# Balanced Performance Between Energy-Delay and Bit Error Rate in UAV Relay
Networks

M khalil, Member, IEEE

###### Abstract

This paper presents a new strategy for simultaneously reducing energy
consumption, transmission delays, and bit error rate in Unmanned Aerial
Vehicle (UAV) networks. A UAV is fitted with a wireless Bidirectional Relay
(BR) to enable coverage network extension and increase transmission
throughput. The downside of the BR advantages is the delay in data
transmission caused by the UAV’s movement. A consequence of this delay is
increased total energy consumption, causing further degradation in bit error
rate performance, especially at high SNR levels. In wireless communication,
the trade-off between delay and energy consumption is, fortunately, possible
to improve performance. Therefore, this study aims to enhance UAV network
performance by reducing energy consumption, data transmission delay, and bit
error rate. A multi-objective algorithm is employed to generate an adaptive
optimal energy allocation strategy based on the balance between energy
consumption and transmission delays. The results of theoretical analysis are
illustrated with several examples. As herein demonstrated, the proposed
solution effectively balances delay and energy efficiency in a customised
system design and improves the bit error rate in UAV networks.

###### Index Terms:

UAV, two-way amplify-and-forward, relay, energy, delay

##  I Introduction

Unmanned aerial vehicles relay network is a communication system employing
drones fitted with wireless relay devices to enhance the scope and flexibility
of communication systems. UAV is currently one of the essential technologies
serving various applications besides telecommunications, such as product
deliveries, aerial photography, policing and surveillance, military and
infrastructure inspections. An example of UAV’s importance has been
illustrated in the recent outbreak of Coronavirus disease, where UAV was
employed to perform surveillance public and persuade them to follow public
health best practices [1]. In a telecommunication application, UAV enables
communications between many land user nodes to take place through UAV nodes.
It is recognized that by fitting Decode-and-Forward (DF) or Amplify-and-
Forward (AF) relay systems, the drone network gives better coverage and better
throughput and energy performance [2]. An AF relay is considered in this paper
as it can theoretically be applied with less complexity than DF relay, which
requires complete processing, including encoding, re-modulating and re-
transmitting the received signal. Such operations processes require
sophisticated power control, which is unnecessary in an AF relay. The relay
node operates as either one-way (unidirectional), or two-way (bidirectional).
This paper focuses on the BR AF type as several studies, such as [3] have
observed that the BR AF system analysis is appropriate for nano-satellite
communication applications, which are expected to be an essential part of the
5G networks.

The drone’s battery unit, named Battery Eliminator Circuit (BEC), supplies the
necessary energy for the drone, the AF relay and other components equipped on
the drone. Using AF relay nodes increases the drain of AR’s BEC, particularly
in the transmitting mode at which the relay node becomes active. In the
transmission mode, the energy model presents energy consumption for each
transmitted bit. Another factor that increases the drone networks’ energy
consumption is the delay due to the UAV movement. Further, transmission
signals in tow-hop significantly impact each node’s energy consumption [4].
Rising energy consumption again degrades bit error rate, particularly at high
levels of signal-to-noise ratio (SNR) [5]. Moreover, in UAV networks, users
and drone nodes often rely on a battery with a limited amount of energy. Thus,
minimizing energy consumption and delay is a fundamental goal in UAV networks.

Unfortunately, the delay, energy and bit error rate objectives cannot be
minimized simultaneously as these metrics having conflict with each other.
Such a conflicting problem, however, can be addressed by using a multi-
objective solution [6]. One of the effective methods in multi-objective is the
weight of the scalarization method [7]. All objective functions in the weight
scalarization method are consolidated into a single part showing a linear
function. Thus, we can combine the delay, energy and bit error rate objectives
into one function. However, solving such a single function for a bidirectional
relay network is challenging as the communication scenario coincides in two
directions. This paper proposes dividing the problem into two sub-problems to
simplify the weight scalarization function analysis in a bidirectional relay
network. In the first problem, energy consumption and transmission delay are
combined in one function. The solution to the first problem is employed to
optimize energy allocation parameters. Such parameters reduce delay and energy
consumption and enhance bit error rate.

Many studies have analysed the balance between energy and delay by using
information theory; it significantly focuses on designing power allocation
under various constraints on the information delay, such as an average delay
constraint for a buffer, queuing delay in [8], per-packet delay constraint
[9], in addition, a multipacket transmission [10]. Study [11] uses a minimum
departure time as a model of packet delay constraints; this scheme can be
applied to model various quality of service (QoS) constraints. Further, the
design of the system algorithm depends on the availability of Channel State
Information (CSI), which can include fading channels and time-variation as

According to Shannon’s capacity theorem the minimum of
$\mathrm{\frac{energy\>per\>bit\>rate(\mathcal{E\mathrm{)}}}{noise\>power\>density\>(\mathcal{N}\mathit{o})}}$
needed to achieve arbitrarily low bit error probability as
$\mathcal{\Bigl{(}E\text{/}\mathrm{\mathcal{N}\mathrm{\mathit{o}}}\Bigr{)}_{\mathit{\mathrm{min}}}\mathrm{=}}\ln(2).$
For a given error probability and code rate with finite bits, the minimum
energy has been studied in [12]. Authors [13] sought to maximize the average
throughput, which is the equivalent of minimizing the average delay-per-bit
for a given number of bits and input power. Study [14] presents a minimal
energy solution for transmuting finite bits without delay constraints, and
[15] a solution for Energy-delay balance over fading channels has been
demonstrated in [15]. In [16], the model to minimize the energy consumption of
the intermediate relay between the source and the receiver was adopted for
wireless terrestrial relay. The locations of wireless relays may, however, be
random in practical networks. Hence, in [16] the effects of randomly
positioned relays were examined, but this study investigated selecting the
best relay location based on linear places between the source and destination.
However, in these analyses, the CSI is assumed to be available at the source
or destination nodes. Nonetheless, the fading channel at mobility relay
locations often varies rapidly, which can make it difficult to estimate,
especially if the relay is moving in space.

For this reason, Doppler effects are considered by [17], who assumed the relay
was flying at a constant speed and the destination node able to estimate and
compensate. The mentioned studies, however, were focused solely on
transmission delay, without taking into consideration network energy
consumption. In contrast, other researchers [2, 18] were mainly focused on
network energy consumption. Studies [19, 6] investigated the energy
consumption with delay constraints for UAV AF relay network. Based on the
results of these studies, power allocation is essential to maximize UAV
throughput. However, in many applications, particularly for energy-limited
appliances like sensor or relay systems, energy is the main parameter that
carries out a specific operation than its power consumption. Generally, the
energy parameter is subject to time delay, and in the meantime, the UAV
network has strict delay regarding safety information transmission delays.
Despite that, [6] recognized that distributing energy value among terrestrial
networks nodes reduces delay and overall energy consumption.

The location of UAV changes periodically, then the received SNR has time-
varying characteristics; thus, bit rate often changes [20]. Accordingly,
several researchers worked on determining the effect of channel
characteristics on bit error rate performance. For instance, study [2] adopted
variable rate protocol to enhance bit error rate performance and achievable
information rate, as the location of relay changes periodically. Another
researcher [18] investigated how to place UAVs to reduce bit error rates
optimally. One paper [21] proposed a path loss model that accommodates both
Line of Sight (LOS) and Non-Line of Sight (NLOS) path loss conditions.
Likewise, the authors of [22] extended their results to include three-
dimensional space. In [23], the optimum location of device-to-device
communications was also considered in UAV to enhance network performance.
However, optimising the bit error rate by selecting the best relay location is
impractical. Therefore, another work [24] optimised trajectory and energy
control at the same time. The UAV trajectory is also optimised jointly with
the device-UAV association and uplink power to minimise the total transmission
power according to the number of updates in [25].

All of the aforementioned studies demonstrated effective schemes to improve
UAV networks’ performance in terms of UAV placement and energy allocation.
However, an essential factor that is largely ignored in these works is that
UAV networks may have slightly larger transmission times, so the data received
from the ground user will have various SNR levels. Furthermore, works that
developed the energy consumption and data transmission delay metrics, jointly
or individually, ignored the relationship between these metrics and the bit
error rate, which is an essential metric for evaluating the performance of UAV
network applications [26]. Thus, [27] presented a specific system model that
can only provide balance energy and delay transmission data for a
unidirectional UAV AF relay flying in a triangle formation. Error-free
reception, however, is very limed in the practical environment, particularly
in multi-hop networks with varying channel conditions. Also, the
unidirectional relay operates in one-way communication between the source and
destination, whereas such communication has limited applicability. To respond
to this gap, this paper considers a general trade-off energy-delay scenario
that enables energy allocation to be an adaptive factor for providing an
optimal bit error rate for UAV bidirectional AF relay networks over mobile
fading channels. The proposed method is achieved by simultaneously optimising
both energy consumption and transmission delay in UAV networks. The optimum
energy allocation is distributed between relay and source nodes in UAV
networks. Such energy allocation solves two problems. First, it enables the
development of a decision-making platform to achieve the best trade-off
between energy consumption and data transmission delay. Second, it enhances
the performance of UAV networks in terms of bit error rate. Hence, the
proposed method simultaneously improves energy consumption, data transmission
delays and bit error rate in UAV networks. The contribution of this paper is
summarised by defining an algorithm allowing to compute an optimal energy
allocation to each node (users and relay) of the UAV communication network,
and further, it enhances the bit error rate. Most notably, the algorithm
strikes a balance between transmission delays and energy as follows:

  * •

Define UAV network by three nodes, which are two terrestrial users $S_{a}$ and
$S_{b}$ and a wireless relay ($\mathit{R}$) fitted on a drone.

  * •

Calculate a flight distance ($d_{a}$) between $S_{a}$ and $\mathit{R}$ and the
path ($d_{b}$) between $S_{b}$ and $\mathit{R.}$

  * •

Consider the distances $d_{a}$ and $d_{b}$ to calculate the end-to-end
$\mathit{SNRs}$ at $S_{a}$ and $S_{b}$.

  * •

Use expressions of $\mathit{SNRs}$ at $S_{a}$ and $S_{b}$ to calculate energy
allocations for $S_{a}$ as $(\alpha_{a}),$ relay $(\alpha_{r})$ and $S_{b}$ as
$(\alpha_{b}).$

  * •

Define the bit energy consumption $(\mathcal{E}_{\mathit{a}})$ for $S_{a}$ and
$(\mathcal{E}_{\mathit{b}})$ for $S_{b}$ as functions of energy allocations
factors, i.e.,
$\mathcal{E}_{\mathit{a}}\mathcal{\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}}$
and
$\mathcal{E}_{\mathit{b}}\mathcal{\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}},$
also

  * •

Define bit transmission times $(q_{a})$ at $S_{a}$ and $(q_{b})$ at $S_{a}$ as
functions of energy allocations as
$q_{a}\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}$ and
$q_{b}\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}.$

  * •

Use Scalarization optimization method demonstrated in [28] to optimize
$\mathcal{E}_{\mathit{a}}\mathcal{\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}}$,
$\mathcal{E}_{\mathit{b}}\mathcal{\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}},$
$q_{a}\mathcal{\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}}$ and
$q_{b}\mathcal{\mathrm{(\alpha_{a},\alpha_{b},\alpha_{r})}},$ simultaneously,
and this provides an effective method to balance delay-energy performance on
customized system design. In other words, the proposed method offers a
decision-making scheme responsible achieving the best trade-off between
transmission delay and energy consumption in UAV networks.

