Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2023.0322000

Corresponding authors: Waqas Khalid<EMAIL_ADDRESS>gmail.com}) and
Ayaz Ahmad<EMAIL_ADDRESS>
This research was supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education(MOE) (NRF-2022R1I1A1A01071807).

# Efficient UAVs Deployment and Resource Allocation in UAV-Relay Assisted
Public Safety Networks for Video Transmission

NAVEED KHAN1  AYAZ AHMAD1  ABDUL WAKEEL2  ZEESHAN KALEEM1  BUSHRA RASHID3  and
WAQAS KHALID4  Department of Electrical and Computer Engineering, COMSATS
University, Islamabad, Wah Campus, Pakistan  Electrical Engineering
Department, Military College of Signals (MCS), National University of Sciences
and Technology (NUST), Islamabad, Pakistan  Department of Biomedical
Engineering, College of Engineering, King Faisal University, Alahsa, Kingdom
of Saudi Arabia  Institute of Industrial Technology, Korea University, Sejong
30019, South Korea

###### Abstract

Wireless communication highly depends on the cellular ground base station
(GBS). A failure of the cellular GBS, fully or partially, during natural or
man-made disasters creates a communication gap in the disaster-affected areas.
In such situations, public safety communication (PSC) can significantly save
the national infrastructure, property, and lives. Throughout emergencies, the
PSC can provide mission-critical communication and video transmission services
in the affected area. Unmanned aerial vehicles (UAVs) as flying base stations
(UAV-BSs) are particularly suitable for PSC services as they are flexible,
mobile, and easily deployable. This manuscript considers a multi-UAV-assisted
PSC network with an observational UAV receiving videos from the affected
area’s ground users (AGUs) and transmitting them to the nearby GBS via a relay
UAV. The objective of the proposed study is to maximize the average utility of
the video streams generated by the AGUs upon reaching the GBS. This is
achieved by optimizing the positions of the observational and relay UAVs, as
well as the distribution of communication resources, such as bandwidth, and
transmit power, while satisfying the system-designed constraints, such as
transmission rate, rate outage probability, transmit power budget, and
available bandwidth. To this end, a joint UAVs placement and resource
allocation problem is mathematically formulated. The proposed problem poses a
significant challenge for a solution. Considering the block coordinate descent
and successive convex approximation techniques, an efficient iterative
algorithm is proposed. Finally, simulation results are provided which show
that our proposed approach outperforms the existing methods.

###### Index Terms:

Public safety communication networks (PSCNs), UAVs, video transmission,
resource allocation.

=-21pt

##  I Introduction

In mobile communications, cellular base stations may fail due to natural or
man-made disasters. Additionally, deploying advanced networks in some areas
can be impractical or too risky for first responders. For example, the Sichuan
(China) earthquake, in May 2008, caused significant damage to
telecommunication infrastructure [1]. Similarly, the World Trade Center attack
on 9/11 led to nearly 3 million users losing cell phone services [2]. In such
situations, individuals require guidance, counsel, and real-time information
about ongoing situations. Often, they must send requests for help or share
their locations with rescue teams. In disaster settings, where cellular
networks often fail, it is particularly beneficial for users to establish
direct communication with each other [3]. To address these challenges, using
an unmanned aerial vehicle (UAV) as a cellular base station (BS) is a key
solution that allows fast, seamless, and reliable cellular communication,
crucial for public safety [4]. UAVs do not need any restricted and costly
calibrations, e.g., cables, and can effortlessly move and dynamically adjust
their positions, to yield on-demand cellular communications to affected area
ground users (AGUs) in undesirable circumstances[5]. Moreover, UAVs offer a
compelling solution for enhancing the functionality of wireless networks in
boasting various applications, including offloading wireless video services in
regions lacking proper infrastructure or during disasters [6].

The demand for mobile video streaming in everyday life as well as in rescue
operations is on the rise, dominating global mobile data traffic. As per the
Ericsson Mobility Report for 2022 [7], video streaming forms a substantial and
swiftly expanding segment of mobile data traffic. In 2022, video traffic
represented 70% of all cellular data, and it is expected to grow to 80% in the
coming years. However, this surge in video streaming popularity is causing
congestion issues, particularly for users at the edge of cellular coverage,
leading to a degradation in the quality of experience (QoE) for these end
users. To address these challenges, recent research efforts have been focused
on developing efficient solutions to meet users’ QoE requirements. One
conventional approach involves deploying small-cell networks comprising
numerous small base stations (SBSs) [8]. However, this strategy may prove
cost-ineffective in scenarios with temporary spikes in user density or highly
dynamic traffic demands. Alternatively, UAVs have emerged as a viable
alternative for extending wireless network coverage and alleviating congestion
issues. UAVs can offload traffic and provide video services, particularly in
areas affected by disasters [9]. Furthermore, compared to direct transmissions
from terrestrial BS over long distances, the use of UAVs as relays offers
several advantages, such as the potential for line-of-sight (LoS) links with
adjustable altitudes, expanded coverage for ground wireless devices, and
reliable uplink and downlink connections for affected ground users [10].

UAVs, when equipped with wireless transceivers, can transmit video footage
from the affected area during disasters. Generally, the effectiveness of these
wireless transceivers is contingent upon the size of the antenna they employ.
Utilizing high-performance larger antennas can enable UAVs to transmit data
over larger distances. However, due to the constrained payload capacities of
UAVs, high-performance antennas tend to be too large and heavy for them to
carry while maintaining flight capabilities. Consequently, when a single UAV
is tasked with observing and transmitting real-time video data, the range of
communication is restricted, thus limiting the scope of real-time observation.
Nevertheless, this constraint can be addressed through the use of multiple
UAVs, which can establish communication with one another via their integrated
wireless transceivers. This interconnected arrangement is commonly referred to
as a UAV network [11].

In this paper, we propose a public safety communication network (PSCN)
consisting of observation UAV and relay UAV for uplink video streaming in a
fading channel wireless environment. The network utilizes an observation UAV
as an aerial base station to receive video streams from AGUs and transmit them
to the nearby functional ground base station (GBS). Simultaneously, the relay
UAV bridges the connection between the observation UAV and the GBS while
extending the coverage of PSCN. Designing such a UAV-based uplink video
streaming system requires careful consideration of parameters, including the
aerial placement of observation and relay UAVs, as well as the allocation of
resources such as transmission power and bandwidth. These factors are crucial
for achieving superior signal strength and ensuring equitable delivery of good
quality video for all AGUs [12].

###  I-A Related Work

UAVs play a pivotal role in cellular communications, particularly in enhancing
public safety operations. Despite their significant impact, UAV deployment
faces key challenges, including energy limitation [13], flight time
constraints, optimal trajectory [14], interference and resource management
[15], and efficient placement. Addressing these issues is crucial for the
efficient utilization of UAVs in public safety scenarios. Different approaches
have been proposed for the optimal deployment of UAVs, focusing on minimizing
UAV transmit power, expanding wireless coverage, and leveraging UAVs for relay
and video streaming services. In the subsequent sections, we discuss some of
the well-known approaches in this context.

####  I-A1 UAV Placement and Recourse Allocation

In [16], the authors have proposed a single UAV-based disaster management
system for indoor employees in a multi-storeyed building to deliver wireless
coverage. The authors have considered the problem of efficient UAV deployment
to cover the entire building with minimum transmit power. The authors in [17]
have applied the Particle Swarm Optimization algorithm in an indoor scenario.
The objective of their study is to optimize the 3-D placement of UAVs to
minimize the overall transmit power. By utilizing the Particle Swarm
Optimization algorithm, the authors aim to find an optimal configuration that
reduces the total power consumption required for wireless communication in the
indoor environment.

The authors in [18] have proposed an optimal UAV placement system for
delivering cellular services to outdoor and indoor users bypassing the load of
the existing wireless network infrastructure with low transmit power. In [19],
the authors introduced a reinforcement learning approach to address the
challenge of enhancing the max-min sum rate in a computationally demanding
scenario by determining the optimal 3D UAV (ABS) positions and power
allocation In [20], Shukla et al. analyzed the resource distribution of
multiple UAV-based systems subject to minimizing the energy consumption and
operation delay taking into account the edge and cloud servers. In [21],
Zhaohui et al. have minimized the sum power of the UAVs by jointly considering
the beam width, height, bandwidth allocation, and position of the UAVs as
optimization variables. in [22], the authors delved into the joint
optimization of resource allocation and relay selection within a decode-and-
forward downlink OFDMA cooperative network, considering the scenario of
outdated CSI at the base station. The work in [23] has analytically assessed
the rotational angle division multiple access (RADMA) performance implemented
using UAV-BS with array antennas. To demonstrate how RADMA performs better in
a low-power wide-area network configuration, a numerical performance
evaluation is carried out. According to the simulation results, RADMA can
reduce the transmission energy needed by 20%, transmission time by 25%, and
the packet loss ratio by 77%.

Besides minimizing the transmit power, the focus is also to maximize the
cellular coverage of the UAVs for serving a large number of users. The authors
in [24] formulated a 3-D UAV placement problem with the objective of cellular
coverage maximization. The authors in [25] have considered the optimal
placement of UAVs using a circle-packing algorithm. They have demonstrated
that the cellular coverage is maximized by fixing the UAVs 3-D location. User-
centric and network-centric approaches were introduced by Kalantari et al. in
[26]. For each of their approaches, the authors determined 3-D coordinates for
optimal deployment of the UAVs that maximize the sum rate and cellular
coverage.

