# Opportunistic Relay Selection over Generalized Fading and Inverse Gamma
Composite Fading Mixed Multicast Channels: A Secrecy Tradeoff

MD. Shakhawat Hossen  A. S. M. Badrudduza  S. M. Riazul Islam  Department of
Computer Science and Engineering, Sejong University, 209 Neungdong-ro,
Gwangjin-gu, Seoul 05006, South Korea  Abu Hanif  Milton kumar kundu
Department of Electrical & Computer Engineering, RUET  Kyung-Sup Kwak  School
of Information and Communication Engineering, Inha University, Incheon 22212,
South Korea

###### Abstract

## Abstract

The secrecy performance of realistic wireless multicast scenarios can be
significantly deteriorated by the simultaneous occurrence of multipath and
shadowing. To resolve this security threat, in this work an opportunistic
relaying-based dual-hop wireless multicast framework is proposed in which the
source dispatches confidential information to a bunch of receivers via
intermediate relays under the wiretapping attempts of multiple eavesdroppers.
Two scenarios, i.e. non-line of sight (NLOS) and line of sight (LOS)
communications along with the multiplicative and LOS shadowing are considered
where the first scenario assumes $\eta-\mu$ and $\eta-\mu$/inverse Gamma (IG)
composite fading channels and the latter one follows $\kappa-\mu$ and
$\kappa-\mu$/IG composite fading channels as the source to relay and relay to
receiver’s as well as eavesdropper’s links, respectively. Secrecy analysis is
accomplished by deriving closed-form expressions of three familiar secrecy
measures i.e. secure outage probability for multicasting, probability of non-
zero secrecy multicast capacity, and ergodic secrecy multicast capacity. We
further capitalize on those expressions to observe the effects of all system
parameters which are again corroborated via Monte-Carlo simulations. Our
observations indicate that a secrecy tradeoff between the number of relays and
number of receivers, eavesdroppers, and shadowing parameters can be
established to maintain the admissible security level by decreasing the
detrimental influences of fading, shadowing, the number of multicast receivers
and eavesdroppers.

## Keywords

Composite channel, inverse Gamma shadowing, multiple eavesdroppers, physical
layer security, wireless multicasting.

##  1 Introduction

Due to the low power requirement and enhanced spatial diversity, relay
communication has been established as a proven technique for expanding the
coverage of wireless networks. However, such cooperative networks are
vulnerable to eavesdropping, where one or more eavesdroppers might overhear
transmissions and can thereby possibly pose security threats [1, 2]. So
maintaining perfect security during data transfer through a wireless medium in
general and cooperative communication, in particular, has become a significant
challenge for wireless researchers. To solve this problem, researchers
introduced the encryption method where the information was first encrypted
using an encryption key and then transmitted to the receiver through the
wireless channel. The technique was simple but became troublesome because of
the difficulty in encryption key distribution [3]. On the other hand, physical
layer security (PLS) has some tremendous advantages over the encryption
method. PLS is more secure as the confidentiality of the transmitted message
can be guaranteed by exploiting the physical characteristics of the wireless
channel as well as the proper coding and signal processing. Shadowing and
multipath fading are two very important characteristics of any wireless
channel which co-exist together and affect the channel simultaneously. So to
successfully characterize the random fluctuations occurring in the channel,
researchers have introduced composite fading models which can efficiently
reflect the practical scenarios which go through both fading and shadowing
[4]. Traditionally, lognormal and Gamma distributions were chosen by the
researchers for designing the composite fading channels [5]. But the
mathematical intractability of the lognormal distribution made it difficult to
use in practical situations. Contrariwise, the inverse Gamma (IG) distribution
is mathematically tractable and also exhibits the semi heavy-tailed
characteristics of both lognormal and Gamma distribution. As a result,
researchers have chosen to design composite fading models for generalized
fading channels like $\kappa-\mu$ [6] or $\eta-\mu$ [7] fading channels
employing the IG distribution. In this paper, we perform a PLS analysis of a
relay-based multicast network in the presence of multiple eavesdroppers with
the consideration of compound fading channels.

###  1.1 Literature Survey

Yacoub first uncovered the potentials of generalized fading distributions and
proposed a set of generalized fading schemes, namely, $\kappa-\mu$,
$\eta-\mu$, $\alpha-\mu$, and $\lambda-\mu$ fading models [8, 9, 10]. Since
then, researchers across the globe have examined various other forms of
compound fading channels based on these distributions. Some of the closely
related and representative variants of the distributions above are
$\alpha-\eta-\mu$, $\alpha-\kappa-\mu$, $\alpha-\eta-\kappa-\mu$,
$\alpha-\lambda-\mu-\eta$ fading channels [11, 12, 13]. Researchers have
investigated various statistical characterizations, such as probability
density function (PDF) and cumulative distribution function (CDF) of the
compound fading channels and their variants under different wireless settings,
including fading parameter variation [14, 15, 16, 17]. As evident in their
works, one can more precisely analyze the performances of many wireless
systems and networks of interest; in terms of channel capacity (CC), outage
probability (OP), coverage probability (CP), and bit error rate (BER). For
example, the compound fading approaches with or without shadowing have been
applied to analyze advanced wireless paradigms, including D2D, amplify and
forward, decode and forward, 5G radio access, and optical communication
systems [18, 19, 20]. As known, PLS can secure information by exploiting the
randomness of the wireless channel [21]. Generalized fading channels and their
variants are therefore also being considered in the analysis of PLS of
wireless systems and different PLS performance metrics such as secure outage
probability (SOP), average secrecy capacity (ASC), and strictly positive
secrecy capacity (SPSC) are evaluated [22, 23, 24, 25].

As mentioned in the early part of this section, IG is mathematically more
tractable. As such, researchers are recently paying attention to the IG
shadowing-based composite fading models. The effects of multipath and
shadowing on the SER performance over $\eta-\mu$ / IG and $\kappa-\mu$ / IG
distribution were analyzed in [26]; the authors therein derived CFEs for the
SER. The expression for the PDF, higher-order moments (HOM), and amount of
fading (AF) considering the $\kappa-\mu$ / IG and $\eta-\mu$ / IG composite
fading models were derived in [4]. Here the authors take into account the line
of sight (LOS) and non-LOS (NLOS) channel conditions. The performance of
wireless communication systems employing MRC diversity and coherent modulation
techniques was investigated over $\eta-\mu$ / IG composite fading channel in
[27] by deriving expressions for SER. The novel analytical expressions for the
envelope and power PDF and OP were derived under $\eta-\mu$/ inverse Gaussian
fading channels in [28] to show the impacts of multipath and shadowing
characteristics. In [7], the feature of the shadowed fading behavior was
inquired over the $\eta-\mu$/ IG composite fading model. Using Kullback-
Leibler divergence, the authors claimed that the $\eta-\mu$/ IG compound
fading channels demonstrate a better suit than the $\kappa-\mu$/ IG composite
fading model. A common approach of characterizing the composite fading
distributions based on IG shadowing was introduced by [29], where the authors
showed the formation of statistical characterization by presenting a composite
IG/two-wave with diffuse power fading model. Performance of the efficient
capacity (EC) was explored in [16] considering the different multipath fading,
delay constraint, and shadowing conditions. In [30], EC was investigated over
$\eta-\mu$/ IG composite fading channel and the authors exhibited the
influence of multipath fading and shadowing severity. ASEP, channel inversion
with fixed rate (CIFR), channel capacity under optimal rate adaption (ORA),
and truncated CIFR (TIFR) were derived in [31] for analyzing the CC and error
probability of wireless model over IG shadowed fading channel.

###  1.2 Motivation

It is noteworthy that the aforesaid researches are confined to the analysis of
the secrecy performance of wireless networks either in unicast or broadcast
scenarios. In many situations, especially in dense networks, however,
multicasting is preferred over broadcasting as it delivers the packet to
intended recipients only. Hence security issue is more important in a
multicast network rather than broadcast one. Moreover, it offers better
utilization of bandwidth and better congestion control. On the other hand,
whereas relays are usually placed as part of planned network installations or
the users experiencing better channel conditions are assigned as relays, the
locations of the mobile users are more random and even sparse. Also, the users
practically experience shadowing phenomena. Consequently, although the
channels between source and relays can be modeled as $\eta-\mu$ and
$\kappa-\mu$ distributions, the channels between relays and the end-users
might be a better fit with IG shadowing-based fading. On that, a secrecy
analysis over $\eta-\mu$ and $\eta-\mu$/ IG, $\kappa-\mu$ and $\kappa-\mu$/ IG
dual-hop link for a multicast framework are worth investigating. In this
study, we are thus motivated to mathematically address the security of
cooperative multicast relay networks over $\eta-\mu$ and $\eta-\mu$/ IG,
$\kappa-\mu$ and $\kappa-\mu$/ IG composite fading channels in the presence of
a set of eavesdroppers.

###  1.3 Contributions

The specific contributions of this paper are as follows:

  * •

We first individually perform the statistical characterizations of the SNR at
each hop and realize the associated PDF expressions. Then the PDF and CDF of
the dual-hop end-to-end SNR are derived. In addition to dual-hop channel
models under two distinct scenarios, our analysis considers the multicast
channel model and eavesdropper channel model by using order statistics.

  * •

Secondly, we study the secrecy capacity of the relay-based multicast system
and derive the closed-form expressions for secure outage probability for
multicasting (SOPM), probability of non-zero secrecy multicast capacity
(PNSMC), and ergodic secrecy multicast capacity (ESMC). Please note that these
expressions are completely novel as no literature has considered the
combination of $\eta-\mu$ and $\eta-\mu$/ IG, $\kappa-\mu$ and $\kappa-\mu$/
IG with multiple receivers as well as eavesdroppers. Moreover, these
expressions can be used to replicate the results of [32, 33, 34, 35, 36] just
by changing the parameter values as given in [4, Table 1].

  * •

Then, we illustrate the results based on the derived expressions of SOPM,
PNSMC, and ESMC. The performances of our dual-hop cooperative relay system are
in agreement with the secrecy performance of various existing models.

  * •

Also, we perform Monte-Carlo simulations and conduct a thorough numerical
evaluation. Our results, which focus on SOPM, PNSMC, and ESMC, demonstrate the
feasibility of adopting PLS in cooperative multicast networks with realistic
composite fading. The results show that the secrecy performance enhances with
fading and shadowing parameters as well as the average SNR of the main
channel. In contrast, secrecy performance degrades with the target secrecy
rate, number of receivers, eavesdroppers, and average SNR of eavesdropper
channel.

###  1.4 Organization

The rest of the paper is embodied as follows. The system model is shown in
section II. The channel models for both LOS and NLOS scenarios are presented
in Sections III-VI. The expressions of the performance metrics are derived in
Sections VII, VIII, and IX. Section X illustrates the numerical results and
finally, section XI draws the ending remarks of this research.

##  2 System Model

Figure 1:  Proposed multicast scenario with opportunistic relaying.

The proposed system network is depicted in Fig 1, where a single antenna
source, $\mathcal{S}$ dispatches secret messages to a group of $\mathcal{M}$
legitimate receivers via $\mathcal{K}$ relays. A bunch of $\mathcal{N}$
eavesdroppers are also present in that network which is presumed to be passive
and attempting to decode the private information. But we are intended to
defend the eavesdroppers from decoding the transmitted message. Here every
relay is furnished with a single antenna, contrariwise each receiver and each
eavesdropper is involved with an antenna, respectively. The distances between
the source and the receivers as well as the eavesdroppers are considered very
large and hence no direct link exists from source to receivers and from source
to eavesdroppers. Hence the total communication occurs only through the
relays. Here the channels between $\mathcal{S}$ and $\mathcal{M}$ are notified
as main channels and the channels between $\mathcal{S}$ to $\mathcal{N}$ are
indicated as eavesdropper channels. Entire communication occurs at two
different phases. In the first phase, the source transmits information to the
relays. Then each relay contends to be the best relay and receivers receive
the signal from only the best relay which is considered as the second phase.
We consider two scenarios depending on the LOS and NLOS communication links.
In scenario I, we assume all the links of the first and second hops are NLOS.
The channels between source to relays i.e $\mathcal{S}\rightarrow\mathcal{K}$
links are assumed to undergo $\eta-\mu$ fading. On the other hand relays to
desire receivers i.e $\mathcal{K}\rightarrow\mathcal{M}$ links undergo
$\eta-\mu$ / IG fading. When all the communication links are LOS i.e. scenario
II, we assume the $\mathcal{S}\rightarrow\mathcal{K}$ links undergo
$\kappa-\mu$ fading whereas the $\mathcal{K}\rightarrow\mathcal{M}$ and
$\mathcal{K}\rightarrow\mathcal{N}$ links undergo $\kappa-\mu$ / IG fading.

The channel coefficients between $\mathcal{S}$ and $p$th
$(p=1,2,3,\cdots,\mathcal{K})$ link is denoted by
$\mathbf{a}_{p,j}\in\mathbb{C}^{1\times 1}$, $\mathcal{K}$ and $m$th
$(m=1,2,3,\cdots,\mathcal{M})$ link is notified by
$\mathbf{b}_{m,j}\in\mathbb{C}^{1\times 1}$ , $\mathcal{K}$ and $t$th
$(t=1,2,3,\cdots,\mathcal{N})$ link is denoted by
$\mathbf{c}_{t,j}\in\mathbb{C}^{1\times 1}$, respectively. Here $j\in[1,2]$
indicates the first and second scenarios, respectively. Hence the
corresponding SNRs of $\mathcal{S}\rightarrow\mathcal{K}$,
$\mathcal{K}\rightarrow\mathcal{M}$, and $\mathcal{K}\rightarrow\mathcal{N}$
links are respectively, given by
$\Upsilon_{sp,j}=\frac{P_{s,j}}{N_{p,j}}\|a_{p,j}\|^{2}$,
$\Upsilon_{pm,j}=\frac{P_{k,j}}{N_{m,j}}\|b_{m,j}\|^{2}$, and
$\Upsilon_{pt,j}=\frac{P_{k,j}}{N_{t,j}}\|c_{t,j}\|^{2}$. Here $P_{s,j}$, and
$P_{k,j}$ are the transmit powers from $\mathcal{S}$, and $\mathcal{K}$. The
additive white Gaussian noise terms at $\mathcal{K}$, $\mathcal{M}$ and
$\mathcal{N}$ are denoted by $N_{p,j}$, $N_{m,j}$, and $N_{t,j}$,
respectively.