  * •

Use optimum $\alpha_{a},\alpha_{b},\alpha_{r}$ to maximise end-to-end SNR and
thus lower bit error rates.

The proposed method has been evaluated for the UAV network, and the analytic
outcomes reveal that the proposed approach enables energy consumption,
transmission delay and bit error rate to be minimized in a well-balanced
scheme.

Though the energy allocation strategy has been used to optimise wireless
network performance, as in [29], this paper emphasises the potential of using
such a strategy to simultaneously improve the data rate, energy consumption
and bit error rate performance in UAV networks. According to the author’s
knowledge, a UAV network has not been optimised using such a schema
previously.

Before further discussion, we provide Table I to summarized notations used in
this manuscript.

The rest of this paper is arranged as follows: Sections II and III describe
the system model and the received signal at destination node analysis,
respectively; Section IV introduces a new method to calculate energy
allocation for UAV users and relay; Section VI presents the simulation
results; and finally, the conclusion is presented in Section VII.

##  II SYSTEM MODEL

Consider two terrestrial wireless users, $S_{a}$ and $S_{b}$, exchanging
information simultaneously via a wireless relay ($\mathit{R}$) node fitted
with an drone, i.e., an UAV network. The relay acts in a two-way amplify-and-
forward (AF) relay mode. It is assumed that all the nodes are equipped with a
single antenna, and all the links among the nodes are half-duplex and use the
same carrier frequency. Flying a drone involves flight along any path in
coordinates of three dimensions ($x$, $y$, $z$), where ($\pm x,\,0,\,0$)
represent the locations of land nodes i.e. $S_{a}$ and $S_{b}$ and $z$
represents the altitude ($\mathcal{H})$ as shown in Fig (1).

For analyzing UAV flight path, we suppose that the drone flies along a path
defined by a series of waypoints, which is assumed to be the drone’s initial
position, defined as
$\mathrm{p_{i}}$$(\mathrm{\mathit{x}_{i},\,\mathit{y}_{i}\,,\mathit{z}_{i}}).$
Thus, the Cartesian coordinates of $\mathrm{p_{i}}$ are determined from the
coordinate transformations as $\mathrm{\mathit{x}_{i}}=r\cos(\theta)$,
$\mathrm{\mathit{y}_{i}}=r\,\sin(\theta$) and
$\mathrm{\mathit{z}_{i}}=r\,\sin(\phi)$. The polar coordinates $r$, $\theta$
and $\mathcal{\phi}$ are measured from the centre $O$$(0,0,z)$ of the flight
path.

The Euclidean distance among $S_{a}$, $O$ and $\mathit{S_{b}}$ are calculated
at $r=0$, as $\mathit{\pm d}$ which represents the separating distance between
$S_{a}$ and $S_{b}$ . On the other hand, when $r>0$, the total distance ($d)$
of $d_{a}$ and $d_{b}$ is varied based on $r$ value. Thus, the Euclidean
distance between $S_{a}$ and $\mathit{R}$ is calculated as

$d_{a}=\left(d^{2}+\frac{r^{2}}{2}\left(3-\cos(2\phi)\right)-\psi\right)^{1/2},$
(1)

and the distance between $R$ and $S_{b}$ is

$d_{b}=\left(d^{2}+\frac{r^{2}}{2}\left(3-\cos(2\phi)\right)+\psi\right)^{1/2},$
(2)

where $\psi=2\,r\,d\,cos\theta$.

TABLE I: Summary of Notations Notations |  Description
---|---
$S_{a}$ and $S_{b}$ |  two terrestrial wireless users
$d_{a}$ and $d_{b}$ |  the distances between source-drone and drone-receiver, respectively.
$p_{a}$,$p_{b}$ and $p_{R}$ |  allocation powers for $S_{a},S_{b}$ and relay, respectively.
$\Re_{a},\Re_{b}$ |  data rates for $S_{a},S_{b}$
$h$ and $g$ |  fading channels for $d_{a}$ and $d_{b}$ , respectively.
$Y_{a}$ and $Y_{b}$ |  received signal by $S_{b}$ and $S_{b}$ , respectively.
$\gamma_{a}$ and $\gamma_{b}$ |  SNR at $S_{b}$ and $S_{b}$.
$\mathit{q_{a}}$ and $\mathit{q_{b}}$ |  data transmission time by $S_{b}$ and $S_{b}$, respectively.
$\mathcal{E}_{\mathit{a}}$ and $\mathcal{E}_{\mathit{b}}$ |  data transmission energies for $S_{b}$ and $S_{b}$, respectively.
$\alpha_{a},$$\alpha_{b}$ and $\alpha_{r}$ |  energies allocation factors for $S_{b}$, $S_{b}$ and the relay, respectively.
$w_{a}$, $w_{b}$ and $w_{r}$ |  weight coefficients for $S_{b}$, $S_{b}$ and the relay, respectively.
$\text{\texthtb}_{e}$ |  bit error rate
Figure 1: System Model

##  III Received signal at detestation node

This part derives an expression for the users $\mathit{SNRs}$ of the proposed
UAV network. We first assume that the destination is entirely compensated by
the Doppler effect due to the UAV’s mobility as the UAV follows a trajectory
with a fixed flying speed [30]. Thereby, both flat fading channels
$\mathit{S_{a}-R}$ and $\mathit{R-S_{b}}$ have power gains following the free-
space path attenuation schemas
$h\,d_{a}^{\frac{-\text{\textramshorns}}{2}}$and
$g\,\,d_{b}^{\frac{-\text{\textramshorns}}{2}},$respectively, where $h$ is the
channel coefficient between $S_{a}$ and $R$, $g$ is the channel coefficient
between $S_{b}$ and $R$,  is the path loss exponent which is commonly
estimated in the range of $2\leq\text{\textramshorns}\leq 4$. It is also
assumed that the channel’s characteristics are available at the user nodes.
Further, user $S_{a}$ is able to adjust the transmit powers in $S_{b}$ and
relay $\mathit{R}$. Similarly, $S_{a}$ can regulate $S_{a}$ and relay
$\mathit{R}$ powers using the power allocation process.

A further assumption is that the users adopt binary phase shift keying
modulation to broadcast their signals $\chi_{a}$ and $\chi_{b}$, with average
power $p_{a}$ and $p_{b}$, from users $S_{a}$ and $S_{b}$, respectively. Both
signals $\chi_{a}$ and $\chi_{b}$ are transmitted during the time interval
$\mathit{0}<\mathit{t}\leq\mathcal{\mathcal{T}},$and each signal follows a
circularly symmetric complex Gaussian distribution $\mathcal{CN}(0,1)$, i.e.
$E\\{\bigl{|}\chi_{a}\,\bigr{|}^{2}\\}=1$ and
$E\\{\bigl{|}\chi_{b}\bigr{|}^{2}\\}=1$, where $E\\{.\\}$ denotes an expected
value and $\bigl{|}.\bigr{|}^{2}$ is the absolute square of a signal.

The $S_{a}$ and $S_{b}$ nodes transmit their bits according to a schedule that
defines the commencing and the duration of each bit transmission. Then,
$\chi_{a}$ and $\chi_{b}$ arrive to their target node in the time interval
$\mathit{0}<\mathit{t}\leq\mathcal{\mathcal{T}}.$ In order to simplify,
$\mathcal{T}$ is discretized into $n$ time slots as
$\delta_{\mathit{t}}\mathcal{\mathrm{=}T}$/$\mathcal{N}$, where
$\delta_{\mathit{t}}$ represents the time slot length. The value of
$\delta_{\mathit{t}}$ is assumed to be small enough that the UAV’s location
can be supposed to be approximately constant within a slot. Thereby, the UAV’s
trajectory
$\left(x_{i}\mathrm{(\mathit{t})},y_{i}\mathrm{(\mathit{t})},z_{i}\mathrm{(\mathit{t})}\right)$
over $\mathcal{T}$ can be specified by $\mathcal{N}$ scope as
$\bigl{\\{}x_{i}[n],y_{i}[n],z_{i}\mathrm{(\mathit{n})}\bigr{\\}}_{n}^{\mathcal{N}}$
$\in\mathbb{R^{\mathrm{3}}},$where $\mathit{n=1,2,...\mathcal{N-\mathrm{1.}}}$

The exchange signals between $S_{a}$ and $S_{b}$ occur in two-hops. In the
first hop, the relay receives signals from both $S_{a}$ and $S_{b}$ as

$Y_{R}[n]=\sqrt{p_{a}\bigl{[}n\bigr{]}}\chi_{a}h+\sqrt{p_{b}\bigl{[}n\bigr{]}}\chi_{b\,}g\,d_{b}^{\frac{-\alpha}{2}}+\mathit{n}_{R},$
(3)

where $h\,d_{a}^{\frac{-\alpha}{2}}$ and $g\,d_{b}^{\frac{-\alpha}{2}}$ are
the fading channels model defined in [31], $n_{R}$ is the Gaussian noise of
the relay with zero mean and variance $(\sigma^{2})$.

The $Y_{R}$ signal is then amplified by the relay amplification factor
($\beta$) given by:

$\beta\bigl{[}n\bigr{]}=\sqrt{p_{R}\bigl{[}n\bigr{]}/(\left|h\right|^{2}d_{a}^{-\alpha}p_{a}\bigl{[}n\bigr{]}\,+\left|g\right|^{2}d_{b}^{-\alpha}p_{b}\bigl{[}n\bigr{]}+\sigma^{2}}),$
(4)

where $p_{R}\bigl{[}n\bigr{]}$ is the allocated power for $R$ node .

After that, each user receives the amplified signals through the same flat
fading channel, during the second time slot. The received signal by $S_{a}$ is
then included in the relay-amplified signal beside the direct signal received
from $S_{b}$ node. This gives

$\displaystyle
Y_{a}\bigl{[}n\bigr{]}=h\,d_{a}^{\frac{-\alpha}{2}}\beta\bigl{[}n\bigr{]}y_{R}+\sqrt{p_{b}}\chi_{b\,}g\,d_{b}^{\frac{-\alpha}{2}}$
$\displaystyle+2\sqrt{p_{b}}\chi_{b\,}gh\,\left|d\right|^{\frac{-\alpha}{2}}+n_{a}.$
(5)

Node $S_{b}$ also involves the relay-amplified signal and the direct $S_{a}$
signal as the following

$\displaystyle
Y_{b}\bigl{[}n\bigr{]}=g\,d_{b}^{\frac{-\alpha}{2}}\beta\bigl{[}n\bigr{]}y_{R}+\sqrt{p_{a}}\chi_{a}\,h\,\,d_{a}^{\frac{-\alpha}{2}}$
$\displaystyle+2\sqrt{p_{a}}\chi_{a\,}gh\,\left|d\right|^{\frac{-\alpha}{2}}+n_{b},$
(6)

where $gh$ is the channel coefficient between user $S_{a}$ and user $S_{b}$
with the separating distance 2$\left|d\right|,$and $n_{a}$ and $n_{b}$ are the
users Gaussian noises with a zero mean and variance $(\sigma^{2})$ for both
user $S_{a}$ and user $S_{b},$ respectively.