The authors in [27] introduced a strategy for positioning multiple UAV-based
base stations in a manner that maximizes cellular access for users while
minimizing inter-symbol interference. Two methods were proposed for the UAVs
placement. In the first method, a linear algorithm was Suggested for the
sequential deployment of the UAVs. In the second method, the authors used
concurrent UAV placement with machine learning algorithms. The experimental
results obtained show that user coverage can be enhanced if the ground users
are spread unevenly. The authors in [28] have maximized the coverage area of
UAV-based cellular communication networks subject to UAV hovering duration and
the average sum of bits transmitted to the users. In [29], the authors
analyzed the potential of deploying multiple UAVs to improve coverage
performance when interference is a factor. The proposed approach suggests that
by adjusting the height of the UAVs in different working environments, it is
possible to optimize coverage and determine the best configuration for a
particular set of UAVs. This method provides valuable design guidelines for
achieving better coverage and minimizing interference among UAVs.

In [30], the authors have studied the placement of UAVs in a catastrophe-
affected zone to handle the high influx of user traffic that typically occurs
during emergencies. The authors aim to identify the optimal UAV coverage that
would ensure the maximum user throughput with fair distribution across all
areas of the topology. The authors in [31] deployed UAVs in a square-shaped
area, assuming that all links within the region have Line-of-Sight (LoS) and
directional antennas are utilized to maximize coverage.

An attractive area of research is to deploy UAVs for video streaming. A
flexible amount of support is provided to the existing cellular networks by
deploying UAVs for multimedia services in real-time scenarios.[32] proposed a
virtual reality (VR) wireless network, where the authors used UAVs to transmit
the acquired video data to a ground BS. The authors in [33] investigated the
power allocation and video bitstream adaptation in the context of video
streaming across multi-node wireless networks where interference varies over
time. [34] and [35] have analyzed energy efficiency in a UAV-enabled video
streaming scenario by keeping the flight height of different UAVs fixed. For
multi-user video transmission, the authors in [36] optimized bandwidth and
transmit power distribution in a UAV relay network to maximize the long-term
QoE of the users.

In [37], the authors have explored UAV video streaming in sensor-augmented
systems and proposed video streaming algorithms for the sensor data. In [38],
the authors proposed an optimized heterogeneous framework for a video cellular
network, where the authors have jointly optimized the rate allocation,
selection of video quality, and time-domain resource portioning. In [39], a
live video streaming forest fire surveillance system was proposed. The system
is low-cost to implement, simple to use, and has a large coverage area. For
adaptive streaming in mobile wireless networks, utility maximization was
proposed in [40]. The authors have shown that maximum utility can be achieved
through efficient resource allocation and dynamic rate adaptability.

UAVs typically rely on limited power sources, primarily batteries, which
constrain their flight duration. For instance, the majority of commercial
rotary-wing UAVs have a maximum flight time of just around 30 minutes [41].
Consequently, various strategies have been proposed to extend UAV flight
durations. These solutions include battery swapping [42], harnessing solar
power for charging [43], RF-based charging [44], laser-based wireless power
transfer (WPT) [45], and optimizing UAV deployment [46]. [47] presented a
750-W 85-kHz band inductive rapid charging system designed for mid-sized UAVs
(drones). This system was intended for opportunistic charging during
continuous industrial operations. Additionally, in [48], the authors explored
the adoption of renewable energy generation and storage equipment integrated
with traditional charging stations. The aim is to minimize reliance on power
sourced from the distribution network. In [49], authors examined the concept
of drone-to-drone opportunity charging as a means to enhance flight duration,
particularly in the context of multi-agent systems.

####  I-A2 UAV Deployment as a Relay

Recently, different authors have worked on the deployment of UAVs as relays
over a region, to deliver cellular coverage to the ground users when a direct
communication link is blocked either due to malfunctioning of the existing
base station or long distance from the BSs. In [50], the authors have
considered multiple pairs of ground user scenarios where direct communication
links are not available either due to long distances or no LOS, i.e.,
blockages. For multiple pairs of ground users, the authors deployed a single
UAV as a relay in a time-division manner, and the minimum average rate is
maximized by allocating optimum time slots to all communication pairs. In
[51], the authors have investigated an optimal 3-D placement of UAVs as relays
when the existing base station malfunctions. Three different models were used
to achieve optimal 3-D placement of the relay UAV. In [52], the authors have
proposed a matching theory to analyze the advantages and difficulties of UAVs
as relay models in bulky UAV communication systems.

The authors in [53] have proposed UAV-based cellular communication systems in
which a relay-based UAV is utilized for bridging a set of distant user
equipment (UE) with the ground base station and vice versa. This two-way
communication is possible when the UAV is relayed by orthogonal frequency (OF)
bands. The UAVs transmission power and placement are jointly optimized to
achieve a high total rate for both down-link and up-link communication subject
to the SNR constraints and communication power constraints on the UAV control
channel. In [54], the authors have studied UAV as a relay for uplink data
transmission to connect as many IoT devices as possible while still meeting
their heterogeneous quality of service needs. In [55], the authors worked on
UAV placing problems in a relay network by integrating the local topological
knowledge, where the aim was to deploy the UAV in a location with a good LOS.
The authors in [56] proposed dynamic positioning systems to build a
communication link between the disjoint nodes by utilizing UAV as a relay. The
authors suggest that their approach could find practical applications in real-
life situations, such as during natural disasters like floods and earthquakes,
as well as to gather data from deployed sensor nodes. In [57], the authors
introduced a UAV-based emergency Wi-Fi system to assist the rescue activities
by supervising the survivors at the nearby rescue camp, whereas, the authors
in [58] have designed UAV-based energy-efficient relay systems that jointly
optimize the BS and UAV transmit power.

###  I-B Contributions

Different from the aforementioned research, herein, we analyze joint
optimization of the observation and relay UAVs locations, transmit power, and
bandwidth distribution for different numbers of AGUs over a fading channel in
PSCNs. The main objective is to augment the average utility of video streams
created by the AGUs when arriving at the GBS over a fading channel. The
primary contributions of this manuscript can be summarised as follows:

  * •

We propose a multi-UAV-based up-link video streaming system for video
transmission in fading channel conditions within PSCNs. Analytical expressions
for the rate outage probability between the observation UAV and AGU over the
fading channel are calculated. Furthermore, we formulate a video streaming
utility function for each AGU. Subsequently, we develop an optimization
problem aimed at maximizing the average streaming utility for all AGUs. The
optimization problem involves a joint optimization of the observation and
relay UAV’s locations, transmit power, and allocation of the bandwidth. The
optimization is subject to the information causality constraint of the relay
UAV and GBS, as well as the outage probability constraint between the
observation UAV and AGUs.

  * •

Due to the complexity of the formulated problem, a direct solution is
challenging. Consequently, an iterative algorithm is proposed to obtain an
efficient solution. This algorithm utilizes successive convex approximation
(SCA) and block coordinate descent techniques. In particular, two auxiliary
problems are optimized iteratively: the allocation of bandwidth and power with
fixed UAV locations, and the positioning of UAVs with fixed bandwidth and
transmit power.

  * •

We present simulation results to demonstrate the trade-off between the data
rate of the observation UAV and the relay UAV, as well as the trade-off
between the relay UAV and the GBS. Additionally, we provide evidence of the
effectiveness of our proposed design by evaluating the average streaming
utility.

The remaining manuscript is organized as follows: Section II provides the
problem formulation and system model of our proposed PSCNs, whereas, the
mathematical solution of the formulated problem is given in Section III.
Section IV presents the system parameters, numerical findings, and simulation
results. Concluding remarks and future research directions are provided in
Section V.

##  II System model and problem formulation

Herein, we present the problem statement and a UAV-based video streaming PSCNs
system model. Table I summarizes the important notations used and the
explanations that go with them.

TABLE I: Parameters definition Notation | Definition
---|---
B | System bandwidth ($Hz$)
$H_{o}$ | Observation UAV elevation ($m$)
$H_{b}$ | Ground BS elevation ($m$)
$U$ | Total number of affected ground users (AGUs)
$P_{o}^{max}$ | Observation UAV maximum transmit power (Watts)
$P_{r}^{max}$ | Relay UAV maximum transmit power (Watts)
$P_{u}^{max}$ | Affected area users’ maximum transmit power (Watts)
$n_{u}$ |  $u$th affected ground user(AGU)
$w_{b}$ | Ground BS’s two-dimensional (2D) coordinates
$w_{u}$ | 2D coordinates of $n_{u}$
$q_{o}$ | 2D coordinates of Observation UAV
$q_{r}$ | 2D coordinates of Relay UAV
$\rho$ | Target outage probability threshold
$x_{u}$ | Part of the total bandwidth allocated to $n_{u}$
$P_{u}^{out}$ | Rate outage probability of user $n_{u}$
$P_{u}$ | Transmit power assigned to the $n_{u}$ user
$R_{u}$ | Uplink transmit rate over AGU to observation
| UAV link for AGU $n_{u}$ (bps/Hz)
$R_{r}$ | Transmit rate over observation UAV
| to relay UAV link (bps/Hz)
$R_{b}$ | Transmission rate of relay UAV to
| the ground BS (bps/Hz)
$D$ | D represent the PSCNs size
$\overline{r}_{u}$ | Required video playback rate (bps) of AGU $n_{u}$
$S_{u}$ | AGU $n_{u}$ streaming utility

###  II-A System model

In our proposed PSCN, we consider an up-link UAV-based video streaming system
where AGUs are connected to the GBS through observation and relay UAVs. These
UAVs are utilized to offload the video streams from the AGUs to the GBS over a
fading channel, as illustrated in Fig. 1. In our proposed PSCN, we employ one
observation UAV and one relay UAV both performing different tasks. The
observation UAV is primarily used to receive and re-transmit the video
generated by the AGUs, while the relay UAV serves as the connection between
the observation UAV and the GBS for real-time video transmission over longer
distances. The observation UAV is not suitable for relaying data between AGUs
and the GBS due to its long distance from the GBS. Furthermore, being far away
from the rely UAV, the AGUs cannot establish direct communication with it.

