# On Effective Secrecy Throughput of Underlay Spectrum Sharing $\alpha-\mu$/
Málaga Hybrid Model under Interference-and-Transmit Power Constraints

Md. Ibrahim  Department of Electrical & Electronic Engineering, Rajshahi
University of Engineering & Technology (RUET), Rajshahi-6204, Bangladesh  A.
S. M. Badrudduza  Md. Shakhawat Hossen  M. K. Kundu  Department of Electrical
& Computer Engineering, RUET  Imran Shafique Ansari  James Watt School of
Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom

###### Abstract

Abstract: The underlay cognitive radio-based hybrid radio frequency / free-
space optical (RF / FSO) systems have been emerged as a promising technology
due to its ability to eliminate spectrum scarcity and spectrum under-
utilization problems. Consequently, this work analyzes the physical layer
security aspects of a cognitive RF / FSO hybrid network that includes a
primary user, a secondary source, a secondary receiver, and an eavesdropper
where the secret communication takes place between two legitimate secondary
peers over the RF and FSO links simultaneously, and the eavesdropper can
overhear the RF link only. In particular, the maximum transmit power
limitation at the secondary user as well as the permissible interference power
restriction at the primary user are also taken into consideration. All the RF
links are modeled with $\alpha$-$\mu$ fading whereas the FSO link undergoes
Málaga ($\mathcal{M}$) turbulence with link blockage and pointing error
impairments. At the receiver, the selection combining diversity technique is
utilized to select the signal with the best electrical signal-to-ratio (SNR).
Moreover, the closed-form expressions for the secrecy outage probability,
probability of strictly positive secrecy capacity, and effective secrecy
throughput are derived to analyze the secrecy performance. Besides, the
impacts of fading, primary-secondary interference, detection techniques, link
blockage probability, atmospheric turbulence, and pointing error are examined.
Finally, Monte-Carlo simulations are performed to corroborate the derived
expressions.

Keywords: Cognitive underlay network, effective secrecy throughput, hybrid RF
/ FSO system, Málaga turbulence, pointing error, secure outage probability.

##  1 Introduction

###  1.1 Background and Related Works

With the rapid increase in wireless devices, the shortage of spectrum owing to
the data traffic reflects a significant threat in contemporary technology [1,
2]. In order to meet the requirements, several optimistic technologies have
been developed recently. Among these, both cognitive radio network (CRN) and
free-space optical (FSO) systems have received considerable attention from the
research communities [3, 4]. Although FSO communication appears with specific
merits, such as high bandwidth, inherent security, and cost-effective
operation, several issues like atmospheric turbulence and pointing error can
affect the secure transmission greatly [5]. Furthermore, the probability of
transmission blockage due to extreme line-of-site (LOS) necessity in FSO link
leads to the system failure [6]. Hence, utilizing radio frequency (RF)
technology as an alternative option along with the FSO system assures
successful transmission since the FSO link delivers high data speed while the
RF link is independent of the atmospheric turbulence and weather conditions
[7].

Due to the increased demand for faster data rates in wireless communication,
hybrid networks, which consist of RF and FSO systems, are deployed utilizing a
variety of techniques. An adaptive coding method in order to enhance the
system performance of RF / FSO hybrid network was considered in [8, 9].
Nonetheless, the performance of these systems is highly dependent on the
existence of feedback information at the receiver [9]. Another attractive
feature is the switching mechanism that uses RF link when FSO link is down
[10]. Since one link is underutilized at a time, there is a significant amount
of bandwidth wasted [11]. To extend the communication range, several existing
works like [12, 13, 14, 15] utilize relaying protocols in mixed RF-FSO models.
However, weather conditions (i.e. fog) can influence the performance of dual-
hop networks badly. Unlike [10, 11], the considered hybrid RF / FSO model
dispatches identical information via two separate links simultaneously and
combines the received signal using the diversity combining method. Again, the
mixed RF-FSO system differs completely from the hybrid RF / FSO model in terms
of system configuration. Recently, a comprehensive analysis on hybrid RF / FSO
model was performed in [16, 17] where the hybrid system exhibits better link
availability compared to the individual links.

In CRN, an unlicensed user gets the permission of accessing the same spectrum
of the licensed user through different spectrum sharing methods [18]. More
particularly, in the underlay approach, the secondary user ($SU$) makes use of
the primary user’s ($PU$) shared spectrum as long as the interference
occurring at SU is less than the permitted threshold value [4]. Considering CR
technology in the RF links, the system performance of mixed RF-FSO systems are
investigated in [19, 20, 18, 21, 22, 4]. In [20], the performance analysis was
accomplished by developing analytical expressions of outage probability (OP)
where RF-FSO links were subjected to Nakagami-$m$ fading and double
generalized Gamma (DGG) turbulence, respectively. However, this model was
extended in [18, 21] where the impact of antenna diversity in RF links was
observed. In [22], the OP analysis was performed for amplify-and-forward (AF)
relaying-based dual-hop network while multi-users and multi-destination were
taken into consideration. The error probability (EP) expression was derived in
[4] where the influence of RF and FSO parameters on the system performance was
inspected.

Security has become a widespread concern in the next-generation wireless
communications (5G and beyond) due to its high risk of information leakage
[23, 24]. In this domain, physical layer security (PLS) has been recognized as
a complementary choice to the traditional cryptography strategy [25, 26].
According to the comprehensive literature analysis, current PLS research has
mainly limited on hybrid RF / FSO models [25, 27, 28] and mixed RF-FSO
networks [29, 30, 31]. In [25], the PLS of a hybrid FSO / RF network was
examined and it was found that secrecy performance is highly dependent on air
turbulence, pointing error, and RF fading parameters. In [27], authors
obtained the secrecy diversity gain for RF backhaul system along with parallel
FSO link whereas the impact of power allocation on secrecy performance was
investigated in [28] for the parallel mmWave and FSO links. Including link
blockage (LB) probability in the FSO link due to the sudden presence of moving
objects (e.g. clouds, birds, insects, etc.), [32] introduced the strategy of
enhancing secrecy behavior for CR-based hybrid RF / FSO system.

###  1.2 Motivation and Contributions

Despite the fact that a CR-based hybrid RF / FSO system configuration is more
robust and practical than a single RF or FSO link [27], there are very few
studies that have looked into the secrecy analysis of such systems and
majority of the previously mentioned research focused on performance analysis
of hybrid RF / FSO systems only. As a result, the investigation of PLS over
the CR-based hybrid model is currently an open concept. To the best of
authors’ knowledge, there are no works that conduct secrecy analysis of a
hybrid CR RF / FSO model considering double power constraints (i.e. the
transmit power constraints at the $SU$ and maximum allowable interference
power at the $PU$). Being motivated by this research gap, in this work, the
security performance for an underlay CR based hybrid RF / FSO fading channel
is demonstrated taking into account the single power constraint (i.e. maximum
permissible interference at the $SU$) and the double power constrains in the
presence of an eavesdropper that can intercept the secure communication via a
RF link. The generalized $\alpha$-$\mu$ fading channel is considered for all
the RF links since it can characterize small scale fading perfectly [33]. The
FSO link, on the other hand, is assumed to follow Málaga fading distribution,
since it is also a generalized model and capable of accurately describing the
pointing error and turbulence scenarios [34]. The contributions of this
research are highlighted as follows:

  1. 1.

Firstly, the expressions of cumulative density function (CDF) for selection
combining (SC) based hybrid cognitive spectrum-sharing RF / FSO network
assuming two different scenarios (e.g. single and double power constraints)
are derived in closed-form. It is noteworthy that most of the works in the
literature explored secrecy behaviour of the CR RF / FSO schemes considering
maximum interference power constrain at $PU$ only, which is just a single part
of the proposed model. In this perspective, considering both single and double
power constraints in such type of hybrid model is certainly a novel and more
practical approach. Moreover, this expressions of CDF also incorporates some
other existing works [16, 17, 25, 27, 32] as special cases.

  2. 2.

Secondly, the secrecy characteristics are derived for the proposed system
model utilizing the underlay approach over the RF links and developing the
analytical expressions for secrecy outage probability (SOP), probability of
strictly positive secrecy capacity (SPSC), and effective secrecy throughout
(EST) for the two scenarios. Since the considered $\alpha-\mu$ and Málaga
models are generalized, and combination of the two in such type of hybrid
model has been considered for the first time in the literature, according to
the authors’ knowledge, the derived expressions are also novel and
generalized.

  3. 3.

The derived expressions are further utilized to realize some numerical
outcomes with some specific figures demonstrating impacts of each system
parameters i.e. fading, primary-secondary interference, LB probability,
pointing error, detection techniques, and various atmospheric turbulence
conditions, etc. It is also notable that both detection techniques, i.e.
heterodyne detection (HD) and intensity modulation/direct detection (IM/DD)
techniques, have not been considered yet in such type of hybrid system model
except in this work. Finally, the accuracy of the deduced analytical
expressions are corroborated via Monte-Carlo (MC) simulations.

###  1.3 Organization

The outline of this paper is arranged as follows: In section-2, 3, and 4, the
proposed framework along with the channel models is described. The novel
analytical expressions of the performance metrics (i.e. SOP, probability of
SPSC, and EST analysis for the two scenarios) are demonstrated in Section-5.
Section-6 incorporates some exclusive numerical results followed by insightful
discussions. Finally, the concluding remarks of this work are presented in
Section-7.

##  2 System Model

Figure 1: Proposed system model for hybrid RF / FSO based cognitive underlay
network.

A hybrid cognitive underlay network (CUN) is illustrated in Fig. 1, which
incorporates two parallel links: cognitive RF and FSO sub-systems,
respectively. It is essential to mention that FSO allows the benefits of using
a license-free spectrum with high data rates [12]. But in the FSO sub-system,
the signal transmission can be substantially disrupted by atmospheric
turbulence conditions and pointing errors. Hence, the RF sub-system is
utilized to transmit the same information concurrently with the FSO sub-system
thereby maintaining a continuous data transmission. To prevent any data loss,
the receiver $R$ exploits selection combining diversity and always picks the
best signal, i.e. the signal with highest (signal-to-noise ratio) SNR, among
the two sub-links. Since the secondary user ($S$) utilizes the given licensed
spectrum of the primary user ($P$), in the proposed underlay technique, the
system requires that there must be no adverse interference at $P$.