Equations (5) and (6) each have their own transmission signal mixed with the
received signal and this is returned to self-interference (SI). Here, we
assume that both users have an SI cancellation circuit that allows for their
own transmitted signal to be cancelled out [32]. Thus, the SNRs at each user
are given by

$\displaystyle\gamma_{a}^{\iota}$
$\displaystyle\bigl{[}n\bigr{]}=\frac{p_{R}\,\gamma_{b}\>p_{b}\,\gamma_{a}\,d_{b}^{-\alpha}d_{h}^{-\alpha}}{\gamma_{a}\>(p_{R}+p_{a})d_{a}^{-\alpha}+p_{b}\,\gamma_{b}d_{b}^{-\alpha}+1}+p_{R}p_{b}\gamma_{ab}\,d_{b}^{-\alpha},$
(7) $\displaystyle\gamma_{b}^{\iota}$
$\displaystyle\bigl{[}n\bigr{]}=\frac{p_{R}\,\gamma_{b}\>p_{a}\,\gamma_{a}\,d_{a}^{-\alpha}d_{b}^{-\alpha}}{p_{a}\,\gamma_{a}\>d_{a}^{-\alpha}+(p_{R}+p_{b})\,\gamma_{b}\,d_{b}^{-\alpha}+1}+p_{R}\,p_{a}\,\gamma_{ab}\,d_{a}^{-\alpha},$
(8)

where $\gamma_{a}^{\iota}$ and $\gamma_{b}^{\iota}$ are the actual line-of-
sight $\textit{$\mathit{SNR}$}s$ for $S_{a}$ and $S_{b}$, respectively,
$\gamma_{ab}=\frac{\left|gh\right|^{2}}{\sigma^{2}},$
$\gamma_{a}=\frac{\left|h\right|^{2}}{\sigma^{2}}$ and
$\gamma_{b}=\frac{\left|g\right|^{2}}{\sigma^{2}}.$

In the proposed UAV, the direct link between $S_{a}$ and user $S_{b}$ is
usually not possible due to long-distance flight and degradation in channel
quality. Therefore, the destination nodes receive only the relayed signals
from the $R$ node, i.e., $\gamma_{ab}=0$. Equations (7) and (8) then become

$\displaystyle\gamma_{a_{e}}$
$\displaystyle\bigl{[}n\bigr{]}=\frac{p_{R}\,\gamma_{b}\>p_{b}\,\gamma_{a}\,d_{a}^{-\alpha}d_{b}^{-\alpha}}{\gamma_{a}\>(p_{R}+p_{a})d_{a}^{-\alpha}+p_{b}\,\gamma_{b}d_{b}^{-\alpha}+1},$
(9) $\displaystyle\gamma_{b_{e}}\bigl{[}n\bigr{]}$
$\displaystyle=\frac{p_{R}\,\gamma_{b}\>p_{a}\,\gamma_{a}\,d_{a}^{-\alpha}d_{b}^{-\alpha}}{p_{a}\,\gamma_{a}\>d_{a}^{-\alpha}+(p_{R}+p_{b})\,\gamma_{b}\,d_{b}^{-\alpha}+1},$
(10)

where $\gamma_{a_{e}}$ and $\gamma_{b_{e}}$ are the exact non-line-of-sight
SNRs for $S_{a}$ and $S_{b}$, respectively

The average power $\overline{p}_{r}$, $\overline{p}_{a}$ and
$\overline{p}_{b}$ should not exceed the overall network power
$(\overline{P})$
$\left\\{\overline{P}:\overline{P}\in\mathbb{R},\overline{P}=\underset{i}{\sum}\overline{p}_{i}\quad
i\in\left(a,r,b\right)\right\\}$. In order to manage $\overline{P}$ value, the
power allocation strategy is used to allocate a specific power value to the
relay and both user nodes. All nodes have inclusive knowledge of information
about channels and maximum transmission power for each related node [33], so
both $S_{a}$ and $S_{b}$ nodes achieve the allocation strategy by employing
the reverse channels for feedback allocated energy factors to the
corresponding node. After that, $S_{a}$, $S_{b}$ and relay nodes adjust their
transmit power based on these feedback factors [34].

It assumes that the energy allocation factors $S_{a}$, $\mathit{R}$ and
$S_{b}$, respectively, are set as

$\displaystyle\alpha_{a}\Rightarrow\left\\{\alpha_{a}:\in\mathbb{R\mathit{,}}\rightarrow
0\leq\alpha_{a}\leq 1\right\\},$ (11)
$\displaystyle\alpha_{r}\Rightarrow\left\\{\alpha_{r}:\in\mathbb{R\mathit{,}}\rightarrow
0\leq\alpha_{r}\leq 1\right\\},$ (12)
$\displaystyle\alpha_{b}\Rightarrow\left\\{\alpha_{b}:\in\mathbb{R\mathit{,}}\rightarrow
0\leq\alpha_{b}\leq 1\right\\}.$ (13)

Once received, the corresponding nodes adjust their powers as
$\overline{P}\>\alpha_{b},\overline{P}\>\alpha_{r}$ and
$\overline{P}\>\alpha_{a}$. So, substituting these regulated powers into (9)
and (10) and considering high $\mathit{SNR}$ domain, we get

$\displaystyle\gamma_{a}[n]=\overline{P}\frac{\alpha_{r}\>\bigl{[}n\bigr{]}\,\gamma_{b}\>\alpha_{b}\bigl{[}n\bigr{]}\,\gamma_{a}d_{a}^{-\text{\textramshorns}}d_{b}^{-\text{\textramshorns}}}{\gamma_{a}\>(\alpha_{r}\>\bigl{[}n\bigr{]}\,+\alpha_{a}\>\bigl{[}n\bigr{]}\,)d_{a}^{-\text{\textramshorns}}+\alpha_{b}\bigl{[}n\bigr{]}\,\gamma_{b}d_{b}^{-\text{\textramshorns}}},$
(14)
$\displaystyle\gamma_{b}[n]=\overline{P}\frac{\alpha_{r}\>\bigl{[}n\bigr{]}\,\,\gamma_{b}\>\alpha_{a}\>\bigl{[}n\bigr{]}\,\,\gamma_{a}d_{a}^{-\text{\textramshorns}}d_{b}^{-\text{\textramshorns}}}{\alpha_{a}\>\bigl{[}n\bigr{]}\,\,\gamma_{a}\>d_{a}^{-\text{\textramshorns}}+(\alpha_{r}\>\bigl{[}n\bigr{]}\,+\alpha_{b}\>\bigl{[}n\bigr{]}\,)\,\gamma_{b}d_{b}^{-\text{\textramshorns}}},$
(15)

where $\gamma_{a}$ and $\gamma_{b}$ are the high SNRs domain for (9) and (10),
respectively

Now, another metric is the channel capacity, which is given by

$C[n]=\frac{1}{2}\log_{2}(1+\phi[n]).\;\quad bits/s,$ (16)

where $\phi[n]=1+\gamma_{a}[n]+\gamma_{b}[n]+\gamma_{a}[n]\gamma_{b}[n].$

To adapt reliably transmitted information rate ($\mathcal{R}$), we have

$\mathcal{R}[\mathit{n}]=\mathfrak{h}\>C[n]\;\left(0<\text{\text{$\mathfrak{h}$}}<1\right)$
(17)

At $\text{\text{$\mathfrak{h}$}}=1,$the maximum data rate in (16) becomes

$\Re_{i}=\Re_{a}+\Re_{b}=\frac{1}{2}\left(\log_{2}(1+\gamma_{a}[n])+\log_{2}(1+\gamma_{b}[n])\right),$
(18)

where $\Re_{i}$ is the total data rate in two way [35].

As mentioned earlier, the data rate $\mathcal{R}[\mathit{n}]$ is passed
through two hops, i.e., two cascade channels. The transmission rate through
two channels is subjected to an Information Cascade (IC) analysis [36], which
is defined as a propagation sequence of data bits transmitting over chaotic
channels. Further, the mutual-information flow along the cascade of channels
cannot exceed each channel individually. Along the cascade channels, the flow
mutual information capacity cannot overpass each channel capacity. Regarding
UAV channels, the first channel delivers a sequence of bits that transmitted
by users nodes to the $\mathit{R}$ node during a time slot
$\mathit{n=\mathrm{1}.}$ The $\mathit{R}$ node requires one-slot processing to
forward the received data to the $S_{a}$ and $S_{b}$ node during the second
time slots, i.e. $\mathit{n}$= 2, 3, 4,…$\mathcal{N}$ [37]. Then, the
information-causality constrain is obtained as:

$\underset{\begin{gathered}S_{a}\rightarrow R\rightarrow S_{b}\\\
S_{b}\rightarrow R\rightarrow
S_{a}\end{gathered}}{\underbrace{\sum_{\mathrm{i=2}}^{\mathrm{n}}\overline{R}\mathrm{[\mathit{i}]}}}\leq\underset{\begin{gathered}R\rightarrow
S_{a}\\\ R\rightarrow
S_{b}\end{gathered}}{\underbrace{\sum_{\mathrm{i=1}}^{\mathrm{n-1}}\overline{R}\mathrm{[\mathit{i}]}}}.$
(19)

$S_{a}\rightarrow R\rightarrow S_{b}$ and $S_{b}\rightarrow R\rightarrow
S_{a}$ represent the transmitted a signal from $S_{a}$ to $S_{b}$ and from
$S_{b}$ to $S_{a}$, respectively, $R\rightarrow S_{a}$ and $R\rightarrow
S_{b}$ typify the amplified signal forwarded by $\mathit{R}$ to $S_{a}$ and
$S_{b}$ through one channel.

Input: $\overline{P},\textit{x(t)},\alpha_{a},\alpha_{b},\text{and
}\alpha_{r}$

Initialization of parameters

while  _$t\neq 0$_ do

$d_{a}\leftarrow\eqref{eq: d1}$

$d_{b}\leftarrow\eqref{eq: d2}$

$\gamma_{a}\leftarrow E{Y_{a}[n]}$ $\triangleright$ expectation value of (9)

$\gamma_{a}\leftarrow E{Y_{b}[n]}$ $\triangleright$ expectation value of (10)
$p_{a}=\overline{P}\>\alpha_{a}$ ; $p_{b}=\overline{P}\>\alpha_{b}$;
$p_{r}=\overline{P}\>\alpha_{r}$

$\gamma_{a}\leftarrow\eqref{eq:Hi-SNR a}$

$\gamma_{b}\leftarrow\eqref{eq:HI-SNR b}$

end while

Output: $\mathit{SNR}$

Algorithm 1 Procedure of $\mathit{SNR}$ analysis

##  IV Energy Allocation

This section will concentrate on an energy allocation problem to manage energy
consumption in the users and relay nodes of UAV networks. Energy consumption
is related to transmission delay [38], so the proposed energy allocation
allows total energy consumption and data transmission time (delay) to be
minimised under a balanced approach.