##  3 Channel Model (Scenario I)

###  3.1 PDFs of SNRs for $\mathcal{S}\rightarrow\mathcal{K}$,
$\mathcal{K}\rightarrow\mathcal{M}$ and $\mathcal{K}\rightarrow\mathcal{N}$
Links

The PDF of SNR for $\mathcal{S}\rightarrow\mathcal{K}$ link denoted as
$\Upsilon_{sp,1}$ is given by [37, eq. 2]

$\displaystyle f_{sp,1}(\Upsilon)$
$\displaystyle=\alpha_{1}\Upsilon^{\mu_{p1}-0.5}e^{-\beta_{s}\Upsilon}I_{\mu_{p1}-0.5}(\varepsilon_{1}\Upsilon),$
(1)

where
$\alpha_{1}=\frac{2\sqrt{\pi}\mu_{p1}^{\mu_{p1}+0.5}h_{1}^{\mu_{p1}}}{\Gamma(\mu_{p1})H_{1}^{\mu_{p1}-0.5}\phi_{p1}^{\mu_{p1}+0.5}}$,
$h_{1}=\frac{1}{1-\eta_{p}^{2}}$, $H_{1}=\frac{\eta_{p}}{1-\eta_{p}^{2}}$,
$\varepsilon_{1}=\frac{2\mu_{p1}H_{1}}{\phi_{p1}}$,
$\beta_{s}=\frac{2\mu_{p1}h_{1}}{\phi_{p1}}$, $\mu_{p1}>$ 0 symbolizes the
number of multipath clusters of the first hop, $\eta_{p}$ which is bounded in
$-1<\eta_{p}<1$ demonstrates the correlation coefficient between the in-phase
and quadrature scattered waves in each multi-path cluster of the first hop
[38], and $\phi_{p1}$ is the average SNR of
$\mathcal{S}\rightarrow\mathcal{K}$ link and $I_{\nu}(.)$ is the modified
Bessel function which is defined in [39, eq 8.440]. Simplifying (1) using [39,
eq 8.445], the PDF of $\Upsilon_{sp,1}$ can be expressed as

$\displaystyle
f_{sp,1}(\Upsilon)=\sum_{n_{1}=0}^{\infty}\alpha_{2}\Upsilon^{2\mu_{p1}-1+2n_{1}}e^{-\beta_{s}\Upsilon},$
(2)

where
$\alpha_{2}=\frac{a_{1}}{n_{1}!\Gamma(\mu_{p1}+0.5+n_{1})}(\frac{\varepsilon_{1}}{2})^{\mu_{p1}-0.5+2n_{1}}$.
Now the PDF of $\Upsilon_{pm,1}$ is given by [4, eq. 13]

$\displaystyle f_{pm,1}(\Upsilon)$
$\displaystyle=\frac{2^{2\mu_{m}}\mu_{m1}^{2\mu_{m1}}}{\left(m_{m1}\phi_{m1}+2h_{2}\mu_{m1}\Upsilon\right){}^{m_{m1}+2\mu_{m1}}}$
$\displaystyle\times\frac{\left(m_{m1}\phi_{m1}\right){}^{m_{m1}}h_{2}^{\mu_{m}}\Upsilon^{2\mu_{m}-1}}{B\left(m_{m1},2\mu_{m1}\right)}$
$\displaystyle\times\,_{2}F_{1}\left(\epsilon_{1},\epsilon_{2};\epsilon_{3};\left(\frac{2H_{2}\mu_{m1}\Upsilon}{m_{m1}\phi_{m1}+2h_{2}\mu_{m1}\Upsilon}\right)^{2}\right),$
(3)

where $h_{2}=\frac{1}{1-\eta_{m}^{2}}$,
$H_{2}=\frac{\eta_{m}}{1-\eta_{m}^{2}}$,
$\epsilon_{1}=\frac{1}{2}\left(m_{m1}+2\mu_{m1}\right)$,
$\epsilon_{2}=\frac{1}{2}\left(m_{m1}+2\mu_{m1}+1\right)$,
$\epsilon_{3}=\frac{1}{2}\left(2\mu_{m1}+1\right)$, $\mu_{m1}>$ 0 is related
to the number of multipath clusters of the $\mathcal{K}\rightarrow\mathcal{M}$
link and $\eta_{m}$ is concerned with the scattered wave power ratio between
the in-phase and quadrature components of each multipath cluster of the
$\mathcal{K}\rightarrow\mathcal{M}$ link, B($\cdot$, $\cdot$) denotes the Beta
function as defined in [39, eq 8.384.1], and ${}_{2}F_{1}$($\cdot$, $\cdot$;
$\cdot$; $\cdot$) notifies the Gaussian hyper-geometric function as defined in
[39, eq 9.111]. Using [39, eq 9.14.1] and modifying (3.1) accordingly, the PDF
of $\Upsilon_{pm,1}$ can be expressed as

$\displaystyle f_{pm,1}(\Upsilon)$
$\displaystyle=\sum_{r_{1}=0}^{\infty}\frac{\lambda_{2}\Upsilon^{2\mu_{m1}+2r_{1}-1}}{\left(1+\beta_{m}\Upsilon\right){}^{m_{m1}+2\mu_{m1}+2r_{1}}},$
(4)

where $\beta_{m}=\frac{2h_{2}\mu_{m1}}{m_{m1}\phi_{m1}}$, $\phi_{m1}$ is the
average SNR of $\mathcal{K}$ to $\mathcal{M}$ link,
$\lambda_{2}=\frac{\lambda_{1}\left(\epsilon_{1}\right)_{r_{1}}\left(\epsilon_{2}\right)_{r_{1}}\left(2H_{2}\mu_{m1}\right){}^{2r_{1}}}{r_{1}!\left(\epsilon_{3}\right)_{r_{1}}\left(m_{m1}\phi_{m1}\right){}^{m_{m1}+2\mu_{m1}+2r_{1}}}$,
and
$\lambda_{1}=\frac{2^{2\mu_{m1}}\mu_{m1}^{2\mu_{m1}}\left(m_{m1}\phi_{m1}\right){}^{m_{m1}}h_{2}^{\mu_{m1}}}{B\left(m_{m1},2\mu_{m1}\right)}$,
Similar to (4), the PDF of $\Upsilon_{pt,1}$ can be expressed as

$\displaystyle f_{pt,1}(\Upsilon)$
$\displaystyle=\sum_{\nu_{1}=0}^{\infty}\frac{\xi_{2}\Upsilon^{2\mu_{t1}+2\upsilon_{1}-1}}{\left(1+\beta_{e}\Upsilon\right){}^{2\mu_{t1}+m_{t1}+2\upsilon_{1}}},$
(5)

where $\beta_{e}=\frac{2h_{3}\mu_{t1}}{m_{t1}\phi_{t1}}$, $\phi_{t1}$ is the
average SNR of $\mathcal{K}\rightarrow\mathcal{N}$ link, $\mu_{t1}>$ 0
symbolizes the number of multipath clusters of the
$\mathcal{K}\rightarrow\mathcal{N}$ link and $\eta_{t}$ is concerned with the
scattered wave power ratio between the in-phase and quadrature components of
each multipath cluster of the $\mathcal{K}\rightarrow\mathcal{N}$ link,
$\xi_{2}=\frac{\xi_{1}\left(\kappa_{1}\right)_{\upsilon_{1}}\left(\kappa_{2}\right)_{\upsilon_{1}}\left(2H_{3}\mu_{t1}\right){}^{2\upsilon_{1}}}{\upsilon_{1}!\left(\kappa_{3}\right)_{\upsilon_{1}}\left(m_{t1}\phi_{t1}\right){}^{2\mu_{t1}+m_{t1}+2\upsilon_{1}}}$,
$\xi_{1}=\frac{2^{2\mu_{t1}}\mu_{t1}^{2\mu_{t1}}h_{3}^{\mu_{t1}}\left(m_{t1}\phi_{t1}\right){}^{m_{t1}}}{B\left(m_{t1},2\mu_{t1}\right)}$,
$h_{3}=\frac{1}{1-\eta_{t}^{2}}$, $H_{3}=\frac{\eta_{t}}{1-\eta_{t}^{2}}$,
$\kappa_{1}=\frac{1}{2}\left(2\mu_{t1}+m_{t1}\right)$,
$\kappa_{2}=\frac{1}{2}\left(2\mu_{t1}+m_{t1}+1\right)$, and
$\kappa_{3}=\frac{1}{2}\left(2\mu_{t1}+1\right)$.

###  3.2 PDFs of Dual-hop SNRs

Denoting the SNR of $\mathcal{S}\rightarrow\mathcal{K}\rightarrow\mathcal{M}$
link by $\Upsilon_{sm,1}$, its PDF can be obtained as

$\displaystyle f_{sm,1}(\Upsilon)$
$\displaystyle=\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\biggl{[}\Upsilon^{-n_{2}+2\mu_{p1}+2n_{1}}e^{-\beta_{s}\Upsilon}$
$\displaystyle+\Upsilon^{-n_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]},$
(6)

where $n_{2}=m_{m1}+r_{2}+1$.

Proof: See Appendix 12.1.

Similarly, by denoting the SNR of
$\mathcal{S}\rightarrow\mathcal{K}\rightarrow\mathcal{N}$ link by
$\Upsilon_{st,1}$, its PDF can be given by

$\displaystyle f_{st,1}(\Upsilon)$
$\displaystyle=\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\biggl{[}\Upsilon^{-n_{3}+2\mu_{p1}+2n_{1}}e^{-\beta_{s}\Upsilon}$
$\displaystyle+\Upsilon^{-n_{3}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]},$
(7)

where $n_{3}=m_{t1}+\nu_{2}+1$.

Proof: See Appendix 12.2.

###  3.3 PDFs of SNRs for the Best Relay

Let,$\Upsilon_{bm,1}$ denotes the SNR between the best relay, and $m$th
receiver which is defined as

$\displaystyle\Upsilon_{bm,1}=arg_{p\varepsilon\varsigma}^{max}\min(\Upsilon_{sp,1},\Upsilon_{pm,1}),$
(8)

where $\varsigma$ = (1,2,…,$\mathcal{K}$) is the relay set. The PDF of
$\Upsilon_{bm,1}$ can be obtained as the following.

$\displaystyle
f_{bm,1}(\Upsilon)=\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\mathcal{K}\biggl{[}\Upsilon^{-n_{2}+2\mu_{p1}+2n_{1}}e^{-\beta_{s}\Upsilon}$
$\displaystyle+\Upsilon^{-n_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\Upsilon^{-m_{m1}-r_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{\mathcal{K}-1}.$
(9)

Proof: See Appendix 13.1.

Similar to (8), notifying the SNR between the best relay and the $t$th
eavesdropper by $\Upsilon_{bt,1}$, we have

$\displaystyle\Upsilon_{bt,1}=arg_{p\varepsilon\varsigma}^{max}\min(\Upsilon_{sp,1},\Upsilon_{pt,1}).$
(10)

The PDF $\Upsilon_{bt,1}$ can be obtained as the following.

$\displaystyle
f_{bt,1}(\Upsilon)=\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\mathcal{K}\biggl{[}\Upsilon^{-n_{3}+2\mu_{p1}+2n_{1}}e^{-\beta_{s}\Upsilon}$
$\displaystyle+\Upsilon^{-n_{3}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\Upsilon^{-m_{t1}-\nu_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{\mathcal{K}-1}.$
(11)

Proof: See Appendix 13.2.

##  4 Channel Model (Scenario II)

###  4.1 PDFs of SNRs for $\mathcal{S}\rightarrow\mathcal{K}$,
$\mathcal{K}\rightarrow\mathcal{M}$ and $\mathcal{K}\rightarrow\mathcal{N}$
Links

The PDF of $\Upsilon_{sp,2}$ is given by [40, Eq. (2)]

$\displaystyle f_{sp,2}(\Upsilon)$
$\displaystyle=\frac{\mu_{p2}(1+\kappa_{p})^{\frac{\mu_{p2}+1}{2}}\Upsilon^{\frac{\mu_{p2}-1}{2}}e^{\frac{-\mu_{p2}(1+\kappa_{p})\Upsilon}{\phi_{p2}}}}{\kappa_{p}^{\frac{\mu_{p2}-1}{2}}\phi_{p2}^{\frac{\mu_{p2}+1}{2}}e^{\mu_{p2}\kappa_{p}}}$
$\displaystyle\times
I_{\mu_{p2}-1}\left(2\mu_{p2}\sqrt{\frac{\kappa_{p}(1+\kappa_{p})\Upsilon}{\phi_{p2}}}\right),$
(12)

where $\kappa_{p}>0$ is the ratio of the total power of the dominant
components to the total power of the scattered waves, $\mu_{p2}>0$ is related
to the number of multipath clusters that is given by
$\mu_{p2}=\frac{\mathbb{E}^{2}(\Upsilon_{sp,2})(1+2\kappa_{p})}{\mathbb{V}(\Upsilon_{sp,2})(1+\kappa_{p})^{2}}$
[40], where $\mathbb{E}(\cdot)$ and $\mathbb{V}(\cdot)$ indicates the
expectation and variance operators, apart from that
$\phi_{p2}=\mathbb{E}(\Upsilon_{sp,2})$, is the average signal-to-noise-ratio
(SNR) of $\mathcal{S}\rightarrow\mathcal{K}$ link and $I_{n}(\cdot)$ is the
modified Bessel function of the first kind and order $n$. Simplifying (4.1) by
using [39, eq. 8.445], the PDF of $\Upsilon_{sp,2}$ can be evaluated as

$\displaystyle f_{sp,2}(\Upsilon)$
$\displaystyle=\sum_{\rho_{1}=0}^{\infty}\delta_{2}\Upsilon^{\mu_{p2}-1+\rho_{1}}e^{\frac{-\mu_{p2}(1+\kappa_{p})\Upsilon}{\phi_{p2}}},$
(13)

where
$\delta_{2}=\delta_{1}\frac{\mu_{p2}^{\mu_{p2}-1+2\rho_{1}}(\frac{\kappa_{p}(1+\kappa_{p})}{\phi_{p2}})^{\frac{\mu_{p2}-1+2\rho_{1}}{2}}}{\rho_{1}!\Gamma(\mu_{p2}+\rho_{1})}$,
$\delta_{1}=\frac{\mu_{p2}(1+\kappa_{p})^{\frac{\mu_{p2}+1}{2}}}{\kappa_{p}^{\frac{\mu_{p2}-1}{2}}\phi_{p2}^{\frac{\mu_{p2}+1}{2}}e^{\mu_{p2}\kappa_{p}}}$,
and $\Gamma(\cdot)$ denotes the Gamma function [39]. Now the PDF of
$\Upsilon_{pm,2}$ is given by [4, Eq. (4)]

$\displaystyle
f_{pm,2}(\Upsilon)=\frac{e^{-\mu_{m2}\kappa_{m}}\mu_{m2}^{\mu_{m2}}}{[\mu_{m2}(\kappa_{m}+1)\Upsilon+m_{m2}\phi_{m2}]^{m_{m2}+\mu_{m2}}}$
$\displaystyle\times\frac{(\kappa_{m}+1)^{\mu_{m2}}(m_{m2}\phi_{m2})^{m_{m2}}\Upsilon^{\mu_{m2}-1}}{B(m_{m2},\mu_{m2})}$
$\displaystyle\times\,{}_{1}F_{1}\left(m_{m2}+\mu_{m2};\mu_{m2};\frac{\mu_{m2}^{2}\kappa_{m}(\kappa_{m}+1)\Upsilon}{\mu_{m2}(\kappa_{m}+1)\Upsilon+m_{m2}\phi_{m2}}\right),$
(14)

where $\phi_{m2}$ is the average SNR of $\mathcal{K}\rightarrow\mathcal{M}$
links, B($\cdot$, $\cdot$) and ${}_{1}F_{1}$($\cdot$; $\cdot$; $\cdot$)
denotes the Beta and the Confluent Hypergeometric functions as defined in [39,
eq. 9.210.1], and $(j_{1})_{p_{1}}=\frac{\Gamma(j_{1}+p_{1})}{\Gamma(j_{1})}$
denotes the Pochhammer symbol. Using [39, eq. 9.14.1], (4.1) can be simplified
as

$\displaystyle
f_{pm,2}(\Upsilon)=\sum_{\rho_{2}=0}^{\infty}\frac{\delta_{5}\Upsilon^{\rho_{2}+\mu_{m2}-1}}{(1+\delta_{4}\Upsilon)^{m_{m2}+\mu_{m2}+\rho_{2}}},$
(15)

where
$\delta_{5}=\delta_{3}\frac{\left(\kappa_{m}\left(\kappa_{m}+1\right)\mu_{m2}^{2}\right){}^{\rho_{2}}\left(m_{m2}+\mu_{m2}\right)_{\rho_{2}}}{\rho_{2}!\left(\mu_{m2}\right)_{\rho_{2}}}$,
$\delta_{4}=\frac{\kappa_{m}\mu_{m2}+\mu_{m2}}{m_{m2}\Phi_{m2}}$, and
$\delta_{3}=\frac{\mu_{m2}^{\mu_{m2}}\left(m_{m2}\Phi_{m2}\right){}^{m}_{m2}e^{\kappa_{m}\left(-\mu_{m2}\right)}\left(\kappa_{m}+1\right){}^{\mu_{m2}}}{B\left(m_{m2},\mu_{m2}\right)\left(m_{m2}\Phi_{m2}\right){}^{m_{m2}+\mu_{m2}+\rho_{2}}}$.
Similarly, the PDF of $\Upsilon_{pt,2}$ for
$\mathcal{K}\rightarrow\mathcal{N}$ link can be illustrated as

$\displaystyle
f_{pt,2}(\Upsilon)=\sum_{\sigma_{2}=0}^{\infty}\frac{\beta_{5}\Upsilon^{\sigma_{2}+\mu_{t2}-1}}{(1+\beta_{4}\Upsilon)^{m_{t2}+\mu_{t2}+\sigma_{2}}},$
(16)

where
$\beta_{3}=\beta_{1}\frac{\left(\kappa_{t}\left(\kappa_{t}+1\right)\mu_{t2}^{2}\right){}^{\sigma_{2}}\left(m_{t2}+\mu_{t2}\right)_{\sigma_{2}}}{\sigma_{2}!\left(\mu_{t2}\right)_{\sigma_{2}}}$,
$\beta_{2}=\frac{\kappa_{t}\mu_{t2}+\mu_{t2}}{m_{t2}\Phi_{t2}}$, and
$\beta_{1}=\frac{\mu_{t2}^{\mu_{t2}}\left(m_{t2}\Phi_{t2}\right){}^{m_{t2}}e^{\kappa_{t}\left(-\mu_{t2}\right)}\left(\kappa_{t}+1\right){}^{\mu_{t2}}}{B\left(m_{t2},\mu_{t2}\right)\left(m_{t2}\Phi_{t2}\right){}^{m_{t2}+\mu_{t2}+\sigma_{2}}}$.