####  2.0.1 Cognitive RF Sub-link

Based on the underlay approach, $S$ transmits its concealed information to $R$
under the following constraints:

  * •

The peak tolerable interference power impinged by $S$ on $P$ cannot exceed a
predefined value $P_{Q}$. In such a case, $S$ is not considered as a power
limited terminal and it has the full freedom to utilize its power depending on
a single power constraint, i.e. interference severity at $P$. This particular
case is denoted as Scenario-I in the rest of this work.

  * •

The CUN in Scenario-I sometimes may not be practical, specifically when the
channel coefficient of $S-P$ link rapidly fluctuates. This may lead to
feedback burden since the instantaneous feedback gain is difficult to track.
To mitigate these types of feedback burden and feedback errors, mean value
power allocation strategy can be adopted [35]. Hence, besides only
interference/single power constraint case, in double power constraint case,
$S$ is assumed to be a power limited terminal and allowed to exploit maximum
transmit power $P_{T}$. This particular case is denoted as Scenario-II in the
remaining manuscript.

For Scenario-II, utilizing mean value power allocation, the transmit power of
$S$ is denoted as

$\displaystyle
P_{t,II}=\min\biggl{(}\frac{P_{Q}}{|g_{p}|^{2}},P_{T}\biggl{)},$ (1)

where $g_{p}$ denotes the channel gain of $S-P$ interference link. Therefore,
the SNR at $R$ is expressed as

$\displaystyle\gamma_{r,II}=\min\biggl{(}\frac{\Psi_{Q}}{|g_{p}|^{2}},\Psi_{T}\biggl{)}|g_{r}|^{2},$
(2)

where $\Psi_{Q}=\frac{P_{Q}}{N_{r}}$, $\Psi_{T}=\frac{P_{T}}{N_{r}}$, $N_{r}$
represents the noise power imposed on $R$, and $g_{r}$ is the channel gain of
$S-R$ RF sub-link. However, when $S$ is not a power-limited terminal
(Scenario-I), the transmit power at $S$ is represented as

$\displaystyle P_{t,I}=\frac{P_{Q}}{|g_{p}|^{2}}.$ (3)

In this case, the SNR of $S-R$ RF sub-link is expressed as

$\displaystyle\gamma_{r,I}=\frac{\Psi_{Q}|g_{r}|^{2}}{|g_{p}|^{2}}.$ (4)

####  2.0.2 FSO Sub-link

The FSO sub-system comprises of $S$ with a transmit aperture, FSO sub-link,
and $R$ with a receive photo-detector. The transmitter dispatches information
in optical form that is further converted into electrical one utilizing the
photo-detector at $R$. Thereby, instantaneous SNR for $S-R$ FSO sub-link is
written as

$\displaystyle\gamma_{o}=\frac{P_{f}}{N_{o}}\|g_{o}\|^{2},$ (5)

where $g_{o}$ signifies the channel gain of $S-R$ FSO sub-link, $N_{o}$
denotes the optical noise power at $R$, and $P_{f}$ is the transmit optical
power at $S$.

Finally, after selection combining at $R$, the received SNR is written as [25,
Eq. (11)]

$\displaystyle\gamma_{f,j}=\max(\gamma_{r,j},\gamma_{o}),$ (6)

where $j\in\\{I,II\\}$.

####  2.0.3 Eavesdropper Link

In the proposed system, an eavesdropper ($E$) targets the RF sub-link to
intercept the confidential data stream. The instantaneous SNR at $E$ is
expressed as

$\displaystyle\gamma_{e,j}=\frac{P_{t,j}}{N_{e}}\|g_{e}\|^{2},$ (7)

where $g_{e}$ denotes the channel gain of $S-E$ link and $N_{e}$ is the noise
power at $E$.

##  3 Channels Realization

In this section, the channel models of the $S-R$ (RF and FSO sub-links),
$S-E$, and $S-P$ links are realized individually for the further utilization
in the mathematical modeling of the proposed CUN system.

###  3.1 PDF and CDF of SNRs for the RF Links

The $S-R$ RF sub-link, $S-P$, and $S-E$ links are assumed to follow
$\alpha-\mu$ distribution, where $\alpha$ denotes the non-linearity parameter
of propagation environment ($\alpha>0$) and $\mu$ represents the number of
multi-path clusters ($\mu>0$). Since $\alpha-\mu$ model exhibits generalized
characteristics [36], several multi-path models [37], e.g. exponential,
Rayleigh, Weibull, Gaussian, Gamma, and Nakagami-$m$, etc., can be obtained by
setting different values of $\alpha$ and $\mu$. The PDF of $\gamma_{i}$,
$i\in\\{r,p,e\\}$, where $r,p,$ and $e$ correspond to the $S-R$ (RF), $S-P$,
and $S-E$ links, respectively, is given as [38, Eq. (2)]

$\displaystyle
f_{\gamma_{i}}(\gamma)=\frac{\alpha_{i}\delta_{i}^{\mu_{i}}}{2\Gamma(\mu_{i})}e^{-\delta_{i}\gamma^{\tilde{\alpha_{i}}}}\gamma^{\Theta_{i}}.$
(8)

Here $\delta_{i}=\Phi_{i}^{-\tilde{\alpha_{i}}}$,
$\Theta_{i}=\tilde{\alpha_{i}}\mu_{i}-1$,
$\tilde{\alpha_{i}}=\frac{\alpha_{i}}{2}$, $\Phi_{i}$ is the average SNR of
the RF links, and $\Gamma(\cdot)$ denotes the Gamma operator. The CDF of
$\gamma_{i}$ is defined as

$\displaystyle
F_{\gamma_{i}}(\gamma)=\int_{0}^{\gamma}f_{\gamma_{i}}(\gamma)d\gamma.$ (9)

Plugging (8) into (9), and after performing integration upon utilizing [39,
Eq. (3.381.8)], the CDF leads to

$\displaystyle
F_{\gamma_{i}}(\gamma)=\frac{\gamma(\mu_{i},\delta_{i}\gamma^{\tilde{\alpha_{i}}})}{\Gamma(\mu_{i})},$
(10)

where $\gamma(,.,)$ denotes the lower incomplete Gamma function. Finally, with
the help of [39, Eq. (8.352.6)], the CDF of $\gamma_{i}$ can be expressed in
an alternative form as

$\displaystyle
F_{\gamma_{i}}(\gamma)=1-e^{-\delta_{i}\gamma^{\tilde{\alpha_{i}}}}\sum_{m_{i}=0}^{\mu_{i}-1}\frac{(\delta_{i}\gamma^{\tilde{\alpha_{i}}})^{m_{i}}}{m_{i}!}.$
(11)

###  3.2 PDF and CDF of SNR for FSO SUB-Link

The $S-R$ FSO sub-link is considered to follow Málaga ($\mathcal{M}$)
turbulence. Since Málaga turbulence is a generalized FSO channel, it
incorporates some other distributions such as Gamma-Gamma, lognormal, Rice-
Nakagami, and Gamma as special cases [31, Table II]. Hence, the PDF of
$\gamma_{o}$ is expressed as [40, Eq. (9)]

$\displaystyle
f_{\gamma_{o}}(\gamma)=\frac{\epsilon^{2}\chi_{o}}{2^{s}\gamma}\sum_{m_{o}=0}^{\beta_{o}}\vartheta_{n}G_{1,3}^{3,0}\left[\varpi\biggl{(}\frac{\gamma}{\mu_{s}}\biggl{)}^{\frac{1}{s}}\biggl{|}\begin{array}[]{c}\epsilon^{2}+1\\\
\epsilon^{2},\alpha_{o},m_{o}\\\ \end{array}\right],$ (14)

where

$\displaystyle\chi_{o}$
$\displaystyle=\frac{2\alpha_{o}^{\frac{\alpha_{o}}{2}}}{g^{1+\frac{\alpha_{o}}{2}}\Gamma(\alpha_{o})}\biggl{(}\frac{g\beta_{o}}{g\beta_{o}+\Omega_{o}}\biggl{)}^{\beta_{o}+\frac{\alpha_{o}}{2}},$
$\displaystyle\varpi$
$\displaystyle=\frac{\epsilon^{2}\alpha_{o}\beta_{o}(g+\Omega_{o})}{(\epsilon^{2}+1)(g\beta_{o}+\Omega_{o})},$
$\displaystyle\vartheta_{n}$
$\displaystyle=\upsilon_{n}\biggl{(}\frac{\alpha_{o}\beta_{o}}{g\beta_{o}+\Omega_{o}}\biggl{)}^{-\frac{\alpha_{o}+m_{o}}{2}},$
$\displaystyle\upsilon_{n}$
$\displaystyle=\binom{\beta_{o}-1}{m_{o}-1}\frac{(g\beta_{o}+\Omega_{o})^{1-\frac{m_{o}}{2}}}{(m_{o}-1)!}\biggl{(}\frac{\Omega_{o}}{g}\biggl{)}^{m_{o}-1}\biggl{(}\frac{\alpha_{o}}{\beta_{o}}\biggl{)}^{\frac{m_{o}}{2}},$

$\alpha_{o}$ and $\beta_{o}$ denote the fading parameters, $g=2b_{o}(1-\rho)$
represents the average power of scattering components received through off-
axis eddies, $2b_{o}$ denotes the total average power of scattering
components, $\epsilon$ ($0\leq\rho\leq 1$) signifies the amount of scattering
power coupled to the LOS component, the average power realized from coherent
contributions is symbolized by
$\Omega_{o}=\acute{\Omega_{o}}+2b_{o}\rho+2\sqrt{2b_{o}\rho\acute{\Omega_{o}}}\cos(\phi_{A}-\phi_{B})$,
$\acute{\Omega_{o}}$ is termed as the average power of LOS components,
$\phi_{A}$ and $\phi_{B}$ represent the deterministic phases of LOS,
$\mu_{1}=\Phi_{o}$,
$\mu_{2}=\frac{\alpha_{o}\epsilon^{2}(\epsilon^{2}+1)^{-2}(\epsilon^{2}+2)(g+\Omega_{o})\Phi_{o}}{(\alpha_{o}+1)[2g(g+2\Omega_{o})+\Omega_{o}^{2}(1+1/\beta_{o})]}$,
$s$ denotes the detection technique (i.e. $s=1$ refers to HD technique and
$s=2$ refers to IM/DD technique), $\Phi_{o}$ denotes the average SNR of FSO
sub-link that is related to electrical SNR $\mu_{s}$, and G$[\cdot]$
represents the Meijer’s G function [41]. Plugging (14) into (9), the CDF of
$\gamma_{o}$ is written as [40, Eq. 11]