In Fig. 1, the group of AGUs is represented by
${U}=\\{n_{1},n_{2},\cdots,n_{u}\\}$, where the observation UAV equipped with
a wireless transceiver is used to receive the video streaming services from
these AGUs. Moreover, the positions of the AGUs are assumed to be known. In a
three-dimensional (3-D) Cartesian coordinate system, the coordinates of the
observation UAV and the relay UAV are represented as $(a_{o},c_{o},H_{o})$ and
$(a_{r},c_{r},H_{r})$, respectively. Moreover, the coordinate of the GBS is
represented as $(a_{b},c_{b},H_{b})$. Likewise, $(a_{u},c_{u},0)$ are the
coordinates of the AGU $n_{u},1\leq u\leq U$. We further define
$w_{b}~{}=~{}[a_{b},c_{b}]^{T}$ and $w_{u}~{}=~{}[a_{u},~{}c_{u}]^{T}$ to
depict the aforementioned 2-D point projection on the ground plane.

We suppose that the observation and relay UAVs are used for the provisioning
of communication services under the communication range/coverage of a single
base station, therefore, UAVs mobility is not required. That is, only the
optimal placement of both UAVs is considered. Both the observation and relay
UAVs are always located at a set elevation above the ground, symbolized by
$H_{o}$ and $H_{r}$, respectively. Furthermore, D in Fig.1 represents the
distance from the center of the observation area to the GBS. Let the ground
plane coordinates of the observation and relay UAVs be represented by
$q_{o}=[a_{o},c_{o}]^{T}\in R^{(2\times 1)}$ and $q_{r}=[a_{r},c_{r}]^{T}\in
R^{(2\times 1)}$, respectively. Based on these positions, we can
mathematically express the distances between the AGU $n_{u}$ and the
observation UAV $d_{uo}$, between the observation UAV and the relay UAV
$d_{or}$, and between the relay UAV and the GBS $d_{rb}$ as follows

$\displaystyle d_{uo}$ $\displaystyle=$
$\displaystyle\sqrt{(H_{o})^{2}~{}+~{}\parallel~{}q_{o}~{}-~{}w_{u}\parallel^{2}}$
$\displaystyle d_{or}$ $\displaystyle=$
$\displaystyle\sqrt{(H_{r}-H_{o})^{2}+\parallel
q_{r}-q_{o}\parallel^{2}},\qquad\mbox{and}$ $\displaystyle d_{rb}$
$\displaystyle=$ $\displaystyle\sqrt{(H_{b}-H_{r})^{2}+\parallel
w_{b}-q_{r}\parallel^{2}}\;,$

respectively.

Figure 1: System overview: Overall geographical scenario

###  II-B Channel model

The communication link between AGUs and the observation UAV is dominated by
large-scale and small-scale fading [34] whereas the channel between the
observation and relay UAVs and the relay UAV and GBS are dominated by the
free-space path loss (FSPL) model. The fading channel coefficient between the
AGU $n_{u}$ and observation UAV can be modeled as
$h_{uo}=\epsilon_{u}\sqrt{\gamma_{u}}$, where $\epsilon_{u}$ is a complex
random variable that accounts for small-scale fading, whereas the large-scale
channel attenuation between the observation UAV and the AGUs is represented as
${\gamma_{u}}$. Specifically, with $d_{uo}$ been the distance between the
observation UAV and the AGU $n_{u}$,then

$\displaystyle\gamma_{u}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{0}}{(d_{uo})^{2}}$ (1) $\displaystyle=$
$\displaystyle\frac{\alpha_{0}}{\left(\sqrt{(H_{o})^{2}+\parallel
q_{o}-w_{u}\parallel^{2}}\right)^{2}}\;,\quad\forall u$

where $\alpha_{0}$ represent the channel gain at $d_{0}=1m$. The small-scale
fading among the AGUs and observation UAV follows the Rician fading model as
the AGUs and observation UAV are in Line-of-Sight (LoS) [59]. The Rician
factor $K_{C}$ reflects the power ratio between the LoS signal constituent and
the dispersed constituents in the Rician fading channel. Consequently, the CDF
of $|\epsilon_{u}|^{2}$ can be written as [34, 59],

$\displaystyle F(z)$ $\displaystyle\cong$ $\displaystyle
p(|\epsilon_{u}|^{2}<z)$ (2) $\displaystyle=$ $\displaystyle
1-Q_{1}\left(\sqrt{2K_{C}},\sqrt{2(K_{C}+1)z}\right)\;,$

where $Q_{1}(\cdot)$ represent the Marcum-Q function. The Marcum-Q function
with modified Bessel function $I_{0}$ of order 0 represented as $Q_{1}(a,b)$
can be written as

$Q_{1}(a,b)=\int_{b}^{\infty}x~{}e^{\frac{-x^{2}+a^{2}}{2}}~{}I_{0}(ax)~{}dx\;.$
(3)

Let $P_{u}$ be the user’s transmit power and $N_{0}^{2}$ be the noise power
density at the AGU, the channel capacity (b/s/Hz) between the AGUs and
observation UAV can then be defined as

$\displaystyle C_{u}$
$\displaystyle=\log_{2}\left(1+\frac{P_{u}|h_{uo}|^{2}}{N_{o}^{2}}\right)$ (4)
$\displaystyle=\log_{2}\left(1+\frac{P_{u}\gamma_{u}|\epsilon_{u}|^{2}}{N_{o}^{2}}\right)$

For simplicity, in our proposed PSCNs, we allocate equal bandwidth to both the
information received by the observation UAV from the AGUs and its transmission
to the GBS. Furthermore, we adopt FDMA as the communication technique,
distributing the total bandwidth $B$ among all the AGUs. Precisely,
designating $x_{u}$ as the fractional bandwidth allotted to users
$n_{u},x_{u}\geq 0$ and $\sum_{u}^{U}x_{u}\leq 1$. Furthermore, FDMA is chosen
for interference mitigation, easy implementation, and spectrum efficiency
[69]. The normalized attainable rate (b/s/Hz) for the AGUs uplink
communication can be calculated as

$\displaystyle R_{u}$ $\displaystyle=$ $\displaystyle
x_{u}\log_{2}\left(1+\frac{P_{u}|h_{uo}|^{2}}{x_{u}BN_{o}}\right)$ (5)
$\displaystyle=$ $\displaystyle
x_{u}\log_{2}\left(1+\frac{P_{u}\mu_{0}|\epsilon_{u}|^{2}}{x_{u}((H_{o})^{2}+\parallel
q_{o}-w_{u}\parallel^{2})}\right),$

where $\mu_{0}=\frac{\alpha_{0}}{BN_{o}}$. We define the rate outage
probability $P_{u}^{out}$ as the probability that the communication between
the AGU and the observation UAV fails, written as

$\displaystyle p_{u}^{out}$ $\displaystyle\triangleq$ $\displaystyle
P(C_{u}<R_{u})$ (6) $\displaystyle=$ $\displaystyle
P\left(|\epsilon_{u}|^{2}<\frac{2^{R_{u}/x_{u}-1}}{P_{u}\gamma_{u}/N_{o}^{2}}\right)$
$\displaystyle=$ $\displaystyle
F\left(\frac{(2^{R_{u}/x_{u}-1})((H_{o})^{2}+\parallel
q_{o}-w_{u}\parallel^{2})}{P_{u}\mu_{0}}\right)$

Due to the CDF definition in Eq. 2, the last equality holds. Subsequently, we
indicate the outage probability as the rate outage probability as expressed in
Eq. 6. Furthermore, it is assumed that the channels between the observation
UAV and relay UAV, as well as the relay UAV and GBS, are primarily
characterized by LoS propagation. As a result, the channel gains for these two
links can be effectively modeled using the FSPL equation.