###  4.2 PDF of Dual-Hop SNRs

Denoting $\Upsilon_{sm,2}$ as the SNR of
$\mathcal{S}\rightarrow\mathcal{K}\rightarrow\mathcal{M}$ link, the PDF of
$\Upsilon_{sm,2}$ can be obtained as the following.

$\displaystyle f_{sm,2}$
$\displaystyle(\Upsilon)=\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{7}\biggl{[}\Upsilon^{-\rho_{3}-m_{m2}-1+\mu_{p2}+\rho_{1}}$
$\displaystyle\times
e^{-\Upsilon\delta_{b_{1}}}+\Upsilon^{-1-\rho_{3}-m_{m2}}\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}.$
(17)

Proof: See Appendix 14.1.

Similarly, denoting $\Upsilon_{st,2}$ as the SNR of
$\mathcal{S}\rightarrow\mathcal{K}\rightarrow\mathcal{N}$ link, the PDF of
$\Upsilon_{st,2}$ can be obtained as the following.

$\displaystyle f_{st,2}(\Upsilon)=$
$\displaystyle\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\biggl{[}e^{-\Upsilon\delta_{b_{1}}}\Upsilon^{-\sigma_{3}-m_{t2}-1+\mu_{p2}+\sigma_{1}}$
$\displaystyle+\Upsilon^{-1-\sigma_{3}-m_{t2}}\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}.$
(18)

Proof: See Appendix 14.2.

###  4.3 PDFs of SNRs for best relay

Let, $\Upsilon_{bm,2}$ denotes the SNR between the best relay and $m$th
receiver which is defined as

$\displaystyle\Upsilon_{bm,2}=arg_{p\in\varsigma}^{max}min(\Upsilon_{sp,2},\Upsilon_{pm,2}).$
(19)

The PDF of $\Upsilon_{bm,2}$ can be obtained as the following.

$\displaystyle
f_{bm,2}(\Upsilon)=\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{7}\mathcal{K}\biggl{[}\Upsilon^{-\rho_{3}-m_{m2}-1+\mu_{p2}+\rho_{1}}$
$\displaystyle\times
e^{-\Upsilon\delta_{b_{1}}}+\Upsilon^{-1-\rho_{3}-m_{m2}}\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{7}\Upsilon^{-m_{m2}-\rho_{3}}\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)\biggl{]}^{\mathcal{K}-1}.$
(20)

Proof: See Appendix 15.1.

Similar to (19), the SNR between best relay and $t\textit{th}$ eavesdropper is
defined by $\Upsilon_{bt,2}$, we have

$\displaystyle\Upsilon_{bt,2}=arg_{p\in\varsigma}^{max}min(\Upsilon_{sp,2},\Upsilon_{pt,2}).$
(21)

The PDF of $\Upsilon_{bt,2}$ can be obtained as

$\displaystyle
f_{bt,2}(\Upsilon)=\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\mathcal{K}\biggl{[}\Upsilon^{-\sigma_{3}-m_{t2}-1+\mu_{p2}+\sigma_{1}}$
$\displaystyle\times
e^{-\Upsilon\delta_{b_{1}}}+\Upsilon^{-1-\sigma_{3}-m_{t2}}\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\Upsilon^{-m_{t2}-\sigma_{3}}\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)\biggl{]}^{\mathcal{K}-1}.$
(22)

Proof: See Appendix 15.2.

##  5 Multicast Channel Model

We consider multiple receivers in the proposed multicast system and the
secrecy analysis is carried out by means of assuming the worst case of the
multicast channels i.e. minimum SNR among $\mathcal{M}$ instantaneous SNRs at
the legitimate receivers. The significance of this assumption is that if the
security can be enhanced for the worst case, then we can clearly declare that
the total system is also secure for any other possible cases of the multicast
channels. Let $\sigma_{min,j}=min_{1\leqslant
m\leqslant\mathcal{M}}\Upsilon_{bm,j}$ denotes the minimum SNR of the
multicast channels. Since $\Upsilon_{b1,j}$, $\Upsilon_{b2,j}$,
$\Upsilon_{b3,j}$, …, $\Upsilon_{b\mathcal{M},j}$ are independent, hence the
PDF of $\sigma_{min,j}$ is defined using order statistics as [36, Eq. 12]

$\displaystyle
f_{\sigma_{min,j}}(\Upsilon_{bm,j})=\mathcal{M}f_{bm,j}(\Upsilon)[1-F_{bm,j}(\Upsilon)]^{\mathcal{M}-1}.$

The following subsections include derivations of the expressions of
$f_{\sigma_{min,1}}(\Upsilon)$ and $f_{\sigma_{min,2}}(\Upsilon)$ for the two
considered scenarios.

###  5.1 Scenario I

The final expression of $f_{\sigma_{min,1}}(\Upsilon)$ can be derived as,

$\displaystyle
f_{\sigma_{min,1}}(\Upsilon)=\sum_{\psi_{r_{6}}}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}$
$\displaystyle\times\biggl{(}\alpha_{3}\Upsilon^{\alpha_{5}}+\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\alpha_{4}\Upsilon^{\alpha_{6}}\biggl{)}e^{-\beta_{4}\Upsilon},$
(23)

where $\alpha_{3}=\Lambda_{2}\Lambda_{6}\Lambda_{9}\mathcal{M}\mathcal{K}$,
$\alpha_{4}=\Lambda_{2}\Lambda_{6}\Lambda_{7}\Lambda_{9}\mathcal{M}\mathcal{K}$,
$\alpha_{5}=-n_{2}+2\mu_{p1}+2n_{1}+\varphi_{\psi_{r_{6}}}$,
$\alpha_{6}=-n_{2}+r_{7}+\varphi_{\psi_{r_{6}}}$, and
$\beta_{4}=\beta_{s}+\phi_{\psi_{r_{6}}}$.

Proof: See Appendix 16.1.

###  5.2 Scenario II

Similarly, the final expression of $f_{\sigma_{min,2}}(\Upsilon_{bm,2})$ can
be derived as

$\displaystyle
f_{\sigma_{min,2}}(\Upsilon)=\sum_{\psi_{\rho_{7}}}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}$
$\displaystyle\times\biggl{(}\varpi_{1}\Upsilon^{\varpi_{4}}+\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\varpi_{2}\Upsilon^{\varpi_{5}}\biggl{)}e^{-\Upsilon\varpi_{3}},$
(24)

where
$\delta_{14}=\sum_{\psi_{\rho_{7}}}\binom{\rho_{7}}{g_{0,0,0,0},\ldots,g_{\rho_{9},\rho_{3}\rho_{2},\rho_{1}},\ldots,n_{\mu_{p2}+\rho_{1}-1,\infty,\infty,\infty}}\\\
\chi_{\psi_{\rho_{7}}}$,
$\varpi_{1}=\mathcal{M}\mathcal{K}\delta_{7}\delta_{10}\delta_{14}$,
$\varpi_{2}=\mathcal{M}\mathcal{K}\delta_{7}\delta_{10}\delta_{12}\delta_{14}$,
$\varpi_{3}=\delta_{b_{1}}+\Theta_{\psi_{\rho_{7}}}$,
$\varpi_{4}=\eta_{\psi_{\rho_{7}}}+\delta_{11}+\mu_{p2}+\rho_{1}$, and
$\varpi_{5}=\eta_{\psi_{\rho_{7}}}+\delta_{11}+\rho_{8}$.

Proof: See Appendix 16.2.

##  6 Eavesdropper Channel Model

Since multiple eavesdroppers exist in the proposed system, the worst secure
condition in that particular case is obtained by maximizing the impacts of the
eavesdroppers i.e. assuming the maximum SNR among $\mathcal{N}$ instantaneous
SNRs at the eavesdropper terminals. Let $\sigma_{max,j}=max_{1\leqslant
t\leqslant\mathcal{N}}\Upsilon_{bt,j}$ denotes the maximum SNR of the
eavesdropper channels. Again, since $\Upsilon_{b1,j}$, $\Upsilon_{b2,j}$,
$\Upsilon_{b3,j}$,…,$\Upsilon_{b\mathcal{N},j}$ are independent, hence the PDF
of $\sigma_{max,j}$ is defined as [36, Eq. 15]

$\displaystyle
f_{\sigma_{max,j}}(\Upsilon)=\mathcal{N}f_{bt,j}(\Upsilon)[F_{bt,j}(\Upsilon)]^{\mathcal{N}-1}.$

The following subsections illustrate the formation of the expressions for
$f_{\sigma_{max,1}}(\Upsilon)$ and $f_{\sigma_{max,2}}(\Upsilon)$ for the both
scenarios (I and II).

###  6.1 Scenario I

The final expression of $f_{\sigma_{max,1}}(\Upsilon)$ can be derived as,

$\displaystyle
f_{\sigma_{max,1}}(\Upsilon)=\sum_{\psi_{\nu_{6}}}\sum_{\nu_{6}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}$
$\displaystyle\times\biggl{(}\alpha_{10}\Upsilon^{\alpha_{12}}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\alpha_{11}\Upsilon^{\alpha_{13}}\biggl{)}e^{-\beta_{5}\Upsilon},$
(25)

where
$\chi_{9}=\sum_{\psi_{\nu_{6}}}\binom{\nu_{6}}{q_{0,0,0,0,\ldots,}q_{\nu_{8},\nu_{2},\nu_{1},n_{1}.\ldots,}n_{2\mu_{p1}+2n_{1}-1,\infty,\infty,\infty}}\\\
\Xi_{\psi_{\nu_{6}}}$,$\alpha_{10}=\chi_{2}\chi_{6}\chi_{9}\mathcal{N}\mathcal{K}$,
$\alpha_{11}=\chi_{2}\chi_{6}\chi_{7}\chi_{9}\mathcal{N}\mathcal{K}$,
$\alpha_{12}=-n_{3}+2\mu_{p1}+2n_{1}+\varphi_{\psi_{\nu_{6}}}$,
$\alpha_{13}=-n_{3}+\nu_{7}+\varphi_{\psi_{\nu_{6}}}$, and
$\beta_{5}=\beta_{s}+\phi_{\psi_{\nu_{6}}}$.

Proof: See Appendix 17.1.

###  6.2 Scenario II

Similarly, the final expression of $f_{\sigma_{max,2}}(\Upsilon_{bt,2})$ can
be derived as,

$\displaystyle
f_{\sigma_{max,2}}(\Upsilon)=\sum_{\psi_{\sigma_{10}}}\sum_{\sigma_{10}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}$
$\displaystyle\times\biggl{(}\omega_{1}\Upsilon^{\omega_{4}}+\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\omega_{2}\Upsilon^{\omega_{5}}\biggl{)}e^{-\Upsilon\omega_{3}},$
(26)

where
$\beta_{18}=\sum_{\psi_{\sigma_{10}}}\binom{\sigma_{10}}{f_{0,0,0,0},\ldots,f_{\sigma_{12},\sigma_{3},\sigma_{2},\rho_{1}},\ldots,n_{\mu+\rho_{1}-1,\infty,\infty,\infty}}\\\
\chi_{\psi_{\sigma_{10}}}$,
$\omega_{1}=\mathcal{N}\mathcal{K}\beta_{7}\beta_{15}\beta_{18}$,
$\omega_{2}=\mathcal{N}\mathcal{K}\beta_{7}\beta_{15}\beta_{16}\beta_{18}$,
$\omega_{3}=\delta_{b_{1}}+\Theta_{\psi_{\sigma_{10}}}$,
$\omega_{4}=\eta_{\psi_{\sigma_{10}}}+\beta_{10}+\mu_{p1}+\rho_{1}$, and
$\omega_{5}=\eta_{\psi_{\sigma_{10}}}+\beta_{10}+\sigma_{11}$.

Proof: See Appendix 17.2.

$\displaystyle
P_{out}(\sigma_{s1})=1-\sum_{\psi_{r_{6}}}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\sum_{\psi_{\nu_{6}}}\sum_{\nu_{6}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\biggl{[}\sum_{r_{9}=0}^{\alpha_{5}}\sum_{r_{11}=0}^{r_{9}}\biggl{(}\frac{\Lambda_{10}\alpha_{10}(r_{11}+\alpha_{12})!}{(\beta_{5}+\beta_{4}q_{\zeta})^{r_{11}+\alpha_{12}+1}}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}$
$\displaystyle\frac{\Lambda_{10}\alpha_{11}(r_{11}+\alpha_{13})!}{(\beta_{5}+\beta_{4}q_{\zeta})^{r_{11}+\alpha_{13}+1}}\biggl{)}-\sum_{r_{10}=0}^{\alpha_{6}}\sum_{r_{12}=0}^{r_{10}}\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\biggl{(}\frac{\Lambda_{11}\alpha_{10}(r_{12}+\alpha_{12})!}{(\beta_{5}+\beta_{4}q_{\zeta})^{r_{12}+\alpha_{12}+1}}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\frac{\Lambda_{11}\alpha_{11}(r_{12}+\alpha_{13})!}{(\beta_{5}+\beta_{4}q_{\zeta})^{r_{12}+\alpha_{13}+1}}\biggl{)}\biggl{]}.$
(29) $\displaystyle
P_{out}(\sigma_{s2})=1-\sum_{\psi_{\rho_{7}}}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\psi_{\sigma_{10}}}\sum_{\sigma_{10}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\biggl{[}\sum_{\sigma_{13}=0}^{\varpi_{4}}\sum_{\sigma_{15}=0}^{\sigma_{13}}\biggl{(}\frac{\delta_{19}\omega_{1}(\sigma_{15}+\omega_{4})!}{(\varpi_{3}+\omega_{3}q_{\theta})^{\sigma_{15}+\omega_{4}+1}}$
$\displaystyle+\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\frac{\delta_{19}\omega_{2}(\sigma_{15}+\omega_{5})!}{(\varpi_{3}+\omega_{3}q_{\theta})^{\sigma_{15}+\omega_{5}+1}}\biggl{)}-\sum_{\sigma_{16}=0}^{\varpi_{5}}\sum_{\sigma_{16}=0}^{\sigma_{14}}\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\biggl{(}\frac{\delta_{20}\omega_{1}(\sigma_{16}+\omega_{4})!}{(\varpi_{3}+\omega_{3}q_{\theta})^{\sigma_{16}+\omega_{4}+1}}+\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\frac{\delta_{20}\omega_{2}(\sigma_{16}+\omega_{5})!}{(\varpi_{3}+\omega_{3}q_{\theta})^{\sigma_{16}+\omega_{5}+1}}\biggl{)}\biggl{]}.$
(32)

##  7 Secure Outage Analysis

Denoting $C_{s,j}$ as the instantaneous secrecy multicast rate [41] and
$\sigma_{sj}$ as the target secrecy rate, the SOPM is defined as [36, Eq. 23]

$\displaystyle
P_{out}(\sigma_{sj})=\Pr(C_{s,j}<\sigma_{sj})=1-\int_{0}^{\infty}\int_{\zeta}^{\infty}f_{\sigma_{min,j}}(\Upsilon_{bm,j})$
$\displaystyle\times
f_{\sigma_{max,j}}(\Upsilon_{bt,j})d\Upsilon_{bm,j}d\Upsilon_{bt,j},$

where $\zeta=2^{\sigma_{sj}}(1+\Upsilon_{bt,j})-1$ and $\sigma_{sj}>0$. The
definition indicates that, the secure transmission is effective only if
$C_{s,j}>\sigma_{sj}$. Here $f_{\sigma_{min,j}}(\Upsilon_{bm,j})$ is obtained
from Eqs. (5.1), and (5.2) and similarly,
$f_{\sigma_{max,j}}(\Upsilon_{bt,j})$ is represented by Eqs. (6.1) and (6.2),
respectively.

