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# Destination Scheduling for Secure Pinhole-Based Power-Line Communication

Chinmoy Kundu, , Ankit Dubey, , Andrea M. Tonello , Arumugam Nallanathan , And
Mark F. Flanagan Chinmoy Kundu is with School of Electrical and Electronic
Engineering, University College Dublin, Belfield, IrelandAnkit Dubey is with
Department of EE, Indian Institute of Technology Jammu, Jammu & Kashmir, India
Andrea M. Tonello is with Institute of Networked and Embedded Systems,
University of Klagenfurt, AustriaArumugam Nallanathan is with School of
Electronic Engineering and Computer Science, Queen Mary University of London,
U.K.This publication has emanated from research supported in part by Science
Foundation Ireland (SFI) under Grant Number 17/US/3445 and 22/IRDIFA/10425 and
by the Department of Science and Technology (DST), India sponsored project
TMD/CERI/BEE/2016/059.

###### Abstract

We propose an optimal destination scheduling scheme to improve the physical
layer security (PLS) of a power-line communication (PLC) based Internet-of-
Things system in the presence of an eavesdropper. We consider a pinhole (PH)
architecture for a multi-node PLC network to capture the keyhole effect in
PLC. The transmitter-to-PH link is shared between the destinations and an
eavesdropper which correlates all end-to-end links. The individual channel
gains are assumed to follow independent log-normal statistics. Furthermore,
the additive impulsive noise at each node is modeled by an independent
Bernoulli-Gaussian process. Exact computable expressions for the average
secrecy capacity (ASC) and the probability of intercept (POI) performance over
many different networks are derived. Approximate closed-form expressions for
the asymptotic ASC and POI are also provided. We find that the asymptotic ASC
saturates to a constant level as transmit power increases. We observe that the
PH has an adverse effect on the ASC. Although the shared link affects the ASC,
it has no effect on the POI. We show that by artificially controlling the
impulsive to background noise power ratio and its arrival rate at the
receivers, the secrecy performance can be improved.

###### Index Terms:

Bernoulli-Gaussian impulsive noise, destination scheduling, log-normal
distribution, physical layer security, power-line communication.

##  I INTRODUCTION

Power-line communication (PLC) can be an excellent candidate for many upcoming
industrial applications, e.g., home automation, energy monitoring systems,
smart grid, and more recently the internet of things (IoT) as it can exploit
the existing power lines to transfer high-speed data content [1, 2, 3]. In a
smart grid environment, devices like smart meters (SMs) communicate with the
data concentrator unit (DCU) to send their data to the head-end system (HES)
for various applications as shown, for example, in Fig. 1 [4, 5].
Communication links between a SM and DCU can be established using PLC. DCUs
can connect to the HES through standard communication technologies [4, 5] for
further storage or processing the data. As we focus on the performance of the
PLC network in this paper, communication between DCU and HES is omitted from
further discussions.

Like any other communication system design, channel modeling is an important
consideration in PLC systems as well. Towards modeling PLC channels, more
specifically channel transfer functions, initially, two categories are
developed, i.e., phenomenological modeling [6] and deterministic modeling [7].
The former model is based on multipath effects due to impedance mismatch. In
this model, the channel parameters are fitted after measuring the channel.
Hence, can not be used to obtain the transfer function a priory. The latter
one is based on transmission line (TL) theory. In both these cases, knowledge
of the whole topology or site-specific information is required for the
computation of the channel transfer function which is difficult to obtain. To
alleviate this problem, a physically meaningful simple statistical
characterization of the PLC channel is proposed across different topologies in
[8, 9]. The conclusion of the model is that multipath in PLC channels gives
rise to lognormally distributed average channel gain statistics similar to
shadow fading in wireless channels. The concept of “distance” in shadow fading
is effectively replaced by the concept of “link topology” in PLC channel. The
authors in [11, 10] also verify the average channel gain statistics of in-home
power line communication channel as log-normal in the wider frequency band
than adopted in [8, 9]. The statistics of the first arriving path, which is
generally the dominant path, is also shown to be log-normally distributed in
[12].

The PLC channel is essentially a frequency selective channel. However, the
authors in [13, 14] have shown that PLC networks with proper branching
(uniform conductors) exhibit large coherence bandwidth. Hence for narrowband
signals, the channel is frequency flat. Moreover, for broadband signals,
multi-carrier modulation (MCM) schemes are employed, e.g., orthogonal
frequency division multiplexing (OFDM) and filterbank multi-carrier (FBMC),
and hence the signals in each sub-channels can be assumed to be undergoing
frequency flat channel [15]. Therefore, it is sufficient to perform analysis
under the assumption of frequency flat channel at the sub-channel level.

The PLC links also suffer from impulsive noise, the effect of which is
captured by modeling it as an additive Bernoulli-Gaussian noise (ABGN) process
[1, 16, 17]. In an ABGN model, background noise is omnipresent and modelled as
white Gaussian noise process like in RF systems; however, impulsive noise
(also modelled as additive white Gaussian noise) occurs occasionally with a
certain probability (controlled by a Bernoulli process).

In a PLC network, a part of the wires is shared among various communication
links (as can be seen from Fig. 1) and form a tree topology. The node up to
which the part of the wire is shared is called a pinhole (PH) [15]. The PH
introduces a keyhole effect between branches. Due to the PH, various links in
PLC become correlated. The keyhole effect has been studied previously in
wireless communication scenarios [18, 19, 20]. An example of a keyhole in a
realistic wireless environment is propagation in a hallway or a tunnel. The
keyhole effect assumes that the equivalent channel between any two nodes
connected via a pinhole is the product of the channel gains between those
individual nodes and the pinhole. The effect of a pinhole in a wireless
multiple-input multiple-output (MIMO) channel is reduced channel rank and
hence lower average capacity. The PH effect was studied in cooperative multi-
hop PLC in [34]; it was shown that due to the keyhole effect, relaying or
multi-hop communications does not increase the diversity gain.

It is highly likely that an eavesdropper, pretending to be a DCU, can desire
to access the information from SMs by taking advantage of the broadcast nature
of the PLC channel [27, 21, 15, 22, 24, 25, 26, 23, 28]. To prevent sensitive
information from being compromised, traditionally, security has been
considered in the higher layers of the communication protocol stack [21].
However, in addition to the traditional higher layer security mechanisms, the
security of a communication system can be further enhanced by using physical
layer security (PLS) [27, 15, 22, 24, 25, 26, 23, 28]. PLS exploits the
physical channel characteristics to achieve security against eavesdropping.

PLS for a PLC system was introduced in [15], wherein the authors studied the
secrecy rate distribution and secrecy rate region for multi-carrier and multi-
user broadcast channel; here, the authors considered both simulated and
experimental channel realizations. The effect of a PH on the secrecy rate was
shown. The average secrecy capacity (ASC) and secrecy outage probability (SOP)
were evaluated in [22] for a cooperative PLC network in the presence of an
eavesdropper when all links were correlated. The authors investigated the
effect of artificial noise power, relay gain, and channel correlation on
secrecy performance. The effect of destination and eavesdropper channel
correlation on the PLS of a simple three-node (source-destination-
eavesdropper) PLC system was analyzed in [23]. In [24], the authors proposed
an artificial noise (AN)-aided PLS for OFDM-based hybrid parallel PLC/wireless
systems. Subsequently, the authors in [25] studied the PLS of a hybrid
PLC/wireless system and concluded that this hybrid communication improved the
PLS of low bit-rate communication. Through extensive experiments, the authors
in [26] studied the security vulnerability of an in-home broadband PLC system
when unshielded power cables carry out data transmission. Further, the same
authors in [27] showed that PLS over PLC networks is possible and provided
appropriate design guidelines for PLC modems. Recently, the SOP was also
evaluated for a non-orthogonal multiple-access (NOMA) scheme in a PLC network
in [28]. The authors observed that a higher rate of impulsive noise degraded
the SOP of NOMA users significantly.

None of the works described above have studied PLS considering shared links
connected via a PH except for [15] where the secrecy rate distribution was
analyzed. Although the authors in [22, 23] studied the effect of correlation
on the PLS performance, they did not consider a pinhole-based model for
correlation; moreover, the system model is simple and consists of a three-node
structure, with or without a relay, and a single destination. The authors also
considered exactly the same impulsive noise at the destination and
eavesdropper, which is an ideal assumption. Further, no work in the literature
(including [15]) has considered node scheduling in a multi-user PLC system for
secrecy improvement. This is achieved in the present work, and moreover,
deviating from an idealistic assumption of the same impulsive noise at the
receivers, we assume independent impulsive noise arrival processes at the
destinations and the eavesdropper. In a large-scale PLC network with many
independent noise sources, independent impulsive noise arrival processes at
different nodes is a practical assumption though some spatial correlation
exists between noises at different nodes in PLC. This makes our analysis more
practical compared to the existing literature. This assumption may lead to the
analysis where artificial impulsive noise is introduced into the system for
secrecy purposes.

Motivated by the above discussion, we analyze the secrecy of a PLC-based
Internet-of-Things (IoT) network where a single legitimate source, multiple
legitimate destinations, and an eavesdropper are connected via a PH. The
transmitter-to-pinhole link is shared between the destinations and the
eavesdropper. With the objective of improving secrecy performance, a
destination among multiple destinations is scheduled optimally for
communication. PLC-based IoT network with distributed low-cost SMs considered
in this paper was the motivation behind applying node scheduling in
particular. It is well-known from wireless research that node scheduling can
improve secrecy without increasing the complexity of the network [35, 36, 37,
38, 39, 40, 41]. As the PLC channel is a wireline channel where Bernoulli-
Gaussian impulsive noise exists and the channel does not undergo fading in
time, results from wireless research do not apply. Hence, our aim in this
paper is to extend the optimal node scheduling technique already prevalent in
wireless networks to the wireline PLC network to improve PLS across different
networks and also to show the detrimental effect of the pinhole on
performance.

Our main contributions are:

  * •

We propose for the first time an optimal destination scheduling scheme in the
pinhole-based PLC network to maximize the secrecy performance.