$\displaystyle
F_{\gamma_{o}}(\gamma)=K\sum_{m_{o}=0}^{\beta_{o}}\varsigma_{n}G_{s+1,3s+1}^{3s,1}\left[\frac{V\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right],$ (17)

where $K=\frac{\epsilon^{2}\chi_{o}}{2^{s}(2\pi)^{s-1}}$,
$\varsigma_{n}=\vartheta_{n}s^{\alpha_{o}+m_{o}-1}$,
$V=\frac{\varpi^{s}}{s^{2s}}$,
$q_{1}=\\{\frac{\epsilon^{2}+1}{s},\cdots,\frac{\epsilon^{2}+s}{s}\\}$
including $s$ number of terms, and
$q_{2}=\\{\frac{\epsilon^{2}}{s},\cdots,\frac{\epsilon^{2}+s-1}{s},\frac{\alpha_{o}}{s},\cdots,\frac{\alpha_{o}+s-1}{s},\frac{m_{o}}{s},\cdots,\frac{m_{o}+s-1}{s}\\}$
including $3s$ number of terms.

The FSO link can be temporarily blocked due to its extreme LOS requirements.
Hence, including link blocking probability ($P_{o}$) in FSO and assuming SNR
of the $S-R$ FSO sub-link as $\gamma_{o^{*}}$, the PDF of $\gamma_{o^{*}}$ is
expressed finally as [6, Eq. (10)]

$\displaystyle
f_{\gamma_{o^{*}}}(\gamma)=P_{o}\delta(\gamma)+(1-P_{o})f_{\gamma_{o}}(\gamma),$
(18)

where $\delta(\cdot)$ denotes a Dirac delta function [42, Eq.
(14.03.02.0001.01)]. Now, the CDF of $\gamma_{o^{*}}$ is expressed as

$\displaystyle F_{\gamma_{o^{*}}}(\gamma)$
$\displaystyle=\int_{0}^{\gamma}f_{\gamma_{o^{*}}}(\gamma)d\gamma\cong
P_{o}+(1-P_{o})F_{\gamma_{o}}(\gamma).$ (19)

Substituting (17) into (19), we obtain

$\displaystyle F_{\gamma_{o^{*}}}(\gamma)$
$\displaystyle=P_{o}+(1-P_{o})K\sum_{m_{o}=0}^{\beta_{o}}\varsigma_{n}\,G_{s+1,3s+1}^{3s,1}\left[\frac{V\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right].$ (22)

##  4 Cognitive Underlay Network

This section demonstrates the CDFs of SNRs for the hybrid RF / FSO CUN
considering both scenarios (Scenarios- I & II).

###  4.1 Scenario I

Considering the interference of $S-P$ link and denoting SNR of the $S-R$ RF
sub-link as $\gamma_{r,I}$, the CDF of $\gamma_{r,I}$ is written as [4, Eq.
(4)]

$\displaystyle
F_{\gamma_{r,I}}(\gamma)=\int_{0}^{\infty}F_{\gamma_{r}}\biggl{(}\frac{\gamma}{\Psi_{Q}}x\biggl{)}f_{\gamma_{p}}(x)dx.$
(23)

Now substituting (11) and (8) into (23), the CDF is expressed further as

$\displaystyle F_{\gamma_{r,I}}(\gamma)$
$\displaystyle=1-\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\alpha_{p}\delta_{r}^{m_{r}}\delta_{p}^{\mu_{p}}}{2\Gamma(\mu_{p})m_{r}!}\biggl{(}\frac{\gamma}{\Psi_{Q}}\biggl{)}^{\tilde{\alpha_{r}}m_{r}}$
$\displaystyle\times\int_{0}^{\infty}e^{-(\delta_{p}x^{\tilde{\alpha_{p}}}+\Xi_{1}^{I}\gamma_{r}^{\tilde{\alpha_{r}}}x^{\tilde{\alpha_{r}}})}x^{\Theta_{p}+\tilde{\alpha_{r}}m_{r}}dx,$
(24)

where $\Xi_{1}^{I}=\delta_{r}\Psi_{Q}^{-\tilde{\alpha_{r}}}$. Now, following
some algebraic manipulations, assuming
$\tilde{\alpha_{p}}=\tilde{\alpha_{r}}$, and performing integration utilizing
[39, Eq. (3.326.2)], the CDF of $S-R$ RF sub-link is finally obtained as

$\displaystyle F_{\gamma_{r,I}}(\gamma)$
$\displaystyle=1-\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\Gamma(\Xi_{2}^{I})\alpha_{p}\delta_{r}^{m_{r}}\delta_{p}^{\mu_{p}}}{\tilde{\alpha_{p}}2\Gamma(\mu_{p})m_{r}!}\biggl{(}\frac{\gamma}{\Psi_{Q}}\biggl{)}^{\tilde{\alpha_{r}}m_{r}}$
$\displaystyle\times(\Xi_{1}^{I}\gamma^{\tilde{\alpha_{r}}}+\Phi_{p}^{-\tilde{\alpha_{p}}})^{-\Xi_{2}^{I}},$
(25)

where
$\Xi_{2}^{I}=\frac{\tilde{\alpha_{r}}m_{r}+\mu_{p}\tilde{\alpha_{p}}}{\tilde{\alpha_{p}}}$.
According to (6), the CDF of end-to-end SNR at $R$ is written as

$\displaystyle F_{\gamma_{f,I}}(\gamma)=F_{\gamma_{r,I}}(\gamma)\times
F_{\gamma_{o^{*}}}(\gamma).$ (26)

By replacing (4.1) and (22) into (26), the CDF of $\gamma_{f,I}$ is derived in
(29).

$\displaystyle F_{\gamma_{f,I}}(\gamma)$
$\displaystyle=P_{o}+(1-P_{o})\sum_{m_{o}=0}^{\beta_{o}}K\varsigma_{n}G_{s+1,3s+1}^{3s,1}\left[\frac{V\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\
\end{array}\right]-P_{o}\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\delta_{r}^{m_{r}}\alpha_{p}\delta_{p}^{\mu_{p}}\Gamma(\Xi_{2}^{I})}{2\Gamma(\mu_{p})\tilde{\alpha_{p}}m_{r}!}\biggl{(}\frac{\gamma}{\Psi_{Q}}\biggl{)}^{\tilde{\alpha_{r}}m_{r}}(\Xi_{1}^{I}\gamma^{\tilde{\alpha_{p}}}+\Phi_{p}^{-\tilde{\alpha_{p}}})^{-\Xi_{2}^{I}}$
(29)
$\displaystyle-(1-P_{o})\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{o}=0}^{\beta_{o}}K\varsigma_{n}\frac{\Gamma(\Xi_{2}^{I})\alpha_{p}\delta_{p}^{\mu_{p}}\delta_{r}^{m_{r}}}{\tilde{\alpha_{p}}2\Gamma(\mu_{p})m_{r}!}\biggl{(}\frac{\gamma}{\Psi_{Q}}\biggl{)}^{\tilde{\alpha_{r}}m_{r}}(\Xi_{1}^{I}\gamma^{\tilde{\alpha_{p}}}+\Phi_{p}^{-\tilde{\alpha_{p}}})^{-\Xi_{2}^{I}}G_{s+1,3s+1}^{3s,1}\left[\frac{V\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right].$ (32)

###  4.2 Scenario II

Assuming double power constraints in the RF sub-link, the CDF of
$F_{\gamma_{r,II}}$ is written as

$\displaystyle F_{\gamma_{r,II}}(\gamma_{r})$
$\displaystyle=\Pr\biggl{\\{}\min\biggl{(}\frac{\Psi_{Q}}{|g_{p}|^{2}},\Psi_{T}\biggl{)}|g_{r}|^{2}\leq\gamma_{r}\biggl{\\}}$
$\displaystyle=\underbrace{\Pr\biggl{\\{}|g_{r}|^{2}\leq\frac{\gamma_{r}}{\Psi_{T}},\frac{\Psi_{Q}}{|g_{p}|^{2}}\geq\Psi_{T}\biggl{\\}}}_{\Lambda_{1}}$
$\displaystyle+\underbrace{\Pr\biggl{\\{}\frac{|g_{r}|^{2}}{|g_{p}|^{2}}\leq\frac{\gamma_{r}}{\Psi_{Q}},\frac{\Psi_{Q}}{|g_{p}|^{2}}\leq\Psi_{T}\biggl{\\}}}_{\Lambda_{2}}.$
(33)

Since $g_{r}$ and $g_{p}$ are completely independent to one other, hence,
$\Lambda_{1}$ is expressed in an alternative form as

$\displaystyle\Lambda_{1}$
$\displaystyle=\Pr\biggl{\\{}|g_{p}|^{2}\leq\frac{\Psi_{Q}}{\Psi_{T}}\biggl{\\}}\Pr\biggl{\\{}|g_{r}|^{2}\leq\frac{\gamma_{r}}{\Psi_{T}}\biggl{\\}}$
$\displaystyle=F_{\gamma_{p}}\biggl{(}\frac{\Psi_{Q}}{\Psi_{T}}\biggl{)}F_{\gamma_{r}}\biggl{(}\frac{\gamma}{\Psi_{T}}\biggl{)}.$
(34)