###  7.1 Scenario I

In the case of scenario-I, SOPM is defined as

$\displaystyle P_{out}(\sigma_{s1})=\Pr(C_{s,1}>\sigma_{s1})=1-$
$\displaystyle\int_{0}^{\infty}\int_{\zeta}^{\infty}f_{\sigma_{min,1}}(\Upsilon_{bm,1})f_{\sigma_{max,1}}(\Upsilon_{bt,1})d\Upsilon_{bm,1}d\Upsilon_{bt,1},$
(27)

where $\zeta=2^{\sigma_{s1}}(1+\Upsilon_{bt,1})-1$ and $\sigma_{s1}>0$.
Substituting (5.1), and (6.1) into (7.1), we have

$\displaystyle
P_{out}(\sigma_{s1})=1-\sum_{\psi_{r_{6}}}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\int_{0}^{\infty}\int_{\zeta}^{\infty}$
$\displaystyle\times\biggl{(}\alpha_{3}\Upsilon_{bm,1}^{\alpha_{5}}+\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\alpha_{4}\Upsilon_{bm,1}^{\alpha_{6}}\biggl{)}e^{-\beta_{4}\Upsilon_{bm,1}}\sum_{\psi_{\nu_{6}}}\sum_{\nu_{6}=0}^{\mathcal{K}\mathcal{N}-1}$
$\displaystyle\times\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\biggl{(}\alpha_{10}\Upsilon_{bt,1}^{\alpha_{12}}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\alpha_{11}\Upsilon_{bt,1}^{\alpha_{13}}\biggl{)}$
$\displaystyle\times
e^{-\beta_{5}\Upsilon_{bt,1}}d\Upsilon_{bm,1}d\Upsilon_{bt,1}.$ (28)

Now, executing integration using [39, eq 3.351.2], the closed-form expression
of SOPM is obtained in 6.2, where $p_{\zeta}=2^{\sigma_{s1}}-1$,
$q_{\zeta}=2^{\sigma_{s1}}$,
$\Lambda_{10}=\frac{\alpha_{3}\alpha_{5}!\binom{r_{9}}{r_{11}}p_{\zeta}^{r_{9}-r_{11}}q_{\zeta}^{r_{11}}}{r_{9}!(\beta_{4})^{\alpha_{5}-r_{9}+1}e^{\beta_{4}p_{\zeta}}}$,
and
$\Lambda_{11}=\frac{\alpha_{4}\alpha_{6}!\binom{r_{10}}{r_{12}}p_{\zeta}^{r_{10}-r_{12}}q_{\zeta}^{r_{12}}}{r_{10}!(\beta_{4})^{\alpha_{6}-r_{10}+1}e^{\beta_{4}p_{\zeta}}}$.

###  7.2 Scenario II

The SOPM in the case of scenario-II is defined as

$\displaystyle P_{out}(\sigma_{s2})=\Pr(C_{s,2}>\sigma_{s2})=1-$
$\displaystyle\int_{0}^{\infty}\int_{\varrho}^{\infty}f_{\sigma_{min,2}}(\Upsilon_{bm,2})f_{\sigma_{max,2}}(\Upsilon_{bt,2})d(\Upsilon_{bm,2})d(\Upsilon_{bt,2}),$
(30)

where $\varrho=2^{\sigma_{s2}}(1+\Upsilon_{bt,2})-1$ and $\sigma_{s2}>0$.
Substituting the values of (5.2) and (6.2) into (7.2), we have

$\displaystyle
P_{out}(\sigma_{s2})=1-\sum_{\psi_{\rho_{7}}}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\int_{0}^{\infty}\int_{\varrho}^{\infty}$
$\displaystyle\times
e^{-\Upsilon_{bm,2}\varpi_{3}}(\varpi_{1}\Upsilon_{bm,2}^{\varpi_{4}}+\varpi_{2}\Upsilon_{bm,2}^{\varpi_{5}})\sum_{\psi_{\sigma_{10}}}\sum_{\sigma_{10}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}$
$\displaystyle\times\sum_{\rho_{1}=0}^{\infty}e^{-\Upsilon_{bt,2}\omega_{3}}(\omega_{1}\Upsilon_{bt,2}^{\omega_{4}}+\omega_{2}\Upsilon_{bt,2}^{\omega_{5}})d(\Upsilon_{bm,2})d(\Upsilon_{bt,2}).$
(31)

Now, performing integration on (7.2) using [39, eq 3.351.2], the closed-form
expression of SOPM is expressed in 6.2, where $p_{\theta}=2^{\sigma_{s2}}-1$,
$q_{\theta}=2^{\sigma_{s2}}$,
$\delta_{19}=\frac{\varpi_{1}\varpi_{4}!\binom{\sigma_{13}}{\sigma_{15}}p_{\theta}^{\sigma_{13}-\sigma_{15}}q_{\theta}^{\sigma_{15}}}{\sigma_{13}!(\varpi_{3})^{\varpi_{4}-\sigma_{13}+1}e^{\varpi_{3}p_{\theta}}}$,
and
$\delta_{20}=\frac{\varpi_{2}\varpi_{5}!\binom{\sigma_{14}}{\sigma_{16}}p_{\theta}^{\sigma_{14}-\sigma_{16}}q_{\theta}^{\sigma_{16}}}{\sigma_{14}!(\varpi_{3})^{\varpi_{5}-\sigma_{14}+1}e^{\varpi_{3}p_{\theta}}}$.

$\displaystyle\Pr(C_{s,1}>0)=\sum_{\psi_{r_{6}}}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\sum_{\psi_{\nu_{6}}}\sum_{\nu_{6}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\biggl{[}\biggl{(}\chi_{10}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\chi_{12}\biggl{)}\alpha_{3}\alpha_{5}!\beta_{4}^{-(\alpha_{5}+1)}$
$\displaystyle+\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\biggl{(}\chi_{10}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\chi_{12}\biggl{)}\frac{\alpha_{4}\alpha_{6}!}{\beta_{4}^{\alpha_{6}+1}}-\sum_{\nu_{9}=0}^{\alpha_{12}}\biggl{(}\frac{\chi_{11}\alpha_{3}(\alpha_{5}+\nu_{9})!}{(\beta_{4}+\beta_{5})^{\alpha_{5}+\nu_{9}+1}}+\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\frac{\chi_{11}\alpha_{4}(\alpha_{6}+\nu_{9})!}{(\beta_{4}+\beta_{5})^{\alpha_{6}+\nu_{9}+1}}\biggl{)}$
$\displaystyle-\sum_{\nu_{10}=0}^{\alpha_{13}}\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\biggl{(}\frac{\chi_{13}\alpha_{3}(\alpha_{5}+\nu_{10})!}{(\beta_{4}+\beta_{5})^{\alpha_{5}+\nu_{10}+1}}+\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\frac{\chi_{13}\alpha_{4}(\alpha_{6}+\nu_{10})!}{(\beta_{4}+\beta_{5})^{\alpha_{6}+\nu_{10}+1}}\biggl{)}\biggl{]}.$
(34)
$\displaystyle\Pr(C_{s,2}>0)=\sum_{\psi_{\rho_{7}}}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\psi_{\sigma_{10}}}\sum_{\sigma_{10}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\biggl{[}\biggl{(}\delta_{15}+\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\delta_{17}\biggl{)}\varpi_{1}\varpi_{4}!\varpi_{3}^{-(\varpi_{4}+1)}$
$\displaystyle+\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\biggl{(}\delta_{15}+\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\delta_{17}\biggl{)}\frac{\varpi_{2}\varpi_{5}!}{\varpi_{3}^{\varpi_{5}+1}}-\sum_{\sigma_{13}=0}^{\omega_{4}}\biggl{(}\frac{\delta_{16}\varpi_{1}(\sigma_{13}+\varpi_{4})!}{(\omega_{3}+\varpi_{3})^{\sigma_{13}+\varpi_{4}+1}}+\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\frac{\delta_{16}\varpi_{2}(\sigma_{13}+\varpi_{5})!}{(\omega_{3}+\varpi_{3})^{\sigma_{13}+\varpi_{5}+1}}\biggl{)}$
$\displaystyle-\sum_{\sigma_{14}=0}^{\omega_{5}}\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\biggl{(}\frac{\delta_{18}\varpi_{1}(\sigma_{14}+\varpi_{4})!}{(\omega_{3}+\varpi_{3})^{\sigma_{14}+\varpi_{4}+1}}+\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\frac{\delta_{18}\varpi_{2}(\sigma_{14}+\varpi_{5})!}{(\omega_{3}+\varpi_{3})^{\sigma_{14}+\varpi_{5}+1}}\biggl{)}\biggl{]}.$
(36)

##  8 Non-zero Secrecy Capacity Analysis

The PNSMC is defined as [36, Eq. 18]

$\displaystyle P_{r}(C_{s,j}>0)=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\Upsilon_{bm,j}}f_{\sigma_{min,j}}(\Upsilon_{bm,j})f_{\sigma_{max,j}}(\Upsilon_{bt,j})d\Upsilon_{bt,j}d\Upsilon_{bm,j}.$

For a secure communication, the secrecy multicast capacity must be a positive
quantity, otherwise the secrecy of the transmitted information can not be
guaranteed. Using the expressions of $f_{\sigma_{min,j}}(\Upsilon_{bm,j})$,
and $f_{\sigma_{max,j}}(\Upsilon_{bt,j})$ from (5.1), (5.2), (6.1), and (6.2),
we derive the expressions of PNSC as the following:

###  8.1 Scenario I

The PNSMC in the case of Scenario-I is defined as

$\displaystyle\Pr(C_{s,1}>0)=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\Upsilon_{bm,1}}f_{\sigma_{min,1}}(\Upsilon_{bm,1})f_{\sigma_{max,1}}(\Upsilon_{bt,1})sd\Upsilon_{bt,1}d\Upsilon_{bm,1}.$
(33)

Substituting (5.1) and (6.1) into (8.1) and performing integration using [39,
eqs (3.351.1, 3.351.3)], the closed-form expression of PNSMC is obtained in
(7.2), where
$\chi_{10}=\frac{\alpha_{10}\alpha_{12}!}{\beta_{5}^{\alpha_{12}+1}}$,
$\chi_{11}=\frac{\alpha_{10}\alpha_{12}!}{\nu_{9}!(\beta_{5})^{\alpha_{12}-\nu_{9}+1}}$,
$\chi_{12}=\frac{\alpha_{11}\alpha_{13}!}{\beta_{5}^{\alpha_{15}+1}}$, and
$\chi_{13}=\frac{\alpha_{11}\alpha_{13}!}{\nu_{10}!(\beta_{5})^{\alpha_{13}-\nu_{10}+1}}$.

###  8.2  For Scenario II:

In the case of Scenario-II mixed, the PNSMC is defined as

$\displaystyle\Pr(C_{s,2}>0)=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\Upsilon_{bm,2}}f_{\sigma_{min,2}}(\Upsilon_{bm,2})f_{\sigma_{max,2}}(\Upsilon_{bt,2})d\Upsilon_{bt,2}d\Upsilon_{bm,2}.$
(35)

Substituting the value of (5.2) and (6.2) into (8.2) and then performing
integration using [39, eqs. 3.351.2, and 3.351.3], we get the final expression
of PNSMC in (7.2), where
$\delta_{15}=\frac{\omega_{1}\omega_{4}!}{\omega_{3}^{\omega_{4}+1}}$,
$\delta_{16}=\frac{\omega_{1}\omega_{4}!}{\sigma_{13}!\omega_{3}^{\omega_{4}-\sigma_{13}+1}}$,
$\delta_{17}=\frac{\omega_{2}\omega_{5}!}{\omega_{3}^{\omega_{5}+1}}$, and
$\delta_{18}=\frac{\omega_{2}\omega_{5}!}{\sigma_{14}!\omega_{3}^{\omega_{5}-\sigma_{14}+1}}$.

