appendix B: Proof of Lemma 3), we obtain (3)
and (3). Lemma 3 is hence completed.

## Appendix C

From (B.3), the upper bound of $z_{k}[n]t_{1k}[n]$ in $r_{1k}^{\rm lb}[n]$ is:

$\displaystyle z_{k}[n]t_{1k}[n]$
$\displaystyle\leq\left(z_{1k}[n]t_{1k}[n]\right)^{\rm ub}$
$\displaystyle\triangleq\frac{z_{1k}^{(j)}[n]\left(t_{1k}[n]\right)^{2}}{2t_{1k}^{(j)}[n]}+\frac{t_{1k}^{(j)}[n]\left(z_{1k}[n]\right)^{2}}{2z_{1k}^{(j)}[n]}.$
(C.1)

By making use of (3), (3), and (Appendix C), the lower bound of $\Phi_{1k}[n]$
and $\Phi_{2k}[n]$ are, respectively

$\displaystyle\Phi_{1k}[n]$ $\displaystyle\geq\bar{\Phi}_{1k}[n]\triangleq
B\Big{(}\Xi_{1}+\Xi_{2}-\Xi_{3}\Big{)},$ (C.2) $\displaystyle\Phi_{2k}[n]$
$\displaystyle\geq\bar{\Phi}_{2k}[n]\triangleq
B\Big{(}\Xi_{4}+\Xi_{5}-\Xi_{6}\Big{)},$ (C.3)

where

$\displaystyle\Xi_{1}$
$\displaystyle\triangleq\log_{2}\Biggl{(}1+\frac{e^{-E}p_{1k}^{(j)}[n]\omega_{0}}{z_{1k}^{(j)}[n]t_{1k}^{(j)}[n]}\Biggr{)}-\frac{e^{-E}p_{1k}^{(j)}[n]\omega_{0}}{z_{1k}^{(j)}[n]t_{1k}^{(j)}[n]\ln
2},$ $\displaystyle\Xi_{2}$ $\displaystyle\triangleq
e^{-E}\omega_{0}\frac{2\sqrt{p_{1k}^{(j)}[n]}\sqrt{p_{1k}[n]}}{z_{1k}^{(j)}[n]t_{1k}^{(j)}[n]\ln
2},$ $\displaystyle\Xi_{3}$
$\displaystyle\triangleq\frac{e^{-E}p_{1k}^{(j)}[n]\omega_{0}}{\big{(}e^{-E}p_{1k}^{(j)}[n]\omega_{0}+z_{1k}^{(j)}[n]t_{1k}^{(j)}[n]\big{)}z_{1k}^{(j)}[n]t_{1k}^{(j)}[n]\ln
2}$ $\displaystyle\times$
$\displaystyle\Bigg{(}e^{-E}p_{1k}[n]\omega_{0}+\frac{z_{1k}^{(j)}[n]\left(t_{1k}[n]\right)^{2}}{2t_{1k}^{(j)}[n]}+\frac{t_{1k}^{(j)}[n]\left(z_{1k}[n]\right)^{2}}{2z_{1k}^{(j)}[n]}\Bigg{)},$
$\displaystyle\Xi_{4}$
$\displaystyle\triangleq\log_{2}\Bigg{(}1+\frac{e^{-E}p_{2k}^{(j)}[n]\omega_{0}}{z_{2k}^{(j)}[n]\sigma^{2}}\Bigg{)}-\frac{e^{-E}p_{2k}^{(j)}[n]\omega_{0}}{z_{2k}^{(j)}[n]\sigma^{2}\ln
2},$ $\displaystyle\Xi_{5}$
$\displaystyle\triangleq\frac{e^{-E}\omega_{0}}{z_{2k}^{(j)}[n]\sigma^{2}\ln
2}2\sqrt{p_{2k}^{(j)}[n]}\sqrt{p_{2k}[n]},$ $\displaystyle\Xi_{6}$
$\displaystyle\triangleq\frac{e^{-E}p_{2k}^{(j)}[n]\omega_{0}}{e^{-E}p_{2k}^{(j)}[n]\omega_{0}+z_{2k}^{(j)}[n]\sigma^{2}}\times\frac{\Big{(}e^{-E}p_{2k}[n]\omega_{0}+z_{2k}[n]\sigma^{2}\Big{)}}{z_{2k}^{(j)}[n]\sigma^{2}\ln
2}.$

## Appendix D: Proof of Proposition 1

For the sake of notational convenience, let us define the feasible set
$\chi^{(j)}$ of (70a) at the initial stage of the $(j+1)$-th iteration, such
that