Substituting (11) into (4.2), $\Lambda_{1}$ can be finally written as

$\displaystyle\Lambda_{1}$
$\displaystyle=1-\sum_{m_{p}=0}^{\mu_{p}-1}\Xi_{1}^{II}-\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{2}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}$
$\displaystyle+\sum_{m_{p}=0}^{\mu_{p}-1}\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{3}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}+\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})},$
(35)

where
$\Xi_{1}^{II}=\frac{\delta_{p}^{m_{p}}}{m_{p}!}\biggl{(}\frac{\Psi_{Q}}{\Psi_{T}}\biggl{)}^{\tilde{\alpha_{p}}m_{p}}e^{-\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}}$,
$\Xi_{2}^{II}=\frac{\delta_{r}^{m_{r}}}{m_{r}!}\Psi_{T}^{-\tilde{\alpha_{r}}m_{r}}$,
and
$\Xi_{3}^{II}=\frac{\delta_{p}^{m_{p}}\delta_{r}^{m_{r}}}{m_{r}!m_{p}!}\left(\frac{\Psi_{Q}}{\Psi_{T}}\right)^{\tilde{\alpha_{p}}m_{p}}\Psi_{T}^{-\tilde{\alpha_{r}}m_{r}}$.
Now, according to the concept of probability theory [43], $\Lambda_{2}$ is
written as

$\displaystyle\Lambda_{2}$
$\displaystyle=\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}f_{\gamma_{p}}(y)\int_{0}^{\frac{\gamma_{r}}{\Psi_{Q}}y}f_{\gamma_{r}}(x)dxdy$
$\displaystyle=\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}f_{\gamma_{p}}(y)F_{\gamma_{r}}\biggl{(}\frac{\gamma
y}{\Psi_{Q}}\biggl{)}dy.$ (36)

Replacing (8) and (11) into (4.2), $\Lambda_{2}$ is obtained as

$\displaystyle\Lambda_{2}$
$\displaystyle=\Xi_{5}^{II}-\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}\binom{m_{3}}{m_{4}}\binom{\Omega+m_{5}-1}{m_{5}}$
$\displaystyle\times\Xi_{6}^{II}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}+\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})},$
(37)

where
$\Xi_{5}^{II}=\frac{\alpha_{p}\delta_{p}^{\mu_{p}}}{2\Gamma(\mu_{p})\tilde{\alpha_{p}}\delta_{p}^{\Xi_{4}^{II}}}\Gamma\left[\Xi_{4}^{II},\delta_{p}\left(\frac{\Psi_{Q}}{\Psi_{T}}\right)^{\tilde{\alpha_{p}}}\right]$,
$\Xi_{4}^{II}=\frac{\Theta_{p}+1}{\tilde{\alpha_{p}}}$,
$\Omega=\frac{\Theta_{p}+\tilde{\alpha_{r}}m_{r}+1}{\tilde{\alpha_{r}}}$, and
$\Xi_{6}^{II}=\frac{\alpha_{p}\delta_{r}^{m_{r}+m_{4}+m_{5}}(\Omega-1)!(-1)^{m_{5}}}{2\Gamma(\mu_{p})m_{r}!\tilde{\alpha_{r}}m_{3}!}\delta_{p}^{-(\Omega+m_{5}-m_{3}+m_{4}-\mu_{p})}\\\
\Psi_{Q}^{-\tilde{\alpha_{r}}(m_{r}-m_{3}+m_{5})}(\Psi_{T}^{-\tilde{\alpha_{r}}})^{m_{3}+m_{4}}$.
For proof, please see the Appendix. Finally, utilizing (4.2) and (4.2) into
(4.2), and performing some simple mathematical operations,
$F_{\gamma_{r,II}}(\gamma)$ is written as shown in (4.2).

$\displaystyle F_{\gamma_{r,II}}(\gamma)$
$\displaystyle=1+\Xi_{5}^{II}-\sum_{m_{p}=0}^{\mu_{p}-1}\Xi_{1}^{II}-\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{2}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}+\sum_{m_{p}=0}^{\mu_{p}-1}\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{3}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}+\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})}$
$\displaystyle-\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}\binom{m_{3}}{m_{4}}\binom{\Omega+m_{5}-1}{m_{5}}\Xi_{6}^{II}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}+\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})}.$
(38)

Now, similar to (26), the CDF of $\gamma_{f,II}$ is expressed as

$\displaystyle F_{\gamma_{f,II}}(\gamma)=F_{\gamma_{r,II}}(\gamma)\times
F_{\gamma_{o^{*}}}(\gamma).$ (39)

Now, substituting (4.2) and (22) into (39), the CDF of $\gamma_{f,II}$ is
obtained as shown in (4.2), where
$\mathcal{X}=1+\Xi_{5}^{II}-\sum_{m_{p}=0}^{\mu_{p}-1}\Xi_{1}^{II}$.

$\displaystyle F_{\gamma_{f,II}}(\gamma)$
$\displaystyle=P_{o}\Biggl{\\{}\mathcal{X}-\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{2}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}+\sum_{m_{p}=0}^{\mu_{p}-1}\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{3}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}+\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})}-\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}$
$\displaystyle\times\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}\binom{m_{3}}{m_{4}}\binom{\Omega+m_{5}-1}{m_{5}}\Xi_{6}^{II}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}+\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})}\Biggl{\\}}+(1-P_{o})K\sum_{m_{o}=0}^{\beta_{o}}\varsigma_{n}$
$\displaystyle\times
G_{s+1,3s+1}^{3s,1}\left[\frac{V\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\
\end{array}\right]\Biggl{\\{}\mathcal{X}-\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{2}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}+\sum_{m_{p}=0}^{\mu_{p}-1}\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{3}^{II}\gamma^{\tilde{\alpha_{r}}m_{r}}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}+\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})}$
(42)
$\displaystyle-\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}\binom{m_{3}}{m_{4}}\binom{\Omega+m_{5}-1}{m_{5}}\Xi_{6}^{II}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}\,e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}+\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}})}\Biggl{\\}}.$
(43)

##  5 Performance Analysis

In this section, we demonstrate the novel analytical expressions for SOP,
probability of SPSC, and EST utilizing (8), (29), and (4.2).

###  5.1 Secure Outage Probability Analysis

In order to investigate the secrecy behaviour of the proposed hybrid system,
SOP is one of the fundamental performance metrics. SOP particularly depends on
whether the target secrecy rate ($\Upsilon_{e}$) is greater than secrecy
capacity ($C_{s}$), where
$C_{s}=[\log_{2}(1+\gamma_{f,j})-\log_{2}(1+\gamma_{e,j})]^{+}$ and $[z]^{+}=$
max$\\{z,0\\}$. So, mathematically, SOP can be described as [44, Eq. (14)]

$\displaystyle SOP$
$\displaystyle=\Pr\left\\{C_{s}(\gamma_{f,j},\gamma_{e,j})\leq\Upsilon_{e}\right\\}$
$\displaystyle=\int_{0}^{\infty}F_{\gamma_{f,j}}(\sigma\gamma+\sigma-1)f_{\gamma_{e,j}}(\gamma)d\gamma,$
(44)

where $\sigma=2^{\Upsilon_{e}}$ and $\Upsilon_{e}>0$. Since Meijer’s $G$
function exists in (29) and (4.2), derivation of SOP utilizing (44) is almost
impossible. Hence, the lower bound of SOP can be derived as

$\displaystyle SOP\geq SOP_{L}$
$\displaystyle=\int_{0}^{\infty}F_{\gamma_{f,j}}(\sigma\gamma)f_{\gamma_{e,j}}(\gamma)d\gamma.$
(45)

####  5.1.1 Scenario I

Substituting (29) and (8) into (45), SOP is written as

$\displaystyle SOP_{L}^{I}$
$\displaystyle=\frac{\alpha_{e}\delta_{e}^{\mu_{e}}P_{o}}{2\Gamma(\mu_{e})}\Im_{1}+(1-P_{o})K\varsigma_{n}$
$\displaystyle\times\sum_{m_{o}=0}^{\beta_{o}}\biggl{\\{}\frac{\alpha_{e}\delta_{e}^{\mu_{e}}}{2\Gamma(\mu_{e})}\Im_{2}-\Xi_{5}^{I}\Im_{4}\biggl{\\}}-P_{o}\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{5}^{I}\Im_{3},$
(46)

where
$\Xi_{5}^{I}=\binom{\Xi_{2}^{I}+m_{2}-1}{m_{2}}\frac{\alpha_{p}\delta_{p}^{\mu_{p}}}{2\Gamma(\mu_{p})}\frac{\alpha_{e}\delta_{e}^{\mu_{e}}}{2\Gamma(\mu_{e})}\frac{\delta_{r}^{m_{r}}}{m_{r}!}\frac{\Gamma(\Xi_{2}^{I})}{\tilde{\alpha_{p}}}\left(\frac{\sigma}{\Psi_{Q}}\right)^{\tilde{\alpha_{r}}m_{r}}\\\
(-1)^{m_{2}}\left(\Xi_{1}^{I}\sigma^{\tilde{\alpha_{r}}}\right)^{m_{2}}\left(\Phi_{p}^{-\tilde{\alpha_{p}}}\right)^{-\Xi_{2}^{I}-m_{2}}$
and the four integral terms, $\Im_{1}$, $\Im_{2}$, $\Im_{3}$, and $\Im_{4}$
are derived as follows.