$\displaystyle\langle
C_{s,1}\rangle=\sum_{\psi_{r_{6}}}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\sum_{\psi_{\nu_{6}}}\sum_{\nu_{6}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\biggl{[}\sum_{r_{13}=0}^{\alpha_{5}}\frac{\alpha_{3}\alpha_{5}!\beta_{4}^{-\alpha_{5}-1}}{(\alpha_{5}-r_{13})!}\biggl{[}\frac{(-1)^{\alpha_{5}-r_{13}-1}\mathbf{Ei}[-\beta_{4}]}{e^{-\beta_{4}}\beta_{4}^{-\alpha_{5}+r_{13}}}$
$\displaystyle+\sum_{r_{14}=1}^{\alpha_{5}-r_{13}}\frac{(r_{14}-1)!}{(-\beta_{4})^{-\alpha_{5}+r_{13}+r_{14}}}\biggl{]}+\sum_{r_{15}=0}^{\alpha_{6}}\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\frac{\alpha_{4}\alpha_{6}!\beta_{4}^{-\alpha_{6}-1}}{(\alpha_{6}-r_{15})!}\biggl{[}\frac{(-1)^{\alpha_{6}-r_{15}-1}\mathbf{Ei}[-\beta_{4}]}{e^{-\beta_{4}}\beta_{4}^{-\alpha_{6}+r_{15}}}+\sum_{r_{16}=1}^{\alpha_{6}-r_{15}}\frac{(r_{16}-1)!}{(-\beta_{4})^{-\alpha_{6}+r_{15}+r_{16}}}\biggl{]}$
$\displaystyle-\sum_{\nu_{11}=0}^{\alpha_{12}}\frac{\alpha_{10}\alpha_{12}!\beta_{5}^{-\alpha_{12}-1}}{(\alpha_{12}-\nu_{11})!}\biggl{[}\frac{(-1)^{\alpha_{12}-\nu_{11}-1}\mathbf{Ei}[-\beta_{5}]}{e^{-\beta_{5}}\beta_{5}^{-\alpha_{12}+\nu_{11}}}+\sum_{\nu_{12}=1}^{\alpha_{12}-\nu_{11}}\frac{(\nu_{12}-1)!}{(-\beta_{5})^{-\alpha_{12}+\nu_{11}+\nu_{12}}}\biggl{]}$
$\displaystyle-\sum_{\nu_{13}=0}^{\alpha_{13}}\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\frac{\alpha_{11}\alpha_{13}!\beta_{5}^{-\alpha_{13}-1}}{(\alpha_{13}-\nu_{13})!}\biggl{[}\frac{(-1)^{\alpha_{13}-\nu_{13}-1}\mathbf{Ei}[-\beta_{5}]}{e^{-\beta_{5}}\beta_{5}^{-\alpha_{13}+\nu_{13}}}+\sum_{\nu_{12}=1}^{\alpha_{13}-\nu_{13}}\frac{(\nu_{14}-1)!}{(-\beta_{5})^{-\alpha_{13}+\nu_{13}+\nu_{14}}}\biggl{]}\biggl{]}.$
(38) $\displaystyle\langle
C_{s,2}\rangle=\sum_{\psi_{\rho_{7}}}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\psi_{\sigma_{10}}}\sum_{\sigma_{10}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\biggl{[}\sum_{\sigma_{15}=0}^{\varpi_{4}}\frac{\varpi_{1}\varpi_{4}!}{\varpi_{3}^{\varpi_{4}+1}(\varpi_{4}-\sigma_{15})!}\biggl{[}\frac{(-1)^{\varpi_{4}-\sigma_{15}-1}\mathbf{Ei}(-\varpi_{3})}{e^{-\varpi_{3}}(\frac{1}{\varpi_{3}})^{\varpi_{4}-\sigma_{15}}}$
$\displaystyle+\sum_{\sigma_{16}=1}^{\varpi_{4}-\sigma_{15}}\frac{(\sigma_{16}-1)!}{(\frac{-1}{\varpi_{3}})^{\varpi_{4}-\sigma_{15}-\sigma_{16}}}\biggl{]}+\sum_{\sigma_{17}=0}^{\varpi_{5}}\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\frac{\varpi_{2}\varpi_{5}!\varpi_{3}^{-(\varpi_{5}+1)}}{(\varpi_{5}-\sigma_{17})!}\biggl{[}\frac{(-1)^{\varpi_{5}-\sigma_{17}-1}\mathbf{Ei}(-\varpi_{3})}{e^{-\varpi_{3}}(\frac{1}{\varpi_{3}})^{\varpi_{5}-\sigma_{17}}}+\sum_{\sigma_{18}=1}^{\varpi_{5}-\sigma_{17}}\frac{(\sigma_{18}-1)!}{(\frac{-1}{\varpi_{3}})^{\varpi_{5}-\sigma_{17}-\sigma_{18}}}\biggl{]}$
$\displaystyle-\sum_{\sigma_{19}=0}^{\omega_{4}}\frac{\omega_{1}\omega_{4}!}{\omega_{3}^{\omega_{4}+1}(\omega_{4}-\sigma_{19})!}\biggl{[}\frac{(-1)^{\omega_{4}-\sigma_{19}-1}\mathbf{Ei}(-\omega_{3})}{e^{-\omega_{3}}(\frac{1}{\omega_{3}})^{\omega_{4}-\sigma_{19}}}+\sum_{\sigma_{20}=1}^{\omega_{4}-\sigma_{19}}\frac{(\sigma_{20}-1)!}{(\frac{-1}{\omega_{3}})^{\omega_{4}-\sigma_{19}-\sigma_{20}}}\biggl{]}$
$\displaystyle-\sum_{\sigma_{21}=0}^{\omega_{5}}\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\frac{\omega_{2}\omega_{5}!}{\omega_{3}^{\omega_{5}+1}(\omega_{5}-\sigma_{21})!}\biggl{[}\frac{(-1)^{\omega_{5}-\sigma_{21}-1}\mathbf{Ei}(-\omega_{3})}{e^{-\omega_{3}}(\frac{1}{\omega_{3}})^{\omega_{5}-\sigma_{21}}}+\sum_{\sigma_{22}=1}^{\omega_{5}-\sigma_{21}}\frac{(\sigma_{22}-1)!}{(\frac{-1}{\omega_{3}})^{\omega_{5}-\sigma_{21}-\sigma_{22}}}\biggl{]}\biggl{]}.$
(40)

##  9 Ergodic Secrecy Multicast Capacity Analysis

The ESMC is defined as the average value of the instantaneous secrecy
capacity. It can be mathematically expressed as [36, Eq. 21]

$\displaystyle\langle C_{s,j}\rangle$
$\displaystyle=\int_{0}^{\infty}log_{2}(1+\Upsilon_{bm,j})f_{\sigma_{min,j}}(\Upsilon_{bm,j})d\Upsilon_{bm,j}$
$\displaystyle-\int_{0}^{\infty}log_{2}(1+\Upsilon_{bt,j})f_{\sigma_{max,j}}(\Upsilon_{bt,j}))d\Upsilon_{bt,j}.$

Utilizing the expressions of $f_{\sigma_{min,j}}(\Upsilon_{bm,j})$, and
$f_{\sigma_{max,j}}(\Upsilon_{bm,j})$ from (5.1), (5.2), (6.1), and (6.2), we
derive the expressions for ESMC in closed-form in the following sub-sections.

###  9.1 Scenario I

The ESMC for the case of Scenario-I is defined as

$\displaystyle\langle C_{s,1}\rangle$
$\displaystyle=\int_{0}^{\infty}log_{2}(1+\Upsilon_{bm,1})f_{\sigma_{min,1}}(\Upsilon_{bm,1})d\Upsilon_{bm,1}$
$\displaystyle-\int_{0}^{\infty}log_{2}(1+\Upsilon_{bt,1})f_{\sigma_{max,1}}(\Upsilon_{bt,1}))d\Upsilon_{bt,1}.$
(37)

Substituting (5.1) and (6.1) into (9.1), and performing integration by making
use of [39, eq 4.222.8], we get the closed-form expression of ESMC is shown in
8.2, where $\mathbf{Ei}[.]$ denotes the exponential integral function.

###  9.2 For Scenario II

In the case of Scenario-II, the ESMC is defined as

$\displaystyle\langle C_{s,2}\rangle$
$\displaystyle=\int_{0}^{\infty}log_{2}(1+\Upsilon_{bm,2})f_{\sigma_{min,2}}(\Upsilon_{bm,2})d\Upsilon_{bm,2}$
$\displaystyle-\int_{0}^{\infty}log_{2}(1+\Upsilon_{bt,2})f_{\sigma_{max,2}}(\Upsilon_{bt,2})d\Upsilon_{bt,2}.$
(39)

Substituting (5.2) and (6.2) into (9.2), and then performing integration
utilizing [39, eq 4.222.8], the closed-form expression of ESMC is obtained as
given in 8.2.

##  10 NUMERICAL RESULTS

Figure 2:  The SOPM versus average SNR of multicast channels varying
$\sigma_{s1}$ and $\sigma_{s2}$ for selected values of $\phi_{t1}$ and
$\phi_{t2}$ with $\mathcal{M}=\mathcal{N}=$ 2, $\mathcal{K}=4$, (a)
$m_{m1}=m_{t1}=5$, $\mu_{p1}=\mu_{m1}=\mu_{t1}=0.5$ and
$\eta_{p}=\eta_{m}=\eta_{t}=0.5$, and (b) $m_{m2}=m_{t2}=5$,
$\mu_{p2}=\mu_{m2}=\mu_{t2}=0.5$, $\kappa_{p}=\kappa_{m}=\kappa_{t}=0.5$.

Figure 3:  The SOPM versus average SNR of case-I and case-II multicast
channels varying $\phi_{t1}$ and $\phi_{t2}$ with $\mathcal{M}=\mathcal{N}$=2,
$\mathcal{K}=$ 2, (a) $m_{m1}=m_{t1}=5$, $\mu_{p1}=\mu_{m1}=\mu_{t1}=2$,
$\eta_{p}=\eta_{m}=\eta_{t}=$ 0.5 and $\sigma_{s1}=0.5$ bits/s/Hz, and (b)
$m_{m2}=m_{t2}=5$, $\mu_{p2}=\mu_{m2}=\mu_{t2}=2$,
$\kappa_{p}=\kappa_{m}=\kappa_{t}=0.5$, $\sigma_{s2}=0.5$ bits/s/Hz.

In this section, we graphically represent the numerical results corresponding
to the derived analytical expressions for the SOPM, PNSMC, and ESMC and
observe how the system parameters, e.g., fading severity, shadowing, the
number of relays, receivers, and eavesdroppers, etc. influence the multicast
system’s secrecy characteristics. Analytical results are obtained using
Mathematica software and for simulation purposes, we averaged $10^{6}$ random
samples of each channel. Note that the infinite series in (6.2), (6.2), (7.2),
(7.2), (8.2) and (8.2) converges quickly with a high level of accuracy just
after fewer terms. Hence we assume the first 25 terms for each infinite
series. Moreover, with a view to obtaining a clear comparison in terms of
secrecy performance between the two considered scenarios, we present a set of
curves corresponding to each scenario in each figure.

The SOPM is depicted as a function of the average SNR of the multicast
channels in Figures 2a and 2b in order to observe the impacts due to variation
in target secrecy rate. In each scenario, we assume two cases with
$\phi_{t1}=\phi_{t2}=$ 5 dB and 10 dB. We can observe as seen from the figure
that the SOPM increases with $\sigma_{s1}$ and $\sigma_{s2}$ which is also
shown in [42]. Note that we assume a passive eavesdropping scenario in this
work, hence the channel state information of eavesdropper networks is not
known. For that reason, we assume a target secrecy rate and send information
at this particular capacity. But if the instantaneous secrecy capacity falls
below this target secure transmission capacity, then there will be a secrecy
outage. Moreover, since the wireless channels are random in nature, an
increase in target secrecy rate increases the probability of instantaneous
secrecy rate falling below the target rate, hence the SOPM, as can be seen
from the figure, will increase.

Figure 4:  The PNSMC versus average SNR of multicast channels varying
$\mu_{m1}$, $\mu_{t1}$, $\mu_{m2}$, $\mu_{t2}$ for selected values of
$\phi_{t1}$= 0 dB, 5 dB, and $\phi_{t2}$= 0 dB, 5 dB with
$\mathcal{M}=\mathcal{N}=$ 2, $\mathcal{K}=$ 2, (a) $m_{m1}=m_{t1}=5$,
$\mu_{p1}=1$ and $\eta_{p}=\eta_{m}=\eta_{t}=$ 0.5, and (b) $m_{m2}=m_{t2}=5$,
$\mu_{p2}=2$, and $\kappa_{p}=\kappa_{m}=\kappa_{t}=0.5$.

In Figures 3a and 3b, we illustrate the SOPM versus average SNR of the
multicast channels showing the impacts of $\phi_{t1}$ and $\phi_{t2}$. It can
be clearly seen from both the scenarios that the outage performance is better
for lower values of the eavesdropper channel’s average SNR. It is obvious that
the eavesdropper links becomes better due to an increased value of $\phi_{t1}$
and $\phi_{t2}$ which lets the eavesdroppers sneak more information from the
multicast channels, and accordingly, the SOPM also increases. Same results
also can be seen in [43, 44] which legibly justify our observations.

Figure 5:  The PNSMC versus average SNR of multicast channels varying
$m_{m1}$, $m_{t1}$, $m_{m2}$, and $m_{t2}$ for selected values of
$\mathcal{M}$ with $\mathcal{N}=2$, $\mathcal{K}=2$, (a) $\phi_{t1}=$ 0 dB,
$\mu_{p1}=\mu_{m1}=\mu_{t1}=$ 2, and $\eta_{p}=\eta_{m}=\eta_{t}=$ 0.5, and
(b) $\phi_{t2}=$ 0 dB, $\mu_{p2}=\mu_{m2}=\mu_{t2}=$ 2, and
$\kappa_{p}=\kappa_{m}=\kappa_{t}=$ 0.5.

The impacts of the number of multipath clusters of the multicast and the
eavesdropper channels are depicted in Figures 4a and 4b in terms of PNSMC. It
is observed that the PNSMC performance becomes better for the higher values of
$\mu_{m1}$ and $\mu_{m2}$. The reason is that with the increase in the number
of multipath clusters in the $\mathcal{K}\rightarrow\mathcal{M}$ link, the
fading of the corresponding link is reduced. On the contrary, the lower amount
of fading assists in improving the secrecy capacity, and thus the PNSMC
performance also enhances as testified in [37]. Similarly, an increase in
$\mu_{t_{1}}$ and $\mu_{t_{2}}$ reduce the fading of the
$\mathcal{K}\rightarrow\mathcal{N}$ which will be beneficial for the
eavesdroppers for wiretapping more confidential data. But if we simultaneously
vary the shadowing severity of both the multicast and the eavesdropper links
with equal values of $\mu_{m_{1}}$, $\mu_{m_{2}}$, $\mu_{t_{1}}$, and
$\mu_{t_{2}}$, the PNSMC performance becomes better.

Figure 6:  The PNSMC versus average SNR of multicast channels varying
$\mathcal{K}$ for selected values of $\mathcal{N}$ with $\mathcal{M}=$ 2, (a)
$\phi_{t1}=$ 0 dB, $m_{m1}=m_{t1}=$ 5, $\mu_{p1}=\mu_{m1}=\mu_{t1}=$ 2, and
$\eta_{p}=\eta_{m}=\eta_{t}=$ 0.5, and (b) $\phi_{t2}=$ 0 dB, $m_{m2}=m_{t2}=$
5, $\mu_{p2}=\mu_{m2}=\mu_{t2}=$ 2, and $\kappa_{p}=\kappa_{m}=\kappa_{t}=$
0.5.

Figure 7:  The PNSMC versus average SNR of multicast channels varying
$\mathcal{N}$ for selected values of $\phi_{t1}$ and $\phi_{t2}$ with
$\mathcal{M}=$ 4, $\mathcal{K}=$ 2, (a) $m_{m1}=m_{t1}=$ 5,
$\mu_{p1}=\mu_{m1}=\mu_{t1}=$ 2, and $\eta_{p}=$$\eta_{m}=$$\eta_{t}=$ 0.5,
and (b) $m_{m2}=m_{t2}=$ 5, $\mu_{p2}=\mu_{m2}=\mu_{t2}=$ 2, and
$\kappa_{p}=\kappa_{m}=\kappa_{t}=$ 0.5.

Figure 8:  The ESMC versus average SNR of multicast channels varying
$\mathcal{M}$ for selected values of $\phi_{t1}$ and $\phi_{t2}$ with
$\mathcal{N}=$ 2, $\mathcal{K}=$ 4, (a) $m_{m1}=m_{t1}=$ 5,
$\mu_{p1}=\mu_{m1}=\mu_{t1}=$ 2, $\eta_{p}=\eta_{m}=\eta_{t}=$ 0.5,
$\sigma_{s1}$ = 0.5 bits/s/Hz, and (b) $m_{m2}=m_{t2}=$ 5,
$\mu_{p2}=\mu_{m2}=\mu_{t2}=$ 2, $\kappa_{p}=\kappa_{m}=\kappa_{t}=$ 0.5, and
$\sigma_{s2}$ = 0.5 bits/s/Hz.

Figure 9:  The ESMC versus average SNR of multicast channel varying
$\eta_{p}$, $\mu_{p1}$, $\kappa_{p}$, and $\mu_{p2}$ for selected values of
$\mathcal{N}$ with $\mathcal{M}$=2, $\mathcal{K}$=4, (a) $\phi_{t1}=$ 0 dB,
$m_{m1}=m_{t1}=5$, $\mu_{m1}=\mu_{t1}=2$ , $\eta_{m}=\eta_{t}=0.5$ and
$\sigma_{s1}=0.5$ bits/s/Hz, and (b) $\phi_{t2}=$ 0 dB, $m_{m2}=m_{t2}=5$,
$\mu_{m2}=\mu_{t2}=2$, $\kappa_{m}=\kappa_{t}=0.5$, and $\sigma_{s2}=0.5$
bits/s/Hz.