$h_{or}=\frac{\alpha_{0}}{(d_{or})^{2}}\;,\quad\mbox{and}\quad
h_{rb}=\frac{\alpha_{0}}{(d_{rb})^{2}}\;,$ (7)

Let $P_{o}$ denote as the transmit power of the observation UAV, then the
normalized attainable rate of the relay UAV (measured in bits per second
hertz, b/s/Hz) can be calculated as

$\displaystyle R_{r}$ $\displaystyle=$
$\displaystyle\log_{2}\left(1+\frac{P_{o}~{}h_{or}}{BN_{o}}\right)$ (8)
$\displaystyle=$
$\displaystyle\log_{2}\left(1+\frac{P_{o}~{}\mu_{0}}{(H_{r}-H_{o})^{2}+\parallel
q_{r}-q_{o}\parallel^{2}}\right)$

Similarly, let $P_{r}$ be the relay UAV transmit power, then the attainable
rate of the ground BS (b/s/Hz) can be written as

$\displaystyle R_{b}$ $\displaystyle=$
$\displaystyle\log_{2}\left(1+\frac{P_{r}~{}h_{rb}}{BN_{o}}\right)$ (9)
$\displaystyle=$
$\displaystyle\log_{2}\left(1+\frac{P_{r}~{}\mu_{0}}{(H_{b}-H_{r})^{2}+\parallel
w_{b}-q_{r}\parallel^{2}}\right)$

###  II-C Video streaming model

For video transmission from the AGUs, we assume the use of adaptive video
streaming. To model the utility of each user, we adopt a simple HTTP video
streaming utility model that solely depends on the video transmission rate.
Specifically, when the GBS receives a higher video rate from an AGU, it
perceives a correspondingly higher video quality for that particular AGU [60].
Moreover, we define the video streaming utility paradigm based on a
logarithmic relationship with the video transmission rate, i.e.,
$S_{u}~{}=~{}\theta\log(\beta R_{u}/\overline{r}_{u})$ as in [61, 62]. The
positive constant factors $\theta$ and $\beta$ vary for different sorts of
applications. $\overline{r}_{u}$ represents the playback rate for AGU $n_{u}$,
which is related to the media outlet’s physical capabilities (e.g., users’
data rate and screen size). Therefore, the AGUs need to use an appropriate
playback rate that matches the video’s frame rate and resolution to ensure
smooth and accurate playback at the BS. If the playback rate is too high or
too low, the video may appear choppy, stutter, or be out of sync with the
audio. The average video streaming utility $\overline{S}$ for all the AGUs can
be written as

$\overline{S}=\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{R_{u}}{\overline{r}_{u}}\right)\;.$
(10)

It is noteworthy to mention that $\overline{S}$ is a positive function that is
concave for $R_{u}$, continuous, and differentiable w.r.t $R_{u}$.

###  II-D Problem formulation

Lets $\mathcal{X}=\\{x_{u},\forall u\\}$,
$\mathcal{P}=\\{P_{u},P_{o},P_{r},\forall u\\}$, and
$\mathcal{Q}=\\{q_{o},q_{r}\\}$.The goal is to maximize the average video
streaming utility for all the AGUs in a UAV-based PSCN by jointly optimizing
the positions of the observation and relay UAVs, along with power allocation
$\mathcal{P}$ and bandwidth distribution $\mathcal{X}$. This optimization is
subject to the constraints of information causalities at the relay UAV and
GBS, as well as the outage probability constraint for the channel between the
observation UAV and AGUs, which should be kept below a specified threshold
$\rho$. The problem is reformulated as

$\mathcal{P}1:\begin{array}[]{c}max\\\
\mathcal{X},\mathcal{P},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{R_{u}(1-\rho)}{\overline{r}_{u}}\right)$

s. t.

$\displaystyle P_{u}^{out}$ $\displaystyle\leq$
$\displaystyle\rho\quad\forall~{}u,$ (11) $\displaystyle R_{r}$
$\displaystyle\geq$ $\displaystyle\sum_{u}^{U}~{}R_{u}$ (12) $\displaystyle
R_{b}$ $\displaystyle\geq$ $\displaystyle R_{r}$ (13)
$\displaystyle\sum_{u}^{U}~{}x_{u}$ $\displaystyle\leq$ $\displaystyle 1$ (14)
$\displaystyle 0\leq$ $\displaystyle x_{u}$ $\displaystyle\leq 1,\qquad\forall
u$ (15) $\displaystyle 0\leq$ $\displaystyle P_{u}\leq$ $\displaystyle
P_{u}^{max}\;,\qquad\forall u$ (16) $\displaystyle 0\leq$ $\displaystyle
P_{o}\leq$ $\displaystyle P_{o}^{max}\;\qquad$ (17) $\displaystyle 0\leq$
$\displaystyle P_{r}\leq$ $\displaystyle P_{r}^{max}\;\qquad$ (18)

In the formulation of the above problem, constraint (12) and (13) are
information causalities constraints through the relay and GBS, thereby the
sums up-link transmission data rate for all the AGUs should not exceed the
relay UAV attainable data rate and the relay UAV transmission data rate should
not exceed the GBS attainable data-rate. Considering the presence of
constraints (12) and (13), it can be observed that the objective function does
not exhibit joint concavity for the variables $\mathcal{P}$, $\mathcal{X}$ and
$\mathcal{Q}$. As a result, problem $\mathcal{P}1$ is classified as a non-
convex problem, posing significant challenges for direct solutions. Therefore,
we use the following approach to solve $\mathcal{P}1$.

##  III Proposed solution

To tackle the challenges posed by the non-convexity of problem $\mathcal{P}1$,
we adopt a strategy of decomposing it into two distinct sub-parts.
Subsequently, we develop an iterative algorithm that utilizes the block
coordinate descent technique. This approach enables us to obtain locally
optimal solutions for each sub-part iteratively. Although block coordinates
descent techniques have been used in other related works [34, 63], the
problems investigated in those works are distinct from our research problem.

By introducing a slack variable
$\tilde{\mathcal{R}}\triangleq\\{\tilde{R_{u}},\forall u\\}$, such that
$\tilde{R_{u}}=R_{u}(1-\rho)$, the problem $\mathcal{P}1$ can be formulated
as:

$\mathcal{P}2:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{X},\mathcal{P},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r}_{u}}\right)$

s. t. (11), (13) - (18)

$\displaystyle R_{u}(1-\rho)$ $\displaystyle\geq$
$\displaystyle\tilde{R_{u}}\qquad\forall u$ (19) $\displaystyle R_{r}$
$\displaystyle\geq$ $\displaystyle\sum_{u}^{U}\tilde{R_{u}}$ (20)

By replacing the inequality in constraint (11) with equality, we get

$\mathcal{P}3:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{X},\mathcal{P},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r}_{u}}\right)$

s. t. (13) - (20)

$P_{u}^{out}=\rho,\quad\forall u$ (21)

_Theorem 1:_ Problem $\mathcal{P}2$ equals problem $\mathcal{P}3$.
_Proof:_ Without losing the optimality of $\mathcal{P}2$, the constraints on
(11) can be changed by the constraints on (21). Moreover, if there is an AGU
$u$ in the optimal solution of $\mathcal{P}2$ for which the constraint (11) is
satisfied, whereas the subsequent constraint (21) is not fulfilled, then, we
can increase $R_{u}$ and the remaining adjustable variables such that
constraint (21) is satisfied without changing the value of the objective, as
$P_{u}^{out}$ does not diminish with $R_{u}$. Consequently, within problem
$\mathcal{P}2$, there exists an optimal solution that ensures the fulfillment
of the constraints stated in (21). Consequently, solving problem
$\mathcal{P}2$ can be regarded as equivalent to solving problem
$\mathcal{P}3$, thus concluding the proof.

To write $R_{u}$ in terms of $F^{-1}(\rho)$, put $P_{u}^{out}=\rho$ in (6) and
rearrange it. Moreover, replacing constraint (21) by $R_{u}$ in
$\mathcal{P}3$, we get the following equivalent problem.

$\mathcal{P}4:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{X},\mathcal{P},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r}_{u}}\right)$

s. t. (14) - (20)

$\displaystyle R_{u}$ $\displaystyle=$ $\displaystyle
x_{u}\log_{2}\left(1+\frac{F^{-1}(\rho)P_{u}\mu_{0}}{x_{u}(H_{o})^{2}+\parallel
q_{o}-w_{u}\parallel^{2}}\right)$ (22) $\displaystyle R_{b}$
$\displaystyle\geq$ $\displaystyle\sum_{u}^{U}\tilde{R_{u}}$ (23)

${F^{-1}(\cdot)}$ denotes the inverse function of $F(\cdot)$.

Note that the Marcum-Q function is difficult to compute, therefore, and as a
result, there is no closed-form expression for calculating $F^{-1}(\rho)$.
However, since $F(\cdot)$ is a non-decreasing function, therefore for
$0\leq\rho\leq 1$ $F^{-1}(\rho)$ is also a non-decreasing function. Given
$\rho$, the value of $F^{-1}(\rho)$ can be calculated using a bisection
numerical approach as used in [64].

Considering the block coordinate descent technique [63], we subsequently
suggest a systematic and effective iterative algorithm for $\mathcal{P}4$. The
primary idea is to split $\mathcal{P}4$ into two sub-categories; firstly power
($\mathcal{P}$) and bandwidth ($\mathcal{X}$) allocation with fixed UAVs
locations and secondly optimizing the UAVs locations ($\mathcal{Q}$) for fixed
power and bandwidth. Then, we optimize the two variable sets, {$\mathcal{X}$,
$\mathcal{P}$} and $\mathcal{Q}=\\{q_{u},q_{r}\\}$ interchangeably through the
block coordinate descent method.

###  III-A Power ($\mathcal{P}$) and bandwidth ($\mathcal{X}$) allocation with
fixed UAVs location

$\mathcal{P}5:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{X},\mathcal{P}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r_{u}}}\right)$

s. t. (14) - (20),(22) - (23) .

Upon observation, it becomes evident that the objective function in terms of
$\tilde{\mathcal{R}_{u}}$ exhibits concavity, while the constraint functions
are all convex in $\mathcal{P}$ and $\mathcal{X}$. Therefore, $\mathcal{P}5$
is considered a standard convex optimization problem that can be effectively
solved using various standard convex optimization techniques or through CVX
[65].