  * •

We show the detrimental effect of correlation between channels due to pinhole
on the secrecy performance.

  * •

Deviating from the existing assumption of the same impulsive noise at both the
destination and eavesdropper, we consider an independent impulsive noise
arrival process in the destinations and eavesdropper.

  * •

We derive the computable expressions for the ASC and probability of intercept
(POI) performance over many different networks.

  * •

In order to provide further insights, we derive the approximate asymptotic ASC
and approximate POI in closed form.

  * •

We show that the introduction of artificial impulsive noise at the nodes may
significantly improve the secrecy performance of the system.

Notation: $\mathbb{E}[\cdot]$ denotes the expectation of its argument,
$\mbox{Pr}[\cdot]$ is the probability of an event, $F_{X}(\cdot)$ represents
the cumulative distribution function (CDF) of the random variable (RV) $X$,
and $f_{X}(\cdot)$ is the corresponding probability density function (PDF).

##  II System Model

Figure 1: Pinhole-based PLC network. Figure 2: Considered model.

We consider a PLC network topology as shown in Fig. 2 which is a part of the
larger pinhole-based PLC network shown in Fig. 1. It consists of $(N+2)$ nodes
connected to a pinhole, PH, where the legitimate SM node $A$ communicates with
one of the scheduled DCUs $\\{B_{n}\\}$ as and when it has any data to update
to the head-end system, where $n\in\\{1,2,\ldots,N\\}$, in the presence of an
eavesdropper $E$. The source or a centralized control unit will schedule the
optimal destination $B_{n^{*}}$, where ${n^{*}}$ denotes the scheduled
destination index among $n\in\\{1,2,\ldots,N\\}$, in order to maximize the
system secrecy during the transmission of data. Here we note that each line
connected to the PH may have a random number of branching thus impedance
discontinuities along the line that are not specifically shown in Fig 2. The
exact knowledge of these impedance discontinuities is very unlikely to be
available, and hence a statistical model already available from the empirical
PLC channel data is adopted for the channel modeling in the next section.

We shall now describe the channel and noise models for the considered system.

###  II-A Channel Model

We assume that the average channel gains of the links $A$-PH, PH-$B_{n}$
(where $n\in\\{1,2,\ldots,N\\}$) and PH-$E$, denoted as $\gamma_{a}$,
$\gamma_{n}$, and $\gamma_{e}$, respectively, follow independent log-normal
statistics. It is well known that signal propagation in the power line is
multipath based. Mismatched termination and random impedance discontinuities
along the line cause successive reflections of the propagation signal. Thus,
path amplitudes in different lines are a function of a product of several
random propagation effects which leads to log-normality in the central limit.
The log-normal distribution is the result of observing the channel gain over
many different networks or over different links in a given network, provided
this network is complex with many branches and nodes. Since log-normality does
not change under power, path gains are log-normally distributed as well. Log-
normal statistics is assumed following [8, 9, 11, 10, 22, 28]. We note that
the distribution of the log-normal RV $\gamma_{\beta}$ where
$\beta\in\\{a,n,e\\}$ with parameters $m_{\beta}$ and $s_{\beta}^{2}$, denoted
as $\mathcal{LN}\left(m_{\beta},s_{\beta}^{2}\right)$, is expressed as [29]

$\displaystyle f_{\gamma_{\beta}}(t)=\frac{1}{t\sqrt{2\pi
s_{\beta}^{2}}}\exp\left(-\frac{1}{2}\left(\frac{\ln{t}-m_{\beta}}{s_{\beta}}\right)^{2}\right),\;t\geq
0.$ (1)

The parameters $m_{\beta}$ and $s_{\beta}^{2}$ are the mean and variance of
the associated Gaussian RV $\ln({\gamma_{\beta}})$, where $\gamma_{\beta}$ is
the log-normal RV. The corresponding CDF is expressed as

$\displaystyle
F_{\gamma_{\beta}}(t)=1-Q\left(\frac{\ln{t}-m_{\beta}}{s_{\beta}}\right),\;t\geq
0,$ (2)

where $Q(t)=\int_{t}^{\infty}(1/{\sqrt{2\pi}})\exp(-t^{2}/2)\mbox{d}t$ is the
Gaussian $Q$-function. The average power of a log-normally distributed link is
then [29]

$\displaystyle\mathbb{E}[\gamma_{\beta}]=\exp\left(m_{\beta}+\frac{1}{2}s_{\beta}^{2}\right).$
(3)

We assume that all the PH-$B_{n}$ links have independent identical
distributions, i.e., $m_{n}=m_{b}$ and $s_{n}=s_{b}$ for each $n$. Since the
PLC network has a PH, the end-to-end channel power gains of the links
$A$-$B_{n}$ and $A$-$E$ are given as $\gamma_{a}\gamma_{n}$ and
$\gamma_{a}\gamma_{e}$, respectively.

It is to be noted that the PLC channels are frequency selective in nature.
Hence, in practice, MCM schemes such as OFDM and FBMC modulation are used to
convert the wideband frequency selective channel into multiple narrowband
frequency flat sub-channels [15]. In this paper, we assume that a suitable MCM
scheme is utilized and therefore, the channel is frequency flat at the sub-
channel level. The analysis in this paper is also valid for narrow-band
single-carrier modulation when operating at a certain frequency.

###  II-B Noise Model

As mentioned, the additive noise of the PLC channel is well modelled by a
Bernoulli-Gaussian process to combine the effect of both the background noise
and the impulsive noise [1]. In this model, the background noise at $B_{n}$
(for any $n$) and $E$ are independently distributed Gaussian RVs with zero
mean and variances $\epsilon_{nW}^{2}$ and $\epsilon_{eW}^{2}$, respectively,
while the impulsive noise at $B_{n}$ (for any $n$) and at $E$ are
independently distributed Gaussian RV with zero mean and variances

$\displaystyle\epsilon_{nI}^{2}=\eta_{n}\epsilon_{nW}^{2}~{}\text{and}~{}\epsilon_{eI}^{2}=\eta_{e}\epsilon_{eW}^{2},$
(4)

respectively, where $\eta_{n}$ and $\eta_{e}$ represent the power ratios of
the impulsive noise to the background noise at $B_{n}$ (for any $n$) and $E$,
respectively. The background noise is omnipresent at the nodes; however, the
impulsive noise occasionally affects the transmission following a Bernoulli
trial and hence, a Bernoulli sample ($0$ or $1$) is multiplied with the
impulsive noise sample. The Bernoulli samples at $B_{n}$ for any $n$ and $E$
occur independently, where $p_{n}$ and $p_{e}$ are the probabilities of
occurrence of the Bernoulli sample $1$ at $B_{n}$ and $E$, respectively. The
parameter $p_{n}$ or $p_{e}$ represents the average rate of occurrence of the
impulsive noise at the corresponding node. Hence, the effective noise variance
at any node considering both background noise and impulsive noise is

$\displaystyle\begin{cases}\epsilon_{\xi W}^{2}&\text{Probability
}(1-p_{\xi})\\\ \epsilon_{\xi W}^{2}+\epsilon_{\xi I}^{2}=\epsilon_{\xi
W}^{2}(1+\eta_{\xi})&\text{Probability }p_{\xi},\end{cases}$ (5)

where $\xi\in\\{n,e\\}$. We assume that every node $B_{n}$ (for
$n=1,2,\ldots,N$) suffers from noise with identical statistics, i.e.,
$p_{n}=p_{b}$, $\epsilon_{nW}^{2}=\epsilon_{bW}^{2}$,
$\epsilon_{nI}^{2}=\epsilon_{bI}^{2}$, and $\eta_{n}=\eta_{b}$, where the
subscript $b$ is used to denote destination nodes or DCUs.

Though all receiving nodes are connected with each other through the power
line, in a large-scale PLC network where nodes are far apart, the undesired
disturbances may be assumed localized and hence independent impulsive noise
arrival process at the selected destination and the eavesdropper can be
assumed. In this case, there can be four possible events with corresponding
probabilities:

$\displaystyle\begin{cases}\text{1. No impulsive noise at either
node}&(1-p_{b})(1-p_{e})\\\ \text{2. Impulsive noise only at
$E$}&(1-p_{b})p_{e}\\\ \text{3. Impulsive noise only at
$B_{n^{*}}$}&p_{b}(1-p_{e})\\\ \text{4. Impulsive noise at both
nodes}&p_{b}p_{e}.\end{cases}$ (6)

These probabilities will be required to analyze the performance metrics for
the system. These performance metrics are defined in the following subsection.

###  II-C Destination Scheduling Scheme

For a given eavesdropping link quality and identical noise statistics in
destination nodes with all nodes connected to the common PH, selecting the
link which maximizes the end-to-end channel power gain of the link $A$-$B_{n}$
among all $n\in\\{1,2,\ldots,N\\}$ provides the maximum security in terms of
the achievable secrecy rate. Hence, we propose to schedule the destination
node with the link that maximizes PH-$B_{n}$ channel power gain. Generally,
channel state information (CSI) is exchanged between nodes before data
communication begins. The transmitter or a centrally designated node with CSI
can decide the intended destination to be scheduled. Node scheduling can be
achieved by adopting an existing node scheduling protocol in a multi-node
network before actual data transmission begins.

We obtain two performance metrics for these kinds of systems, the ASC and POI,
over many networks. We first define these metrics first.

###  II-D Secrecy Capacity

Secrecy capacity is defined as the difference between the achievable rates of
the source-to-destination link and the source-to-eavesdropper link [15].
However, the exact achievable rate of a point-to-point link affected by
Bernoulli-Gaussian impulsive noise is extremely difficult to obtain in closed
form; thus, the authors in [30] proposed bounds for the same. These bounds are
shown to be very tight in all signal-to-noise (SNR) ranges. In this paper, we
use an upper bound from [30] to approximate the achievable rate in bits per
channel use (bpcu).