Figure 10:  The ESMC versus average SNR of multicast channels for selected
values of $\mu_{p1}$, $\mu_{m1}$, $\mu_{t1}$, $\mu_{p2}$, $\mu_{m2}$,
$\mu_{t2}$, $\eta_{p}$, $\eta_{m}$, $\eta_{t}$, $\kappa_{p}$, $\kappa_{m}$,
$\kappa_{t}$ with $\mathcal{M}=\mathcal{N}=\mathcal{K}=2$, (a)
$m_{m1}=m_{t1}=35$, $\phi_{t1}=$ 0 dB, and $\sigma_{s1}=0.5$ bits/s/Hz, (b)
$m_{m2}=m_{t2}=$ 35, $\phi_{t2}=$ 0 dB, and $\sigma_{s2}=0.5$ bits/s/Hz.

Figures 5a and 5b show the impacts of shadowing severity of multicast and
eavesdropper channels on the physical layer security performance. We consider
two groups with $\mathcal{M}$= 2, and 6. In the first group, we vary the
shadowing severity of the $\mathcal{K}\rightarrow\mathcal{M}$ and
$\mathcal{K}\rightarrow\mathcal{N}$ links simultaneously while in the second
group we vary the severity of shadowing of the
$\mathcal{K}\rightarrow\mathcal{M}$ links. It can clearly be observed that
with the equally increasing values of $m_{m1}$ and $m_{t1}$, and $m_{m2}$ and
$m_{t2}$, the secrecy performance is enhanced, and for increasing values of
$m_{m1}$ and $m_{m2}$ individually, the performance also becomes better. This
is because increasing $m_{m1}$ and $m_{m2}$ from 0 to $\infty$ indicates the
shadowing severity changing from a stronger to weaker conditions while an
increasing $m_{t1}$ and $m_{t2}$ reduces the shadowing severity of the
eavesdropper links. The authors in [45] also observed some similar results as
ours which is a clear indication that the results corresponding to our
proposed dual-hop model are valid.

Figures 6a and 6b are depicted to explain the impacts of the number of relays
on the PNSMC performance. We consider two cases with $\mathcal{N}=2$ and
$\mathcal{N}=10$. The numerical results reveal that for both cases the
proposed system exhibits an enhanced level of secrecy with the increasing
number of relays $\mathcal{K}$. This result is also supported by [46]. An
increased $\mathcal{K}$ represents an improved cooperative diversity and at
the same time, the selection of the opportunistic channels in terms of best
relaying ensures maximum capacity at the receive terminals thereby increasing
the secrecy capacity.

In Figures 7a and 7b, the PNSMC is shown against the average SNR of multicast
channels. The effects of the number of eavesdroppers $\mathcal{N}$ are shown
assuming two groups with $\phi_{t1}=\phi_{t2}=$ 0 dB and 5 dB. As we consider
maximum impacts of the eavesdroppers (maximum SNR at the eavesdropper) among
$\mathcal{N}$ eavesdroppers (i.e. the worst case), it is evident that an
increase in $\mathcal{N}$ will increase the probability of the existence of a
strong eavesdropper channel in the system and hence the system’s security will
be degraded as testified in [43].

To demonstrate the effects of the number of multicast receivers in the
proposed model, we depict Figures 8a and 8b by showing ESMC against the
average SNR of multicast channels. It can be observed that the secrecy
capacity deteriorates with the number of receivers. Note that in wireless
multicasting, a specific bandwidth is allocated for a group of multicast
receivers which is divided among them equally according to the multicast
scheme. With the increase in numbers of multicast receivers within the same
allocated bandwidth, the bandwidth per receiver is reduced. Hence the SNR and
capacity at the receiver’s terminals are also degraded which leads to a
deteriorated secrecy performance as shown in [45].

In Figures 9a and 9b, the impacts of the shape parameters of the first-hop are
depicted by showing ESMC against the average SNR of the multicast channels. We
can observe that the ESMC decreases with $\eta_{p}$, but increases with
$\mu_{p_{1}}$, $\mu_{p_{2}}$, and $\kappa_{p}$. The authors of [47, 26] also
obtained similar results that clearly verify our analysis.

Generalization of the Existing Works:

It is noteworthy that the $\kappa-\mu$/IG and $\eta-\mu$/IG models exhibits
extreme versatility as generalized $\kappa-\mu$ and $\eta-\mu$ models are the
special cases of those IG models. Moreover, $\kappa-\mu$ and $\eta-\mu$ models
also unify the performance evaluation of some classical multipath models. In
the IG models, only the shadowing parameters (i.e. $m_{m1}$, $m_{t1}$,
$m_{m2}$, and $m_{t2}$) determine the amount of shadowing of the mean signal
power. If $m_{m1}$, $m_{t1}$, $m_{m2}$, and $m_{t2}$ $\rightarrow\infty$, then
we can assume that both the dominant and scattered components suffer from
severe shadowing. On the other hand, if $m_{m1}$, $m_{t1}$, $m_{m2}$, and
$m_{t2}$ $\rightarrow 0$, then no shadowing is observed in the channels and
hence the mean signal power can be considered to be the deterministic one. For
that particular case, $\kappa-\mu$/IG and $\eta-\mu$/IG models takes the form
of $\kappa-\mu$ and $\eta-\mu$ fading models, respectively.

In Figure 10a and 10b the generic nature of the proposed scenarios has been
shown. For $\eta-\mu$ fading channel, setting $\eta_{p}$ = $\eta_{m}$ =
$\eta_{t}$ = 0.9 and $\mu_{p1}$ = $\mu_{m1}$ = $\mu_{t1}$ = 1 with $m_{m1}$,
$m_{t1}$, $m_{m2}$, and $m_{t2}$ $\rightarrow\infty$, we can obtain
Nakagami-$m$ channel where $\mu_{m1}$, $\mu_{m2}$, $\mu_{t1}$, and $\mu_{t2}$
becomes equivalent to the Nakagami fading parameter $m$. Based on this,
letting $\mu_{m1}$, $\mu_{m2}$, $\mu_{t1}$, and $\mu_{t2}$ equal to 1, we get
Rayleigh distribution. Similarly, Gaussian channels can be obtained with
$\eta_{p}$ = $\eta_{m}$ = $\eta_{t}$ = 0.9 and $\mu_{p1}$ = $\mu_{m1}$ =
$\mu_{t1}$ = 0.25, and Nakagami-$q$ fading channel can be obtained with
$\eta_{p}$ = $\eta_{m}$ = $\eta_{t}$ = 0.25 and $\mu_{p1}$ = $\mu_{m1}$ =
$\mu_{t1}$ = 0.5. In the case of $\kappa-\mu$ model, we can generate Rician
fading channel letting $\kappa_{p}$ = $\kappa_{m}$ = $\kappa_{t}$ = 5.4,
$\mu_{p2}$ = $\mu_{m2}$ = $\mu_{t2}$ = 0.9, where $\kappa_{p}$, $\kappa_{m}$,
and $\kappa_{t}$ become equivalent to the Rician-$K$ factor. Following those
conditions, Rayleigh fading is obtained setting $\kappa_{p}$ = $\kappa_{m}$ =
$\kappa_{t}$ = 0. Nakagami-$m$ is generated considering $\kappa_{p}$ =
$\kappa_{m}$ = $\kappa_{t}$ = 0.001 and $\mu_{p2}$ = $\mu_{m2}$ = $\mu_{t2}$ =
1.5, where $\mu_{p2}$, $\mu_{m2}$ and $\mu_{t2}$ is equivalent to Nakagami
fading parameter $m$. Likewise, we can obtain one-sided Gaussian channel
setting $\kappa_{p}$ = $\kappa_{m}$ = $\kappa_{t}$ = 0 and $\mu_{p2}$ =
$\mu_{m2}$ = $\mu_{t2}$ = 0.5, and shadowed Rician with $\kappa_{p}$ =
$\kappa_{m}$ = $\kappa_{t}$ = 6 , $\mu_{p2}$ = $\mu_{m2}$ = $\mu_{t2}$ = 1.
Besides the proposed scenarios also provide a good match to the secure
shadowed Rician- Rayleigh [48, 49], and shadowed Rician- Nakagami-$m$ [50, 51]
models. Hence, it can clearly be seen from the figures that the proposed
scenarios (I and II) exhibits enormous versatility than any dual-hop multicast
networks of the literature since a wide range of classical dual-hop multicast
and broadcast models can be shown as special cases of the proposed work.

##  11 CONCLUSION

This work focuses on the secrecy analysis of dual-hop multicast communication
systems in the presence of multiple eavesdroppers exploiting the best relay
selection scheme. With a view to observing the impacts of each system
parameter, we derive expressions for SOPM, PNSMC, and ESMC in closed-form
which are further authenticated via Monte-Carlo simulations. It can clearly be
seen from the numerical results that although fading, shadowing (both LOS and
multiplicative), number of multicast receivers and eavesdroppers impose
detrimental impacts on the secrecy performance, an acceptable secrecy level
can till be maintained by first increasing the number of relays and then
exploiting the best relay selection strategy. Moreover, the proposed model
provides immense versatility since it can approximate a wide range of
composite/multipath fading models of the literature.  In future, our plan is
to analyse the secrecy performance of multi-hop-multicast networks over IG
composite fading models incorporating asymptotic analysis at high SNR regime.

##  12 Proof of Dual-hop SNRs (Scenario I)

###  12.1 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{M}$ link

The PDF of $\Upsilon_{sm,1}$ and $\Upsilon_{st,1}$ is defined as [52]

$\displaystyle f_{sm,1}(\Upsilon)$
$\displaystyle=\frac{dF_{sm,1}(\Upsilon)}{d\Upsilon},$ (41) $\displaystyle
f_{st,1}(\Upsilon)$ $\displaystyle=\frac{dF_{st,1}(\Upsilon)}{d\Upsilon},$
(42)

where $F_{sm,1}(\Upsilon)$ and $F_{st,1}(\Upsilon)$ denote the CDFs of
$\Upsilon_{sm,1}$ and $\Upsilon_{st,1}$. Now, the CDF of $\Upsilon_{sm,1}$ is
defined as [53, eq. 12]

$\displaystyle F_{sm,1}(\Upsilon)$
$\displaystyle=1-\Pr(\Upsilon_{sp,1}>\Upsilon)\Pr(\Upsilon_{pm,1}>\Upsilon),$
(43)

where $\Pr(\Upsilon_{sp,1}>\Upsilon_{sm,1})$ and
$\Pr(\Upsilon_{pm,1}>\Upsilon_{sm,1})$ are the complementary cumulative
distribution functions (CCDFs) of $\Upsilon_{sp,1}$ and $\Upsilon_{pm,1}$, and
the CCDFs are, respectively defined by [54]

$\displaystyle\Pr(\Upsilon_{sp,1}>\Upsilon_{sm,1})$
$\displaystyle=\int_{\Upsilon_{sm,1}}^{\infty}f_{sp,1}(\Upsilon)d\Upsilon,$
(44) $\displaystyle\Pr(\Upsilon_{pm,1}>\Upsilon_{sm,1})$
$\displaystyle=\int_{\Upsilon_{sm,1}}^{\infty}f_{pm,1}(\Upsilon)d\Upsilon.$
(45)

Substituting (2) into (44) and performing integration using [39, eq 3.351.2],
we have

$\displaystyle\Pr(\Upsilon_{sp,1}>\Upsilon_{sm,1})$
$\displaystyle=\sum_{n_{1}=0}^{\infty}\frac{\alpha_{2}}{\beta_{s}^{2\mu_{p1}+2n_{1}}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon_{sm,1}).$ (46)

Again substituting (4) into (45) and executing integration using [39, eq
3.194.2], we obtain

$\displaystyle\Pr(\Upsilon_{pm,1}>\Upsilon_{sm,1})=\sum_{r_{1}=0}^{\infty}\frac{\lambda_{2}\Upsilon_{sm,1}^{-m_{m1}}}{\beta_{m}^{m_{m1}+2\mu_{m1}+2r_{1}}m_{m1}}$
$\displaystyle\times\,_{2}F_{1}\left(m_{m1}+2\mu_{m1}+2r_{1},m_{m1};m_{m1}+1;\frac{-1}{\beta_{m}\Upsilon_{sm,1}}\right)$
$\displaystyle=\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\frac{\lambda_{2}\Upsilon_{sm,1}^{-m_{m1}}}{\beta_{m}^{m_{m1}+2\mu_{m1}+2r_{1}}m_{m1}}$
$\displaystyle\times\frac{\left(m_{m1}\right)_{r_{2}}\left(m_{m1}+2\mu_{m1}+2r_{1}\right)_{r_{2}}}{r_{2}!\left(m_{m1}+1\right)_{r_{2}}\left(-\beta_{m}\right)^{r_{2}}}.$
(47)

After a few terms, the infinite series here quickly converges. [55]. Now,
substituting (12.1) and (12.1) into (43), the CDF of $\Upsilon_{sm,1}$ is
obtained as

$\displaystyle F_{sm,1}(\Upsilon)$
$\displaystyle=1-\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\Upsilon^{-m_{m1}-r_{2}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon),$ (48)

where
$\Lambda_{2}=\frac{\Lambda_{1}\left(m_{m1}\right)_{r_{2}}\left(m_{m1}+2\mu_{m1}+2r_{1}\right)_{r_{2}}}{r_{2}!\left(m_{m1}+1\right)_{r_{2}}\left(-\beta_{m}\right)^{r_{2}}}$,
and
$\Lambda_{1}=\frac{\alpha_{2}\lambda_{2}\beta_{s}^{-2\mu_{p1}-2n_{1}}}{\beta_{m}^{m_{m1}+2\mu_{m1}+2r_{1}}m_{m1}}$.
Substituting (12.1) into (41) and performing differentiation with respect to
$\Upsilon_{sm,1}$, the PDF of $\Upsilon_{sm,1}$ is obtained in (3.2).

###  12.2 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{N}$ link

Similar to (12.1), the CDF of $\Upsilon_{st,1}$ is given by

$\displaystyle F_{st,1}(\Upsilon)$
$\displaystyle=1-\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\Upsilon^{-m_{t1}-\nu_{2}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon),$ (49)

where
$\chi_{2}=\frac{\chi_{1}\left(m_{t1}\right)_{\nu_{2}}\left(m_{t1}+2\mu_{t1}+2\nu_{1}\right)_{\nu_{2}}}{\nu_{2}!\left(m_{t1}+1\right)_{\nu_{2}}\left(-\beta_{e}\right)^{\nu_{2}}}$,
and
$\chi_{1}=\frac{\xi_{2}\beta_{s}^{-2\mu_{p1}-2n_{1}}}{\beta_{e}^{m_{t1}+2\mu_{t1}+2\nu_{1}}m_{t1}}$,
Substituting (12.2) into (42) and performing differentiation with respect to
$\Upsilon_{st,1}$, the PDF of $\Upsilon_{st,1}$ is obtained as shown in (3.2).

##  13 Proof of Best Relay Selection (Scenario I)

###  13.1 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{M}$ link

Mathematically, the CDF of $\Upsilon_{bm,1}$ is defined as

$\displaystyle F_{bm,1}(\Upsilon)=[F_{sm,1}(\Upsilon)]^{\mathcal{K}}.$ (50)

Substituting (12.1) into (50), the CDF of $\Upsilon_{bm,1}$ is obtained as

$\displaystyle F_{bm,1}(\Upsilon)=$
$\displaystyle\biggl{[}1-\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\Upsilon^{-m_{m1}-r_{2}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{\mathcal{K}}.$
(51)

Differentiating (50) with respect to $\Upsilon_{sm,1}$, we get the PDF of
$\Upsilon_{bm,1}$ as

$\displaystyle f_{bm,1}(\Upsilon)$
$\displaystyle=\mathcal{K}f_{sm,1}(\Upsilon)[F_{sm,1}(\Upsilon)]^{\mathcal{K}-1}.$
(52)

Substituting (12.1) and (3.2) into (52), we obtain (3.3).