###  III-B Observation and relay UAV position ($\mathcal{Q}$) optimization
with fixed power and bandwidth

For a fixed allocated bandwidth $\mathcal{X}$ and transmit power
$\mathcal{P}$, the next challenge is to optimize the observation and relay
UAVs positions i.e., $\mathcal{Q}=\\{q_{o},q_{r}\\}$, elaborated as

$\mathcal{P}6:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r_{u}}}\right)$

s. t. (19), (22), and (23)

Problem $\mathcal{P}6$ can also be written by expanding the constraint
equations as follows,

$\mathcal{P}6:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r_{u}}}\right)$

s. t.

$x_{u}\log_{2}\left(1+\frac{\mu_{u}}{x_{u}(H_{o})^{2}+\parallel
q_{o}-w_{u}\parallel^{2}}\right)(1-\rho)\geq\tilde{R_{u}}$ (24)
$\log_{2}\left(1+\frac{\mu_{ob}}{(H_{r}-H_{o})^{2}+\parallel
q_{r}-q_{o}\parallel^{2}}\right)\geq\sum_{u}^{U}\tilde{R_{u}}$ (25)
$\log_{2}\left(1+\frac{\mu_{r}}{(H_{b}-H_{r})^{2}+\parallel
w_{b}-q_{r}\parallel^{2}}\right)\geq\sum_{u}^{U}\tilde{R_{u}}\;,$ (26)

where $\mu_{u}=P_{u}\mu_{0}\textfractionsolidus x_{u}$,
$\mu_{ob}=P_{o}\mu_{0}$, and $\mu_{r}=P_{r}\mu_{0}$ are constants with given
$\mathcal{P}$ and $\mathcal{X}$. The problem $\mathcal{P}6$ is non-convex,
however, successive convex approximation of constraints (24), (25), and (26)
through first-order Taylor approximation based iterative approach can make the
problem tractable. That is the constraints (24), (25), and (26) at a given
local point $\mathcal{Q}^{i}$ (where
$\mathcal{Q}^{i}=\\{q_{o}^{i},q_{r}^{i}\\}$, at the $i$th iteration) are
replaced by their convex approximations which make the overall problem convex
and is solved. This whole process is continued iteratively until the algorithm
converges. Therefore, we can utilize the first-order Taylor approximation to
approximate the logarithm function of (24), (25), and (26) and considering the
lower bounds, we have

$\displaystyle x_{u}\log_{2}\left(1+\frac{\mu_{u}}{x_{u}(H_{o})^{2}+\parallel
q_{o}-w_{u}\parallel^{2}}\right)\geq R_{u}^{LB}$ $\displaystyle
R_{u}^{LB}\triangleq x_{u}\left(c_{u}^{i}-D_{u}^{i}\right)\left(\parallel
q_{o}-w_{u}\parallel^{2}-\parallel q_{o}^{i}-w_{u}\parallel^{2}\right)\;,$
(27)

where

$\displaystyle D_{u}^{i}$ $\displaystyle=$
$\displaystyle\frac{\mu_{u}\log_{2}e}{\left(H_{o}^{2}+\parallel
q_{o}^{i}-w_{u}\parallel^{2}\right)\left(H_{o}^{2}+\parallel
q_{o}^{i}-w_{u}\parallel^{2}+\mu_{u}\right)}$ $\displaystyle C_{u}^{i}$
$\displaystyle=$
$\displaystyle\log_{2}\left(1+\frac{\mu_{u}}{H_{o}^{2}+\parallel
q_{o}^{i}-w_{u}\parallel^{2}}\right)\;.$ (28)

For (25)

$R_{o}=\log_{2}\left(1+\frac{\mu_{ob}}{(H_{r}-H_{o})^{2}+\parallel
q_{r}-q_{o}\parallel^{2}}\right)\geq R_{o}^{LB}$ (29)
$R_{o}^{LB}\triangleq\left(C_{o}^{i}-D_{o}^{i}\right)\left(\parallel
q_{r}-q_{o}\parallel^{2}-\parallel q_{r}^{i}-q_{o}^{i}\parallel^{2}\right)\;,$
(30)

where

$\displaystyle D_{o}^{i}=\frac{\mu_{ob}\log_{2}e}{((H_{r}-H_{o})^{2}+\parallel
q_{r}^{i}-q_{o}^{i}\parallel^{2})((H_{r}-H_{o})^{2}+\parallel
q_{r}^{i}-q_{o}^{i}\parallel^{2}+\mu_{ob})}\;,$
$C_{o}^{i}=\log_{2}\left(1+\frac{\mu_{ob}}{(H_{r}-H_{o})^{2}+\parallel
q_{r}^{i}-q_{o}^{i}\parallel^{2}}\right)\;.$ (31)

For (26)

$R_{r}=\log_{2}\left(1+\frac{\mu_{r}}{(H_{b}-H_{r})^{2}+\parallel
w_{b}-q_{r}\parallel^{2}}\right)\geq R_{r}^{LB}$ (32)
$R_{r}^{LB}\triangleq\left(c_{r}^{i}-D_{r}^{i}\right)\left(\parallel
q_{r}-q_{b}\parallel^{2}-\parallel q_{r}^{i}-w_{b}^{i}\parallel^{2}\right)\;,$
(33)

where

$\displaystyle D_{r}^{i}=\frac{\mu_{r}\log_{2}e}{((H_{b}-H_{r})^{2}+\parallel
q_{r}^{i}-w_{b}^{i}\parallel^{2})((H_{b}-H_{r})^{2}+\parallel
q_{r}^{i}-w_{b}^{i}\parallel^{2}+\mu_{r})}\;,$
$C_{r}^{i}=\log_{2}\left(1+\frac{\mu_{r}}{(H_{b}-H_{r})^{2}+\parallel
q_{o}^{i}-w_{b}^{i}\parallel^{2}}\right)\;.$ (34)

For the given $\mathcal{Q}^{i}=\\{q_{o}^{i},q_{r}^{i}\\}$, replacing the lower
bounds by $R_{u}^{LB}$, $\forall u$, $R_{o}^{LB}$ and $R_{r}^{LB}$, in (27),
(30), (33), $\mathcal{P}6$ is approximated as

$\mathcal{P}7:\begin{array}[]{c}\mbox{max}\\\
\tilde{\mathcal{R}},\mathcal{Q}\end{array}\frac{\theta}{U}\sum_{u}^{U}\log\left(\beta\frac{\tilde{R_{u}}}{\overline{r_{u}}}\right)$

s. t.

$\displaystyle R_{u}^{LB}$ $\displaystyle\geq$
$\displaystyle\tilde{R_{u}}\;,\qquad\forall u$ (35) $\displaystyle R_{o}^{LB}$
$\displaystyle\geq$ $\displaystyle\sum_{u}^{U}\tilde{R_{u}}\;,$ (36)
$\displaystyle R_{r}^{LB}$ $\displaystyle\geq$
$\displaystyle\sum_{u}^{U}\tilde{R_{u}}\;.$ (37)

Constraints (35), (36), and (37) are all concave with respect to $\mathcal{Q}$
and $\tilde{R_{u}}$ jointly, and therefore problem $\mathcal{P}7$ is a convex
optimization problem. Standard convex optimization methods and solvers like
CVX can be used to solve this problem within a polynomial-time complexity. It
is important to note that the lower bounds specified in Equations (27), (30),
and (33), can be easily validated. If the constraints (35), (36), and (37) in
$\mathcal{P}7$ are satisfied, it can be inferred that the constraints defined
in Equations (24), (25), and (26) in $\mathcal{P}6$ are also satisfied.
However, the reverse is not necessarily true. Consequently, the feasible
solution space of $\mathcal{P}7$ is a subspace of that of $\mathcal{P}6$,
making the optimum solution of $\mathcal{P}7$ a lower bound to that of
$\mathcal{P}6$. To solve the fundamental non-convex problem $\mathcal{P}6$, we
adopt an iterative approach that involves optimizing $\mathcal{P}7$
iteratively using the specified local point $\mathcal{Q}^{i}$ at each
iteration.

Drawing from the aforementioned outcomes, by utilizing the block coordinate
descent method, we introduce an iterative algorithm to address problem
$\mathcal{P}4$. This algorithm involves partitioning the optimization
variables from the original problem $\mathcal{P}4$ into two blocks, namely
$\mathcal{Q}=\\{q_{o},q_{r}\\}$ and {$\mathcal{X}$, $\mathcal{P}$}.
Subsequently, we iteratively optimize the transmit power and bandwidth
allotment {$\mathcal{X}$, $\mathcal{P}$} as well as the UAV (observation and
relay) locations $\mathcal{Q}$. In each iteration, we solve either problem
$\mathcal{P}5$ or $\mathcal{P}7$ accordingly, and the resulting solution is
then utilized as the input for the subsequent iteration. The specific steps of
this algorithm are summarized in the PSCNs Algorithm 1 below.