In Shannon’s theorem, prolonging the transmission time reduces transmitting
power. Thus, the total energy consumption in UAV networks is minimised by
maximising transmission time. However, increasing transmission delay in
transmitting the information directly affects the user’s service. Hence,
transmission time $(q)$ and power networks must be designed with a trade-off
scheme. $\mathit{q}$ is the amount of time required by a user to send out a
single packet of bits; it depends on the network’s bandwidth and length of the
packet, as $q$ = (Data size/bandwidth) (sec). By using low-latency algorithms
or low-delay transmission protocols, data transmission amounts can be
adjusted, and delays can be reduced. Such approaches allow the data
transmission amount to be managed according to the change in delay performance
due to a change in transmission amount. Each data bit delivered at
$\mathit{q}$ can consume energy ($\mathcal{E})$ as $\mathcal{E}=q\>P$. Then,
the total energy model in UAV is defined from (14), (15) and (17) as follows

$\displaystyle\mathcal{E}_{\mathit{a}}\bigl{[}n\bigr{]}=q[n]\left(\>(\alpha_{r}\>\bigl{[}n\bigr{]}\,+\alpha_{a}\>\bigl{[}n\bigr{]}\,)H+\alpha_{b}\bigl{[}n\bigr{]}\,G\right)\left(2^{\frac{2}{q[n]}}-1\right),$
(20)
$\displaystyle\mathcal{\mathcal{E}_{\mathit{b}}}\bigl{[}n\bigr{]}=q[n]\left(\alpha_{a}\>\bigl{[}n\bigr{]}\,\,H+(\alpha_{r}\>\bigl{[}n\bigr{]}\,+\alpha_{b}\>\bigl{[}n\bigr{]}\,)\,G\right)\left(2^{\frac{2}{q[n]}}-1\right),$
(21)

where $G=\gamma_{b}d_{b}^{-\text{\textramshorns}}$,
$H=\gamma_{a}d_{a}^{-\text{\textramshorns}}$.

Concurrent with the information exchanging between $S_{a}$ and $S_{b},$ the
overall energy consumption is typically generated by including (20) and (21),
i.e.,
$\,\mathcal{E}\bigl{[}n\bigr{]}=\,\mathcal{E}_{\mathit{a}}\bigl{[}n\bigr{]}+\mathcal{\mathcal{E}_{\mathit{b}}}\bigl{[}n\bigr{]}$,
as in [2]. The region $\mathcal{E}\bigl{[}n\bigr{]}$ can also be specified by
the union of all the possible sets of
$\left(\mathcal{E}_{\mathit{a}},\mathcal{E}_{\mathit{b}}\right),$ as in [30].
Hence, if the region $\mathcal{E}\bigl{[}n\bigr{]}$ is managed by varying
$\alpha_{a},\alpha_{b},\alpha_{r},$then the union of the two sets of energies
in (20) and (21) are subject to definition 1.

Definition 1. The regions of two sets $\,\mathcal{E}_{\mathit{a}}$ and
$\,\mathcal{E}_{\mathit{b}}$ is the collection of all objects that are in
either set. Then, the union of $\,\mathcal{E}_{\mathit{a}}$ and
$\,\mathcal{E}_{\mathit{b}}$ is defined as:
$\,\bigcup\mathcal{E}_{i}\bigl{[}n\bigr{]}=\mathcal{\,E}_{\mathit{a}}\cup\mathcal{\,E}_{\mathit{b}}\left\\{\alpha_{i}:\left(\alpha_{i}\in\mathcal{\,E}_{\mathit{a}}\right)\lor\left(\alpha_{i}\in\mathcal{\,E}_{\mathit{a}}\right)\right\\}$$\quad
i\in\left\\{a,\,b,r\right\\}$.

Now, the total energy consumption can be expressed in regard to $\alpha_{i}$
as the following

$\,\mathcal{E}_{i}(\alpha_{i})\bigl{[}n\bigr{]}=\,\mathcal{E}_{\mathit{a}}\bigl{[}n\bigr{]}+\mathcal{\mathcal{E}_{\mathit{b}}}\bigl{[}n\bigr{]}\quad\quad
i\in\left\\{a,b,r\right\\}.$ (22)

For each transmitted bit, the value of $q$ can be defined as
$q\bigl{[}n\bigr{]}=1/\mathcal{R}[n]$, so from (17) we have

$\mathit{q_{a}}(\alpha_{i})[\mathit{n}]=2\>\log_{2}\left(1+\,G\overline{P}\frac{\alpha_{r}\>\alpha_{b}}{\alpha_{r}+\alpha_{a}\>+\alpha_{b}\,\frac{G}{H}}\right)^{-1},$
(23)

$\mathit{q_{b}}(\alpha_{i})[\mathit{n}]=2\>\log_{2}\left(1+\>G\overline{P}\frac{\alpha_{r}\>\alpha_{a}}{\alpha_{a}\>+(\alpha_{r}\,+\alpha_{b}\,)\frac{G}{H}}\right)^{-1}i\in\left\\{a,\,b,r\right\\},$
(24)

where (.)-1 indicates reciprocal action.

In two-way relay networks, the powers design of relay and users nodes are
defined as $\left(p_{a}[n]+p_{r}[n]+p_{b}[n]\right)\leq P\bigl{[}n]$ [32]. In
this case, the proposed allocated power is expressed considering (19) as

$\underset{\mathrm{\mathit{\begin{gathered}\mathrm{the\;}\mathrm{total\>transmit\;power\;of}\\\
\mathrm{two\;hops}\end{gathered}}}}{\underbrace{\sum_{\mathrm{\mathit{\mathrm{n=2}}}}^{\mathit{\mathcal{N}}}\left(p_{a}[n]+p_{r}[n]+p_{b}[n]\right)}}\leq
P\bigl{[}n\bigr{]}.$ (25)

The purpose of this paper is to optimize $\alpha_{i}$ i.e.,
$\alpha_{a},\alpha_{b}$ and $\alpha_{\mathit{r}}$ in order to regulate the
energy consumption of (20) and (21) and the transmission delay of (23) and
(24) in a balanced way. Hence, the optimisation problem can be formulated as
follows

$\displaystyle\mathcal{\mathrm{min}\>\>E}_{a}(\alpha_{i})[n],$ (26)
$\displaystyle\mathcal{\mathrm{min}\>\>E}_{b}(\alpha_{i})[n],$ (27)
$\displaystyle\mathit{\mathcal{\mathrm{min}\>\>}}\left(q_{a}(\alpha_{i}[\mathit{n}]\mathit{+q_{b}}(\alpha_{i})[\mathit{n}]\right),$
(28) $\displaystyle s.t\qquad\forall
n\quad\alpha_{a}[\mathit{n}]+\alpha_{b}[\mathit{n}]+\alpha_{r}[\mathit{n}]\leq
1.$ (29)

All objectives (26)-(28) must be minimized at once under the constraint of
(29); however, the issue is that both (26) and (27) are in contras to (28).
Multi-objective optimisation techniques, particularly the weight scalarization
method [7], are some of the most reliable methods for resolving such an issue.
In the weight scalarization method, all of the objective functions are
consolidated into a single function that appears as a linear function. Many
studies have adopted the scalarization approach to optimise different
mathematical functions such as quadratic and logarithmic functions. By
employing the scalarization approach to minimise (26)-(28) under constraint
(29), the following expression is obtained

$\displaystyle
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)[n]=\sum_{\mathrm{\mathit{\mathrm{n=2}}}}^{\mathit{\mathcal{N}}}\left(w_{a}\mathcal{\>E}_{a}(\alpha_{i})[n]+w_{b}\mathcal{\>E}_{a}(\alpha_{i})[n]\right.$
$\displaystyle\left.+w_{r}\>\left(\mathit{q_{a}}(\alpha_{i})[\mathit{n}]+\mathit{q_{b}}(\alpha_{i})[\mathit{n}]\right)\right)\quad\quad\forall
n,$ (30)

where $w_{a}\left\\{w_{a}:\,\in\mathbb{R},\;0<w_{a}\leq 1\right\\}$,
$w_{b}\left\\{w_{b}:\,\in\mathbb{R},\;0<w_{b}\leq 1\right\\}$ and
$w_{r}\left\\{w_{r}:\,\in\mathbb{R},\;0<w_{r}\leq 1\right\\}.$

The weight coefficients are limited to the following constraint:
$\sum_{\mathrm{\mathit{i}}}^{\mathit{m}}w_{i}\leq 1,$where $\mathit{m}$ is the
number of functions and $i\in\left\\{a,b,r\right\\}.$ Equation (30) reveals
that both $\mathit{q_{a}}(\alpha_{i})[\mathit{n}]$ and
$\mathit{q_{b}}(\alpha_{i})[\mathit{n}]$ express as linear with a single
weight coefficient; this is because the proposed system model assumes that
$S_{a}$ and $S_{b}$, are exchanging information simultaneously with each other
through $\mathit{R}.$

The minimum solution of the objective function
$F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)$ is obtained by satisfying the
conditions
$\begin{bmatrix}{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{a}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{b}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{r}}}\end{bmatrix}^{\mathrm{T}}=0$
and
$\begin{bmatrix}{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{a}\alpha_{b}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{b}\alpha_{r}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{r}\alpha_{a}}}\end{bmatrix}^{\mathrm{T}}>0$.
Thus, it is required to calculate the following equations

$\displaystyle\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}}=q[n]\;H\;\left(2^{\frac{2}{q[n]}}-1\right)\left(w_{a}+w_{b}\right)$
$\displaystyle-
w_{r}G\overline{P}\left(\frac{\alpha_{b}H^{2}\alpha_{r}}{\left(\alpha_{b}G+H\alpha_{r}+H\alpha_{a}\right)^{2}}+\frac{GH\alpha_{r}(\alpha_{b}+\alpha_{r})}{\left(\alpha_{b}G+G\alpha_{r}+H\alpha_{a}\right)^{2}}\right).$
(31) $\displaystyle\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{b}}=q[n]\;G\;\left(2^{\frac{2}{q[n]}}-1\right)\left(w_{a}+w_{b}\right)$
$\displaystyle+w_{r}G\overline{P}\left(\frac{H^{2}\alpha_{r}(\alpha_{a}+\alpha_{r})}{\left(\alpha_{b}G+H\alpha_{r}+H\alpha_{a}\right)^{2}}-\frac{\alpha_{a}GH\alpha_{r}}{\left(\alpha_{b}G+G\alpha_{r}+H\alpha_{a}\right)^{2}}\right),$
(32) $\displaystyle\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial
r}=q[n]\;H\;\left(2^{\frac{2}{q[n]}}-1\right)\left(H\,w_{a}+G\,w_{b}\right)$
$\displaystyle+w_{r}G\overline{P}\left(\frac{\alpha_{b}H(\alpha_{a}H+\alpha_{b}G)}{\left(\alpha_{b}G+H\alpha_{r}+H\alpha_{a}\right)^{2}}+\frac{\alpha_{a}H(\alpha_{a}H+\alpha_{b}G)}{\left(\alpha_{b}G+G\alpha_{r}+H\alpha_{a}\right)^{2}}\right).$
(33)