###  13.2 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{N}$ link

Similar to (13.1), the CDF of $\Upsilon_{bt,1}$ is given by

$\displaystyle F_{bt,1}(\Upsilon)=$
$\displaystyle\biggl{[}1-\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\Upsilon^{-m_{t1}-\nu_{2}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{\mathcal{K}}.$
(53)

Differentiating (13.2) with respect to $\Upsilon_{st,1}$, the PDF of
$\Upsilon_{bt,1}$ can be obtained that is given by (3.3).

##  14 Proof of Dual-hop SNRs (Scenario II)

###  14.1 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{M}$ link

Denoting $\Upsilon_{sm,2}$, and $\Upsilon_{st,2}$ as the SNRs of
$\mathcal{S}\rightarrow\mathcal{K}\rightarrow\mathcal{M}$, and
$\mathcal{S}\rightarrow\mathcal{K}\rightarrow\mathcal{N}$ links, respectively,
the PDFs of $\Upsilon_{sm,2}$ and $\Upsilon_{st,2}$ are given by

$\displaystyle f_{sm,2}(\Upsilon)=\frac{dF_{sm,2}(\Upsilon)}{d\Upsilon},$ (54)
$\displaystyle f_{st,2}(\Upsilon)=\frac{dF_{st,2}(\Upsilon)}{d\Upsilon},$ (55)

where $F_{sm,2}(\Upsilon)$ and $F_{st,2}(\Upsilon)$ denote the CDFs of
$\Upsilon_{sm,2}$ and $\Upsilon_{st,2}$. Similar to (43), the CDF of
$\Upsilon_{sm,2}$ is defined by

$\displaystyle
F_{sm,2}(\Upsilon)=1-\Pr(\Upsilon_{sp,2}>\Upsilon)\Pr(\Upsilon_{pm,2}>\Upsilon),$
(56)

where $\Pr(\Upsilon_{sp,2}>\Upsilon_{sm,2})$ and
$\Pr(\Upsilon_{pm,2}>\Upsilon_{sm,2})$ are the complementary cumulative
distribution functions (CCDFs) of $\Upsilon_{sp,2}$ and $\Upsilon_{pm,2}$
which are, respectively defined by

$\displaystyle\Pr(\Upsilon_{sp,2}>\Upsilon_{sm,2})=\int_{\Upsilon_{sm,2}}^{\infty}f_{sp,2}(\Upsilon)d\Upsilon,$
(57)
$\displaystyle\Pr(\Upsilon_{pm,2}>\Upsilon_{sm,2})=\int_{\Upsilon_{sm,2}}^{\infty}f_{pm,2}(\Upsilon)d\Upsilon.$
(58)

Substituting (13) into (57) and performing integration using [39, eq.
3.351.2], we get

$\displaystyle\Pr(\Upsilon_{sp,2}>\Upsilon_{sm,2})$
$\displaystyle=\sum_{\rho_{1}=0}^{\infty}\delta_{2}\delta_{b_{1}}^{-\mu_{p2}-\rho_{1}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon_{sm,2}),$
(59)

where $\delta_{b_{1}}=\frac{\mu_{p2}(1+\kappa_{p})}{\phi_{p2}}$. Again
substituting (15) into (58) and performing integration, we get

$\displaystyle\Pr(\Upsilon_{pm,2}>\Upsilon_{sm,2})=\sum_{\rho_{2}=0}^{\infty}\frac{\delta_{5}\Upsilon_{sm,2}^{-m_{m2}}}{\delta_{4}^{m_{m2}+\mu_{m}2+\rho_{2}}m_{m2}}$
$\displaystyle\times\,_{2}F_{1}\left(m_{m2}+\mu_{m2}+\rho_{2},m_{m2};m_{m2}+1;\frac{-1}{\delta_{4}\Upsilon_{sm,2}}\right)$
$\displaystyle=\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\frac{\delta_{5}\Upsilon_{sm,2}^{-m_{m2}}}{\delta_{4}^{m_{m2}+\mu_{m}2+\rho_{2}}m_{m2}}$
$\displaystyle\times\frac{\left(m_{m2}+\mu_{m2}+\rho_{2}\right)_{\rho_{3}}\left(m_{m2}\right)_{\rho_{3}}}{(-\delta_{4})^{\rho_{3}}\rho_{3}!\left(m_{m2}+1\right)_{\rho_{3}}}.$
(60)

Where the infinite series converges rapidly as described in [28]. Substituting
(14.1), and (14.1) into (56), the CDF of $\Upsilon_{sm,2}$ is obtained as

$\displaystyle F_{sm,2}(\Upsilon)$
$\displaystyle=1-\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{7}\Upsilon^{-m_{m2}-\rho_{3}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon),$ (61)

where
$\delta_{7}=\delta_{6}\frac{\left(m_{m2}+\mu_{m2}+\rho_{2}\right)_{\rho_{3}}\left(m_{m2}\right)_{\rho_{3}}}{(-\delta_{4})^{\rho_{3}}\rho_{3}!\left(m_{m2}+1\right)_{\rho_{3}}}$,
and
$\delta_{6}=\frac{\delta_{2}\delta_{5}\delta_{b_{1}}^{-\mu_{p2}-\rho_{1}}}{\delta_{4}^{m_{m2}+\mu_{m2}+\rho_{2}}m_{m2}}$.
Substituting (14.1) into (54) and performing differentiation with respect to
$\Upsilon_{sm,2}$, the PDF of $\Upsilon_{sm,2}$ is obtained in (4.2).

###  14.2 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{N}$ link

Similar to (14.1), the CDF of $\Upsilon_{st,2}$ is given by

$\displaystyle F_{st,2}(\Upsilon)$
$\displaystyle=1-\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\sigma_{3}=0}^{\infty}\beta_{7}\Upsilon^{-m_{t2}-\sigma_{3}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon),$ (62)

where
$\beta_{7}=\beta_{6}\frac{\left(m_{t2}+\mu_{t2}+\sigma_{2}\right)_{\sigma_{3}}\left(m_{t2}\right)_{\sigma_{3}}}{(-\beta_{2})^{\sigma_{3}}\sigma_{3}!\left(m_{t2}+1\right)_{\sigma_{3}}}$,
and
$\beta_{6}=\frac{\delta_{2}\beta_{3}\delta_{b_{1}}^{-\mu_{p2}-\sigma_{1}}}{\beta_{2}^{m_{t2}+\mu_{t2}+\sigma_{2}}m_{t2}}$.
Now, substituting (14.2) into (55) and performing differentiation with respect
to $\Upsilon_{st,2}$, we obtain (4.2).

##  15 Proof of Best Relay Selection (Scenario II)

###  15.1 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{M}$ link

The CDF of $\Upsilon_{bm,2}$ is defined as

$\displaystyle F_{bm,2}(\Upsilon)=[F_{sm,2}(\Upsilon)]^{\mathcal{K}}.$ (63)

Now, substituting (14.1) into (63), the CDF of $\Upsilon_{bm,2}$ is obtained
as

$\displaystyle F_{bm,2}(\Upsilon)=$
$\displaystyle[1-\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{7}\Upsilon_{sm,2}^{-m_{m2}-\rho_{3}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon_{sm,2})]^{\mathcal{K}}.$
(64)

Differentiating (63) with respect to $\Upsilon_{sm,2}$, we get the PDF of
$\Upsilon_{bm,2}$ as

$\displaystyle
f_{bm,2}(\Upsilon)=\mathcal{K}f_{sm,2}(\Upsilon)[F_{sm,2}(\Upsilon)]^{\mathcal{K}-1}.$
(65)

Now substituing (14.1) , and (4.2) into (65), we get $f_{bm,2}(\Upsilon)$ as
shown in (4.3).

###  15.2 $\mathcal{S}\rightarrow\mathcal{k}\rightarrow\mathcal{N}$ link

Similar to (15.1), the CDF of $\Upsilon_{bt,2}$ is given by

$\displaystyle F_{bt,2}(\Upsilon)$
$\displaystyle=[1-\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\Upsilon^{-m_{t2}-\sigma_{3}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)]^{\mathcal{K}}.$
(66)

Differentiating (15.2) with respect to $\Upsilon_{st,2}$, the PDF of
$\Upsilon_{bt,2}$ is shown in (4.3).

##  16 Proof of Multicast Channel Models

###  16.1 Scenario I

In the case of mixed $\eta-\mu$ and $\eta-\mu$ /IG composite fading channel,
the PDF of $\sigma_{min,1}$ is given by

$\displaystyle
f_{\sigma_{min,1}}(\Upsilon)=\mathcal{M}f_{bm,1}(\Upsilon)[1-F_{bm,1}(\Upsilon)]^{\mathcal{M}-1}.$
(67)

Plugging (13.1) and (3.3) into (67) leads to

$\displaystyle
f_{\sigma_{min,1}}(\Upsilon)=\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\mathcal{M}\mathcal{K}\biggl{[}e^{-\beta_{s}\Upsilon}$
$\displaystyle\times\Upsilon^{-n_{2}+2\mu_{p1}+2n_{1}}+\Upsilon^{-n_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\Upsilon^{-m_{m1}-r_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{\mathcal{K}-1}$
$\displaystyle\times\biggl{[}1-\biggl{(}1-\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{2}\Upsilon^{-m_{m1}-r_{2}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{)}^{\mathcal{K}}\biggl{]}^{\mathcal{M}-1}.$
(68)

Using binomial expansion of [39, eq 1.111], (16.1) can be expressed as

$\displaystyle
f_{\sigma_{min,1}}(\Upsilon)=\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\mathcal{M}\mathcal{K}\Lambda_{6}\Lambda_{2}$
$\displaystyle\times\biggl{[}\Upsilon^{-n_{2}+2\mu_{p1}+2n_{1}}e^{-\beta_{s}\Upsilon}+\Upsilon^{-n_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}$
$\displaystyle\times\biggl{[}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\sum_{r_{2}=0}^{\infty}\Lambda_{2}\Upsilon^{-m_{m1}-r_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{r_{6}},$
(69)

where
$\Lambda_{6}={(-1)^{r_{5}+r_{6}}}\binom{\mathcal{M}-1}{r_{5}}\binom{\mathcal{K}+\mathcal{K}r_{5}-1}{r_{6}}$.
Making use of [39, eq 8.352.7], (16.1) can be further simplified as

$\displaystyle
f_{\sigma_{min,1}}(\Upsilon)=\sum_{r_{5}=0}^{\mathcal{M}-1}\sum_{r_{6}=0}^{\mathcal{K}+\mathcal{K}r_{5}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\mathcal{M}\mathcal{K}\Lambda_{6}\Lambda_{2}$
$\displaystyle\times\biggl{[}\Upsilon^{-n_{2}+2\mu_{p1}+2n_{1}}+\sum_{r_{7}=0}^{2\mu_{p1}+2n_{1}-1}\Lambda_{7}\Upsilon^{-n_{2}+r_{7}}\biggl{]}$
$\displaystyle\times e^{-\beta_{s}\Upsilon}[\Theta_{1}(\Upsilon)]^{r_{6}},$
(70)

where $\Lambda_{7}=\frac{\Gamma(2\mu_{p1}+2n_{1})}{r_{7}!\beta_{s}^{-r_{7}}}$.
Here,

$\displaystyle\Theta_{1}(\Upsilon)=\sum_{r_{8}=0}^{2\mu_{p1}+2n_{1}-1}\sum_{r_{2}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\Lambda_{8}\Upsilon^{-m_{m1}-r_{2}+r_{8}}e^{-\beta_{s}\Upsilon},$
(71)

where
$\Lambda_{8}=\frac{\Lambda_{2}\Gamma(2\mu_{p1}+2n_{1})}{r_{8}!\beta_{s}^{-r_{8}}}$.
Using the multinomial theorem of [17, eq. 7], we get

$\displaystyle[\Theta_{1}(\Upsilon)]^{r_{6}}=\Xi_{\psi_{r_{6}}}e^{-\phi_{\psi_{r_{6}}}\Upsilon}\Upsilon^{\varphi_{\psi_{r_{6}}}}$
$\displaystyle\times\sum_{\psi_{r_{6}}}\binom{r_{6}}{p_{0,0,0,0,\ldots,}p_{r_{8},r_{2},r_{1},n_{1},\ldots,}n_{2\mu_{p1}+2n_{1}-1,\infty,\infty,\infty}},$
(72)

where
$\binom{b}{b_{1},b_{2},...,b_{m}}=\frac{b!}{b_{1}!,b_{2}!,\ldots,b_{m}!}$
denotes the multinomial coefficients,
$\Xi_{\psi_{r_{6}}}=\Pi_{r_{8},r_{2},r_{1},n_{1}}\Lambda_{8}^{p_{r_{8},r_{2},r_{1},n_{1}}}$,
$\phi_{\psi_{r_{6}}}=\sum_{r_{8}}\sum_{r_{2}}\sum_{r_{1}}\sum_{n_{1}}\beta_{s}p_{r_{8},r_{2},r_{1},n_{1}}$,
and
$\varphi_{\psi_{r_{6}}}=\sum_{r_{8}}\sum_{r_{2}}\sum_{r_{1}}\sum_{n_{1}}(-m_{m1}-r_{2}+r_{8})p_{r_{8},r_{2},r_{1},n_{1}}$.
The sum of (16.1) is to be carried out for each element of $\psi_{r_{6}}$,
which is defined by

$\displaystyle\psi_{r_{6}}$
$\displaystyle=\\{(p_{0,0,0,0,\ldots,}p_{r_{8},r_{2},r_{1},n_{1},\ldots,}n_{2\mu_{p1}+2n_{1}-1,\infty,\infty,\infty}):$
$\displaystyle p_{r_{8},r_{2},r_{1},n_{1}}\in N0\leqslant r_{8}\leqslant
2\mu_{p1}+2n_{1}-1,0\leqslant r_{2}\leqslant\infty,$ $\displaystyle 0\leqslant
r_{1}\leqslant\infty,0\leqslant
n_{1}\leqslant\infty;\sum_{r_{8},r_{2},r_{1},n_{1}}p_{r_{8},r_{2},r_{1},n_{1}}=r_{6}\\}.$
(73)

After modifying (16.1), we get

$\displaystyle[\Theta_{1}(\Upsilon)]^{r_{6}}=\sum_{\psi_{r_{6}}}\Lambda_{9}e^{-\phi_{\psi_{r_{6}}}\Upsilon}\Upsilon^{\varphi_{\psi_{r_{6}}}},$
(74)

where
$\Lambda_{9}=\binom{r_{6}}{p_{0,0,0,0,\ldots,}p_{r_{8},r_{2},r_{1},n_{1},\ldots,}n_{2\mu_{p1}+2n_{1}-1,\infty,\infty,\infty}}\Xi_{\psi_{r_{6}}}$.
Substituting (74) into (16.1), we get the final expression of
$f_{\sigma_{min,1}}(\Upsilon)$ in (5.1).