Considering both background noise and impulsive noise at the receiver, the
approximate capacity at $B_{n^{*}}$ is defined as

$\displaystyle
C_{{n^{*}}}=\delta_{1,b}C_{1,{n^{*}}}+\delta_{2,b}C_{2,{n^{*}}},$ (7)

where

$\displaystyle\delta_{1,b}$
$\displaystyle=(1-p_{b}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\delta_{2,b}=p_{b},$
$\displaystyle C_{1,{n^{*}}}$
$\displaystyle=\log_{2}\left(1+\alpha_{1,b}\gamma_{a}\gamma_{n^{*}}\right),$
$\displaystyle C_{2,{n^{*}}}$
$\displaystyle=\log_{2}\left(1+\alpha_{2,b}\gamma_{a}\gamma_{n^{*}}\right),$
$\displaystyle\alpha_{1,b}$
$\displaystyle=\frac{P}{\epsilon_{bW}^{2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha_{2,b}=\frac{P}{\epsilon_{bW}^{2}(1+\eta_{b})},$
$\displaystyle\gamma_{n^{*}}$
$\displaystyle=\max\\{\gamma_{1},\gamma_{2},\ldots,\gamma_{N}\\}.$ (8)

In (7), the subscript ‘1’ indicates the presence of only background noise
while subscript ‘2’ indicates the presence of both the background and
impulsive noise, $P$ is the transmit power at node $A$, $\alpha_{1,b}$ and
$\alpha_{2,b}$ are the transmit power to the background noise and transmit
power to the background and impulsive noise ratios, respectively,
$C_{1,{n^{*}}}$ and $C_{2,{n^{*}}}$ are the achievable rates when only the
background noise is present with probability $\delta_{1,b}$, and when the
background noise with the impulsive noise is present with probability
$\delta_{2,b}$, respectively. The achievable rate at $E$ can be formulated
simply by replacing $n^{*}$ and $b$ with $e$ in (7) and (II-D).

Taking into account the four possibilities for the manner in which the
Bernoulli-Gaussian noise can affect the selected destination and the
eavesdropper as in (6), the secrecy capacity of the proposed system can be
written with the help of (7) while imposing the positive secrecy constraint as

$\displaystyle C_{S}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\max\\{\left(C_{j,{n^{*}}}-C_{k,e}\right),0\\}.$
(9)

In the above equation, the summation index $j$ and $k$ are to add the secrecy
capacities under all four conditions given in (6).

###  II-E Probability of Intercept (POI)

The POI is defined as the probability of the event when the eavesdropper can
successfully decode $A$’s transmission. This happens when the channel capacity
of the link $A$-$B_{n}$ is lower than that of the link $A$-$E$. The POI
derivation needs to incorporate the different probabilities of arrival of the
impulsive noise at $B_{n^{*}}$ and $E$. Incorporating the four ways that
Bernoulli-Gaussian noise can affect the destination and eavesdropper as in
(6), the POI for the system with the optimal destination scheduling scheme can
be expressed similarly to the expression in (9) for the secrecy capacity as

$\displaystyle{\cal P}_{I}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\mbox{Pr}\left[\alpha_{j,{b}}\gamma_{a}\gamma_{n^{*}}<\alpha_{k,e}\gamma_{a}\gamma_{e}\right].$
(10)

In the above equation, the summation indices $j$ and $k$ are used to add the
POI for each of the four cases in (6).

The following section presents the proposed destination scheduling scheme and
its impact on performance.

##  III Optimal Destination Scheduling Scheme

In this section, we derive the performance metrics for the optimal destination
scheduling scheme to maximize the secrecy performance. Towards this goal, we
first derive the distribution of $\gamma_{n^{*}}$ in (II-D) as the optimal
secrecy capacity is dependent on $\gamma_{n^{*}}$. Due to the independent and
identically distributed PH-$B_{n}$ links, the CDF of $\gamma_{n^{*}}$ can be
obtained using the CDF of $\gamma_{n}$ in (2) as

$\displaystyle F_{\gamma_{n^{*}}}(x)$
$\displaystyle=\mbox{Pr}\left[\max_{n=1,2,\ldots,N}\\{\gamma_{n}\\}\leq
x\right]=\mbox{Pr}\left[\gamma_{n}\leq x\right]^{N}$
$\displaystyle=\left(1-Q\left(\frac{\ln{x}-m_{b}}{s_{b}}\right)\right)^{N}.$
(11)

The corresponding PDF can be obtained by differentiating (11) as

$\displaystyle f_{\gamma_{n^{*}}}(x)$
$\displaystyle=N\left(1-Q\left(\frac{\ln{x}-m_{b}}{s_{b}}\right)\right)^{N-1}$
$\displaystyle\times\frac{1}{x\sqrt{2\pi
s_{b}^{2}}}\exp\left(-\frac{1}{2}\left(\frac{\ln{x}-m_{b}}{s_{b}}\right)^{2}\right).$
(12)

With the help of (11) and (12), we will find the ASC and POI of the proposed
optimal destination scheduling scheme in the subsequent sections.

###  III-A Average Secrecy Capacity (ASC)

The ASC performance across different networks for the proposed destination
scheduling scheme can be evaluated by averaging the secrecy capacity in (9)
with respect to $\gamma_{a}$, $\gamma_{e}$, and $\gamma_{n^{*}}$. Here, we
point out that due to the presence of the common link $A$-PH, the links
$A$-$B_{n^{*}}$ and $A$-$E$ are correlated. As a result, the averaging is
carried out conditioned on $\gamma_{a}$, and then finally we average the
result with respect to the same. The ASC $\bar{C}_{S}$ can be evaluated by
imposing the positive secrecy rate constraint
$\alpha_{j,{b}}\gamma_{n^{*}}>\alpha_{k,e}\gamma_{e}$ for all combinations of
$j\in\\{1,2\\}$ and $k\in\\{1,2\\}$ in (9) as

$\displaystyle\bar{C}_{S}=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\int_{x=0}^{\infty}\left(I_{j,k,n^{*}}(x)-I_{k,j,e}(x)\right)f_{\gamma_{a}}(x)\mbox{d}x,$
(13)

where

$\displaystyle I_{j,k,{n^{*}}}(x)$
$\displaystyle=\int_{y=0}^{\infty}\int_{z=0}^{\frac{\alpha_{j,{b}}}{\alpha_{k,e}}y}\log_{2}\left(1+\alpha_{j,{b}}xy\right)f_{\gamma_{e}}(z)f_{\gamma_{n^{*}}}(y)\mbox{d}z\mbox{d}y$
$\displaystyle=\int_{y=0}^{\infty}\log_{2}\left(1+\alpha_{j,{b}}xy\right)F_{\gamma_{e}}\left(\frac{\alpha_{j,{b}}}{\alpha_{k,e}}y\right)f_{\gamma_{n^{*}}}(y)\mbox{d}y,$
(14) $\displaystyle I_{k,j,e}(x)$
$\displaystyle=\int_{z=0}^{\infty}\int_{y=\frac{\alpha_{k,e}}{\alpha_{j,{b}}}z}^{\infty}\log_{2}\left(1+\alpha_{k,e}xz\right)f_{\gamma_{n^{*}}}(y)f_{\gamma_{e}}(z)\mbox{d}y\mbox{d}z$
$\displaystyle=\int_{z=0}^{\infty}\log_{2}\left(1+\alpha_{k,e}xz\right)f_{\gamma_{e}}(z)$
$\displaystyle\times\left(1-F_{\gamma_{n^{*}}}\left(\frac{\alpha_{k,e}}{\alpha_{j,{b}}}z\right)\right)\mbox{d}z.$
(15)

$I_{j,k,{n^{*}}}(x)$ and $I_{k,j,e}(x)$ are obtained by averaging
$C_{j,{n^{*}}}$ and $C_{k,e}$ in (9), respectively, with respect to
$\gamma_{n^{*}}$ and $\gamma_{e}$ conditioned on $\gamma_{a}$. The integration
limits over $z$ and $y$ in (III-A) and (15), respectively, are due to the
positive secrecy constraint
$\alpha_{j,{b}}\gamma_{n^{*}}>\alpha_{k,e}\gamma_{e}$. By replacing
$f_{\gamma_{e}}(\cdot)$, $F_{\gamma_{e}}(\cdot)$, $F_{\gamma_{n^{*}}}(\cdot)$,
and $f_{\gamma_{n^{*}}}(\cdot)$ from (1), (2), (11), and (12), respectively,
we can express the above integrals as

$\displaystyle
I_{j,k,{n^{*}}}(x)=\int_{y=0}^{\infty}N\log_{2}\Bigg{(}1+\alpha_{j,{b}}xy\Bigg{)}\times$
$\displaystyle\Bigg{(}1-Q\Bigg{(}\frac{\ln\left(\frac{\alpha_{j,{b}}}{\alpha_{k,e}}y\right)-m_{e}}{s_{e}}\Bigg{)}\Bigg{)}\left(1-Q\left(\frac{\ln{y}-m_{b}}{s_{b}}\right)\right)^{N-1}$
$\displaystyle\times\frac{1}{y\sqrt{2\pi
s_{b}^{2}}}\exp\left(-\frac{1}{2}\left(\frac{\ln{y}-m_{b}}{s_{b}}\right)^{2}\right)\mbox{d}y,$
(16)

and

$\displaystyle I_{k,j,e}(x)$
$\displaystyle=\int_{z=0}^{\infty}\log_{2}\left(1+\alpha_{k,e}xz\right)$
$\displaystyle\times\Bigg{(}1-\Bigg{(}1-Q\Bigg{(}\frac{\ln\left(\frac{\alpha_{k,e}}{\alpha_{j,{b}}}z\right)-m_{b}}{s_{b}}\Bigg{)}\Bigg{)}^{N}\Bigg{)}$
$\displaystyle\times\frac{1}{z\sqrt{2\pi
s_{b}^{2}}}\exp\left(-\frac{1}{2}\left(\frac{\ln
z-m_{e}}{s_{e}}\right)^{2}\right)\mbox{d}z\,.$ (17)

In general, integrals such as (16) and (17) do not admit a closed-form
solution. For this reason, our approach in the following will be to transform
the integrals in such a way that the Gauss-Hermite quadrature rule can be
applied to obtain a computable form.