Also we have

$\displaystyle\begin{bmatrix}{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{a}\alpha_{b}}}\\\
{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{b}\alpha_{r}}}\\\
{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{r}\alpha_{a}}}\end{bmatrix}=$
$\displaystyle\begin{bmatrix}\frac{-H^{2}r(\alpha_{a}H-G\alpha_{b}+H\alpha_{r})G\overline{P}}{(\alpha_{a}H+G\alpha_{b}+H\alpha_{r})^{3}}&\frac{GHr(\alpha_{a}H-G(\alpha_{r}+\alpha_{b}))G\overline{P}}{(\alpha_{a}H+G(\alpha_{r}+\alpha_{b}))^{3}}\\\
\frac{G\overline{P}H^{2}\left(\alpha_{a}^{2}H+\alpha_{a}bG+aH\alpha_{r}+2\alpha_{b}G\alpha_{r}\right)}{(H(\alpha_{a}+\alpha_{r})+\alpha_{b}G)^{3}}&\frac{-\alpha_{a}GH(\alpha_{a}H+\alpha_{b}G-G\alpha_{r})G\overline{P}}{(\alpha_{a}H+\alpha_{b}G+G\alpha_{r})^{3}}\\\
\frac{\alpha_{b}H^{2}(\alpha_{b}G+H(\alpha_{a}-\alpha_{r}))G\overline{P}}{(\alpha_{b}G+H(\alpha_{r}+\alpha_{a}))^{3}}&\frac{2\alpha_{a}H^{2}(\alpha_{a}H+\alpha_{b}G)G\overline{P}}{(\alpha_{b}G+G\alpha_{r}+H\alpha_{a})^{3}}\end{bmatrix}.$
(34)

Equation (34) illustrates that the second partial derivative test
$\begin{bmatrix}{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{a}\alpha_{b}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{b}\alpha_{r}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{r}\alpha_{a}}}\end{bmatrix}^{\mathrm{T}}>0$.
Thus, the minimum local points of equations (31) to (33) are obtained after
setting the first conditions, i.e., $\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}}=0$,
$\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{b}}=0$ and
$\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{r}}=0$.

Before continuing with this analysis, another test should be applied to
$F\mathit{\mathrm{[}n}\bigr{]}(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r})$.
This test aims to find whether the $F$ function is a convex function or not,
as the weigh scalarization method is inefficient to find a solution with a
non-convex function [39]. To achieve such a test, the Bordered Hessian
($\mathit{H_{b}}$) matrix is employed as follows:

$\displaystyle\mathtt{z}^{\mathtt{T}}\mathit{H_{b}}\mathtt{z}=\begin{bmatrix}\mathtt{z_{1}}&\mathtt{z_{2}}&\mathtt{z_{3}}&\mathtt{z_{4}}\end{bmatrix}$
$\displaystyle\left[\begin{array}[]{ccc}0&\frac{\partial
F\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{a}}&\frac{\partial
F\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{b}}\\\
\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}}&\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}^{2}}&\frac{\partial
F^{2}(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)^{2}}{\partial\alpha_{a}\partial\alpha_{b}}\\\
\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{b}}&\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}\partial\alpha_{b}}&\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{b}^{2}}\\\
\frac{\partial
F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{r}}&\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}\partial\alpha_{r}}&\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{b}\partial\alpha_{r}}\end{array}\right.$
(39) $\displaystyle\left.\begin{array}[]{c}\frac{\partial
F\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{r}}\\\
\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{a}\partial\alpha_{r}}\\\
\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{b}\partial\alpha_{r}}\\\
\frac{\partial^{2}F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)}{\partial\alpha_{r}^{2}}\end{array}\right]\begin{bmatrix}\mathtt{z_{1}}\\\
\mathtt{z_{2}}\\\ \mathtt{z_{3}}\\\ \mathtt{z_{4}}\end{bmatrix}.$ (44)

Substituting (31)-(34) into (44) leads to the following results:
$\mathtt{z}^{\mathtt{T}}\mathit{H_{b}}\mathtt{z}\geq 0$, i.e.,
$\mathit{H_{b}}$ is non-negative. Therefore, the Hessian matrix is positive
semidefinite; hence $F(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w)$ is a set
of convex.

Now, we evaluate
$\begin{bmatrix}{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{a}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{b}}}&{\normalcolor\frac{\partial\left(\alpha_{a},\alpha_{\mathit{b}},\alpha_{r},w\right)}{\partial\alpha_{r}}}\end{bmatrix}^{\mathrm{T}}=0$
from (31) to (33). This gives

$\displaystyle\overset{*}{\alpha}_{a}(w_{j})=\frac{1}{\psi_{a}}\left(\psi_{b}\pm\sqrt{\psi_{b}^{2}+2\psi_{a}\psi_{c}}\right)\quad\forall
n,$ (45)
$\displaystyle\overset{*}{\alpha}_{\mathit{b}}(w_{j})=\left(\frac{1}{\psi_{a}}\left(\psi_{b}\pm\sqrt{\psi_{b}^{2}+2\psi_{a}\psi_{c}}\right)\left(\frac{H}{G}-1\right)+1\right)$
$\displaystyle\sqrt{\frac{q\left(2^{\frac{2}{q[n]}}-1\right)\left[\;H-G\right]}{2HG\overline{P}\left(w_{3}\right)}\left(1-w_{3}\right)}\quad\forall
n,$ (46)
$\displaystyle\overset{*}{\alpha}_{\mathit{r}}(w_{j})=1-\frac{1}{\psi_{a}}\left(\psi_{b}\pm\sqrt{\psi_{b}^{2}+2\psi_{a}\psi_{c}}\right)$
$\displaystyle-\left(\frac{1}{\psi_{a}}\left(\psi_{b}\pm\sqrt{\psi_{b}^{2}+2\psi_{a}\psi_{c}}\right)\left(\frac{H}{G}-1\right)+1\right)$
$\displaystyle\sqrt{\frac{q\left(2^{\frac{2}{q[n]}}-1\right)\left[\;H-G\right]}{2HG\overline{P}\left(w_{3}\right)}\left(1-w_{3}\right)}\quad\forall
n,$ (47)

where $(w_{j}):j\in\left\\{a\vee\,b,r\right\\}$, $\vee$ is ’or’ symbol,
$\psi_{a}=2\left(G-H\right)$

$\displaystyle\left(\mathrm{1}-\left(\frac{q\left(2^{\frac{2}{q[n]}}-1\right)\left[H-G\right]\left(w_{2}\right)}{2HG\overline{P}\left(w_{3}\right)}-\frac{1}{\left(G-H\right)+H}\right)\mathrm{(G-H)}\right)$
,

$\psi_{b}={\color[rgb]{0,0,0}\left(\frac{H}{G}-1\right)}\sqrt{G\frac{q\left(2^{\frac{2}{q[n]}}-1\right)\left[\;H-G\right]}{2H\overline{P}\left(w_{3}\right)}\left(1-w_{3}\right)}$+2G
and
$\psi_{c}=\sqrt{G\frac{q\left(2^{\frac{2}{q[n]}}-1\right)\left[\;H-G\right]}{2H\overline{P}\left(w_{3}\right)}\left(1-w_{3}\right)}-G$.

The solution of (45)-(47) depend on any variation of two-weight coefficient,
and this gives an expected result because the proposed system model has two
sources, $S_{a}$ and $S_{b}$, and both sources contribute to adjusting the
energy allocation parameters. Based on several $w_{j}$, (45)-(47) provide a
trade-off between energy consumption and transmission time. This agrees with
the analysis in [40], which demonstrated that the weight scalarization method
produce trade-off solution by repeating the analysis process for several
weight coefficients.

Thus, the solution (28) is obtained by using (45)-(47) as follows:

$\overset{*}{q_{i}}\bigl{[}n\bigr{]}(w_{j})=2/\left(1+\log_{2}\left(1+\varPhi\bigl{[}n\bigr{]}\right)\right).$
(48)

By adding (26) and (27) together, the total energy consumption solution is
calculated in terms of (45)-(47) as follows:

$\displaystyle\mathcal{\mathcal{\overset{*}{E}}}_{i}\bigl{[}n\bigr{]}(w_{j})=\overset{*}{q_{i}}\bigl{[}n\bigr{]}\left(2^{\frac{2}{q[n]}}-1\right)\left(\left(\overset{*}{\alpha}_{\mathit{r}}(w_{j})\,+2\overset{*}{\alpha}_{a}(w_{j})\right)H\right.$
$\displaystyle\left.+\left(\overset{*}{\alpha}_{\mathit{r}}(w_{j})+2\overset{*}{\alpha}_{\mathit{b}}(w_{j})\right)\,G\right)\quad
j\in\left\\{a\,\vee\,b,r\right\\},$ (49)

where,
$\varPhi\bigl{[}n\bigr{]}=1+\log_{2}\left(1+\,G\overline{P}\frac{\overset{*}{\alpha}_{\mathit{r}}(w_{j})\>\overset{*}{\alpha}_{\mathit{b}}(w_{j})}{\overset{*}{\alpha}_{\mathit{r}}(w_{j})+\overset{*}{\alpha}_{a}(w_{j})+\overset{*}{\alpha}_{\mathit{b}}(w_{j})\,\frac{G}{H}}\right)$

$+\log_{2}\left(1+\,G\overline{P}\frac{\overset{*}{\alpha}_{\mathit{r}}(w_{j})\overset{*}{\alpha}_{\mathit{b}}(w_{j})}{\overset{*}{\alpha}_{\mathit{r}}(w_{j})+\overset{*}{\alpha}_{a}(w_{j})+\overset{*}{\alpha}_{\mathit{b}}(w_{j})\,\frac{G}{H}}\right)$
$\left(1+\>G\overline{P}\frac{\overset{*}{\alpha}_{\mathit{r}}(w_{j})\>\overset{*}{\alpha}_{a}(w_{j})}{\overset{*}{\alpha}_{a}(w_{j})+(\overset{*}{\alpha}_{\mathit{r}}(w_{j})+\overset{*}{\alpha}_{\mathit{b}}(w_{j}))\frac{G}{H}}\right)$

$+\log_{2}\left(1+\>G\overline{P}\frac{\overset{*}{\alpha}_{\mathit{r}}(w_{j})\>\overset{*}{\alpha}_{a}(w_{j})}{\overset{*}{\alpha}_{a}(w_{j})+(\overset{*}{\alpha}_{\mathit{r}}(w_{j})+\overset{*}{\alpha}_{\mathit{b}}(w_{j}))\frac{G}{H}}\right).$

The (48) and (49) procedures are listed in Algorithm 2.