###  16.2 Scenario II

For the mixed $\kappa-\mu$ and $\kappa-\mu$ /IG composite fading channel, the
PDF of $\sigma_{min,2}$ is given by

$\displaystyle
f_{\sigma_{min,2}}(\Upsilon)=\mathcal{M}f_{bm,2}(\Upsilon)[1-F_{bm,2}(\Upsilon)]^{\mathcal{M}-1}.$
(75)

Substituting (15.1) and (4.3) into (75), we get

$\displaystyle
f_{\sigma_{min,2}}(\Upsilon)=\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{7}\mathcal{M}\mathcal{K}\biggl{[}e^{-\Upsilon\delta_{b_{1}}}$
$\displaystyle\times\Upsilon^{-\rho_{3}-m_{m2}-1+\mu_{p2}+\rho_{1}}+\Upsilon^{-1-\rho_{3}-m_{m2}}$
$\displaystyle\times\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\rho_{3}=0}^{\infty}\delta_{7}\Upsilon^{-m_{m2}-\rho_{3}}\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)\biggl{]}^{\mathcal{K}-1}$
$\displaystyle\times\biggl{[}1-\biggl{(}1-\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\rho_{3}=0}^{\infty}\delta_{7}\Upsilon^{-m_{m2}-\rho_{3}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)\biggl{)}^{\mathcal{K}}\biggl{]}^{\mathcal{M}-1}.$
(76)

Using [39, eq. 1.111], (16.2) can be simplified as

$\displaystyle
f_{\sigma_{min,2}}(\Upsilon)=\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\sum_{\rho_{3}=0}^{\infty}\mathcal{M}\mathcal{K}\delta_{10}\delta_{7}$
$\displaystyle\times\biggl{[}e^{-\Upsilon\delta_{b_{1}}}\Upsilon^{-\rho_{3}-m_{m2}-1+\mu_{p2}+\rho_{1}}$
$\displaystyle+\Upsilon^{-1-\rho_{3}-m_{m2}}\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}\times\biggl{[}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}$
$\displaystyle\times\sum_{\rho_{3}=0}^{\infty}\delta_{7}\Upsilon^{-m_{m2}-\rho_{3}}\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)\biggl{]}^{\rho_{7}},$
(77)

where
$\delta_{10}=(-1)^{\rho_{6}+\rho_{7}}\binom{\mathcal{M}-1}{\rho_{6}}\binom{\mathcal{K}+\mathcal{K}\rho_{6}-1}{\rho_{7}}$.
Again using [39, eq. 8.352.7], (16.2) can be further simplified as

$\displaystyle
f_{\sigma_{min,2}}(\Upsilon)=\sum_{\rho_{6}=0}^{\mathcal{M}-1}\sum_{\rho_{7}=0}^{\mathcal{K}+\mathcal{K}\rho_{6}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\mathcal{M}\mathcal{K}\delta_{10}\delta_{7}$
$\displaystyle\times
e^{-\Upsilon\delta_{b_{1}}}\biggl{[}\Upsilon^{\delta_{11}+\mu_{p2}+\rho_{1}}+\sum_{\rho_{8}=0}^{\mu_{p2}+\rho_{1}-1}\delta_{12}\Upsilon^{\delta_{11}+\rho_{8}}\biggl{]}$
$\displaystyle\times\biggl{[}\Theta_{2}(\Upsilon)\biggl{]}^{\rho_{7}},$ (78)

where $\delta_{11}=-m_{m2}-\rho_{3}-1$ , $\delta_{11}+1=-m_{m2}-\rho_{3}$, and
$\delta_{12}=\frac{\Gamma(\mu_{p2}+\rho_{1})}{(\rho_{8})!\delta_{b_{1}}^{-\rho_{8}}}$.
Here

$\displaystyle\Theta_{2}(\Upsilon)=\sum_{\rho_{9}=0}^{\mu_{p2}+\rho_{1}-1}\sum_{\rho_{3}=0}^{\infty}\sum_{\rho_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\delta_{13}\Upsilon^{\delta_{11}+\rho_{9}+1}e^{-\Upsilon\delta_{b_{1}}},$
(79)

where
$\delta_{13}=\frac{\delta_{7}\Gamma(\mu_{p2}+\rho_{1})}{\rho_{9}!(\delta_{b_{1}})^{-\rho_{9}}}$.
Using the multinomial theorem of [17, eq. 7], we get

$\displaystyle[\Theta_{2}(\Upsilon)]^{\rho_{7}}=\chi_{\psi_{\rho_{7}}}e^{-\Theta_{\psi_{\rho_{7}}}\Upsilon}\Upsilon^{\eta_{\psi_{\rho_{7}}}}$
$\displaystyle\times\sum_{\psi_{\rho_{7}}}\binom{\rho_{7}}{g_{0,0,0,0},\ldots,g_{\rho_{9},\rho_{3}\rho_{2},\rho_{1}},\ldots,n_{\mu_{p2}+\rho_{1}-1,\infty,\infty,\infty}},$
(80)

where $\binom{a}{a_{1},a_{2},\ldots,a_{m}}=\frac{a!}{a_{1}!a_{2}!\ldots
a_{m}!}$ denotes the multinomial coefficients,
$\chi_{\psi_{\rho_{7}}}=\Pi_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}\delta_{13}^{g_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}}$,
$\eta_{\psi_{\rho_{7}}}=\sum_{\rho_{9}}\sum_{\rho_{3}}\sum_{\rho_{2}}\sum_{\rho_{1}}^{(\delta_{11}+\rho_{9}+1){g_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}}}$
and
$\Theta_{\psi_{\rho_{7}}}=\sum_{\rho_{9}}\sum_{\rho_{3}}\sum_{\rho_{2}}\sum_{\rho_{1}}\delta_{b_{1}}{g_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}}$.
For each element of $\psi_{\rho_{7}}$, the sum in (16.2) is to be performed,
which can be defined as

$\displaystyle\psi_{\rho_{7}}=$
$\displaystyle(g_{0,0,0,0},\ldots,g_{\rho_{9},\rho_{3}\rho_{2},\rho_{1}},\ldots,n_{\mu_{p2}+\rho_{1}-1,\infty,\infty,\infty}):$
$\displaystyle
g_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}\in\mathbb{N}0\leq\rho_{9}\leq\mu_{p2}+\rho_{1}-1,0\leq\rho_{3}\leq\infty,$
$\displaystyle
0\leq\rho_{2}\leq\infty,0\leq\rho_{1}\leq\infty;\sum_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}g_{\rho_{9},\rho_{3},\rho_{2},\rho_{1}}=\rho_{7}.$
(81)

Substituting (16.2) into (16.2), we get the final expression of
$f_{\sigma_{min,2}}(\Upsilon_{bm,2})$ in (5.2).

##  17 Proof of Eavesdropper Channel Models

###  17.1 Scenario I

The PDF of $\sigma_{max,1}$ in the case of mixed $\eta-\mu$ and $\eta-\mu$ /IG
composite fading channel is given by

$\displaystyle
f_{\sigma_{max,1}}(\Upsilon)=\mathcal{N}f_{bt,1}(\Upsilon)[F_{bt,1}(\Upsilon)]^{\mathcal{N}-1}.$
(82)

Substituting (13.2), and (3.3) into (82), we get

$\displaystyle
f_{\sigma_{max,1}}(\Upsilon)=\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\sum_{\nu_{2}=0}^{\infty}\chi_{2}\mathcal{N}\mathcal{K}\biggl{[}e^{-\beta_{s}\Upsilon}$
$\displaystyle\times\Upsilon^{-n_{3}+2\mu_{p1}+2n_{1}}+\Upsilon^{-n_{3}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\Upsilon^{-m_{t1}-\nu_{2}}\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)\biggl{]}^{\mathcal{K}-1}$
$\displaystyle\times\biggl{[}[1-\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{2}\Upsilon^{-m_{t1}-\nu_{2}}$
$\displaystyle\times\Gamma(2\mu_{p1}+2n_{1},\beta_{s}\Upsilon)]^{\mathcal{K}}\biggl{]}^{\mathcal{N}-1}.$
(83)

Simplifying (17.1), we get

$\displaystyle
f_{\sigma_{max,1}}(\Upsilon)=\sum_{\nu_{6}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\mathcal{N}\mathcal{K}\chi_{6}\chi_{2}e^{-\beta_{s}\Upsilon}$
$\displaystyle\times\biggl{[}\Upsilon^{-n_{3}+2\mu_{p1}+2n_{1}}+\sum_{\nu_{7}=0}^{2\mu_{p1}+2n_{1}-1}\chi_{7}\Upsilon^{-n_{3}+\nu_{7}}\biggl{]}$
$\displaystyle\times\biggl{[}\Theta_{3}(\Upsilon)\biggl{]}^{\nu_{6}},$ (84)

where $\chi_{6}=\binom{\mathcal{K}\mathcal{N}-1}{\nu_{6}}(-1)^{\nu_{6}}$, and
$\chi_{7}=\frac{\Gamma(2\mu_{p1}+2n_{1})}{\nu_{7}!\beta_{s}^{-\nu_{7}}}$. Here

$\displaystyle\Theta_{3}(\Upsilon)=\sum_{\nu_{8}=0}^{2\mu_{p1}+2n_{1}-1}\sum_{\nu_{2}=0}^{\infty}\sum_{\nu_{1}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\chi_{8}\Upsilon^{-m_{t1}-\nu_{2}+\nu_{8}}e^{-\beta_{s}\Upsilon},$
(85)

where
$\chi_{8}=\frac{\chi_{2}\Gamma(2\mu_{p1}+2n_{1})}{\nu_{8}!\beta_{s}^{-\nu_{8}}}$.
Applying multinomial theorem of [17, eq. 7], we obtain

$\displaystyle[\Theta_{3}(\Upsilon)]^{\nu_{6}}=\Xi_{\nu_{6}}e^{-\phi_{\nu_{6}}\Upsilon}\Upsilon^{\varphi_{\nu_{6}}}$
$\displaystyle\times\sum_{\psi_{\nu_{6}}}\binom{\nu_{6}}{q_{0,0,0,0,\ldots,}q_{\nu_{8},\nu_{2},\nu_{1},n_{1}.\ldots,}n_{2\mu_{p1}+2n_{1}-1,\infty,\infty,\infty}},$
(86)

where
$\Xi_{\psi_{\nu_{6}}}=\Pi_{\nu_{8},\nu_{2},\nu_{1},n_{1}}\chi_{8}^{q_{\nu_{8},\nu_{2},\nu_{1},n_{1}}}$,
$\phi_{\psi_{\nu_{6}}}=\sum_{\nu_{8}}\sum_{\nu_{2}}\sum_{\nu_{1}}\sum_{n_{1}}\beta_{s}q_{\nu_{8},\nu_{2},\nu_{1},n_{1}}$,and
$\varphi_{\psi_{\nu_{6}}}=\sum_{\nu_{8}}\sum_{\nu_{2}}\sum_{\nu_{1}}\sum_{n_{1}}(-m_{t1}-\nu_{2}+\nu_{8})q_{\nu_{8},\nu_{2},\nu_{1},n_{1}}$.
Substituting (17.1) into (17.1), we get we get the final expression of
$f_{\sigma_{max,1}}(\Upsilon)$ in (6.1).

###  17.2 Scenario II

In the case of mixed $\kappa-\mu$ and $\kappa-\mu$ /IG composite fading
channel, the PDF of $\sigma_{max,2}$ is given by

$\displaystyle
f_{\sigma_{max}}(\Upsilon)=\mathcal{N}f_{bt,2}(\Upsilon)[F_{bt,2}(\Upsilon)]^{\mathcal{N}-1}.$
(87)

Substituting (15.2) and (4.3) into (87), we get

$\displaystyle
f_{\sigma_{max,2}}(\Upsilon)=\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\mathcal{N}\mathcal{K}\biggl{[}e^{-\Upsilon\delta_{b_{1}}}$
$\displaystyle\times\Upsilon^{-\sigma_{3}-m_{t2}-1+\mu_{p2}+\sigma_{1}}+\Upsilon^{-1-\sigma_{3}-m_{t2}}$
$\displaystyle\times\Gamma[\mu_{p2}+\rho_{1},\Upsilon\delta_{b_{1}}]\biggl{]}$
$\displaystyle\times\biggl{[}1-\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\Upsilon^{-m_{t2}-\sigma_{3}}\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)\biggl{]}^{\mathcal{K}-1}$
$\displaystyle\times\biggl{[}[1-\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{7}\Upsilon^{-m_{t2}-\sigma_{3}}$
$\displaystyle\times\Gamma(\mu_{p2}+\rho_{1},\delta_{b_{1}}\Upsilon)]^{\mathcal{K}}\biggl{]}^{\mathcal{N}-1}.$
(88)

Simplifying (17.2), we get

$\displaystyle
f_{\sigma_{max,2}}(\Upsilon)=\sum_{\sigma_{10}=0}^{\mathcal{K}\mathcal{N}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\mathcal{N}\mathcal{K}\beta_{15}\beta_{7}e^{-\Upsilon\delta_{b_{1}}}$
$\displaystyle\times\biggl{[}\Upsilon^{\beta_{10}+\mu_{p2}+\rho_{1}}+\sum_{\sigma_{11}=0}^{\mu_{p2}+\rho_{1}-1}\beta_{16}\Upsilon^{\beta_{10}+\sigma_{11}}\biggl{]}$
$\displaystyle\times\biggl{[}\Theta_{4}(\Upsilon)\biggl{]}^{\sigma_{10}},$
(89)

where $\beta_{10}=-m_{t2}-\sigma_{3}-1$, $\beta_{10}+1=-m_{t2}-\sigma_{3}$,
$\beta_{15}=\binom{\mathcal{K}\mathcal{N}-1}{\sigma_{10}}(-1)^{\sigma_{10}}$,
and
$\beta_{16}=\frac{\Gamma(\mu_{p2}+\rho_{1})}{(\sigma_{11})!\delta_{b_{1}}^{-\sigma_{11}}}$.
Here

$\displaystyle\theta_{4}(\Upsilon)=\sum_{\sigma_{12}=0}^{\mu_{p2}+\rho_{1}-1}\sum_{\sigma_{3}=0}^{\infty}\sum_{\sigma_{2}=0}^{\infty}\sum_{\rho_{1}=0}^{\infty}\beta_{17}\Upsilon^{\beta_{10}+\sigma_{12}+1}e^{-\Upsilon\delta_{b_{1}}},$
(90)

where
$\beta_{17}=\frac{\beta_{7}\Gamma(\mu_{p2}+\rho_{1})}{\sigma_{12}!(\delta_{b_{1}})^{-\sigma_{12}}}$.
Using the multinomial theorem of [17, eq. 7], we get

$\displaystyle[\Theta_{4}(\Upsilon)]^{\sigma_{10}}=\chi_{\psi_{\sigma_{10}}}e^{-\Theta_{\psi_{\sigma_{10}}}\Upsilon}\Upsilon^{\eta_{\psi_{\sigma_{10}}}}$
$\displaystyle\times\sum_{\psi_{\sigma_{10}}}\binom{\sigma_{10}}{f_{0,0,0,0},\ldots,f_{\sigma_{12},\sigma_{3},\sigma_{2},\rho_{1}},\ldots,n_{\mu+\rho_{1}-1,\infty,\infty,\infty}},$
(91)

where
$\chi_{\psi_{\sigma_{10}}}=\Pi_{\sigma_{12},\sigma_{3},\sigma_{2},\rho_{1}}\beta_{17}^{f_{\sigma_{12},\sigma_{3},\sigma_{2},\rho_{1}}}$,
$\eta_{\psi_{\sigma_{10}}}=\sum_{\sigma_{12}}\sum_{\sigma_{3}}\sum_{\sigma_{1}}\sum_{\rho_{1}}(\beta_{10}+\sigma_{12}+1){f_{\sigma_{12},\sigma_{3},\sigma_{2},\rho_{1}}}$,
and
$\Theta_{\psi_{\sigma_{10}}}=\sum_{\sigma_{12}}\sum_{\sigma_{3}}\sum_{\sigma_{2}}\sum_{\rho_{1}}\delta_{b_{1}}{f_{\sigma_{12},\sigma_{3},\sigma_{2},\rho_{1}}}$.
Substituting (17.2) into (17.2), we get the final expression of
$f_{\sigma_{max,2}}(\Upsilon)$ in (6.2).

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