Algorithm 1 PSCN Iterative Algorithm for $\mathcal{P}4$

1:Initialize: $P_{u}^{\text{max}}$, $P_{o}^{\text{max}}$,
$P_{r}^{\text{max}}$, $\mathcal{Q}^{o}$, $\rho$, $P_{u}^{o}$, $P_{o}^{o}$, and
$P_{r}^{o}$

2:Let $i=1$

3:Use bisection numerical technique to obtain $F^{-1}(\rho)$

4:repeat

5: Solve problem $\mathcal{P}5$ for given $\mathcal{Q}^{o}$ to get $x^{i+1}$,
$P^{i+1}$

6: Resolve problem $\mathcal{P}7$ for the obtained $x^{i+1}$ and $P^{i+1}$ to
get $\mathcal{Q}^{i+1}$

7: $i=i+1$

8:until convergence

In PSCNs Algorithm 1, it is important to note that during each iteration,
problem $\mathcal{P}7$ is solved optimally using the given local point
$\mathcal{Q}^{i}$. It is observed that the objective value of $\mathcal{P}7$
does not decrease as the iterations progress. Moreover, the objective value is
always bounded above by a finite value. As a result, we can establish the
guarantee of convergence for PSCN Algorithm 1. Furthermore, both
$\mathcal{P}5$ and $\mathcal{P}7$ are convex optimization problems, and
Algorithm 1 exhibits polynomial complexity, as elucidated below. In line 5 of
the algorithm, the resolution of $\mathcal{P}5$ entails solving a convex
problem with $3U+2$ variables and $4U+5$ constraints, where U represents the
number of users. The accuracy of this solution is guaranteed, and its
computational complexity is polynomial, specifically
$O\left(\sqrt{3U+2}(4U+5)\log\frac{1}{\mu}\right)$ where $\mu>0$ is the
accuracy tolerance of the solution [66]. In line 6, $\mathcal{P}7$ is
addressed, which is another convex problem characterized by $U+4$ variables
and $U+2$ constraints, with a computational complexity of
$O\left(\sqrt{U+4}(U+2)\log\frac{1}{\mu}\right)$ [66]. When considering the
overall complexity of Algorithm I, it can be expressed as
$O\left(\sqrt{3U+2}(4U+5)\log\frac{1}{\mu}+\sqrt{U+4}(U+2)\log\frac{1}{\mu}\right)$.
Disregarding constant terms, the complexity in terms of the number of users,
number of constraints, and the accuracy tolerance simplifies to approximately
$O(U^{3/2}\log\frac{1}{\mu})$.

##  IV Simulation Results

This section contains the performance evaluation of the presented solution via
extensive simulations using Matlab R2022a. Moreover, we consider a Rician
fading channel between the AGUs and the observation UAV channel, whereas, the
channel between the observation UAV and the relay UAV, as well as between the
relay UAV and the GBS, are dominated by the LoS links. Furthermore, a
$500\times 500~{}m^{2}$ sized geographic area is considered with the uniform
deployment of the AGUs. The parameters used in video streaming utility
function are $\beta=100$, $\overline{r}_{u}$ = 1 Mbps and $\theta=0.8$ for
every AGU $n_{u}$ as mentioned in [35]. The simulation parameters are
summarized in Table II, outlining the various parameters employed in the
simulations. The GBS is positioned at $w_{b}=[-2500,~{}0]^{T}$, and the
observation and relay UAVs maximum transmission power is taken as $0.1$ Watt.

TABLE II: Parameters notation and definition Variable | Definition | Used
---|---|---
Name | Description | Value
$H_{o}$ | Observation UAV height (m) | 100 m
$H_{r}$ | Relay UAV height (m) | 100 m
$H_{b}$ | Ground BS height (m) | 20 m
$K_{c}$ | Rician fading factor | 4
$\rho$ | Target outage probability threshold | $10^{-2}$
| Maximum allowed transmission power for |
$P_{o}^{max}$ | observation UAV (Watts) | 0.1Watt
| Maximum allowed transmission power for |
$P_{r}^{max}$ | relay UAV (Watts) | 0.1 Watt
| Maximum transmit power of the |
$P_{u}^{max}$ | ground users (Watts) | 0.2 Watt
$B$ | Signal bandwidth (Hz) | 1MHz
$N_{0}$ | Noise power spectral density |  $-170$ dBm / Hz
$\alpha_{0}$ | Channel power gain at 1m reference distance | -60 dB
Figure 2: Proposed iterative algorithm-1 convergence curve

Primarily, we consider the convergence of the presented technique. In the
suggested methodology, the observation UAV and relay UAV initial locations are
considered as $q_{o}$ and $q_{r}$, respectively. Fig. 2 shows the convergence
of PSCNs Algorithm with $U=30$ and $P_{u}^{max}=P_{o}^{max}=P_{r}^{max}=0.1$
Watt. More specifically, we compare the lower bound value of the objective
function obtained through the PSCNs Algorithm 1, with the actual value of the
average streaming video utility computed using ($\mathcal{P}1$). Fig. 2
clearly shows that the two curves are overlapping, which clarifies the fact
that the lower bound of the video streaming utility obtained using PSCNs
Algorithm 1 is tight. In addition, Fig. 2 also depicts that the presented
technique converges fast, which validates the efficiency of the proposed
technique.

Figure 3: Average streaming utility vs the number of AGUs

The performance metric employed in this study is the average video streaming
utility. To assess the efficacy of our proposed solution, we conduct a
comparison with four benchmark schemes: (1) a benchmark scheme involving
resource optimization, (2) a benchmark scheme focusing on UAV position
optimization, (3) the average streaming utility with relay UAVs, and (4) the
average streaming utility without relay UAVs. In the benchmark with resource
optimization, we maintain the UAV position as the initial position in
Algorithm 1, and subsequently optimize the allocation of transmission
bandwidth and transmit power through iterative solutions to problem
$\mathcal{P}5$.

In the benchmark with UAVs position optimization for fixed power and bandwidth
allocation, we iteratively solve problem $\mathcal{P}7$ to obtain an optimal
solution. These benchmark schemes serve as reference points for comparison,
allowing us to assess the performance of the presented solution. In the
calculation of average streaming utility, we consider the approach as proposed
in [34] and [68] respectively, with and without a relay UAV. Fig. 3
illustrates the effect of the AGUs count on the performance for the above-
mentioned techniques with the maximum power of observation UAV, relay UAV, and
AGUs, i.e., $P_{o}^{max}=P_{r}^{max}=0.1W$ and $P_{u}^{max}=0.2W$. The result
reveals a decrease in the average streaming utility as the number of AGUs
increases, which is expected as more AGUs will compete for the limited
communication resources, especially when U is large. However, our presented
solution outperforms the other schemes as the performance is improving with an
increase in the number of AGUs, as shown in Fig. 3.

Figure 4: Average Steaming Utility vs $P_{u}^{max}$, $P_{o}^{max}$ and
$P_{r}^{max}$

In Fig. 4, the average video streaming utility is compared while varying the
maximum transmit powers $P_{u}^{max}$, $P_{o}^{max}$ and $P_{r}^{max}$ of the
AGUs, observation, and relay UAVs, respectively, with a total of $U=10$ users.
As anticipated, an increase in $P_{u}^{max}$, $P_{o}^{max}$, and $P_{r}^{max}$
results in an improved average video streaming utility for all four techniques
considered. This is due to the higher transmission power, which allows
increased transmission rates for each AGU, resulting in improved video
streaming performance. Moreover, as obvious from Fig. 4, our proposed solution
still has improved performance in terms of average video streaming utility
among all. The saturation of the average streaming utility with higher
$P_{u}^{max}$, $P_{o}^{max}$ and $P_{r}^{max}$ is also observed. This is owing
to the streaming utility function definition taking into account the value of
transmission rate that provides a validation for the perception that user
utility saturation occurs with a rise in the rate of transmission.

Figure 5: Average streaming utility vs $\rho$

Fig. 5 illustrates that the average streaming utility experiences a rise
followed by a fall as the target outage probability threshold $\rho$
increases. This trend is particularly noticeable when the value of $\rho$ is
small. An increase in $\rho$ leads to an increase in the transmission rate,
resulting in an improvement in the streaming utility. However, as the outage
probability threshold becomes large due to the effects of fading channels, the
quality of experience (QoE) suffers, leading to video play-out stalling. This
trade-off between streaming utility and QoE is clearly illustrated in Fig. 5.

Figure 6: Average streaming utility vs D

Fig. 6 illustrates the benefits of optimal relay placement by varying the
network size (D) from 1500m to 3200m while keeping the observation area radius
fixed at 500m and the maximum power for both observation and relay at 0.1W. As
the network size increases, the observation area moves further away from the
GBS. However, it is clear from Fig. 6 that our solution outperforms the
remaining solutions.

Figure 7: Proposed method with different video scenarios and controlling
constant parameters $\theta$, $\beta$

Fig. 7 evaluates the utility of video streaming in various video scenarios to
better understand the performance of our proposed method in real-world video
scenarios. Each of these videos has different QoE parameters [67].
Furthermore, we evaluate the effectiveness of our proposed approach across
these video scenarios by employing various comparative criteria, providing
clear evidence of its superiority over competing methods.

##  V Conclusion

In this work, we focused on the joint optimization of observation and relay
UAV positions, as well as bandwidth and transmit power allocation in UAV-
assisted PSCNs, specifically for uplink video stream transmission. We proposed
a unified design approach to maximize the average streaming quality for all
users, taking into account the impact of fading channels in PSCNs. By
leveraging the block coordinate descent and successive convex approximation
techniques, we developed an efficient iterative algorithm and analyzed its
convergence properties. Our simulation results demonstrated that the proposed
approach can significantly enhance the maximum average video streaming quality
in PSCNs.