By applying variable substitutions $t=({\ln{y}-m_{b}})/{s_{b}}$ and
$t=({\ln{z}-m_{e}})/{s_{e}}$ in (16) and (17) respectively, we transform these
expressions into the simpler form

$\displaystyle
I_{q}(x)=\int_{t=-\infty}^{\infty}\Psi_{q}(x,t)f_{Q}(t)\mbox{d}t,$ (18)

where $q\in\\{(j,k,n^{*}),(k,j,e)\\}$,

$\displaystyle\Psi_{j,k,n^{*}}(x,t)$
$\displaystyle=\log_{2}\left(1+\alpha_{j,{b}}x\exp(s_{b}t+m_{b})\right)N\left(1-Q\left(t\right)\right)^{N-1}$
$\displaystyle\times\Bigg{(}1-Q\Bigg{(}\frac{s_{b}t+m_{b}+\ln\left(\frac{\alpha_{j,{b}}}{\alpha_{k,e}}\right)-m_{e}}{s_{e}}\Bigg{)}\Bigg{)},$
(19)

and

$\displaystyle\Psi_{k,j,e}(x,t)=\log_{2}\left(1+\alpha_{k,e}x\exp(s_{e}t+m_{e})\right)$
$\displaystyle\times\Bigg{(}1-\Bigg{(}1-Q\Bigg{(}\frac{s_{e}t+m_{e}+\ln\left(\frac{\alpha_{k,e}}{\alpha_{j,{b}}}\right)-m_{b}}{s_{b}}\Bigg{)}\Bigg{)}^{N}\Bigg{)}.$
(20)

Note that the expression in (18) is actually evaluating the expectation of an
arbitrary function $\Psi_{q}(x,t)$ with respect to the Gaussian probability
density function. It has a numerical solution using the Gauss-Hermite
quadrature method [31] and hence $I_{q}(x)$ can be computed as

$\displaystyle I_{q}(x)$
$\displaystyle=\sum_{\ell=1}^{L}\omega_{\ell}\Psi_{q}(x,\theta_{\ell}),$ (21)

where $L$ is the number of points, $\omega_{\ell}$ are the weights, and
$\theta_{\ell}$ is the abscissa of the Gauss-Hermite quadrature method.

Next, applying (21) in (13), the ASC can be presented in a single integral
form as

$\displaystyle\bar{C}_{S}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\int_{x=0}^{\infty}\frac{I_{j,k,n^{*}}(x)-I_{k,j,e}(x)}{x\sqrt{2\pi
s_{a}^{2}}}$
$\displaystyle\times\exp\left(-\frac{1}{2}\left(\frac{\ln{x}-m_{a}}{s_{a}}\right)^{2}\right)\mbox{d}x.$
(22)

The above equation still does not admit a closed-form solution and we again
resort to the Gauss-Hermite quadrature method. Using the change of variable
$t=({\ln{x}-m_{a}})/{s_{a}}$, we can write

$\displaystyle\bar{C}_{S}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,k,{b}}\delta_{k,j,e}\int_{t=-\infty}^{\infty}\left[I_{j,k,n^{*}}(\exp\left(ts_{a}+m_{a}\right))\right.$
$\displaystyle\left.-I_{k,j,e}(\exp\left(ts_{a}+m_{a}\right))\right]f_{Q}(t)\mbox{d}t.$
(23)

Finally, we see that (23) is in the same form as (18); hence, using the Gauss-
Hermite quadrature rule in (21), the ASC can be obtained in a numerically
computable form.

###  III-B Asymptotic ASC

The computable expression presented in (23) is difficult to analyze with
respect to the parameters $N$, $P$, $m_{\rho}$ and $s_{\rho}$ for
$\rho\in\\{a,b,e\\}$, hence, we provide a closed-form asymptotic ASC in the
high-SNR regime, i.e., when $P$ tends to infinity.

######  Proposition 1.

The approximate asymptotic ASC of the destination scheduling scheme is

$\displaystyle\bar{C}_{S}$
$\displaystyle\approx\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\left((I_{j,k,n^{*}}^{(+)}+I_{j,k,n^{*}}^{(-)})\right.$
$\displaystyle\left.-(I^{(0)}_{{k,j,e}}+I^{(+)}_{{k,j,e}}+I^{(-)}_{{k,j,e}})\right),$
(24)

where

$\displaystyle
I_{j,k,n^{*}}^{(+)}=\frac{N\mathcal{D}_{n^{*}}^{(N-1)}}{\ln{2}}\Bigg{[}\frac{\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}\right)Q\left(\mathcal{B}_{n^{*}}^{(N-1)}\right)}{\mathcal{A}_{n^{*}}^{(N-1)}}\Bigg{.}$
$\displaystyle\Bigg{.}-\frac{s_{b}\exp{\left(-\frac{(\mathcal{B}_{n^{*}}^{(N-1)})^{2}}{2}\right)}}{(\mathcal{A}_{n^{*}}^{(N-1)})^{2}\sqrt{2\pi}}+\frac{s_{b}\mathcal{B}_{n^{*}}^{(N-1)}Q\left(\mathcal{B}_{n^{*}}^{(N-1)}\right)}{(\mathcal{A}_{n^{*}}^{(N-1)})^{2}}\Bigg{]},$
(25) $\displaystyle
I_{j,k,n^{*}}^{(-)}=\frac{N}{\ln{2}}\sum_{n=0}^{N-1}\binom{N-1}{n}(-1)^{n}\mathcal{D}_{n^{*}}^{(n)}\times$
$\displaystyle\Big{[}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}\right)\frac{1-Q\left(\bar{\mathcal{B}}_{n^{*}}^{(n)}\right)}{\mathcal{A}_{n^{*}}^{(n)}}+\frac{s_{b}\exp{\left(-\frac{(\bar{\mathcal{B}}_{n^{*}}^{(n)})^{2}}{2}\right)}}{(\mathcal{A}_{n^{*}}^{(n)})^{2}\sqrt{2\pi}}\Big{.}$
$\displaystyle\Big{.}+\frac{s_{b}\bar{\mathcal{B}}_{n^{*}}^{(n)}\left(1-Q\left(\bar{\mathcal{B}}_{n^{*}}^{(n)}\right)\right)}{(\mathcal{A}_{n^{*}}^{(n)})^{2}}\Big{]},$
(26)

$\displaystyle I_{k,j,e}^{(0)}$
$\displaystyle=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}\right),$
(27) $\displaystyle I^{(+)}_{{k,j,e}}$
$\displaystyle=\frac{\mathcal{D}_{k,j,e}^{(N)}}{\phi_{e}\ln{2}}\Bigg{[}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}\right)\frac{Q\left(\mathcal{B}_{k,j,e}^{(N)}\right)}{\mathcal{A}_{e}^{(N)}}\Bigg{.}$
$\displaystyle\Bigg{.}-\frac{s_{e}}{\phi_{e}(\mathcal{A}_{e}^{(N)})^{2}\sqrt{2\pi}}\exp{\left(-\frac{(\mathcal{B}_{k,j,e}^{(N)})^{2}}{2}\right)}\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{s_{e}\mathcal{B}_{k,j,e}^{(N)}Q\left(\mathcal{B}_{k,j,e}^{(N)}\right)}{\phi_{e}(\mathcal{A}_{e}^{(N)})^{2}}\Bigg{]},$
(28) $\displaystyle I^{(-)}_{{k,j,e}}$
$\displaystyle=\frac{1}{\phi_{e}\ln{2}}\sum_{n=0}^{N}\binom{N}{n}(-1)^{n}\bar{\mathcal{D}}_{k,j,e}^{(n)}$
$\displaystyle\times\Bigg{[}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}\right)\frac{1-Q\left(\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)}{\mathcal{A}_{e}^{(n)}}\Big{.}$
$\displaystyle\Bigg{.}+\frac{s_{e}}{\phi_{e}}\frac{1}{(\mathcal{A}_{e}^{(n)})^{2}\sqrt{2\pi}}\exp{\Big{(}-\frac{(\bar{\mathcal{B}}_{k,j,e}^{(n)})^{2}}{2}\Big{)}}\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{s_{e}}{\phi_{e}}\frac{\bar{\mathcal{B}}_{k,j,e}^{(n)}\left(1-Q\left(\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)\right)}{(\mathcal{A}_{e}^{(n)})^{2}}\Bigg{]},$
(29) $\displaystyle\phi_{e}$ $\displaystyle=\frac{s_{e}}{s_{b}},$ (30)
$\displaystyle\mathcal{A}_{n^{*}}^{(n)}$
$\displaystyle=\sqrt{2nK_{1}+1},~{}\mathcal{B}_{n^{*}}^{(n-1)}=\frac{(n-1)K_{2}}{\mathcal{A}_{n^{*}}^{(n-1)}},$
$\displaystyle\bar{\mathcal{B}}_{n^{*}}^{(n)}$
$\displaystyle=-\frac{nK_{2}}{\mathcal{A}_{n^{*}}^{(n)}},~{}\mathcal{C}_{n^{*}}^{(n)}=2nK_{3},$
$\displaystyle\mathcal{D}_{n^{*}}^{(n)}$
$\displaystyle=\exp\left(-\frac{1}{2}\left(\mathcal{C}_{n^{*}}^{(n)}-(\bar{\mathcal{B}}_{n^{*}}^{(n)})^{2}\right)\right),$
(31) $\displaystyle\mathcal{A}_{e}^{(n)}$
$\displaystyle=\sqrt{2nK_{1}+\frac{1}{\phi_{e}^{2}}},~{}\mathcal{B}_{k,j,e}^{(n)}=\frac{nK_{2}+\frac{\lambda_{k,j,e}}{\phi_{e}^{2}}}{\mathcal{A}_{e}^{(n)}},$
$\displaystyle\bar{\mathcal{B}}_{k,j,e}^{(n)}$
$\displaystyle=\frac{-nK_{2}+\frac{\lambda_{k,j,e}}{\phi_{e}^{2}}}{\mathcal{A}_{e}^{(n)}},~{}\mathcal{C}_{k,j,e}^{(n)}=2nK_{3}+\frac{\lambda_{k,j,e}^{2}}{\phi_{e}^{2}},$
$\displaystyle\mathcal{D}_{k,j,e}^{(n)}$
$\displaystyle=\exp{\left(-\frac{1}{2}\left(\mathcal{C}_{k,j,e}^{(n)}-\left(\mathcal{B}_{k,j,e}^{(n)}\right)^{2}\right)\right)},$
$\displaystyle\bar{\mathcal{D}}_{k,j,e}^{(n)}$
$\displaystyle=\exp{\left(-\frac{1}{2}\left(\mathcal{C}_{k,j,e}^{(n)}-(\bar{\mathcal{B}}_{k,j,e}^{(n)})^{2}\right)\right)},$
(32)

and $K_{1},K_{2},K_{3}\in\mathbb{R}$ are fitting parameters for the
approximate $Q$-function [42].