$\displaystyle\vspace{-0.01cm}\chi^{(j)}\triangleq\\{\boldsymbol{\Psi}^{(j)}|\text{s.t.
\eqref{eq:P13:b}-\eqref{eq:P13:g} are feasible }\\}.$ (D.1)

First, we recall that the approximate functions presented in Section III
satisfy properties of IA algorithm [25, 43]. Let
${\mathbb{F}}(\boldsymbol{\Psi})$ and
$\widetilde{\mathbb{F}}(\boldsymbol{\Psi})$ denote the objective function of
(43a) and (70a), respectively. Following IA principles, the feasible region of
approximated convex function (70a) is a subset of the feasible region of
relaxed problem (43a) [43, Property i of Lemma 2.2]. Thus, it is true that

$\displaystyle{\mathbb{F}}(\boldsymbol{\Psi})$
$\displaystyle\geq\widetilde{\mathbb{F}}(\boldsymbol{\Psi}),\;\forall\boldsymbol{\Psi},$
(D.1) $\displaystyle{\mathbb{F}}(\boldsymbol{\Psi}^{(j)})$
$\displaystyle=\widetilde{\mathbb{F}}(\boldsymbol{\Psi}^{(j)}),\;\forall\boldsymbol{\Psi}.$
(D.2)

Thus, it follows that

$\displaystyle{\mathbb{F}}(\boldsymbol{\Psi}^{(j+1)})\geq\widetilde{\mathbb{F}}(\boldsymbol{\Psi}^{(j+1)})\geq\widetilde{\mathbb{F}}(\boldsymbol{\Psi}^{(j)})={\mathbb{F}}(\boldsymbol{\Psi}^{(j)}),$
(D.3)

where the first inequality is due to (D.1). The second inequality is
attributed to the fact that $\boldsymbol{\Psi}^{(j+1)}$ is a better solution
for (43a) than $\boldsymbol{\Psi}^{(j)}$ [43, Property iv of Lemma 2.2].
Moreover, the sequence $\\{{\mathbb{F}}(\boldsymbol{\Psi}^{(j)})\\}$ will
converge, as shown in [43, Corollary 2.3 ], and each accumulation point
$\boldsymbol{\Psi}^{\star}$ of the sequence $\\{\boldsymbol{\Psi}^{(j)}\\}$ is
a Karush-Kuhn-Tucker point as in [25, Theorem 1] and [43, Proposition 3.2].
Furthermore, since the feasible set $\chi^{(j)}$ is a convex connected set due
to the convexity of (70a) [56]. Moreover, it is closed and bounded because of
power constraints (35a) and (35a), bandwidth constraints (35a) and (35a), and
limited flying time. Consequently, we can obtain a locally optimal solution to
(43a) according to [25, Corollary 1], which completes the proof.