Input: ${\gamma_{a},\text{and }\gamma_{b}}$

Initialization of parameters

while  _$t\neq 0$_ do

$\mathcal{R}\leftarrow\eqref{eq: R-rate}$

$\mathcal{E}_{a}(\alpha_{a},\alpha_{b},\alpha_{r})\leftarrow\eqref{eq: Ea}$

$\mathcal{E}_{b}(\alpha_{b},\alpha_{b},\alpha_{r})\leftarrow\eqref{eq: Eb}$

$\mathit{q}_{a}(\alpha_{a},\alpha_{b},\alpha_{r})\leftarrow\eqref{eq: qa}$

$\mathit{q}_{b}(\alpha_{a},\alpha_{b},\alpha_{r})\leftarrow\eqref{eq: qb}$

for  _$w_{i}=0:$ to $1$_ do

for  _$w_{j}=0:$ to $1$_ do

$F(\alpha_{a},\alpha_{b},\alpha_{r},w_{i,j})\leftarrow\eqref{eq: 1G-Form-1}$

$\overset{*}{\alpha_{a}}(w_{i,j})\leftarrow\eqref{eq:opt-Alph-A}$

$\overset{*}{\alpha_{b}}(w_{i,j})\leftarrow\eqref{eq:opt-Alph-B}$

$\overset{*}{\alpha_{r}}(w_{i,j})\leftarrow\eqref{eq:opt-Alph-r}$

$\overset{*}{\mathit{q}_{i}}(\overset{*}{\alpha_{a}},\overset{*}{\alpha_{b}},\overset{*}{\alpha_{r}})\leftarrow\eqref{eq:
opt_qt}$

$\overset{*}{\mathcal{E}_{i}}(\overset{*}{\alpha_{a}},\overset{*}{\alpha_{b}}\overset{*}{\alpha_{r}})\leftarrow\eqref{eq:
Opt-Ei}$

end for

end for

end while

Output:
$\mathcal{E}_{a}(\alpha_{a},\alpha_{b},\alpha_{r})$;$\mathcal{E}_{b}(\alpha_{b},\alpha_{b},\alpha_{r})$;
$\mathit{q}_{a}(\alpha_{b},\alpha_{b},\alpha_{r})$;$\mathit{q}_{b}(\alpha_{b},\alpha_{b},\alpha_{r})$

Algorithm 2 Proposed energy-delay trade-off

##  V Bit error rate performance

This section discusses the optimum bit error rate
($\overset{*}{\text{\texthtb}_{e}}$) behaviour of the UAV network based on
Equation (49). The bit error metric defines the number of errors that occur
during data transmission on the UAV network. It can be reformatted for each
transmitted bit (i.e. $\mathcal{\overline{R}}=1)$ as follows

$\frac{1}{\overset{*}{\gamma}(w_{j})[n]}=\frac{1}{\left(\left(\overset{*}{\alpha}_{\mathit{r}}(w_{j})\,+2\overset{*}{\alpha}_{a}(w_{j})\right)H\overline{P}\right.}+\,\frac{1}{\left(\overset{*}{\alpha}_{\mathit{r}}(w_{j})+2\overset{*}{\alpha}_{\mathit{b}}(w_{j})\right)\overline{P}G},$
(50)

where $\overset{*}{\gamma}(w_{j})$ the optimal SNR.

Equation (50) reveals that the $\mathit{SNR}$ is increasing function of
optimal allocation factor $(\varphi_{i}),$
$\varphi_{i}\in\left(\overset{*}{\alpha_{a}}(w_{j}),\overset{*}{\alpha}_{b}(w_{j}),\overset{*}{\alpha}_{r}(w_{j})\right)$.
Further, according to [41], bit error rate decreases when the overall received
SNR is maximised. Thus, increasing $\varphi_{i}$ in (50) allows bit error rate
to be minimised as demonstrate in the following analysis.

In the proposed UAV network, the transmitted signal from a $\mathit{S_{a}}$ to
the $\mathit{\mathit{S_{b}}}$ node propagates through two cascaded channels,
as illustrated in (1). Each channel is a Rayleigh fading and, in such a link,
SNR follows a negative exponential distribution. Then the total SNR obtained
from two i.i.d channels follows a negative exponential distribution. To find
the total probability density function (pdf) of the $\mathit{S_{a}-R}$ and
$\mathit{R-S_{b}}$ channels, distribution of the harmonic mean of two i.i.d.
gamma random variables demonstrated in [42] is applied as follows: first
define pdf of the $\mathit{S_{a}-R}$ as

$pdf^{h}=\frac{e^{-\frac{\eta}{\gamma_{h^{p}}[n]}}}{\gamma_{h^{p}}[n]},$ (51)

and pdf of the $\mathit{R-S_{b}}$ link as

$pdf^{g}=\frac{e^{-\frac{\eta}{\gamma_{g^{p}}[n]}}}{\gamma_{g^{p}}[n]},$ (52)

where
$\gamma_{h^{p}}[n]=\overset{*}{\alpha}_{\mathit{r}}(w_{j})\>H\overline{P}$,
$\gamma_{g^{p}}[n]=\frac{\overset{*}{\alpha}_{\mathit{r}}(w_{j})\>\overset{*}{\alpha}_{\mathit{b}}(w_{j})}{\overset{*}{\alpha}_{\mathit{r}}(w_{j})+\overset{*}{\alpha}_{a}(w_{j})}G\overline{P}$,
and $\eta$ is the harmonic mean according to a gamma distribution defined as
$\eta=\mu_{H}(\gamma_{h^{p}}[n],\gamma_{g^{p}}[n])$ [43].

To joint (51) and (52), the modified harmonic mean demonstrated in [44] is
applied as follows

$\displaystyle
pdf^{t}(\eta)=\left(\frac{2}{\gamma_{h^{p}}[n]\,\gamma_{g^{p}}[n]}k_{0}\left(\frac{2\eta}{\sqrt{\gamma_{h^{p}}[n]\,\gamma_{g^{p}}[n]}}\right)\right.$
$\displaystyle\frac{\gamma_{h^{p}}[n]+\gamma_{g^{p}}[n]}{\bigl{(}\gamma_{h^{p}}[n]\gamma_{g^{p}}[n]\bigr{)}^{3/2}}\>k_{1}\left(\frac{2\eta}{\sqrt{\gamma_{h^{p}}[n]\,\gamma_{g^{p}}[n]}}\right)2\eta\,e^{-\eta\,(\frac{\gamma_{h^{p}}[n]+\gamma_{g^{p}}[n]}{\gamma_{h^{p}}[n]\,\gamma_{g^{p}}[n]})},$
(53)

where $pdf^{t}(\eta)$ is the total pdf, $k_{0}$$(.)$ and $k_{1}(.)$ are the
first and the second order modified Bessel function of the second kind.

Equation (53) is simplified by applying the modified Bessel function
properties as $k_{0}(\eta)\rightarrow 0$ and $k_{1}(\underset{\eta\rightarrow
0}{\eta})\rightarrow 1/\eta$. This gives
$pdf^{t}(\eta)\approx\left(\frac{1}{\gamma_{h^{p}}[n]}+\frac{1}{\gamma_{g^{p}}[n]}\right)e^{-\eta\left(\frac{1}{\gamma_{h^{p}}[n]}+\frac{1}{\gamma_{g^{p}}[n]}\right)}.$
By integrating $pdf^{t}(\eta)$ relative to $\eta$, the cumulative distribution
function is obtained

$\mathcal{F}(\eta)=1-e^{-\eta\left(\frac{1}{\gamma_{h^{p}}[n]}+\frac{1}{\gamma_{g^{p}}[n]}\right)},$
(54)

where $\mathcal{F}(\eta)$ is the cumulative distribution function for
$pdf^{t}(\eta).$

At high SNR domain, the first order expansion of $\mathcal{F}(\eta)$ is given
by

$\mathcal{F}(\eta)=\eta\left(\frac{1}{\gamma_{h^{p}}[n]}+\frac{1}{\gamma_{g^{p}}[n]}\right)+o\left(x^{1+\varepsilon}\right),0<\varepsilon<1$
(55)

Now, the approximate bit error rate of the UAV at a high SNR can be roughly
estimated by using [45] as

$\mathrm{Bit\,error\>rate}=\mathbb{E}\left\\{Q\left(\sqrt{2\gamma}\right)\right\\}=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\eta}}{\sqrt{\eta}}\mathcal{F}(\eta)\>d\eta.$
(56)

Substituting (55) in (56) and evaluating the result lead to obtaining the
optimal bit error rate of the UAV as

$\overset{*}{\text{\texthtb}}[n]=\frac{\Gamma\left(\frac{3}{2}\right)}{\overset{*}{\alpha}_{\mathit{r}}(w_{j})\overline{P}\sqrt{\pi}}\\\
\left(\frac{1}{\left(\overset{*}{\alpha}_{\mathit{r}}(w_{j})\,+2\overset{*}{\alpha}_{a}(w_{j})\right)H}+\frac{1}{\left(\overset{*}{\alpha}_{\mathit{r}}(w_{j})+2\overset{*}{\alpha}_{\mathit{b}}(w_{j})\right)G}\right),$
(57)

where $\Gamma(.)$ is gamma function.

In Algorithm 3, the (57) procedure is demonstrated.

Input: $\overset{*}{\varphi_{i}}$

Initialization of parameters

while  _$t\neq 0$_ do

for  _$w_{i}=0:$ to $1$_ do

$\overset{*}{\gamma}(w_{i})[n]\leftarrow\eqref{eq: opt-SNR}$

$pdf^{h}\leftarrow\eqref{eq: pdf1}$

$pdf^{g}\leftarrow\eqref{eq:pdf2}$

$\mathcal{F}(\eta)\leftarrow\eqref{eq: exp}$

end for

end while

Output: $\text{\texthtb}_{e}[n]$

Algorithm 3 Bit error rate performance

##  VI Simulation Results

In this section, numerical simulations are carried out to assess transmission
delay and energy savings for the proposed UAV network. The analytical outcomes
are validated by utilizing Monte Carlo simulations that use $10^{5}$ samples.
The rest of the simulation specifications are set as: $100\leq d\leq 700$
meters and the maximum powers for users and relay nodes are specified by 2
watts.