###### Proof.

By neglecting unity within $\log(\cdot)$ from $C_{j,n^{*}}$ and $C_{k,e}$ as
$P\rightarrow\infty$, we can approximate (9) as

$\displaystyle
C_{S}\approx\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\max\left\\{\log_{2}\left(\frac{\alpha_{j,b}\gamma_{a}\gamma_{n^{*}}}{\alpha_{k,e}\gamma_{a}\gamma_{e}}\right),0\right\\}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\max\left\\{\left(\log_{2}\left(\tilde{\alpha}_{j,b}\gamma_{n^{*}}\right)-\log_{2}\left(\tilde{\alpha}_{k,e}\gamma_{e}\right)\right),0\right\\},$
(33)

where $\tilde{\alpha}_{1,b}=\frac{1}{\epsilon_{bW}^{2}}$,
$\tilde{\alpha}_{2,b}=\frac{1}{\epsilon_{bW}^{2}(1+\eta_{b})}$,
$\tilde{\alpha}_{1,e}=\frac{1}{\epsilon_{eW}^{2}}$, and
$\tilde{\alpha}_{2,e}=\frac{1}{\epsilon_{eW}^{2}(1+\eta_{e})}$ are independent
of $P$. We observe that (III-B) is independent of $\gamma_{a}$ which means
that at high SNR, the correlation between destination-eavesdropper channels
does not have any effect on the secrecy. The asymptotic ASC is derived
following (13) as

$\displaystyle\bar{C}_{S}=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\left(I_{j,k,n^{*}}-I_{k,j,e}\right),$
(34)

where $I_{j,k,n^{*}}$ and $I_{k,j,e}$ are modified from $I_{j,k,n^{*}}(x)$ and
$I_{k,j,e}(x)$ following (III-A) and (15), respectively, and are now
independent of $x$ (i.e., $\gamma_{a}$). Thus, $I_{j,k,n^{*}}$ and $I_{k,j,e}$
are expressed as

$\displaystyle I_{j,k,n^{*}}$
$\displaystyle=\int_{y=0}^{\infty}\log_{2}\left(\tilde{\alpha}_{j,{b}}y\right)F_{\gamma_{e}}\left(\frac{\tilde{\alpha}_{j,{b}}}{\tilde{\alpha}_{k,e}}y\right)f_{\gamma_{n^{*}}}(y)\mbox{d}y,$
(35)

and

$\displaystyle I_{k,j,e}$
$\displaystyle=\int_{z=0}^{\infty}\log_{2}\left(\tilde{\alpha}_{k,e}z\right)f_{\gamma_{e}}(z)\left(1-F_{\gamma_{n^{*}}}\left(\frac{\tilde{\alpha}_{k,e}}{\tilde{\alpha}_{j,{b}}}z\right)\right)\mbox{d}z,$
(36)

respectively. Following manipulations similar to (16)-(18), we can show that
(18) is modified to

$\displaystyle I_{q}=\int_{t=-\infty}^{\infty}\Psi_{q}(t)f_{Q}(t)\mbox{d}t,$
(37)

where $\Psi_{j,k,n^{*}}(t)$ and $\Psi_{k,j,e}(t)$ correspond to
$\Psi_{j,k,n^{*}}(x,t)$ and $\Psi_{k,j,e}(x,t)$ respectively from (19) and
(20); however, these are now independent of $x$. $\Psi_{j,k,n^{*}}(t)$ is
obtained as

$\displaystyle\Psi_{j,k,n^{*}}(t)$
$\displaystyle=\frac{N}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}+s_{b}t\right)$
$\displaystyle\times\left(Q\left(-t\right)\right)^{N-1}Q\left(-\left(\phi_{n^{*}}t+\lambda_{j,k,n^{*}}\right)\right),$
(38)

where $\phi_{n^{*}}=\frac{s_{b}}{s_{e}}$, and
$\lambda_{j,k,n^{*}}=\frac{m_{b}-m_{e}+\ln(\tilde{\alpha}_{j,b}/\tilde{\alpha}_{k,e})}{s_{e}}$.
The above equation is still difficult to handle to get a closed-form solution,
as we note that this should be averaged over the Gaussian PDF in (37). Hence,
we approximate (III-B) further by neglecting the factor
$Q\left(-\left(\phi_{n^{*}}t+\lambda_{j,k,n^{*}}\right)\right)$ when the
quality of the destination channel is far better than that of the eavesdropper
channel, i.e., when $s_{b}\gg s_{e}$ and $m_{b}\gg m_{e}$ as

$\displaystyle\Psi_{j,k,n^{*}}(t)$
$\displaystyle\approx\frac{N}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}+s_{b}t\right)\left(Q\left(-t\right)\right)^{N-1}.$
(39)

As $\phi_{n^{*}}$ and $\lambda_{j,k,n^{*}}$ have a high positive value when
$s_{b}\gg s_{e}$ and $m_{b}\gg m_{e}$ for a given $t$, the factor
$Q\left(-\left(\phi_{n^{*}}t+\lambda_{j,k,n^{*}}\right)\right)$ is close to
unity, and thus $\Psi_{j,k,n^{*}}(t)$ is well-approximated by (39). We also
notice that the factor
$Q\left(-\left(\phi_{n^{*}}t+\lambda_{j,k,n^{*}}\right)\right)$ does not
depend on $N$. Hence, the approximation cannot be improved by varying $N$.

Subsequently, $\Psi_{k,j,e}(t)$ is written as

$\displaystyle\Psi_{k,j,e}(t)=\frac{1}{\ln{2}}\left(\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}+s_{e}t\right)\right.$
$\displaystyle\left.\times\left(1-\left(Q\left(-\left(\phi_{e}t+\lambda_{k,j,e}\right)\right)\right)^{N}\right)\right)$
$\displaystyle=\frac{1}{\ln{2}}\left[\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}+s_{e}t\right)-\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}+s_{e}t\right)\right.$
$\displaystyle\left.\times\left(Q\left(-\left(\phi_{e}t+\lambda_{k,j,e}\right)\right)\right)^{N}\right],$
(40)

where $\phi_{e}=\frac{s_{e}}{s_{b}}$, and
$\lambda_{k,j,e}=\frac{m_{e}-m_{b}+\ln(\tilde{\alpha}_{k,e}/\tilde{\alpha}_{j,b})}{s_{b}}$.

The solutions of $I_{j,k,n^{*}}$ and $I_{k,j,e}$ following (37) with the help
of (39) and (III-B), respectively, are written as

$\displaystyle I_{j,k,n^{*}}$
$\displaystyle\approx\int_{t=-\infty}^{\infty}{\Psi}_{j,k,n^{*}}(t)f_{Q}(t)\mbox{d}t,$
(41) $\displaystyle I_{k,j,e}$
$\displaystyle=\int_{t=-\infty}^{\infty}\Psi_{k,j,e}(t)f_{Q}(t)\mbox{d}t.$
(42)

The solutions of $I_{j,k,n^{*}}$ and $I_{k,j,e}$ are carried out in Appendix
-A and Appendix -B, respectively. By substituting the solutions of
$I_{j,k,n^{*}}$ and $I_{k,j,e}$ from (1) and (1), respectively, into the
asymptotic ASC definition in (34), we obtain the final result in (1). ∎

######  Remark 1.

When $N$ is large, $\mathcal{B}_{n^{*}}^{(N-1)}$ in (30) becomes large and
hence, $Q\left(\mathcal{B}_{n^{*}}^{(N-1)}\right)$ and
$\exp{\left(-\frac{(\mathcal{B}_{n^{*}}^{(N-1)})^{2}}{2}\right)}$ in (1)
become negligible. Thus, (1) tends to zero.

######  Remark 2.

As $n$ increases, $\bar{\mathcal{B}}_{n^{*}}^{(n)}$ defined in (30) tends to a
large negative number and hence,
$Q\left(\bar{\mathcal{B}}_{n^{*}}^{(n)}\right)$ tends to unity and
$\exp{\left(-\frac{(\bar{\mathcal{B}}_{n^{*}}^{(n)})^{2}}{2}\right)}$ tends to
zero. Thus, only lower order terms of $n$ in the summation expressed in (1)
dominate. In particular, $n=0$ is the dominant term in this equation. This
suggests that (1) is non-zero even as $N$ increases.

######  Remark 3.

Following the same logic as in Remark 1, $I^{(+)}_{{k,j,e}}$ given by (1)
tends to zero.

######  Remark 4.

Again, following Remark 2, we can show that as $N$ increases, the higher order
summation terms (large $n$) in (1) tend to zero. In this case, only lower
order summation terms (small $n$) remain. Now, if $m_{b}\gg m_{e}$, it is
evident from (1) that $\bar{\mathcal{B}}_{k,j,e}^{(n)}$ tends to a large
negative number as
$\lambda_{k,j,e}=\frac{m_{e}-m_{b}+\ln(\alpha_{k,e}/\alpha_{j,b})}{s_{b}}$
tends to a large negative number. Hence, all lower order terms of $n$ in (1)
also tend to have low values. Therefore, we can conclude that when $m_{b}\gg
m_{e}$, $I^{(-)}_{{k,j,e}}$ tends to zero as $N$ increases.