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|  Naveed Khan received his Bachelor’s degree in Telecommunication engineering
from University of engineering and technology (UET), Peshawar and Master’s
degree in Electronics Engineering from International Islamic University
Islamabad (IIUI) in 2009 and 2013, respectively. He is a PhD student at
COMSATS University, Islamabad, Wah Campus, Pakistan. He is currently an
Assistant Professor in the Department of Electrical Engineering at Abasyn
University Peshawar, Pakistan. His research interests include unmanned aerial
vehicle communications, image processing, and multimedia communications.
---|---
|  Ayaz Ahmad (S’08, M’15, SM’16) is currently an Associate Professor with the
Department of Electrical and Computer Engineering, COMSATS University
Islamabad – Wah Campus, Pakistan. Prior to that, he was an Assistant Professor
at the same university. He received a B.Sc. degree in electrical engineering
from the University of Engineering and Technology, Peshawar, Pakistan, in
2006, and the M.S. and Ph.D. degrees in Wireless Communication from Ecole
Superieure d’Electricite (Supelec), Gif-sur-Yvette, France, in 2008 and 2011,
respectively. From 2006 to 2007, he was a Faculty Member with the Department
of Electrical Engineering, FAST-NUCES, Peshawar, Pakistan. He is the recipient
of the best research paper award from the Higher Education Commission,
Pakistan for the years 2015-2016, the National Research Productivity Award
from Pakistan Council of Science and Technology (PSCT) in 2017, the best
research paper award in IEEE IEMCON 2018 held in Canada, and Publon Top Peer
Reviewer award in 2019. He has several years of research experience and has
authored or co-authored several scientific publications in various refereed
international journals and conferences. He has also authored or co-authored
several book chapters and is the leading Co-Editor of the book Smart Grid as a
Solution for Renewable and Efficient Energy published in 2016. He is Associate
Editor with IEEE Access, Human-centric Computing and Information Sciences,
PeerJ Computer Science, and Frontiers in Smart Grids. He regularly serves as a
TPC member for several international conferences, including IEEE GLOBECOM,
IEEE ICC, and IEEE PIMRC, and as a reviewer for several renowned international
journals. He is a Senior Member IEEE, and Associate Fellow of Advance HE, UK.
He is a member of the IEEE Communication Society. His research interests
include resource allocation in wireless communication systems, energy
management in smart grids, and the application of optimization methods to
engineering problems.
---|---
|  Abdul Wakeel received his Ph.D. from Jacobs University Bremen, Bremen,
Germany, in 2016. He joined the National University of Sciences and Technology
(NUST) in 2016 as an Assistant Professor. His research areas are PAPR
reduction for OFDM-based SISO/ MIMO systems, Error Control Coding, Signal
Processing, Active noise control, Physical layer security, and wireless
communications.
---|---
|  Zeeshan Kaleem (Senior Member, IEEE) is serving as an Associate Professor
in the Electrical and Computer Engineering Department, COMSATS University
Islamabad, Wah Campus. He received his BS from UET Peshawar in 2007. He
received MS and Ph.D. in Electronics Engineering from South Korea in 2010 and
2016, respectively. Dr. Zeeshan consecutively received the National Research
Productivity Award (RPA) awards from the Pakistan Council of Science and
Technology (PSCT) in 2017 and 2018. We won the Runner-up Award in the National
Hackathon 23 competition for Project to develop a drone detection system. He
also won the Higher Education Commission (HEC) Best Innovator Award in 2017.
He is the recipient of the 2021 Top Reviewer Recognition Award for IEEE
TRANSACTIONS on VEHICULAR TECHNOLOGY. He has published around 70 technical
Journal papers, Book Chapters, and conference papers in reputable
journals/venues and holds 21 US and Korean Patents, and got several research
grants. He is a co-recipient of the best research proposal award from SK
Telecom, Korea. He is currently serving as Technical Editor of several
prestigious Journals/Magazines like IEEE Transactions on Vehicular Technology,
Elsevier Computer and Electrical Engineering, Springer Human-centric Computing
and Information Sciences, and Journal of Information Processing Systems,
Frontiers in Communications and Networks. He has served/serving as Guest
Editor for special issues in IEEE Wireless Communications, IEEE Communications
Magazine, IEEE Access, Sensors, IEEE/KICS Journal of Communications and
Networks, and Physical Communications, and also served as TPC Member for IEEE
Globecom, IEEE VTC, IEEE ICC, and IEEE PIMRC.
---|---
|  Bushra Rashid is currently serving as an Assistant Professor in the
biomedical engineering department, King Faisal University, KSA. She did B.S.
in Electrical Engineering from Wah Engineering College, Wah Cantt, Pakistan in
2013 and M.S. and Ph.D. in Electrical Engineering from Comsats University,
Islamabad, Wah Campus, Pakistan on topics related to multimedia, signal
processing, data security, and AI.
---|---
|  Waqas Khalid received a B.S. degree in Electronics Engineering from GIK
Institute of Engineering Sciences and Technology, KPK, Pakistan, in 2011. He
received M.S. and Ph.D. degrees in Information and Communication Engineering
from Inha University, Incheon, South Korea, and Yeungnam University,
Gyeongsan, South Korea, in 2016 and 2019, respectively. He is currently
working as a research professor at the Institute of Industrial Technology,
Korea University, Sejong, South Korea. His areas of interest include physical
layer modeling, signal processing for wireless communications, and emerging
solutions and technologies for 5G/6G networks such as reconfigurable
intelligent surfaces, energy harvesting, physical layer security, NOMA,
cognitive radio, UAVs, and IoTs.
---|---


& \leq \max \{\decwidth(T_1), \decwidth(T_2), \rank(L \mid R)\}\\
& \codefn \decwidth(T)
\qedhere
\end{align*}

In order to obtain the subgraphs of the desired shape we need to add some extra connections to the boundaries.
This can be done thanks to <Ref>, by taking \(M = \idmat{}\).
We are finally able to prove the lower bound for monoidal width.

Let \(g = \fullboundariesgraph{G}{L}{R}{P}{F}\) in \(\propGraph\) and \(d \in \decset{g}\).
Let \(\Gamma = \danglinggraph{G}{(L \mid R)}\).
Then, there is an inductive rank decomposition \(\mTordec(d)\) of \(\Gamma\) s.t.
\(\decwidth(\mTordec(d)) \leq 2 \cdot \max \{\decwidth(d), \rank(L), \rank(R)\}\).

Proceed by induction on the decomposition tree \(d\).
If it is just a leaf with label \(g\), then its width is defined to be the number \(k\) of vertices of \(g\), \(\decwidth(d) \defn k\).
Pick any inductive rank decomposition of \(\Gamma\) and define \(\mTordec(d) \defn T\).
Surely, \(\decwidth(T) \leq k \codefn \decwidth(d)\)

If \(d = \nodegenerator{d_1}{\dcomp_j}{d_2}\), then \(g\) is the composition of two morphisms: \(g = g_1 \dcomp g_2\), with \(g_i = \fullboundariesgraph{G_i}{L_i}{R_i}{P_i}{F_i}\).
Given the partition of the vertices determined by \(g_1\) and \(g_2\), we can decompose \(g\) in another way, by writing \(\adjeqclass{G} = \adjeqclass{\begin{psmallmatrix} \overline{G}_1 & C \\ \zeromat & \overline{G}_2\end{psmallmatrix}}\) and \(B = (L \mid R) = \begin{psmallmatrix} \overline{L}_1 & \overline{R}_1 \\ \overline{L}_2 & \overline{R}_2 \end{psmallmatrix}\).
Then, we have that \(\overline{G}_1 = G_1\), \(\overline{L}_1 = L_1\), \(P = P_2 \cdot P_1\), \(C = R_1 \cdot \transpose{L}_2\), \(\overline{R}_1 = R_1 \cdot \transpose{P}_2\), \(\overline{L}_2 = L_2 \cdot P_1\), \(\overline{R}_2 = R_2 + L_2 \cdot (F_1 + \transpose{F}_1) \cdot \transpose{P}_2\), \(\overline{G}_2 = G_2 + L_2 \cdot F_1 \cdot \transpose{L}_2\), and \(F = F_2 + P_2 \cdot F_1 \cdot \transpose{P}_2\).
This corresponds to the following diagrammatic rewriting using the equations of \(\propGraph\).

\[\mwdrankwidthlowerproofFigFuture{}\]