The simulation results, which are plotted simultaneously using equations
(48)-(49) refer to the Proposed Algorithm $\mathrm{(\mathit{PA})}$ scheme. The
results of (20)-(24) are plotted under the name Sub-optimal Network
$\mathit{\mathrm{(}SN\mathrm{).}}$

Fig. 2 illustrates optimal trade-off curves between energy and delay
corresponding to various values of $\overset{*}{\alpha_{i}}(w_{\mathit{j}})$.
Each point on the curve corresponds to different energy delay levels, and it
is calculated by adjusting $\overset{*}{\alpha_{i}}(w_{\mathit{j}})$ within
the range between 0 and 1. Based on the trade-off relationship between energy
and delay, we find that the energy decreases monotonically with delay. By
increasing $\overset{*}{\alpha_{i}}(w_{\mathit{j}})$, the optimum delay
decreases under the same energy, as the
$\overset{*}{\alpha_{i}}(w_{\mathit{j}})$ for the delay objective is
decreased. In the same manner, increasing
$\overset{*}{\alpha_{i}}(w_{\mathit{j}})$ leads to reduce energy consumption
due to the long delay in data delivery. As observed, any curve associated with
a specific $\overset{*}{\alpha_{i}}(w_{\mathit{j}}),$ follows an exponential
decay unit approaches to a steady when the transmission delay leans to
infinity. In this case, the energy converges to a constant as delay approaches
$\overset{*}{\alpha_{i}}(w_{\mathit{j}})$ constraint value. The energy, on the
other hand, approaches $\overset{*}{\alpha_{i}}(w_{\mathit{j}})$ constraint
value when the delay tends to infinity. This result is consistent with earlier
researchers results such as [46], which reveals that there is always a trade-
off between energy and delay that enables the performance of the transmission
network to be enhanced. As a continuation of these studies, our proposal in
$\mathit{PA}$ provides a novel trade-off scheme, adopting an energy allocation
strategy to achieve the optimal energy distribution between relay and user
nodes, which, in turn, improves UAV transmission network performance by
reducing the transmission delay (i.e., high data rate) or energy consumption
(i.e., higher energy efficiency).

Figure 2: Energy and Delay trade-off for the $\mathit{PA}$ scheme  Figure 3:
Energy consumption vs link distance for the $\mathit{PA}$ and $\mathit{SN}$
schemes  Figure 4: Energy consumption vs total power for the $\mathit{PA}$ and
$\mathit{SN}$ schemes

Consider the following practical example to illustrate how the framework
proposed can be implemented for UAV networks: transmitting signals, by Sa or
Sb, over poor channel conditions consumes more energy than broadcasting over
normal channel conditions. Under such a scenario, using the proposed method
enables a user’s node to determine whether to send packets or leave the users
idle based on the energy consumption. Then, the proposed approach ensures that
energy consumption is conserved in UAV networks by creating a decision-making
scheme. In order to clarify the characteristic of this decision-making
platform, Fig. 3 illustrates the energy consumption curves versus total
distance $d$, which represents the sum of $d_{a}$ and $d_{b}$. It is clearly
shown that $\mathit{d}$ has a significant impact on energy consumption output
as it decreases significantly with increasing $\mathit{d}$. This is because
increasing $d_{a}$ requires a higher transmit power by $\mathit{S_{a}}$ node
to suppress channel-fading growth, and, hence, higher energy consumption. This
result agrees with what is observed in [47]. In addition, Fig. 3 shows that
the proposed scheme allows energy consumption to be decreased by about 15%
more than the $\mathit{SA}$ scheme.

Fig. 4 depicts that with the rise in power, $\mathcal{E}$ initially decreases
and then reaches a constant. This is because $\mathcal{R}$ only increases
logarithmically with $\mathrm{\mathit{\overline{P}}}$ while the transmission
power consumption increases linearly with the transmission power [48].
Consequently, increasing power divides energy consumption into two stages. In
the first stage, the energy is decreased; then, in the second stages, it
approaches a constant rate. This reveals that there is a transition point that
allows for the improvement of energy consumption. Fig. 4, illustrates that the
domain of the transition point is between 13 and 17 dBm; and a further
reduction of energy consumption is shown by the proposed method of
$\mathit{PA,}$ which corresponds to previous studies [48, 49]. Further
validation is given by Fig. 5, which compares the average data rate and the
maximum transmit power. It is clear that the average data rate in (48) tends
to become a constant at the high range of the transmit power (range between 14
and 16 dBm). This is because the power allocation strategy enables the
transmit power to be regulated [50]. By using the $\mathit{PA}$ scheme, a
higher average system rate is achieved, as the $\mathit{PA}$ minimises both
$\mathcal{E}$ and $\mathit{q}$ (i.e., increases $\mathcal{R}$) simultaneously.
Therefore, $\mathcal{R}$ increases logarithmically, while the transmission
power consumption increases linearly with the transmission power [48]. In
other words, the transition point is improved significantly, and such
enhancement is expected to enhance other performance parameters in UAV
networks. For example, Fig. 6 illustrates the relationship between distance
and achievable data rate. It shows that increasing distance between
$\mathit{S_{a}}$ and $\mathit{\mathit{S_{b}}}$ nodes gradually reduces
achievable data rates due to high channel gain degradation. However, the
$\mathit{PA}$ is always better than the $\mathit{SN}$, which agrees with other
results obtained by [29].

Figure 5: Achievable data rate vs total power for the $\mathit{PA}$ and
$\mathit{SN}$ schemes

In the $\mathit{PA}$, the rate of distance change depends on flight altitude,
and this is required to explain the effect of the drone altitude on data rate
performance, as illustrated in Fig. 7. As depicted in the figure, when the
drone takes off from the $\mathit{S_{a}}$ node level at 100 m, the data rate
decreases slightly with the distance travelled by the drone remains within a
low path loss range [51]. By increasing the altitude between $\mathit{S_{a}}$
node and drone, the path loss increases significantly and the data rate, in
turn, starts to decrease gradually. The results of the proposed algorithm in
the $\mathit{PA}$ show a marked improvement because increasing the altitude
extend $d_{a}$ and, then, the arrival data rate at the relay is decreased.
Hence, in addition to reducing transmission delay, both user nodes allocation
of a higher power to each other in order to motivate information growth. The
data rate curve is enhanced compared with other studies that depend on energy
allocation only, as in [52]. To explain how the proposed
$\overset{*}{\alpha_{i}}(w_{j})$ is allocated for the UAV network, Fig. 8
plots the relationship between $\overset{*}{\alpha_{i}}(w_{j})$ and both
$\overline{P}$ and $w_{j}.$ It can be seen that increasing $\overline{P}$
allows $\overset{*}{\alpha_{i}}(w_{j})$ to be increased while notably
decreasing $w_{j}$. The user nodes manage such a high $\overline{P}$ based on
many factors, such as channel conditions or distance $d_{a}$ and $d_{b}$.
Thus, if $d_{a}$ is higher than $d_{b},$ the path loss of $d_{a}$ is large and
the rate at which data arrives at the relay is low. Therefore, the user nodes
allocate higher $\overset{*}{\alpha_{i}}(w_{j})$ to stimulate an increase in
data transmission by each user when there is an increase in delay time.

To evaluate UAV performance in (57), Fig. 9. shows results of
$\text{\texthtb}_{e}[n]$ versus $\varphi_{i}$, and for various SNRs values,
where
$\overset{*}{\gamma}(w_{j})[n_{3}]>\overset{*}{\gamma}(w_{j})[n_{2}]>\overset{*}{\gamma}(w_{j})[n_{1}].$
Each $\overset{*}{\gamma}(w_{j})[n]$ is obtained from a specific value of
$\varphi_{i}$ ($0<\varphi_{i}<1).$ It is clearly seen that the highest SNR, at
$\overset{*}{\gamma}(w_{j})[n_{3}]$, enhanced the bit error rate result, and
this result confirms the theoretical analysis, as high SNR calculates from
optimal power allocation. This result agrees with previous studies [53], which
indicate that the power allocation $\varphi_{i}$ is an effective metric for
enhancing UAV performance.

Fig. 9. compares between the $SN$ and $PA$ system in term of $\varphi_{i}$ and
bit error rate. Using energy allocation in (49) rises SNR[$\mathit{n}$] which
in turn increases data rate, and the result enhancing bit error rate. Fig. 10.
also shows that the highest SNR, i.e.
$\textit{{SNR}}[n_{3}]>\textit{{SNR}}[n_{2}]>\textit{{SNR}}[n_{1}]$
$\textit{{SNR}}[n]\in\left\\{\gamma(w_{j})[n],\overset{*}{\gamma}(w_{j})[n]\right\\},$
gives the lower bit error rate. This result is consistent with previous
results [54] for terrestrial communication systems. Obviously, a higher SNR
results in better performance for the $PA$ as compare in $\mathit{SN}$,
consequently, the proposed $\overset{*}{\text{\texthtb}}[n]$ can improve UAV
performance.

Figure 6: Achievable data rate vs link distance for the $\mathit{PA}$ and
$\mathit{SN}$ schemes  Figure 7: Achievable data rate vs altitude flight for
the $\mathit{PA}$ and $\mathit{SN}$ schemes  Figure 8: Achievable data rate vs
total power for the $\mathit{PA}$ scheme Figure 9: Power allocation vs bit
error rate for the $\mathit{PA}$ scheme Figure 10: Achievable data rate vs bit
error rate for the $\mathit{PA}$ and $\mathit{SN}$ schemes

##  VII Conclusion and Future Work

This paper offers a new method for drawing out the most effective energy-delay
curve for a UAV layout by optimising the energy allotments for users and relay
nodes. A multi-objective technique for optimising the energy allocation
factors, weight scalarization optimisation, is appraised in this paper. The
signal transmission distance is taken into consideration to evaluate UAVs.
Simulations confirm the results originating from analytical expressions, and a
real-world application scenario demonstrates how the proposed structure will
be used. The paper concludes that the proposed technique effectively executes
optimal decision-making and presents a compromise between energy and delay in
UAVs. It would be interesting to extend our proposed study by considering
future works’ weighted interval scheduling problem. Also, different system
scenarios such as multiple UAVs could be employed instead of a single UAV.
Besides that, statistical channel state information estimation can be used
instead of instantaneous channel state information. Further, computational
complexity is another direction that would be required to evaluate in future
studies.