######  Corollary 1.

The approximate asymptotic ASC of the destination scheduling scheme when
$N\rightarrow\infty$ and $m_{b}\gg m_{e}$ is

$\displaystyle\bar{C}_{S}\approx\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\left(I_{j,k,n^{*}}^{(-)}-I^{(0)}_{{k,j,e}}\right),$
(43)

where $I_{j,k,n^{*}}^{(-)}$ and $I^{(0)}_{{k,j,e}}$ are defined in (1) and
(27), respectively.

###### Proof.

The proof easily follows from Remarks 1-4. ∎

We can easily conclude from (43) how the channel parameters affect the
asymptotic ASC performance. We can infer this from the case when $N=1$. When
$N=1$, (1) becomes

$\displaystyle I_{j,k,n^{*}}^{(-)}$
$\displaystyle=\frac{1}{2\ln{2}}{\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}\right)}.$
(44)

In addition, the second term

$\displaystyle I_{k,j,e}^{(0)}$
$\displaystyle=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}\right),$
(45)

is independent of $N$. Note that (44) and (45) clearly show that the
performance improves as the difference of $m_{b}$, $m_{e}$, and the ratio of
$\tilde{\alpha}_{j,{b}}$, $\tilde{\alpha}_{k,e}$ increases. If the impulsive
noise arrival rates are low, i.e. $p_{b}$ and $p_{e}$ are low, the ratio of
$\tilde{\alpha}_{j,{b}}$, $\tilde{\alpha}_{k,e}$ does not have much effect,
instead the difference between $m_{b}$ and $m_{e}$ becomes the critical design
parameter for secrecy rather than the impulsive noise parameters
$\tilde{\alpha}_{j,{b}}$ and $\tilde{\alpha}_{k,e}$. In general, the dominant
term in $I_{j,k,n^{*}}^{(-)}$ is for $n=0$ and other terms diminish as $n$
increases. Additionally, as $N$ increases, $I_{j,k,n^{*}}^{(-)}$ increases,
hence, the secrecy performance improves with $N$.

###  III-C Probability of Intercept (POI)

In this section, we find the approximate closed-form POI using the definition
provided in (10) with the help of the CDF of $\gamma_{n^{*}}$ from (11) and
the PDF of $\gamma_{e}$ from (1) as

$\displaystyle{\cal
P}_{I}=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\mbox{Pr}\left[\gamma_{n^{*}}\leq\frac{\alpha_{k,e}}{\alpha_{j,{b}}}\gamma_{e}\right]$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\int_{0}^{\infty}\left(1-Q\left(\frac{\ln\left(\frac{\alpha_{k,e}}{\alpha_{j,{b}}}x\right)-m_{b}}{s_{b}}\right)\right)^{N}$
$\displaystyle\times\frac{1}{x\sqrt{2\pi
s_{e}^{2}}}\exp\left(-\frac{1}{2}\left(\frac{\ln(x)-m_{e}}{s_{e}}\right)^{2}\right)\mbox{d}x.$
(46)

As
$\frac{\alpha_{k,e}}{\alpha_{j,{b}}}=\frac{\tilde{\alpha}_{k,e}}{\tilde{\alpha}_{j,{b}}}$
is independent of $P$, the POI is independent of $P$. Using the change of
variable $t=({\ln{x}-m_{e}})/{s_{e}}$, this can be simplified as

$\displaystyle{\cal P}_{I}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\int_{t=-\infty}^{\infty}\left(1-Q\left(\phi_{e}t+\lambda_{k,j,e}\right)\right)^{N}f_{Q}(t)\mbox{d}t$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\int_{t=-\infty}^{\infty}\left(Q\left(-\left(\phi_{e}t+\lambda_{k,j,e}\right)\right)\right)^{N}f_{Q}(t)\mbox{d}t.$
(47)

We notice that integration in (47) is in the same form as (18), where a
function is averaged over the standard normal PDF; hence, its numerical
solution can be achieved using the Gauss-Hermite quadrature rule as in (21).
However, we will also provide an approximate closed-form solution following a
similar mathematical approach to that we used to analyze the approximate
asymptotic ASC in Subsection III-B. We use the approximate $Q$-function given
by (49). We observe that the integration is in the same form as the first
integration evaluated in (-B) in Appendix -B. Hence, we use the results
already derived therein. Following (-B)-(-B) and the integral solutions
provided in Appendix -C, we get the approximate closed-form POI as

$\displaystyle{\cal P}_{I}$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\delta_{j,{b}}\delta_{k,e}\frac{1}{\phi_{e}}\left(\int_{-\infty}^{0}\frac{\mathcal{D}_{k,j,e}^{(N)}}{\sqrt{2\pi}}\right.$
$\displaystyle\left.\times\exp{\left(-\frac{1}{2}\left(\mathcal{A}_{e}^{(N)}u-\mathcal{B}_{k,j,e}^{(N)}\right)^{2}\right)}\mbox{d}u\right.$
$\displaystyle\left.+\sum_{n=0}^{N}\binom{N}{n}(-1)^{n}\int_{0}^{\infty}\frac{\bar{\mathcal{D}}_{k,j,e}^{(n)}}{\sqrt{2\pi}}\right.$
$\displaystyle\left.\times\exp{\left(-\frac{1}{2}\left(\mathcal{A}_{e}^{(n)}u-\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)^{2}\right)}\right)\mbox{d}u$
$\displaystyle=\sum_{j=1}^{2}\sum_{k=1}^{2}\frac{\delta_{j,{b}}\delta_{k,e}}{\phi_{e}}\left(\mathcal{D}_{k,j,e}^{(N)}\frac{Q\left(\mathcal{B}_{k,j,e}^{(N)}\right)}{\mathcal{A}_{e}^{(N)}}\right.$
$\displaystyle\left.+\sum_{n=0}^{N}\binom{N}{n}(-1)^{n}\bar{\mathcal{D}}_{k,j,e}^{(n)}\frac{1-Q\left(\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)}{\mathcal{A}_{e}^{(n)}}\right),$
(48)

where $\mathcal{D}_{k,j,e}^{(n)}$, $\mathcal{B}_{k,j,e}^{(n)}$,
$\mathcal{A}_{e}^{(n)}$, $\bar{\mathcal{D}}_{k,j,e}^{(n)}$,
$\bar{\mathcal{B}}_{k,j,e}^{(n)}$, and $\mathcal{A}_{e}^{(n)}$ are defined in
(1).

######  Remark 5.

Following similar logic as in Remarks 3 and 4, it is easy to show that as
$N\rightarrow\infty$, $Q\left(\mathcal{B}_{k,j,e}^{(N)}\right)$ and
$1-Q\left(\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)$ both tend to zero and thus,
the POI tends to zero.

##  IV Results and Discussion

In this section, we present the numerical results for the ASC and its
approximate asymptotic limit along with the numerical results of POI. The
parameters $s_{\rho}$, $m_{\rho}$, $\forall\rho\in\\{a,b,e\\}$, are assumed to
have the same values as in [9, 12]. The channel parameters, $s_{\rho}$ and
$m_{\rho}$, in general depend on the power distribution network and high
values of these indicate high fluctuations in the received signal power. The
computable expression of the ASC and its closed-form approximate asymptotic
limit derived in this paper are plotted against the transmit power $P$ in
Figs. 3-7. The computable expression of the POI and its closed-form
approximate expression are plotted with $N$ in Fig 8. The horizontal straight
lines in ASC plots depict the closed-form approximate asymptotic expression
derived in the paper. This is indicated as “Approx. asymptote” in the figures.
The marker “$\times$” with the same color as the corresponding analytical
curve indicates numerical results. We assume $K_{1}=0.3842$, $K_{2}=0.7640$,
and $K_{3}=0.6964$ in the $Q$-function approximation, as was adopted in [42].

Figure 3: Average secrecy capacity versus $P$ for varying values of $s_{a}$
and $N$. Here $s_{b}=s_{e}=6$ dB, $m_{a}=m_{b}=-20$ dB, $m_{e}=-40$ dB,
$p_{b}=p_{e}=10^{-1}$, and $\eta_{b}=\eta_{e}=10$.

Fig. 3 compares the ASC performance of the PH-based system, denoted as “PH”,
with the system having only direct source to $N$ destination links without a
PH node, denoted as “No PH”, to show the performance degradation of the system
which is caused by the PH. We have assumed that the average SNR per link for
the PH and No PH systems are the same for a fair comparison. The analytical
performance of the No PH system is evaluated using our proposed analysis just
by adopting $\gamma_{a}=1$ and performing some trivial changes for the same
average SNR per link in both systems. In particular, the effect of the log-
normal variance $s_{a}$ of the shared link in the PH system is shown for a
given set of parameters as presented in the figure caption. The performance
varying $N$ is also plotted. The legend only shows the markers for $N=40$ in
the colour black, however, the same markers in the colour red are also adopted
for $N=10$. Markers in the colour red are omitted in the legend to avoid
overcrowding of markers.

We observe that the performance of the No PH system is always better than the
PH system. The existence of the PH correlates the signal at the destination
and the eavesdropper and thus the secrecy rate decreases. Careful observation
also reveals that as $s_{a}$ increases, the performance degradation increases
in the PH system due to the increased correlation. An increase in $s_{a}$
increases the correlation between end-to-end links. In contrast, as $N$ does
not affect the correlation, the performance degradation in the PH system is
independent of $N$. This can be confirmed from the figure as the performance
gap between the PH and No PH systems remains the same as $N$ increases from 10
to 40.

We further observe that the ASC performance asymptotically saturates to a
constant value. This happens as the ratio of the destination and the
eavesdropper channel SNRs becomes independent of $P$ as $P$ increases. The
asymptotic performance derived in the paper also matches well with the
saturation value, thus verifying the correctness of our approximate asymptotic
analysis. In this particular case, the destination channel quality is far
better than the eavesdropping channel quality ($m_{b}\gg m_{e}$). We notice
that as $P$ increases, the asymptotic behaviour of the No PH and PH systems
becomes the same. This is reasonable, as we observed in the asymptotic
analysis that the ASC performance is independent of $\gamma_{a}$ when
$P\rightarrow\infty$. This suggests that the asymptotic performance is
unaffected by variations in $s_{a}$ and $m_{a}$, which is also confirmed from
the figure. However, different saturation levels occur with different $N$. As
an increase in $N$ increases the number of choices for selecting the best
destination node from, the saturation level improves with it.