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|  Dinh-Hieu Tran (S’20) was born and grew up in Gia Lai, Vietnam (1989). He
received the B.E. degree in Electronics and Telecommunication Engineering
Department from Ho Chi Minh City University of Technology, Vietnam, in 2012.
In 2017, he finished the M.Sc degree in Electronics and Computer Engineering
from Hongik University (Hons.), South Korea. He is currently pursuing the
Ph.D. degree at the Interdisciplinary Centre for Security, Reliability and
Trust (SnT), University of Luxembourg, under the supervision of Prof. Symeon
Chatzinotas and Prof. Björn Ottersten. His research interests include UAVs,
IoTs, Mobile Edge Computing, Caching, Backscatter, B5G for wireless
communication networks. He was a recipient of the IS3C 2016 best paper award.
---|---
|  Van-Dinh Nguyen (Member, IEEE) received the B.E. degree in electrical
engineering from the HoChi Minh City University of Technology, Vietnam, in
2012, and the M.E. and Ph.D. degrees in electronic engineering from Soongsil
University, Seoul, South Korea, in 2015 and 2018, respectively. He was a Post-
Doctoral Researcher and a Lecturer with Soongsil University, a Post-Doctoral
Visiting Scholar with the University of Technology Sydney, Australia, from
July 2018 to August 2018, and a Ph.D. Visiting Scholar with Queen’s University
Belfast, U.K., from June 2015 to July 2015 and in August 2016. He is currently
a Research Associate with the Interdisciplinary Centre for Security,
Reliability and Trust (SnT), University of Luxembourg. His current research
interests include fog/edge computing, the Internet of Things, 5G networks, and
machine learning for wireless communications. He received several best
conference paper awards, IEEE TRANSACTIONS ON COMMUNICATIONS Exemplary
Reviewer 2018, and IEEE GLOBECOM Student Travel Grant Award 2017. He has
authored or co-authored 40 papers published in international journals and
conference proceedings. He has served as a reviewer for many top-tier
international journals on wireless communications and has also been a
technical programme committee member for several flag-ship international
conferences in the related fields. He is an Editor of the IEEE OPEN JOURNAL OF
THE COMMUNICATIONS SOCIETY and IEEE COMMUNICATIONS LETTERS
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|  Symeon Chatzinotas , (S’06-M’09-SM’13) is currently Full Professor / Chief
Scientist I in Satellite Communications and Head of the SIGCOM Research Group
at SnT, University of Luxembourg. He is coordinating the research activities
on communications and networking, acting as a PI for more than 20 projects and
main representative for 3GPP, ETSI, DVB. In the past, he has been a Visiting
Professor at the University of Parma, Italy, lecturing on “5G Wireless
Networks”. He was involved in numerous R$\&$D projects for NCSR Demokritos,
CERTH Hellas and CCSR, University of Surrey. He was the co-recipient of the
2014 IEEE Distinguished Contributions to Satellite Communications Award and
Best Paper Awards at EURASIP JWCN, CROWNCOM, ICSSC. He has (co-)authored more
than 450 technical papers in refereed international journals, conferences and
scientific books. He is currently in the editorial board of the IEEE
Transactions on Communications, IEEE Open Journal of Vehicular Technology and
the International Journal of Satellite Communications and Networking.
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|  Thang X. Vu (M’15) was born in Hai Duong, Vietnam. He received the B.S. and
the M.Sc., both in Electronics and Telecommunications Engineering, from the
VNU University of Engineering and Technology, Vietnam, in 2007 and 2009,
respectively, and the Ph.D. in Electrical Engineering from the University
Paris-Sud, France, in 2014. In 2010, he received the Allocation de Recherche
fellowship to study Ph.D. in France. From September 2010 to May 2014, he was
with the Laboratory of Signals and Systems (LSS), a joint laboratory of CNRS,
CentraleSupelec and University Paris-Sud XI, France. From July 2014 to January
2016, he was a postdoctoral researcher with the Information Systems Technology
and Design (ISTD) pillar, Singapore University of Technology and Design
(SUTD), Singapore. Currently, he is a research scientist at the
Interdisciplinary Centre for Security, Reliability and Trust (SnT), University
of Luxembourg, Luxembourg. His research interests are in the field of wireless
communications, with particular interests of 5G networks and beyond, machine
learning for communications and cross-layer resources optimization. He was a
recipient of the SigTelCom 2019 best paper award.
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|  Björn Ottersten , (S’87-M’89-SM’99-F’04) was born in Stockholm, Sweden, in
1961. He received the M.S. degree in electrical engineering and applied
physics from Linköping University, Linköping, Sweden, in 1986, and the Ph.D.
degree in electrical engineering from Stanford University, Stanford, CA, USA,
in 1990. He has held research positions with the Department of Electrical
Engineering, Linköping University, the Information Systems Laboratory,
Stanford University, the Katholieke Universiteit Leuven, Leuven, Belgium, and
the University of Luxembourg, Luxembourg. From 1996 to 1997, he was the
Director of Research with ArrayComm, Inc., a start-up in San Jose, CA, USA,
based on his patented technology. In 1991, he was appointed Professor of
signal processing with the Royal Institute of Technology (KTH), Stockholm,
Sweden. Dr. Ottersten has been Head of the Department for Signals, Sensors,
and Systems, KTH, and Dean of the School of Electrical Engineering, KTH. He is
currently the Director for the Interdisciplinary Centre for Security,
Reliability and Trust, University of Luxembourg. He is a recipient of the IEEE
Signal Processing Society Technical Achievement Award and the European
Research Council advanced research grant twice. He has co-authored journal
papers that received the IEEE Signal Processing Society Best Paper Award in
1993, 2001, 2006, 2013, and 2019, and 8 IEEE conference papers best paper
awards. He has been a board member of IEEE Signal Processing Society, the
Swedish Research Council and currently serves of the boards of EURASIP and the
Swedish Foundation for Strategic Research. He has served as an Associate
Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the Editorial Board
of the IEEE Signal Processing Magazine. He is currently a member of the
editorial boards of IEEE Open Journal of Signal Processing, EURASIP Signal
Processing Journal, EURASIP Journal of Advances Signal Processing and
Foundations and Trends of Signal Processing. He is a fellow of EURASIP.
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