#### Derivation of $\Im_{1}$

Utilizing the formula of [39, Eq. (3.326.2)], $\Im_{1}$ is expressed as

$\displaystyle\Im_{1}=\int_{0}^{\infty}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}\gamma^{\Theta_{e}}d\gamma=\frac{\Gamma\left(\frac{\Theta_{e}+1}{\tilde{\alpha_{e}}}\right)}{\tilde{\alpha_{e}}}\delta_{e}^{-\frac{\Theta_{e}+1}{\tilde{\alpha_{e}}}}.$
(47)

#### Derivation of $\Im_{2}$

$\Im_{2}$ is written as

$\displaystyle\Im_{2}=$
$\displaystyle\int_{0}^{\infty}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}\gamma^{\Theta_{e}}G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]d\gamma.$ (50)

Now, with the help of [41, Eqs. (2.24.1.1) and (8.4.3.1)], $\Im_{2}$ is
converted as

$\displaystyle\Im_{2}$
$\displaystyle=\int_{0}^{\infty}\gamma^{\Theta_{e}}G_{0,1}^{1,0}\left[\delta_{e}\gamma^{\tilde{\alpha_{e}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\
\end{array}\right]G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]d\gamma$ (55)
$\displaystyle=\frac{\tilde{\alpha_{e}}^{s(2\Theta_{e}+1)}\left(\frac{V\sigma}{\mu_{s}}\right)^{-(\Theta_{e}+1)}}{(2\pi)^{s(\tilde{\alpha_{e}}-1)}}$
$\displaystyle\times
G_{\tilde{\alpha_{e}}(3s+1),1+\tilde{\alpha_{e}}(s+1)}^{1+\tilde{\alpha_{e}},\tilde{3s\alpha_{e}}}\left[\left.\frac{\delta_{e}\tilde{\alpha_{e}}^{2s\tilde{\alpha_{e}}}}{\left(\frac{V\sigma}{\mu_{s}}\right)^{\tilde{\alpha_{e}}}}\right|\begin{array}[]{c}\zeta_{1},\Delta(\tilde{\alpha_{e}},-\Theta_{e})\\\
0,\zeta_{2},\zeta_{3}\\\ \end{array}\right],$ (58)

where $\zeta_{1}=\Delta(\tilde{\alpha_{e}},-\Theta_{e}-q_{2})$,
$\zeta_{2}=\Delta(\tilde{\alpha_{e}},-\Theta_{e}-1)$,
$\zeta_{3}=\Delta(\tilde{\alpha_{e}},-\Theta_{e}-q_{1})$, and
$\Delta(c,r)=\frac{r}{c},\frac{r+1}{c},...,\frac{r+c-1}{c}$ as defined in [45,
Eq. (22)].

#### Derivation of $\Im_{3}$

$\Im_{3}$ is expressed as

$\displaystyle\Im_{3}=$
$\displaystyle\int_{0}^{\infty}(\Xi_{1}^{I}\sigma^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{p}}}+\Phi_{p}^{-\tilde{\alpha_{p}}})^{-\Xi_{2}^{I}}\,\gamma^{\Theta_{e}+\tilde{\alpha_{r}}m_{r}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}d\gamma.$
(59)

Utilizing binomial expansion and [39, Eq. (3.326.2)], $\Im_{3}$ is expressed
in alternative form as

$\displaystyle\Im_{3}=\sum_{m_{2}=0}^{\infty}\int_{0}^{\infty}\gamma^{\Xi_{3}^{I}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}d\gamma=\sum_{m_{2}=0}^{\infty}\frac{\Gamma(\Xi_{4}^{I})}{\tilde{\alpha_{e}}}\delta_{e}^{-\Xi_{4}^{I}},$
(60)

where $\Xi_{3}^{I}=\Theta_{e}+\tilde{\alpha_{r}}m_{r}+\tilde{\alpha_{p}}m_{2}$
and $\Xi_{4}^{I}=\frac{\Xi_{3}^{I}+1}{\tilde{\alpha_{e}}}$.

#### Derivation of $\Im_{4}$

Following the similar approach of $\Im_{2}$ and $\Im_{3}$, $\Im_{4}$ is
illustrated as

$\displaystyle\Im_{4}$
$\displaystyle=\sum_{m_{2}=0}^{\infty}\int_{0}^{\infty}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}\gamma^{\Xi_{3}^{I}}G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]d\gamma$ (63)
$\displaystyle=\sum_{m_{2}=0}^{\infty}\frac{\tilde{\alpha_{e}}^{s(2\,\Xi_{3}^{I}+1)}\left(\frac{V\sigma}{\mu_{s}}\right)^{-(\Xi_{3}^{I}+1)}}{(2\pi)^{s(\tilde{\alpha_{e}}-1)}}$
$\displaystyle\times
G_{\tilde{\alpha_{e}}(3s+1),1+\tilde{\alpha_{e}}(s+1)}^{1+\tilde{\alpha_{e}},\tilde{3s\alpha_{e}}}\left[\frac{\delta_{e}\tilde{\alpha_{e}}^{2s\tilde{\alpha_{e}}}}{\left(\frac{V\sigma}{\mu_{s}}\right)^{\tilde{\alpha_{e}}}}\Biggl{|}\begin{array}[]{c}\zeta_{4},\Delta(\tilde{\alpha_{e}},-\Xi_{3}^{I})\\\
0,\zeta_{5},\zeta_{6}\\\ \end{array}\right].$ (66)

where $\zeta_{4}=\Delta(\tilde{\alpha_{e}},-\Xi_{3}^{I}-q_{2})$,
$\zeta_{5}=\Delta(\tilde{\alpha_{e}},-\Xi_{3}^{I}-1)$, and
$\zeta_{6}=\Delta(\tilde{\alpha_{e}},-\Xi_{3}^{I}-q_{1})$.

####  5.1.2 Scenario II

Substituting (8) and (4.2) into (45), SOP for double power constraints
scenario is derived as (5.1.2), where the eight integral terms,
$\mathcal{R}_{1}$, $\mathcal{R}_{2}$, $\mathcal{R}_{3}$, $\mathcal{R}_{4}$,
$\mathcal{R}_{5}$, $\mathcal{R}_{6}$, $\mathcal{R}_{7}$, and $\mathcal{R}_{8}$
are derived as follows.

$\displaystyle
SOP_{L}^{II}=\frac{\alpha_{e}\delta_{e}^{\mu_{e}}}{2\Gamma(\mu_{e})}\Biggl{[}P_{o}\Biggl{\\{}\mathcal{X}\mathcal{R}_{1}-\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{2}^{II}\sigma^{\tilde{\alpha_{r}}m_{r}}\mathcal{R}_{2}+\sum_{m_{p}=0}^{\mu_{p}-1}\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{3}^{II}\sigma^{\tilde{\alpha_{r}}m_{r}}e^{-\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}}\mathcal{R}_{3}-\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}$
$\displaystyle\times\binom{\Omega+m_{5}-1}{m_{5}}\binom{m_{3}}{m_{4}}\Xi_{6}^{II}\sigma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}e^{-\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}}\mathcal{R}_{4}\Biggl{\\}}+(1-P_{o})K\sum_{m_{o}=0}^{\beta_{o}}\varsigma_{n}\Biggl{\\{}\mathcal{X}\mathcal{R}_{5}-\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{2}^{II}\mathcal{R}_{6}\,\sigma^{\tilde{\alpha_{r}}m_{r}}+\sum_{m_{p}=0}^{\mu_{p}-1}$
$\displaystyle\times\sum_{m_{r}=0}^{\mu_{r}-1}\Xi_{3}^{II}\mathcal{R}_{7}\sigma^{\tilde{\alpha_{r}}m_{r}}e^{-\delta_{p}\Psi_{Q}^{\tilde{\alpha_{p}}}\Psi_{T}^{-\tilde{\alpha_{p}}}}-\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}\binom{m_{3}}{m_{4}}\binom{\Omega+m_{5}-1}{m_{5}}\Xi_{6}^{II}\mathcal{R}_{8}\,\sigma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}e^{-\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}}\Biggl{\\}}\Biggl{]}.$
(67)

#### Derivation of $\mathcal{R}_{1}$

Utilizing [39, Eq. (3.326.2)], $\mathcal{R}_{1}$ is expressed as

$\displaystyle\mathcal{R}_{1}$
$\displaystyle=\int_{0}^{\infty}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}\gamma^{\Theta_{e}}d\gamma=\frac{\Gamma\left(\frac{\Theta_{e}+1}{\tilde{\alpha_{e}}}\right)}{\tilde{\alpha_{e}}}\delta_{e}^{-\frac{\Theta_{e}+1}{\tilde{\alpha_{e}}}}.$
(68)

#### Derivation of $\mathcal{R}_{2}$ and $\mathcal{R}_{3}$

$\mathcal{R}_{2}$ and $\mathcal{R}_{3}$ are expressed as

$\displaystyle\mathcal{R}_{2}=\mathcal{R}_{3}=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}m_{r}+\Theta_{e}}e^{-\delta_{r}\sigma^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}d\gamma.$
(69)

Following some mathematical manipulations defined in [46] and utilizing
identities [41, Eqs. (8.4.3.1) and (2.24.1.1)], $\mathcal{R}_{2}$ and
$\mathcal{R}_{3}$ are further derived as

$\displaystyle\mathcal{R}_{2}$
$\displaystyle=\mathcal{R}_{3}=\int_{0}^{\infty}\gamma^{\frac{\tilde{\alpha_{r}}m_{r}+\Theta_{e}}{\tilde{\alpha_{r}}}}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma}e^{-\delta_{e}\gamma^{\frac{\tilde{\alpha_{e}}}{\tilde{\alpha_{r}}}}}d\left(\gamma^{\frac{1}{\tilde{\alpha_{r}}}}\right)$
$\displaystyle=\frac{1}{\tilde{\alpha_{r}}}\int_{0}^{\infty}\gamma^{\Xi_{7}^{II}-1}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma}e^{-\delta_{e}\gamma^{\frac{\tilde{\alpha_{e}}}{\tilde{\alpha_{r}}}}}d\gamma$
$\displaystyle=\frac{1}{\tilde{\alpha_{r}}}\int_{0}^{\infty}\gamma^{\Xi_{7}^{II}-1}G_{0,1}^{1,0}\left[\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma\biggl{|}\begin{array}[]{c}-\\\
0\\\ \end{array}\right]$ (72) $\displaystyle\times
G_{0,1}^{1,0}\left[\delta_{e}\gamma^{\frac{\tilde{\alpha_{e}}}{\tilde{\alpha_{r}}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\
\end{array}\right]d\gamma=\frac{\tilde{\alpha_{e}}^{\Xi_{7}^{II}-\frac{1}{2}}(\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}})^{-\Xi_{7}^{II}}}{\tilde{\alpha_{r}}^{\frac{1}{2}}(2\pi)^{\frac{1}{2}(\tilde{\alpha_{e}}+\tilde{\alpha_{r}}-2)}}$
(75) $\displaystyle\times
G_{\tilde{\alpha_{e}},\tilde{\alpha_{r}}}^{\tilde{\alpha_{r}},\tilde{\alpha_{e}}}\left[\frac{\delta_{e}^{\tilde{\alpha_{r}}}\tilde{\alpha_{r}}^{-\tilde{\alpha_{r}}}}{(\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}})^{\tilde{\alpha_{e}}}\tilde{\alpha_{e}}^{-\tilde{\alpha_{e}}}}\biggl{|}\begin{array}[]{c}\Delta(\tilde{\alpha_{e}},1-\Xi_{7}^{II})\\\
0\\\ \end{array}\right],$ (78)

where
$\Xi_{7}^{II}=\frac{\tilde{\alpha_{r}}m_{r}+\Theta_{e}+\tilde{\alpha_{r}}}{\tilde{\alpha_{r}}}$.