Figure 4: Average secrecy capacity versus $P$ for varying values of $m_{a}$
and $N$. Here $s_{a}=s_{b}=s_{e}=6$ dB, $m_{b}=-20$ dB, $m_{e}=-40$ dB,
$p_{b}=p_{e}=10^{-1}$, and $\eta_{b}=\eta_{e}=10$.

Fig. 4 compares the performance of the PH and No PH systems when the log-
normal mean $m_{a}$ of the shared link improves. Markers in the colour red are
omitted here as well in the legend. We notice that as $m_{a}$ increases, the
performance degradation due to the PH exhibits a similar trend to that due to
$s_{a}$ in Fig. 3. The reasoning is the same as for $s_{a}$ in Fig. 3.
However, the performance degradation due to increasing $m_{a}$ is not much as
large as in Fig. 3. Moreover, the approximate asymptote is unaffected by
$m_{a}$ as the ASC performance is independent of $\gamma_{a}$ at high $P$. The
performance improvement due to an increase in $N$ can also be seen in this
figure. Furthermore, Fig. 3 and Fig. 4 reveal that an improvement in the
shared channel quality ($s_{a}$ and $m_{a}$) improves the ASC performance.
This is counterintuitive, as one might expect that the shared link between the
eavesdropping and destination channel would not have any effect on the
secrecy.

Figure 5: Average secrecy capacity versus $P$ for varying values of $s_{b}$,
$s_{e}$, and $N$. Here $s_{a}=6$ dB, $m_{a}=m_{b}=-20$ dB, $m_{e}=-40$ dB,
$p_{b}=p_{e}=10^{-1}$, and $\eta_{b}=\eta_{e}=10$.

Fig. 5 shows the effect of $s_{b}$ and $s_{e}$ on the ASC performance for a
given set of parameters as depicted in the figure caption. Different channel
quality variations are shown, for example, when $s_{b}>s_{e}$ and
$s_{b}<s_{e}$ while $m_{b}\gg m_{e}$. The approximate asymptote accurately
predicts the saturated ASC in these situations even though the destination and
eavesdropper channel qualities vary significantly, which verifies the
correctness of our approximation. We also notice that as $s_{b}$ improves, the
performance improves, however, as $s_{e}$ improves, the performance degrades.
This is intuitively plausible since the improvement in the destination channel
quality improves the secrecy and the improvement in the eavesdropping channel
quality degrades the secrecy.

Figure 6: Average secrecy capacity versus $P$ for varying values of $m_{b}$.
Here $N=10$, $s_{a}=s_{b}=s_{e}=6$ dB, $m_{a}=-20$ dB, $m_{e}=-40$ dB,
$p_{b}=p_{e}=10^{-1}$, and $\eta_{b}=\eta_{e}=10$.

Fig. 6 examines how the accuracy of the proposed approximate asymptotic ASC
varies with the difference between $m_{b}$ and $m_{e}$. We find that when
$m_{b}=-20$ dB while $m_{e}=-40$ dB, the approximate asymptote is very
accurate; however, there is a performance gap between the approximate
asymptotic and the actual ASC when $m_{b}=-30$ dB while $m_{e}=-40$ dB. This
gap widens as the difference between $m_{b}$ and $m_{e}$ reduces.

Figure 7: Average secrecy capacity versus $P$ for varying values of
$\eta_{b},\eta_{e},p_{b}$, and $p_{e}$. Here $N=10$, $s_{a}=s_{b}=s_{e}=6$ dB,
$m_{a}=m_{b}=-20$ dB, and $m_{e}=-40$ dB.

Fig. 7 shows the effect of the independent noise assumption at the
destinations and the eavesdropper on the ASC performance. An independent
impulsive noise arrival process at the destination and eavesdropper with
probabilities $p_{b}$ and $p_{e}$, respectively, are considered assuming each
taking values $0.1$ and $0.9$. In addition, different strengths of impulsive
noise at these nodes, i.e., $\eta_{b}$ and $\eta_{e}$ each taking values 10
and 100 are assumed. Two extreme values of $\eta_{\zeta}$ and $p_{\zeta}$,
where $\zeta\in\\{b,e\\}$, are considered to infer the performance of other
possible conditions in between these limits. When both nodes have low
impulsive noise arrival rates (black curves), whether $\eta_{e}>\eta_{b}$ or
$\eta_{e}<\eta_{b}$ does not make any difference to the performance. However,
when both nodes have a high probability of impulsive noise arrival (red
curves), the case where $\eta_{e}>\eta_{b}$ has a far better performance
compared to the case where $\eta_{e}<\eta_{b}$. This suggests that by
artificially controlling the power ratios of the impulsive noise to the
background noise independently at the destinations and at the eavesdropper,
the secrecy performance can be significantly improved.

Figure 8: POI versus $N$ for varying values of $s_{b},s_{e}$, and $m_{b}$.
Here $m_{e}=-40$ dB, $p_{b}=p_{e}=10^{-1}$, and $\eta_{b}=\eta_{e}=10$.

Fig. 8 compares the POI performance for different values of $s_{b}$, $m_{b}$,
and $s_{e}$ as a function of $N$ (note that the POI does not depend on $P$).
The approximate closed-form solution along with the computable solution using
the Gauss-Hermite method derived in the paper is shown with the corresponding
numerical results. The Gauss-Hermite computable results are plotted with the
marker “+”. All of these plots merge with each other, validating the
correctness of our analysis. As $N$ increases, the performance improves due to
the increased number of choices of selection. Further, as $m_{b}$ improves for
a given $s_{b}$ and $s_{e}$ combination, the POI improves.

##  V Conclusion

In this paper, for the first time, an optimal destination scheduling mechanism
is proposed to improve the PLS of a pinhole-based PLC network. Computable
expressions for both the ASC and POI performance across many different
networks are derived along with a closed-form approximate asymptotic ASC and
an approximate POI. We conclude that the shared link in the pinhole-based
system adversely affects the average secrecy rate compared to the system
without a pinhole, however, it has no effect on the probability of intercept.
We observe that the degradation in ASC increases as the shared link log-normal
mean and variance increase. From the asymptotic analysis, we find that the ASC
asymptotically saturates to a constant value as the transmit power increases,
while the POI can be decreased by increasing the number of destination nodes.
We also show that by controlling the artificial impulsive noise power and its
arrival rate, the secrecy rate can be improved.

###  -A Solution of $I_{j,k,n^{*}}$ given by (41).

We will find an approximate closed-form solution of (41) with the help the
following approximation to the $Q$-function [42]

$\displaystyle
Q(t)=\exp\left(-\left(K_{1}t^{2}+K_{2}t+K_{3}\right)\right)~{}~{}\mbox{where}~{}~{}t\geq
0,$ (49)

where $K_{1},K_{2},K_{3}\in\mathbb{R}$ are fitting parameters [42]. The
approximation is valid for $t\geq 0$, hence we divide the integration region
in (41) as

$\displaystyle I_{j,k,n^{*}}$
$\displaystyle=I_{j,k,n^{*}}^{(+)}+I_{j,k,n^{*}}^{(-)},$ (50)

where

$\displaystyle I_{j,k,n^{*}}^{(+)}$
$\displaystyle=\int_{-\infty}^{0}\tilde{\Psi}_{j,k,n^{*}}(t)f_{Q}(t)\mbox{d}t,$
(51) $\displaystyle I_{j,k,n^{*}}^{(-)}$
$\displaystyle=\int_{0}^{\infty}\tilde{\Psi}_{j,k,n^{*}}(t)f_{Q}(t)\mbox{d}t.$
(52)

In (39), $Q$-function in $\tilde{\Psi}_{j,k,n^{*}}(t)$ has a positive argument
when $-\infty\leq t<0$; hence, we rewrite $I_{j,k,n^{*}}^{(+)}$ by applying
the approximate $Q$-function as

$\displaystyle
I_{j,k,n^{*}}^{(+)}=\int_{-\infty}^{0}\frac{N}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}+s_{b}t\right)$
$\displaystyle\times\exp\left(-(N-1)\left(K_{1}t^{2}-K_{2}t+K_{3}\right)\right)f_{Q}(t)\mbox{d}t.$
(53)

After substituting for $f_{Q}(t)$ and performing some further manipulations we
obtain

$\displaystyle
I_{j,k,n^{*}}^{(+)}=\int_{-\infty}^{0}\frac{N}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}+s_{b}t\right)$
$\displaystyle\times\frac{\mathcal{D}_{n^{*}}^{(N-1)}}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\mathcal{A}_{n^{*}}^{(N-1)}t-\mathcal{B}_{n^{*}}^{(N-1)}\right)^{2}\right)\mbox{d}t,$
(54)

where

$\displaystyle\mathcal{A}_{n^{*}}^{(N-1)}=\sqrt{2(N-1)K_{1}+1},~{}~{}~{}\mathcal{B}_{n^{*}}^{(N-1)}=\frac{(N-1)K_{2}}{\mathcal{A}_{n^{*}}^{(N-1)}},~{}~{}~{}$
$\displaystyle\mathcal{C}_{n^{*}}^{(N-1)}=2(N-1)K_{3},$
$\displaystyle\mathcal{D}_{n^{*}}^{(N-1)}=\exp\left(-\frac{1}{2}\left(\mathcal{C}_{n^{*}}^{(N-1)}-(\mathcal{B}_{n^{*}}^{(N-1)})^{2}\right)\right).$
(55)

The solution of (-A) is obtained with the help of (71) and (-C) in Appendix -C
as