## References

  * [1] M. I. Khalil, “Energy efficiency maximization of relay aerial robotic networks,” _IEEE Transactions on Green Communications and Networking_ , vol. 4, no. 4, pp. 1081–1090, 2020.
  * [2] F. Ono, H. Ochiai, and R. Miura, “A wireless relay network based on unmanned aircraft system with rate optimization,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 11, pp. 7699–7708, Nov 2016.
  * [3] W. Zeng, J. Zhang, D. W. K. Ng, B. Ai, and Z. Zhong, “Two-way hybrid terrestrial-satellite relaying systems: Performance analysis and relay selection,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 7, pp. 7011–7023, 2019.
  * [4] R. Zhang and J. Gorce, “Optimal transmission range for minimum energy consumption in wireless sensor networks,” in _2008 IEEE Wireless Communications and Networking Conference_ , 2008, pp. 757–762.
  * [5] B. Zhou, L. Jiang, S. Zhao, and C. He, “BER analysis of TDD downlink multiuser MIMO systems with imperfect channel state information,” _EURASIP Journal on Applied Signal Processing_ , vol. 2011, no. 1, Dec. 2011\.
  * [6] A. Zappone, L. Sanguinetti, and M. Debbah, “Energy-delay efficient power control in wireless networks,” _IEEE Transactions on Communications_ , vol. 66, no. 1, pp. 418–431, 2018.
  * [7] A. Brandt, _Optimization Methods for Material Design of Cement-based Composites_ , ser. Modern Concrete Technology. Taylor & Francis, 1998. [Online]. Available: https://books.google.com.au/books?id=RZTPi0V1JKEC
  * [8] P. Nuggehalli, V. Srinivasan, and R. R. Rao, “Delay constrained energy efficient transmission strategies for wireless devices,” in _Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies_ , vol. 3, 2002, pp. 1765–1772 vol.3.
  * [9] Xiliang Zhong and Cheng-Zhong Xu, “Delay-constrained energy-efficient wireless packet scheduling with qos guarantees,” in _GLOBECOM ’05. IEEE Global Telecommunications Conference, 2005._ , vol. 6, 2005, pp. 5 pp.–3340.
  * [10] E. Uysal-Biyikoglu, B. Prabhakar, and A. El Gamal, “Energy-efficient packet transmission over a wireless link,” _IEEE/ACM Transactions on Networking_ , vol. 10, no. 4, pp. 487–499, 2002.
  * [11] M. A. Zafer and E. Modiano, “A calculus approach to minimum energy transmission policies with quality of service guarantees,” in _Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies._ , vol. 1, 2005, pp. 548–559 vol. 1.
  * [12] C. E. Shannon, “Probability of error for optimal codes in a gaussian channel,” _The Bell System Technical Journal_ , vol. 38, no. 3, pp. 611–656, 1959.
  * [13] Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” _IEEE Transactions on Information Theory_ , vol. 56, no. 5, pp. 2307–2359, 2010.
  * [14] ——, “Minimum energy to send $k$ bits through the gaussian channel with and without feedback,” _IEEE Transactions on Information Theory_ , vol. 57, no. 8, pp. 4880–4902, 2011.
  * [15] R. A. Berry and R. G. Gallager, “Communication over fading channels with delay constraints,” _IEEE Transactions on Information Theory_ , vol. 48, no. 5, pp. 1135–1149, 2002.
  * [16] O. Waqar, M. Imran, M. Dianati, and R. Tafazolli, “Energy consumption analysis and optimization of ber-constrained amplify-and-forward relay networks,” _Vehicular Technology, IEEE Transactions on_ , vol. 63, no. 3, pp. 1256–1269, March 2014.
  * [17] Y. Zeng, R. Zhang, and T. J. Lim, “Throughput maximization for uav-enabled mobile relaying systems,” _IEEE Transactions on Communications_ , vol. 64, no. 12, pp. 4983–4996, Dec 2016.
  * [18] Y. Chen, W. Feng, and G. Zheng, “Optimum placement of uav as relays,” _IEEE Communications Letters_ , vol. 22, no. 2, pp. 248–251, Feb 2018.
  * [19] X. Jiang, Z. Wu, Z. Yin, and Z. Yang, “Power and trajectory optimization for uav-enabled amplify-and-forward relay networks,” _IEEE Access_ , vol. 6, pp. 48 688–48 696, 2018.
  * [20] K. Govindan, K. Zeng, and P. Mohapatra, “Probability density of the received power in mobile networks,” _IEEE Transactions on Wireless Communications_ , vol. 10, no. 11, pp. 3613–3619, 2011.
  * [21] A. Al-Hourani, S. Kandeepan, and S. Lardner, “Optimal lap altitude for maximum coverage,” _IEEE Wireless Communications Letters_ , vol. 3, no. 6, pp. 569–572, 2014.
  * [22] M. Alzenad, A. El-Keyi, F. Lagum, and H. Yanikomeroglu, “3-d placement of an unmanned aerial vehicle base station (uav-bs) for energy-efficient maximal coverage,” _IEEE Wireless Communications Letters_ , vol. 6, no. 4, pp. 434–437, 2017.
  * [23] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Unmanned aerial vehicle with underlaid device-to-device communications: Performance and tradeoffs,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 6, pp. 3949–3963, June 2016.
  * [24] S. Zhang, H. Zhang, Q. He, K. Bian, and L. Song, “Joint trajectory and power optimization for uav relay networks,” _IEEE Communications Letters_ , vol. 22, no. 1, pp. 161–164, 2018.
  * [25] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Mobile unmanned aerial vehicles (uavs) for energy-efficient internet of things communications,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 11, pp. 7574–7589, 2017.
  * [26] M. Luo, G. Villemaud, J. M. Gorce, and J. Zhang, “Realistic prediction of ber and amc for indoor wireless transmissions,” _IEEE Antennas and Wireless Propagation Letters_ , vol. 11, pp. 1084–1087, 2012.
  * [27] M. I. Khalil, “Delay and energy balance for unmanned aerial vehicle networks,” in _2021 IEEE 11th Annual Computing and Communication Workshop and Conference (CCWC)_ , 2021, pp. 1453–1458.
  * [28] O. Weck, “Multiobjective optimization : History and promise,” in _MULTIOBJECTIVE OPTIMIZATION : HISTORY AND PROMISE_ , 2004.
  * [29] C. Tian, Z. Qian, X. Wang, and L. Hu, “Analysis of joint relay selection and resource allocation scheme for relay-aided d2d communication networks,” _IEEE Access_ , vol. 7, pp. 142 715–142 725, 2019.
  * [30] D. Yang, Q. Wu, Y. Zeng, and R. Zhang, “Energy tradeoff in ground-to-uav communication via trajectory design,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 7, pp. 6721–6726, July 2018.
  * [31] Y. Li, X. Zhang, M. Peng, and W. Wang, “Power provisioning and relay positioning for two-way relay channel with analog network coding,” _IEEE Signal Processing Letters_ , vol. 18, no. 9, pp. 517–520, Sept 2011\.
  * [32] M. Khalil, S. Berber, and K. Sowerby, “High snr approximation for performance analysis of two-way multiple relay networks,” _Physical Communication_ , vol. 24, pp. 62 – 70, 2017. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1874490716301446
  * [33] A. S. Y. Poon, D. N. C. Tse, and R. W. Brodersen, “An adaptive multiantenna transceiver for slowly flat fading channels,” _IEEE Transactions on Communications_ , vol. 51, no. 11, pp. 1820–1827, 2003.
  * [34] J. Zander, “Radio resource management in future wireless networks: requirements and limitations,” _IEEE Communications Magazine_ , vol. 35, no. 8, pp. 30–36, 1997.
  * [35] Y. Zhang, Y. Ma, and R. Tafazolli, “Power allocation for bidirectional af relaying over rayleigh fading channels,” _Communications Letters, IEEE_ , vol. 14, no. 2, pp. 145–147, 2010, iD: 1.
  * [36] A. B. Kiely and J. T. Coffey, “On the capacity of a cascade of channels,” _IEEE Transactions on Information Theory_ , vol. 39, no. 4, pp. 1310–1321, 1993.
  * [37] S. Salehkalaibar, M. Wigger, and L. Wang, “Hypothesis testing over cascade channels,” in _2017 IEEE Information Theory Workshop (ITW)_ , 2017, pp. 369–373.
  * [38] J. Ren, G. Yu, Y. He, and G. Y. Li, “Collaborative cloud and edge computing for latency minimization,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 5, pp. 5031–5044, 2019.
  * [39] J. Koski, “Defectiveness of weighting method in multicriterion optimization of structures,” _Communications in Applied Numerical Methods_ , vol. 1, no. 6, pp. 333–337, 1985. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.1630010613
  * [40] T. B. To and U. Korn, “An evolution strategy for the multiobjective optimization,” _The Second International Conference on Genetic Algorithms, Brno, Czech Republic_ , 01 1996.
  * [41] G. Farhadi and N. C. Beaulieu, “On the performance of amplify-and-forward cooperative systems with fixed gain relays,” _Wireless Communications, IEEE Transactions on_ , vol. 7, no. 5, pp. 1851–1856, 2008.
  * [42] M. Hasna and M.-S. Alouini, “Harmonic mean and end-to-end performance of transmission systems with relays,” _Communications, IEEE Transactions on_ , vol. 52, no. 1, pp. 130–135, Jan 2004.
  * [43] ——, “End-to-end performance of transmission systems with relays over rayleigh-fading channels,” _Wireless Communications, IEEE Transactions on_ , vol. 2, no. 6, pp. 1126–1131, Nov 2003.
  * [44] R. H. Louie, Y. Li, H. Suraweera, and B. Vucetic, “Performance analysis of beamforming in two hop amplify and forward relay networks with antenna correlation,” _Wireless Communications, IEEE Transactions on_ , vol. 8, no. 6, pp. 3132–3141, June 2009.
  * [45] Z. Wang and G. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” _Communications, IEEE Transactions on_ , vol. 51, no. 8, pp. 1389–1398, Aug 2003.
  * [46] B. Gurakan, O. Ozel, and S. Ulukus, “Optimal energy and data routing in networks with energy cooperation,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 2, pp. 857–870, 2016.
  * [47] K. Yang, S. Martin, C. Xing, J. Wu, and R. Fan, “Energy-efficient power control for device-to-device communications,” _IEEE Journal on Selected Areas in Communications_ , vol. 34, no. 12, pp. 3208–3220, 2016.
  * [48] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy-efficient resource allocation in ofdma networks,” _IEEE Transactions on Communications_ , vol. 60, no. 12, pp. 3767–3778, 2012.
  * [49] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation in ofdma systems with large numbers of base station antennas,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 9, pp. 3292–3304, 2012.
  * [50] ——, “Energy-efficient resource allocation in multi-cell ofdma systems with limited backhaul capacity,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 10, pp. 3618–3631, 2012.
  * [51] W. Khawaja, I. Guvenc, D. W. Matolak, U. Fiebig, and N. Schneckenburger, “A survey of air-to-ground propagation channel modeling for unmanned aerial vehicles,” _IEEE Communications Surveys Tutorials_ , vol. 21, no. 3, pp. 2361–2391, 2019.
  * [52] L. Sboui, H. Ghazzai, Z. Rezki, and M. Alouini, “Achievable rates of uav-relayed cooperative cognitive radio mimo systems,” _IEEE Access_ , vol. 5, pp. 5190–5204, 2017.
  * [53] M. Mondelli, Q. Zhou, X. Ma, and V. Lottici, “A cooperative approach for amplify-and-forward differential transmitted reference ir-uwb relay systems,” in _2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2012, pp. 2905–2908.
  * [54] W. Al-Hanafy and S. Weiss, “Discrete rate maximisation power allocation with enhanced bit error ratio,” _IET Communications_ , vol. 6, pp. 1019–1024(5), June 2012.