#### Derivation of $\mathcal{R}_{4}$

By following the similar approach as of $\mathcal{R}_{2}$ and
$\mathcal{R}_{3}$, $\mathcal{R}_{4}$ is expressed as

$\displaystyle\mathcal{R}_{4}=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})+\Theta_{e}}e^{-\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}d\gamma$
$\displaystyle=\int_{0}^{\infty}\gamma^{\frac{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})+\Theta_{e}}{\tilde{\alpha_{r}}}}e^{-\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma}e^{-\delta_{e}\gamma^{\frac{\tilde{\alpha_{e}}}{\tilde{\alpha_{r}}}}}d\left(\gamma^{\frac{1}{\tilde{\alpha_{r}}}}\right)$
$\displaystyle=\frac{1}{\tilde{\alpha_{r}}}\int_{0}^{\infty}\gamma^{\Xi_{8}^{II}-1}e^{-\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma}e^{-\delta_{e}\gamma^{\frac{\tilde{\alpha_{e}}}{\tilde{\alpha_{r}}}}}d\gamma$
$\displaystyle=\frac{1}{\tilde{\alpha_{r}}}\int_{0}^{\infty}\gamma^{\Xi_{8}^{II}-1}G_{0,1}^{1,0}\left[\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma\biggl{|}\begin{array}[]{c}-\\\
0\\\ \end{array}\right]$ (81) $\displaystyle\times
G_{0,1}^{1,0}\left[\delta_{e}\gamma^{\frac{\tilde{\alpha_{e}}}{\tilde{\alpha_{r}}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\
\end{array}\right]d\gamma=\frac{\tilde{\alpha_{e}}^{\Xi_{8}^{II}-\frac{1}{2}}(\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}})^{-\Xi_{8}^{II}}}{\tilde{\alpha_{r}}^{\frac{1}{2}}(2\pi)^{\frac{1}{2}(\tilde{\alpha_{e}}+\tilde{\alpha_{r}}-2)}}$
(84) $\displaystyle\times
G_{\tilde{\alpha_{e}},\tilde{\alpha_{r}}}^{\tilde{\alpha_{r}},\tilde{\alpha_{e}}}\left[\frac{\delta_{e}^{\tilde{\alpha_{r}}}\tilde{\alpha_{r}}^{-\tilde{\alpha_{r}}}}{(\delta_{r}\Psi_{T}^{-\tilde{2\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}})^{\tilde{\alpha_{e}}}\tilde{\alpha_{e}}^{-\tilde{\alpha_{e}}}}\biggl{|}\begin{array}[]{c}\Delta(\tilde{\alpha_{e}},1-\Xi_{8}^{II})\\\
0\\\ \end{array}\right],$ (87)

where
$\Xi_{8}^{II}=\frac{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})+\Theta_{e}+\tilde{\alpha_{r}}}{\tilde{\alpha_{r}}}$.

#### Derivation of $\mathcal{R}_{5}$

With some mathematical manipulations and following identities [41, Eqs.
(8.4.3.1) and (2.24.1.1)], $\mathcal{R}_{5}$ is derived as

$\displaystyle\mathcal{R}_{5}=\int_{0}^{\infty}\gamma^{\Theta_{e}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]d\gamma$ (90)
$\displaystyle=\int_{0}^{\infty}\gamma^{\Theta_{e}}G_{0,1}^{1,0}\left[\delta_{e}\gamma^{\tilde{\alpha_{e}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\
\end{array}\right]G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]\ d\gamma$ (95)
$\displaystyle=\frac{\tilde{\alpha_{e}}^{s(2\Theta_{e}+1)}\left(\frac{V\sigma}{\mu_{s}}\right)^{-(\Theta_{e}+1)}}{(2\pi)^{s(\tilde{\alpha_{e}}-1)}}$
$\displaystyle\times
G_{\tilde{\alpha_{e}}(3s+1),1+\tilde{\alpha_{e}}(s+1)}^{1+\tilde{\alpha_{e}},\tilde{3s\alpha_{e}}}\left[\frac{\delta_{e}\tilde{\alpha_{e}}^{2s\tilde{\alpha_{e}}}}{\left(\frac{V\sigma}{\mu_{s}}\right)^{\tilde{\alpha_{e}}}}\Biggl{|}\begin{array}[]{c}\zeta_{1},\Delta(\tilde{\alpha_{e}},-\Theta_{e})\\\
0,\zeta_{2},\zeta_{3}\\\ \end{array}\right].$ (98)

#### Derivation of $\mathcal{R}_{6}$ and $\mathcal{R}_{7}$

$\mathcal{R}_{6}$ and $\mathcal{R}_{7}$ are derived by following [41, Eq.
(8.4.3.1)] as

$\displaystyle\mathcal{R}_{6}=\mathcal{R}_{7}$
$\displaystyle=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}m_{r}+\Theta_{e}}e^{-\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}$
$\displaystyle\times
G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]d\gamma$ (101)
$\displaystyle=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}m_{r}+\Theta_{e}}G_{0,1}^{1,0}\left[\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\ \end{array}\right]$ (104) $\displaystyle\times
G_{0,1}^{1,0}\left[\delta_{e}\gamma^{\tilde{\alpha_{e}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\
\end{array}\right]G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\ \end{array}\right]d\gamma.$ (109)

Now, for tractable analysis, transforming Meijer’s $G$ functions into Fox’s
$H$ functions via utilizing [47, Eqs. (6.2.3) and (6.2.8)] and performing
integration by means of [48, Eq. (2.3)] and [36, Eq. (3)], $\mathcal{R}_{6}$
and $\mathcal{R}_{7}$ are obtained as

$\displaystyle\mathcal{R}_{6}=\mathcal{R}_{7}=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}m_{r}+\Theta_{e}}H_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}(1,1),(q_{1},1)\\\
(q_{2},1),(0,1)\\\ \end{array}\right]$ (112) $\displaystyle\times
H_{0,1}^{1,0}\left[\Xi_{10}^{II}\gamma^{\tilde{\alpha_{r}}}\biggl{|}\begin{array}[]{c}-\\\
(0,1)\\\
\end{array}\right]H_{0,1}^{1,0}\left[\delta_{e}\gamma^{\tilde{\alpha_{e}}}\biggl{|}\begin{array}[]{c}-\\\
(0,1)\\\
\end{array}\right]d\gamma=\frac{\delta_{e}^{-\frac{\Xi_{9}^{II}}{\tilde{\alpha_{e}}}}}{\tilde{\alpha_{e}}}$
(117) $\displaystyle\times
H_{1,0:0,1:s+1,3s+1}^{1,0:1,0:3s,1}\biggl{[}\begin{array}[]{c}J_{1}\\\
J_{2}\\\ \end{array}\biggl{|}\begin{array}[]{c}-\\\ (0,1)\\\
\end{array}\biggl{|}\begin{array}[]{c}(1,1),(q_{1},1)\\\ (q_{2},1),(0,1)\\\
\end{array}\biggl{|}J_{3},J_{4}\biggl{]},$ (124)

where $\Xi_{9}^{II}=\tilde{\alpha_{r}}m_{r}+\Theta_{e}+1$,
$\Xi_{10}^{II}=\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}$,
$J_{1}=\left(1-\frac{\Xi_{9}^{II}}{\tilde{\alpha_{e}}};\frac{\tilde{\alpha_{r}}}{\tilde{\alpha_{e}}},\frac{1}{\tilde{\alpha_{e}}}\right)$,
$J_{2}=(1;-)$,
$J_{3}=\Xi_{10}^{II}\delta_{e}^{-\frac{\tilde{\alpha_{r}}}{\tilde{\alpha_{e}}}}$,
$J_{4}=V\sigma\mu_{s}^{-1}\delta_{e}^{-\frac{1}{\tilde{\alpha_{e}}}}$, and
$H_{p,q}^{m,n}[\cdot]$ is the Fox’s $H$ function introduced in [49, Eq.
(1.2)], and
$H_{c_{1},d_{1}:c_{2},d_{2}:c_{3},d_{3}}^{x_{1},y_{1}:x_{2},y_{2}:x_{3},y_{3}}[\cdot]$
is the extended generalized bivariate Fox’s $H$ function (EGBFHF) as explained
in [49, Eq. (2.57)].

#### Derivation of $\mathcal{R}_{8}$

$\mathcal{R}_{8}$ is derived by following the similar identities from
$\mathcal{R}_{6}$ and $\mathcal{R}_{7}$ as

$\displaystyle\mathcal{R}_{8}=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})+\Theta_{e}}e^{-\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}\gamma^{\tilde{\alpha_{r}}}}e^{-\delta_{e}\gamma^{\tilde{\alpha_{e}}}}$
$\displaystyle\times
G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\
\end{array}\right]d\gamma=\int_{0}^{\infty}\gamma^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})+\Theta_{e}}$
(127) $\displaystyle\times
G_{0,1}^{1,0}\left[\Xi_{12}^{II}\gamma^{\tilde{\alpha_{r}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\
\end{array}\right]\,G_{0,1}^{1,0}\left[\delta_{e}\gamma^{\tilde{\alpha_{e}}}\biggl{|}\begin{array}[]{c}-\\\
0\\\ \end{array}\right]$ (132) $\displaystyle\times
G_{s+1,3s+1}^{3s,1}\left[\frac{V\sigma\gamma}{\mu_{s}}\biggl{|}\begin{array}[]{c}1,q_{1}\\\
q_{2},0\\\
\end{array}\right]d\gamma=\tilde{\alpha_{e}}^{-1}\delta_{e}^{-\frac{\Xi_{11}^{II}}{\tilde{\alpha_{e}}}}$
(135) $\displaystyle\times
H_{1,0:0,1:s+1,3s+1}^{1,0:1,0:3s,1}\biggl{[}\begin{array}[]{c}J_{5}\\\
J_{2}\\\ \end{array}\biggl{|}\begin{array}[]{c}-\\\ (0,1)\\\
\end{array}\biggl{|}\begin{array}[]{c}(1,1),(q_{1},1)\\\ (q_{2},1),(0,1)\\\
\end{array}\biggl{|}J_{6},J_{4}\biggl{]},$ (142)

where $\Xi_{11}^{II}=\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})+\Theta_{e}+1$,
$\Xi_{12}^{II}=\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\sigma^{\tilde{\alpha_{r}}}$,
$J_{5}=(1-\frac{\Xi_{11}^{II}}{\tilde{\alpha_{e}}};\frac{\tilde{\alpha_{r}}}{\tilde{\alpha_{e}}},\frac{1}{\tilde{\alpha_{e}}})$,
and
$J_{6}=\Xi_{12}^{II}\,\delta_{e}^{-\frac{\tilde{\alpha_{r}}}{\tilde{\alpha_{e}}}}$.