We define \(\overline{B}_1 \defn (\overline{L}_1 \mid \overline{R}_1 \mid C)\) and \(\overline{B}_2 \defn (\overline{L}_2 \mid \overline{R}_2 \mid \transpose{C})\).
In order to build an inductive rank decomposition of \(\Gamma\), we need rank decompositions of \(\overline{\Gamma}_i = \danglinggraph{\overline{G}_i}{\overline{B}_i}\).
We obtain these in three steps.
Firstly, we apply induction to obtain inductive rank decompositions \(\mTordec(d_i)\) of \(\Gamma_i = \danglinggraph{G_i}{(L_i \mid R_i)}\) such that \(\decwidth(\mTordec(d_i)) \leq 2 \cdot \max \{\decwidth(d_i), \rank(L_i), \rank(R_i)\}\).
Secondly, we apply <Ref> to obtain an inductive rank decomposition \(T'_2\) of \(\Gamma'_2 = \danglinggraph{G_2 + L_2 \cdot F_1 \cdot \transpose{L_2}}{(L_2 \mid R_2 + L_2 \cdot (F_1 + \transpose{F_1}) \cdot \transpose{P_2})}\) such that \(\decwidth(T'_2) \leq \decwidth(\mTordec(d_2))\).
Lastly, we observe that \((\overline{R}_1 \mid C) = R_1 \cdot (\transpose{P_2} \mid \transpose{L_2})\) and \((\overline{L}_2 \mid \transpose{C}) = L_2 \cdot (P_1 \mid \transpose{R_1})\).
Then we obtain that \(\overline{B}_1 = (L_1 \mid R_1) \cdot \begin{psmallmatrix} \id{n} & \zeromat & \zeromat\\ \zeromat &  \transpose{P_2} & \transpose{L_2} \end{psmallmatrix}\) and \(\overline{B}_2 = (L_2 \mid R_2 + L_2 \cdot (F_1 + \transpose{F_1}) \cdot \transpose{P_2}) \cdot \begin{psmallmatrix} P_1 & \zeromat & \transpose{R}_1 \\ \zeromat & \id{m} & \zeromat \end{psmallmatrix}\), and we can apply <Ref>, with \(M = \idmat{}\), to get inductive rank decompositions \(T_i\) of \(\overline{\Gamma}_i\) such that \(\decwidth(T_1) \leq \decwidth(\mTordec(d_1))\) and \(\decwidth(T_2) \leq \decwidth(T'_2) \leq \decwidth(\mTordec(d_2))\).
If \(k_1, k_2 > 0\), then we define \(\mTordec(d) \defn \nodegenerator{T_1}{\Gamma}{T_2}\), which is an inductive rank decomposition of \(\Gamma\) because \(\overline{\Gamma}_i\) satisfy the two conditions in <Ref>.
If \(k_1 = 0\), then \(\Gamma = \overline{\Gamma}_2\) and we can define \(\mTordec(d) \defn T_2\).
Similarly, if \(k_2 = 0\), then \(\Gamma = \overline{\Gamma}_1\) and we can define \(\mTordec(d) \defn T_1\).
In any case, we can compute the width of \(\mTordec(d)\) (if \(k_i = 0\) then \(T_i = \emptydec\) and \(\decwidth(T_i) = 0\)) using the inductive hypothesis, <Ref>, <Ref>, the fact that \(\rank(L) \geq \rank(L_1)\), \(\rank(R) \geq \rank(R_2)\) and \(j \geq \rank(R_1), \rank(L_2)\) because \(R_1 \colon j \to k_1\) and \(L_2 \colon j \to k_2\).

\begin{align*}
& \decwidth(T) \\
& \defn \max \{\decwidth(T_1), \decwidth(T_2), \rank(L \mid R)\}\\
& \leq \max \{\decwidth(\mTordec(d_1)), \decwidth(T'_2), \rank(L \mid R)\}\\
& \leq \max \{\decwidth(\mTordec(d_1)), \decwidth(\mTordec(d_2)), \rank(L \mid R)\}\\
& \leq \max \{\decwidth(\mTordec(d_1)), \decwidth(\mTordec(d_2)), \rank(L) + \rank(R)\}\\
& \leq \max \{2 \cdot \decwidth(d_1), 2 \cdot \rank(L_1), 2 \cdot \rank(R_1), 2 \cdot \decwidth(d_2), 2 \cdot \rank(L_2), 2 \cdot \rank(R_2), \rank(L) + \rank(R)\}\\
& \leq 2 \cdot \max \{\decwidth(d_1), \rank(L_1), \rank(R_1), \decwidth(d_2), \rank(L_2), \rank(R_2), \rank(L), \rank(R)\}\\
& \leq 2 \cdot \max \{\decwidth(d_1), \decwidth(d_2), j, \rank(L), \rank(R)\}\\
& \codefn 2 \cdot \max \{\decwidth(d), \rank(L), \rank(R)\}
\end{align*}

If \(d = \nodegenerator{d_1}{\tensor}{d_2}\), then \(g\) is the monoidal product of two morphisms: \(g = g_1 \tensor g_2\), with \(g_i = \fullboundariesgraph{G_i}{L_i}{R_i}{P_i}{F_i} \colon n_i \to m_i\).
By exlicitly computing the monoidal product, we obtain that \(\adjeqclass{G} = \adjeqclass{\begin{psmallmatrix} G_1 & \zeromat \\ \zeromat & G_2 \end{psmallmatrix}}\), \(L = \begin{psmallmatrix} L_1 & \zeromat \\ \zeromat & L_2 \end{psmallmatrix}\), \(R = \begin{psmallmatrix} R_1 & \zeromat \\ \zeromat & R_2 \end{psmallmatrix}\), \(P = \begin{psmallmatrix} P_1 & \zeromat \\ \zeromat & P_2 \end{psmallmatrix}\) and \(F = \begin{psmallmatrix} F_1 & \zeromat \\ \zeromat & F_2 \end{psmallmatrix}\).
By induction, we have inductive rank decompositions \(\mTordec(d_i)\) of \(\Gamma_i \defn \danglinggraph{G_i}{B_i}\), where \(B_i = (L_i \mid R_i)\), of bounded width: \(\decwidth(\mTordec(d_i)) \leq 2 \cdot \max \{\decwidth(d_i), \rank(L_i), \rank(R_i)\}\).
Let \(\overline{B}_1 \defn (L_1 \mid \zeromat_{n_2} \mid R_1 \mid \zeromat_{m_2} \mid \zeromat_{k_2}) = B_1 \cdot \begin{psmallmatrix} \id{n_1} & \zeromat & \zeromat & \zeromat & \zeromat \\ \zeromat & \zeromat & \id{m_1} & \zeromat & \zeromat \end{psmallmatrix}\) and  \(\overline{B}_2 \defn (\zeromat_{n_1} \mid L_2 \mid \zeromat_{m_1} \mid R_2 \mid \zeromat_{k_1}) = B_2 \cdot \begin{psmallmatrix} \zeromat & \id{n_2} & \zeromat & \zeromat & \zeromat \\ \zeromat & \zeromat & \zeromat & \id{m_2} & \zeromat \end{psmallmatrix}\).
By taking \(M = \idmat{}\) in <Ref>, we can obtain inductive rank decompositions \(T_i\) of \(\overline{\Gamma}_i \defn \danglinggraph{G_i}{\overline{B}_i}\) such that \(\decwidth(T_i) \leq \decwidth(\mTordec(d_i))\).
If \(k_1, k_2 > 0\), then we define \(\mTordec(d) \defn \nodegenerator{T_1}{\Gamma}{T_2}\), which is an inductive rank decomposition of \(\Gamma\) because \(\overline{\Gamma}_i\) satisfy the two conditions in <Ref>.
If \(k_1 = 0\), then \(\Gamma = \overline{\Gamma}_2\) and we can define \(\mTordec(d) \defn T_2\).
Similarly, if \(k_2 = 0\), then \(\Gamma = \overline{\Gamma}_1\) and we can define \(\mTordec(d) \defn T_1\).
In any case, we can compute the width of \(\mTordec(d)\) (if \(k_i = 0\) then \(T_i = \emptydec\) and \(\decwidth(T_i) = 0\)) using the inductive hypothesis and <Ref>.

\begin{align*}
& \decwidth(T) \\
& \defn \max \{\decwidth(T_1), \decwidth(T_2), \rank(L \mid R)\}\\
& \leq \max \{\decwidth(\mTordec(d_1)), \decwidth(\mTordec(d_2)), \rank(L \mid R)\}\\
& \leq \max \{\decwidth(\mTordec(d_1)), \decwidth(\mTordec(d_2)), \rank(L) + \rank(R)\}\\
& \leq \max \{2 \cdot \decwidth(d_1), 2 \cdot \rank(L_1), 2 \cdot \rank(R_1), 2 \cdot \decwidth(d_2), 2 \cdot \rank(L_2), 2 \cdot \rank(R_2), \rank(L) + \rank(R)\}\\
& \leq 2 \cdot \max \{\decwidth(d_1), \rank(L_1), \rank(R_1), \decwidth(d_2), \rank(L_2), \rank(R_2), \rank(L), \rank(R)\}\\
& \leq 2 \cdot \max \{\decwidth(d_1), \decwidth(d_2), \rank(L), \rank(R)\}\\
& \codefn 2 \cdot \max \{\decwidth(d), \rank(L), \rank(R)\}
\qedhere
\end{align*}

From <Ref>, <Ref> and <Ref>, we obtain the main result of this section.

Let \(G\) be a graph and let \(g = \fullboundariesgraph{G}{\initmap{}}{\initmap{}}{\emptymat}{\emptymat}\) be the corresponding morphism in \(\propGraph\).
Then, \(\frac{1}{2} \cdot \rankwidth(G) \leq \mwd(g) \leq 2 \cdot \rankwidth(G)\).

§ CONCLUSION AND FUTURE WORK

We defined monoidal width for measuring the complexity of morphisms in monoidal categories.
The concrete examples that we aimed to capture are tree width and rank width.
In fact, we have shown that, by choosing suitable categorical algebras, monoidal width is equivalent to these widths.
We have also related monoidal width to the rank of matrices over the natural numbers.

Our future goal is to leverage the generality of monoidal categories to study other examples outside the graph theory literature.
In the same way Courcelle's theorem gives fixed-parameter tractability of a class of problems on graphs with parameter tree width or rank width, we aim to obtain fixed-parameter tractability of a class of problems on morphisms of monoidal categories with parameter monoidal width.
This result would rely on Feferman-Vaught-Mostowski type theorems specific to the operations of a particular monoidal category \(\cat{C}\) or particular class of monoidal categories, which would ensure that the problems at hand respect the compositional structure of these categories.

Computing a compositional problem on the set of morphisms \(\cat{C}_k(X,Y)\) with \(k\)-bounded monoidal width with a compositional algorithm is linear in \(\nodeweight\).
Explicitly, computing the solution on \(f \in \cat{C}_{k}(X,Y)\) takes \(\mathcal{O}(c(k) \cdot \nodeweight(f))\), for some more than exponential function \(c \colon \naturals \to \naturals\).