###  5.2 Strictly Positive Secrecy Capacity Analysis

In order to ensure secrecy in a wiretap prone model, the probability of SPSC
acts as one of the crucial performance metric that ensures uninterrupted data
stream conveyance only when secrecy behaviour of the system remains up to
scratch ($\gamma_{f,j}>\gamma_{e,j}$). Mathematically, the analytical
expression of probability of SPSC can be easily deducted from the SOP that is
given as [31, Eq. (25)]

$\displaystyle SPSC$ $\displaystyle=\Pr\left\\{C_{s}>0\right\\}$
$\displaystyle=1-\Pr\left\\{C_{s}\leq 0\right\\}.$ $\displaystyle\text{Hence,
}SPSC^{I}$ $\displaystyle=1-\text{SOP}_{L}^{I}|_{\Upsilon_{e}=0},\text{
(Scenario-I)}$ (48) $\displaystyle\text{and }SPSC^{II}$
$\displaystyle=1-\text{SOP}_{L}^{II}|_{\Upsilon_{e}=0}.\text{ (Scenario-II)}$
(49)

###  5.3 Effective Secrecy Throughput Analysis

EST is another secrecy performance metric that ensures achievable secure
average throughput measurements. When the reliability issue is considered
equally important along with the security issue then this parameter is
considered. Mathematically, EST is formulated as the product of target secrecy
rate and the probability of successful transmission that is given as [50, Eq.
(5)]

$\displaystyle EST$ $\displaystyle=\Upsilon_{e}(1-SOP).$
$\displaystyle\text{Hence, }EST^{I}$
$\displaystyle=\Upsilon_{e}(1-SOP_{L}^{I}),\text{ (Scenario-I)}$ (50)
$\displaystyle\text{and }EST^{II}$
$\displaystyle=\Upsilon_{e}(1-SOP_{L}^{II}).\text{ (Scenario-II)}$ (51)

##  6 Numerical Results

In this section, the impacts of all the system parameters (e.g., $\alpha_{i}$,
$\mu_{i}$, $\Phi_{i}$, $\Psi_{Q}$, $\Psi_{T}$, $P_{o}$, $\alpha_{o}$,
$\beta_{o}$, $\epsilon$, and $s$) on the secrecy behaviour of the proposed
scenarios are illustrated graphically utilizing the expressions in (5.1.1),
(5.1.2), (48), (49), (50), and (51). Note that the univariate Meijer’s $G$ and
Fox’s $H$ functions can be easily computed utilizing standard Mathematica
packages whereas the EGBFHF is computed utilizing [36, Table I]. Moreover, all
the mathematical expressions are also validated via MC simulations averaging
$10^{6}$ random samples of $\alpha-\mu$ and Málaga random variables. It can
clearly be seen that MC simulation results are in a good agreement with the
analytical results that strongly corroborate the analysis of this paper. All
the numerical results are demonstrated assuming ($\alpha_{o}$, $\beta_{o}$)
$=$ ($2.296,2$) for strong, ($\alpha_{o}$, $\beta_{o}$) $=$ ($4.2,3$) for
moderate, and ($\alpha_{o}$, $\beta_{o}$) $=$ ($8,4$) for weak turbulences,
$r=1$ for HD and $r=2$ for IM/DD techniques, and $\Upsilon_{e}=0.05$
bits/sec/Hz.

The impacts of $\alpha_{i}$ on the EST and SPSC are demonstrated in Figs. 2
and 3 for Scenario-I.

Figure 2:  The EST versus $\Psi_{Q}$ for selected values of $\alpha_{p}$ and
$\alpha_{e}$ with $\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$,
$s=2$, $\epsilon=1$, $P_{o}=0.5$, $\alpha_{r}=2$, $\mu_{r}=\mu_{p}=\mu_{e}=2$,
$\Phi_{r}=15dB$, $\Phi_{p}=10dB$, $\Phi_{e}=\mu_{s}-5dB$, and
$\Upsilon_{e}=0.05$ bits/sec/Hz.

Figure 3:  The SPSC versus $\Phi_{o}$ for selected values of $\alpha_{r}$ and
$s$ with $\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$,
$\epsilon=1$, $P_{o}=0.5$, $\alpha_{p}=\alpha_{e}=5$,
$\mu_{r}=\mu_{p}=\mu_{e}=6$, $\Phi_{r}=\Phi_{p}=15dB$, $\Phi_{e}=0dB$, and
$\Psi_{Q}=15dB$.

Figure 4: The SOP versus $\Phi_{p}$ for selected values of $\mu_{e}$ and
$\Phi_{e}$ with $\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$,
$s=1$, $\epsilon=1$, $\alpha_{r}=\alpha_{p}=\alpha_{e}=2$,
$\mu_{r}=\mu_{p}=2$, $P_{o}=0.1$, $\Phi_{r}=15dB$, $\mu_{r}=12dB$,
$\Psi_{Q}=-5dB$, and $\Upsilon_{e}=0.05$ bits/sec/Hz.

It is observed that EST gradually increases with $\Psi_{Q}$. Similar
observations are also seen in [51]. Again, since with the increase in non-
linearity parameter, the communication link becomes better [5] (i.e. the link
will undergo less amount of fading with higher $\alpha_{i}$), the EST becomes
better for lower $\alpha_{p}$ and $\alpha_{e}$. On the other hand, as
expected, the SPSC is enhanced as $\Phi_{o}$ and $\alpha_{r}$ increase from a
lower to a higher value.

The SOP is plotted against $\Phi_{p},\Psi_{Q}$, and $\Phi_{r}$ considering
Scenario-I in Figs. 4, 5, and 6, respectively, with a view to observe the
impacts of the number of multipath clusters.

Figure 5:  The SOP versus $\Psi_{Q}$ for selected values of $\mu_{p}$ with
$\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$, $\epsilon=1$, $s=1$,
$\alpha_{r}=\alpha_{p}=\alpha_{e}=2$, $\mu_{r}=\mu_{e}=2$, $P_{o}=0.1$,
$\Phi_{p}=10dB$, $\Phi_{r}=15dB$ $\Phi_{o}=10dB$, $\Phi_{e}=10dB$, and
$\Upsilon_{e}=0.05$ bits/sec/Hz.

Figure 6:  The SOP versus $\Phi_{r}$ for selected values of $\mu_{r}$ with
$\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$, $\epsilon_{o}=1$,
$s=1$, $\alpha_{r}=\alpha_{p}=\alpha_{e}=2$, $\mu_{p}=\mu_{e}=2$, $P_{o}=0.1$,
$\Phi_{p}=15dB$, $\Phi_{o}=10dB$, $\Phi_{e}=15dB$, $\Psi_{Q}=-5dB$, and
$\Upsilon_{e}=0.05$ bits/sec/Hz.

It can clearly be noted that the SOP performance improves with $\Psi_{Q}$ and
$\Phi_{r}$, and degrades with $\Phi_{p}$ as testified in [4]. This is as
expected because the larger $\Psi_{Q}$ indicates the higher transmitting power
at $S$. On the other hand, $\Phi_{r}$ improves the $S-R$ RF sub-link whereas
$\Phi_{p}$ strengths the $S-P$ link. It is obvious that an increase in
$\mu_{i}$ will increase the channel diversity thereby mitigating the channel
fading. Hence, the SOP decreases with $\mu_{r}$, and increases with $\mu_{p}$
and $\mu_{e}$. Similar results were also demonstrated in [5] that corroborate
the outcomes in this work. In Fig. 7, impact of $\mu_{p}$ is demonstrated on
SOP considering Scenario-II. Similar to Scenario-I, in this particular case,
it is also observed that the SOP drastically degrades with $\mu_{p}$.

Figure 7:  The SOP versus $\Psi_{T}$ for selected values of $\mu_{p}$ with
$\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$, $\epsilon=6.7$,
$P_{o}=0.2$, $s=1$, $\alpha_{r}=\alpha_{p}=\alpha_{e}=2$, $\mu_{r}=\mu_{e}=2$,
$\Phi_{p}=-5dB$, $\mu_{s}=10dB$, $\Phi_{r}=-5dB$, $\Psi_{Q}=5dB$,
$\Phi_{e}=5dB$, and $\Upsilon_{e}=0.05$ bits/sec/Hz.

In Fig. 8, the probability of link blockage is observed via plotting SOP
against average SNR of FSO link under various levels of pointing error.

Figure 8:  The SOP versus $\mu_{s}$ for selected values of $P_{o}$ and
$\epsilon$ with $\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$,
$s=1$, $\alpha_{r}=\alpha_{p}=\alpha_{e}=2$, $\mu_{r}=\mu_{p}=\mu_{e}=6$,
$\Phi_{r}=\Phi_{p}=15dB$, $\Phi_{e}=-5dB$, $\Psi_{Q}=-10dB$, and
$\Upsilon_{e}=0.05$ bits/sec/Hz.

It is observed that probability of link blockage impose significant impact on
the SOP performance, that is, the SOP degrades with the increase in $P_{o}$
(increased $P_{o}$ signifies a stronger link blockage). It is obvious because
similar to the observations in [6], the atmospheric turbulence degrades with
the increase in link blockage, which yields to a degraded secrecy performance.

Figure 9 depicts the impact of $\Psi_{T}$ on the EST performance for Scenario-
II.

Figure 9:  The EST versus $\Phi_{p}$ for selected values of $\Psi_{T}$ with
$\alpha_{o}=2.296$, $\beta_{o}=2$, $\Omega_{o}=1$, $g=2$, $s=2$,
$\epsilon=6.7$, $P_{o}=0.1$, $\alpha_{r}=\alpha_{p}=\alpha_{e}=2$,
$\mu_{r}=\mu_{p}=\mu_{e}=2$, $\Phi_{o}=-5dB$, $\Phi_{r}=10dB$,
$\Psi_{Q}=-10dB$, $\Phi_{e}=-5dB$, and $\Upsilon_{e}=0.05$ bits/sec/Hz.

Figure 10:  The SOP versus $\Phi_{o}$ for selected values of $\alpha_{o}$,
$\beta_{o}$, and $s$ with $\Omega_{o}=1$, $g=2$, $\epsilon=6.7$, $P_{o}=0.1$,
$\alpha_{r}=\alpha_{p}=\alpha_{e}=2$, $\mu_{r}=\mu_{p}=\mu_{e}=6$,
$\Phi_{r}=\Phi_{p}=15dB$, $\Phi_{e}=-5dB$, $\Psi_{T}=\Psi_{Q}=-10dB$, and
$\Upsilon_{e}=0.05$ bits/sec/Hz.

It can be noted that an increased $\Psi_{T}$ is beneficial for EST
performance, since for that case the increased transmit power yields a better
received SNR at the destination. After $15dB$, an error floor is observed, as
expected. This occurs as the transmit power at $S$ is limited by a threshold
value.

The FSO link performance is significantly affected by the atmospheric
turbulence. Based on the values of $\alpha_{o}$ and $\beta_{o}$, three
turbulence conditions (e.g. strong, moderate, and weak) are considered in this
work. Figures 10 and 11 demonstrate how the secrecy performance is affected
with various turbulence conditions under Scenarios- I & II.

Figure 11:  The SOP versus $\Phi_{o}$ for selected values of $\alpha_{o}$,
$\beta_{o}$ and $s$ with $\Omega_{o}=1$, $g=2$, $\epsilon=1$, $P_{o}=0.1$,
$\alpha_{r}=\alpha_{p}=\alpha_{e}=2$,
$\mu_{r}=\mu_{p}=\mu_{e}=6$,$\Phi_{r}=\Phi_{p}=15dB$, $\Phi_{e}=-5dB$,
$\Psi_{Q}=-10dB$, and $\Upsilon_{e}=0.05$ bits/sec/Hz.

Figure 12:  The SOP versus $\Phi_{o}$ for selected values of $\alpha_{o}$,
$\beta_{o}$ and $\epsilon$ with $\Omega_{o}=1$, $g=2$, $s=2$, $P_{o}=0.1$,
$\alpha_{r}=\alpha_{p}=\alpha_{e}=2$,
$\mu_{r}=\mu_{p}=\mu_{e}=6$,$\Phi_{r}=\Phi_{p}=15dB$, $\Phi_{e}=-5dB$,
$\Psi_{Q}=-10dB$, and $\Upsilon_{e}=0.05$ bits/sec/Hz.

Since stronger turbulence yields a weaker SNR at the destination, the weaker
the turbulence, the better the secrecy performance that is also demonstrated
in [31]. Moreover, a comparison between two detection techniques is also
presented that clearly reveals that HD technique overcomes the atmospheric
turbulence more appropriately than the IM/DD technique. The results in [50]
also completely match with this outcome.

Besides the atmospheric turbulence, pointing error also plays a vital role on
the FSO link performance that is illustrated in Fig. 12 by depicting SOP as a
function of the average SNR of the FSO channel experiencing Scenario-I.

The result exhibits that higher pointing error (lower value of $\epsilon$) is
more detrimental for better SOP than the lower pointing error (higher value of
$\epsilon$). This happens since lower pointing error designates a better
pointing accuracy that yields a better detection of the received signal. This
statement can be easily justified with similar results presented in [50].

##  7 Conclusions

In this work, the secrecy performance of a CUN based hybrid RF / FSO network
has been investigated wherein both maximum transmit power and interference
power constraints within the CR spectrum are taken into consideration.
Assuming all the RF and FSO links were subjected to $\alpha-\mu$ fading and
Málaga ($\mathcal{M}$) turbulence, respectively, the expressions of SOP,
probability of SPSC, and EST were derived in closed-form. The tightness of the
simulation and analytical results were also demonstrated via MC simulations.
Furthermore, the impacts of the system parameters on the security performance
were analyzed capitalizing on the derived expressions. Numerical outcomes
revealed that with decrease in fading severity of primary interference link
($S-P$ link) and wiretap link ($S-E$ link), and increase in link blockage
probability, atmospheric turbulence and pointing error, the secrecy
performance deteriorates significantly where the HD technique overcomes those
secrecy threats more strongly relative to the IM/DD technique.

## Appendix

###  7.1 Proof of $\Lambda_{2}$

$\Lambda_{2}$ is given as

$\displaystyle\Lambda_{2}$
$\displaystyle=\underbrace{\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}\frac{\alpha_{p}\delta_{p}^{\mu_{p}}}{2\Gamma(\mu_{p})}y^{\Theta_{p}}e^{-\delta_{p}y^{\tilde{\alpha_{p}}}}dy}_{\mathcal{P}_{1}}$
$\displaystyle-\underbrace{\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\alpha_{p}\delta_{p}^{\mu_{p}}}{2\Gamma(\mu_{p})}\frac{\delta_{r}^{m_{r}}}{m_{r}!}\frac{\Psi_{Q}^{-\tilde{\alpha_{r}}m_{r}}\gamma_{r}^{\tilde{\alpha_{r}}m_{r}}y^{\Theta_{p}+\tilde{\alpha_{r}}m_{r}}}{e^{(\delta_{p}y^{\tilde{\alpha_{p}}}+\delta_{r}\Psi_{Q}^{-\tilde{\alpha_{r}}}\gamma_{r}^{\tilde{\alpha_{r}}}y^{\tilde{\alpha_{r}}})}}dy}_{\mathcal{P}_{2}}.$

By using [39, Eq. (3.381.9)], $\mathcal{P}_{1}$ is expressed as

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{\alpha_{p}\delta_{p}^{\mu_{p}}}{2\Gamma(\mu_{p})}\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}y^{\Theta_{p}}e^{-\delta_{p}y^{\tilde{\alpha_{p}}}}dy=\Xi_{5}^{II}.$

$\mathcal{P}_{2}$ is written as

$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\alpha_{p}\delta_{p}^{\mu_{p}}\delta_{r}^{m_{r}}}{2\Gamma(\mu_{p})m_{r}!}\Psi_{Q}^{-\tilde{\alpha_{r}}m_{r}}\gamma_{r}^{\tilde{\alpha_{r}}m_{r}}\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}y^{\Theta_{p}+\tilde{\alpha_{r}}m_{r}}$
$\displaystyle\times
e^{-\delta_{p}y^{\tilde{\alpha_{p}}}}e^{-\delta_{r}\Psi_{Q}^{-\tilde{\alpha_{r}}}\gamma_{r}^{\tilde{\alpha_{r}}}y^{\tilde{\alpha_{r}}}}dy.$
(52)

Due to tractable analysis, assuming $\tilde{\alpha_{p}}=\tilde{\alpha_{r}}$
and using similar formula as for $\mathcal{P}_{1}$, $\mathcal{P}_{2}$ is
expressed as

$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\alpha_{p}\delta_{p}^{\mu_{p}}\delta_{r}^{m_{r}}}{2\Gamma(\mu_{p})m_{r}!}\Psi_{Q}^{-\tilde{\alpha_{r}}m_{r}}\gamma_{r}^{\tilde{\alpha_{r}}m_{r}}\int_{\frac{\Psi_{Q}}{\Psi_{T}}}^{\infty}y^{\Theta_{p}+\tilde{\alpha_{r}}m_{r}}$
$\displaystyle\times
e^{-y^{\tilde{\alpha_{r}}}(\delta_{p}+\delta_{r}\Psi_{Q}^{-\tilde{\alpha_{r}}}\gamma_{r}^{\tilde{\alpha_{r}}})}dy=\sum_{m_{r}=0}^{\mu_{r}-1}\frac{\alpha_{p}\delta_{p}^{\mu_{p}}\delta_{r}^{m_{r}}}{2\Gamma(\mu_{p})m_{r}!}$
$\displaystyle\times\Psi_{Q}^{-\tilde{\alpha_{r}}m_{r}}\gamma_{r}^{\tilde{\alpha_{r}}m_{r}}\frac{\Gamma\left[\Omega,\left(\delta_{p}+\delta_{r}\Psi_{T}^{-\tilde{\alpha_{r}}}\gamma_{r}^{\tilde{\alpha_{r}}}\right)\left(\frac{\Psi_{Q}}{\Psi_{T}}\right)^{\tilde{\alpha_{r}}}\right]}{\tilde{\alpha_{r}}(\delta_{p}+\delta_{r}\Psi_{Q}^{-\tilde{\alpha_{r}}}\gamma_{r}^{\tilde{\alpha_{r}}})^{\Omega}}.$

Utilizing the integral formula of [39, Eq. (8.352.7)] and performing some
mathematical manipulations, $\mathcal{P}_{2}$ is demonstrated in an
alternative form as

$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\sum_{m_{r}=0}^{\mu_{r}-1}\sum_{m_{3}=0}^{\Omega-1}\sum_{m_{4}=0}^{m_{3}}\sum_{m_{5}=0}^{\infty}\binom{m_{3}}{m_{4}}\binom{\Omega+m_{5}-1}{m_{5}}$
$\displaystyle\times\Xi_{6}^{II}\gamma_{r}^{\tilde{\alpha_{r}}(m_{r}+m_{4}+m_{5})}e^{-(\delta_{p}\Psi_{Q}^{\tilde{\alpha_{r}}}\Psi_{T}^{-\tilde{\alpha_{r}}}+\delta_{r}\Psi_{T}^{-2\tilde{\alpha_{r}}}\Psi_{Q}^{\tilde{\alpha_{r}}}\gamma_{r}^{\tilde{\alpha_{r}}})}.$
(53)

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