$\displaystyle
I_{j,k,n^{*}}^{(+)}=\frac{N\mathcal{D}_{n^{*}}^{(N-1)}}{\ln{2}}\Bigg{[}\frac{\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}\right)Q\left(\mathcal{B}_{n^{*}}^{(N-1)}\right)}{\mathcal{A}_{n^{*}}^{(N-1)}}\Bigg{.}$
$\displaystyle\Bigg{.}-\frac{s_{b}\exp{\left(-\frac{(\mathcal{B}_{n^{*}}^{(N-1)})^{2}}{2}\right)}}{(\mathcal{A}_{n^{*}}^{(N-1)})^{2}\sqrt{2\pi}}+\frac{s_{b}\mathcal{B}_{n^{*}}^{(N-1)}Q\left(\mathcal{B}_{n^{*}}^{(N-1)}\right)}{(\mathcal{A}_{n^{*}}^{(N-1)})^{2}}\Bigg{]}.$
(56)

Similarly, using approximation of the $Q$-function in (52), substituting for
$f_{Q}(t)$, and performing some further manipulations, we can write
$I_{j,k,n^{*}}^{(-)}$ as

$\displaystyle
I_{j,k,n^{*}}^{(-)}=\int_{0}^{\infty}\frac{N}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}+s_{b}t\right)$
$\displaystyle\times\left(1-\exp\left(-\left(K_{1}t^{2}+K_{2}t+K_{3}\right)\right)\right)^{(N-1)}f_{Q}(t)\mbox{d}t$
$\displaystyle=\int_{0}^{\infty}\frac{N}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}+s_{b}t\right)\sum_{n=0}^{N-1}\binom{N-1}{n}(-1)^{n}$
$\displaystyle\times\frac{\mathcal{D}_{n^{*}}^{(n)}}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\mathcal{A}_{n^{*}}^{(n)}t-\bar{\mathcal{B}}_{n^{*}}^{(n)}\right)^{2}\right)\mbox{d}t,$
(57)

where

$\displaystyle\mathcal{A}_{n^{*}}^{(n)}$
$\displaystyle=\sqrt{2nK_{1}+1},~{}~{}~{}\bar{\mathcal{B}}_{n^{*}}^{(n)}=-\frac{nK_{2}}{\mathcal{A}_{n^{*}}^{(n)}},~{}~{}~{}\mathcal{C}_{n^{*}}^{(n)}=2nK_{3},~{}~{}~{}$
$\displaystyle\mathcal{D}_{n^{*}}^{(n)}$
$\displaystyle=\exp\left(-\frac{1}{2}\left(\mathcal{C}_{n^{*}}^{(n)}-(\bar{\mathcal{B}}_{n^{*}}^{(n)})^{2}\right)\right).$
(58)

The solution of (-A) is obtained with the help of (73) and (-C) in Appendix -C
as

$\displaystyle
I_{j,k,n^{*}}^{(-)}=\frac{N}{\ln{2}}\sum_{n=0}^{N-1}\binom{N-1}{n}(-1)^{n}\mathcal{D}_{n^{*}}^{(n)}$
$\displaystyle\times\Bigg{[}\left(\ln\left(\tilde{\alpha}_{j,{b}}\right)+m_{b}\right)\frac{1-Q\left(\bar{\mathcal{B}}_{n^{*}}^{(n)}\right)}{\mathcal{A}_{n^{*}}^{(n)}}\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{s_{b}\exp{\left(-\frac{(\bar{\mathcal{B}}_{n^{*}}^{(n)})^{2}}{2}\right)}}{(\mathcal{A}_{n^{*}}^{(n)})^{2}\sqrt{2\pi}}+\frac{s_{b}\bar{\mathcal{B}}_{n^{*}}^{(n)}\left(1-Q\left(\bar{\mathcal{B}}_{n^{*}}^{(n)}\right)\right)}{(\mathcal{A}_{n^{*}}^{(n)})^{2}}\Bigg{]}.$
(59)

Finally, $I_{j,k,n^{*}}^{(+)}$ and $I_{j,k,n^{*}}^{(-)}$ from (-A) and (-A)
are expressed in (1) and (1), respectively, in Proposition 1.

###  -B Solution of $I_{k,j,e}$ given by (42).

As in Appendix -A, we will find the approximate closed-form solution of
$I_{k,j,e}$ in (42) with the help of the approximate $Q$-function. First, we
simplify the equation as

$\displaystyle I_{k,j,e}$ $\displaystyle=I_{k,j,e}^{(0)}+I^{(1)}_{{k,j,e}},$
(60)

where

$\displaystyle I_{k,j,e}^{(0)}$
$\displaystyle=\frac{1}{\ln{2}}\int_{-\infty}^{\infty}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}+s_{e}t\right)f_{Q}(t)\mbox{d}t$
$\displaystyle=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}\right),$
(61)

and

$\displaystyle I^{(1)}_{{k,j,e}}$
$\displaystyle=\frac{1}{\ln{2}}\int_{-\infty}^{\infty}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}+s_{e}t\right)$
$\displaystyle\times
Q^{N}\left(-\left(\phi_{e}t+\lambda_{k,j,e}\right)\right)f_{Q}(t)\mbox{d}t.$
(62)

After the change of variable $u=\phi_{e}t+\lambda_{k,j,e}$, and upon some
further manipulations, we obtain

$\displaystyle I^{(1)}_{{k,j,e}}$
$\displaystyle=\frac{1}{\ln{2}}\int_{-\infty}^{\infty}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}+\frac{s_{e}u}{\phi_{e}}\right)$
$\displaystyle\times\frac{1}{\phi_{e}}Q^{N}\left(-u\right)\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\left(u-\lambda_{k,j,e}\right)^{2}}{2\phi_{e}^{2}}\right)\mbox{d}u.$
(63)

Next, following Appendix -A, to apply the approximate $Q$-function in (-B) we
divide the integration region appropriately as

$\displaystyle I^{(1)}_{{k,j,e}}$
$\displaystyle=I^{(+)}_{{k,j,e}}+I^{(-)}_{{k,j,e}},$ (64)

where

$\displaystyle I^{(+)}_{{k,j,e}}$
$\displaystyle=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}+\frac{s_{e}u}{\phi_{e}}\right)$
$\displaystyle\times\frac{1}{\phi_{e}}\int_{-\infty}^{0}\frac{Q^{N}\left(-u\right)}{\sqrt{2\pi}}\exp\left(-\frac{\left(u-\lambda_{k,j,e}\right)^{2}}{2\phi_{e}^{2}}\right)\mbox{d}u,$
(65) $\displaystyle I^{(-)}_{{k,j,e}}$
$\displaystyle=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}+\frac{s_{e}u}{\phi_{e}}\right)$
$\displaystyle\times\frac{1}{\phi_{e}}\int_{0}^{\infty}\frac{Q^{N}\left(-u\right)}{\sqrt{2\pi}}\exp\left(-\frac{\left(u-\lambda_{k,j,e}\right)^{2}}{2\phi_{e}^{2}}\right)\mbox{d}u.$
(66)

After using the approximate $Q$-function expression in (-B) and performing
further manipulations, we make use of (71) and (-C) in Appendix -C to find the
approximate closed-form solution of $I^{(+)}_{{k,j,e}}$ as

$\displaystyle
I^{(+)}_{{k,j,e}}=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}+\frac{s_{e}u}{\phi_{e}}\right)$
$\displaystyle\times\frac{1}{\phi_{e}}\int_{-\infty}^{0}\frac{\mathcal{D}_{k,j,e}^{(N)}}{\sqrt{2\pi}}\exp{\left(-\frac{1}{2}\left(\mathcal{A}_{e}^{(N)}u-\mathcal{B}_{k,j,e}^{(N)}\right)^{2}\right)}\mbox{d}u$
$\displaystyle=\frac{\mathcal{D}_{k,j,e}^{(N)}}{\phi_{e}\ln{2}}\Bigg{[}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}\right)\frac{Q\left(\mathcal{B}_{k,j,e}^{(N)}\right)}{\mathcal{A}_{e}^{(N)}}-\Bigg{.}$
$\displaystyle\Bigg{.}\frac{s_{e}}{\phi_{e}(\mathcal{A}_{e}^{(N)})^{2}\sqrt{2\pi}}\exp{\left(-\frac{(\mathcal{B}_{k,j,e}^{(N)})^{2}}{2}\right)}+\frac{s_{e}\mathcal{B}_{k,j,e}^{(N)}Q\left(\mathcal{B}_{k,j,e}^{(N)}\right)}{\phi_{e}(\mathcal{A}_{e}^{(N)})^{2}}\Bigg{]},$
(67)

where

$\displaystyle\mathcal{A}_{e}^{(N)}$
$\displaystyle=\sqrt{2NK_{1}+\frac{1}{\phi_{e}^{2}}},~{}~{}~{}\mathcal{B}_{k,j,e}^{(N)}=\frac{NK_{2}+\frac{\lambda_{k,j,e}}{\phi_{e}^{2}}}{\mathcal{A}_{e}^{(N)}},~{}~{}~{}$
$\displaystyle\mathcal{C}_{k,j,e}^{(N)}$
$\displaystyle=2NK_{3}+\frac{\lambda_{k,j,e}^{2}}{\phi_{e}^{2}},$
$\displaystyle\mathcal{D}_{k,j,e}^{(N)}$
$\displaystyle=\exp{\left(-\frac{1}{2}\left(\mathcal{C}_{k,j,e}^{(N)}-\left(\mathcal{B}_{k,j,e}^{(N)}\right)^{2}\right)\right)}.$
(68)

Similarly, after replacing the approximate $Q$-function in (-B) and using (73)
and (-C) in Appendix -C, we find the approximate closed-form solution of
$I^{(-)}_{{k,j,e}}$ as

$\displaystyle
I^{(-)}_{{k,j,e}}=\frac{1}{\ln{2}}\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}+\frac{s_{e}u}{\phi_{e}}\right)$
$\displaystyle\times\frac{1}{\phi_{e}}\sum_{n=0}^{N}\binom{N}{n}(-1)^{n}\int_{0}^{\infty}\frac{\bar{\mathcal{D}}_{k,j,e}^{(n)}}{\sqrt{2\pi}}$
$\displaystyle\times\exp{\left(-\frac{1}{2}\left(\mathcal{A}_{e}^{(n)}u-\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)^{2}\right)}\mbox{d}u$
$\displaystyle=\frac{1}{\phi_{e}\ln{2}}\sum_{n=0}^{N}\binom{N}{n}(-1)^{n}\bar{\mathcal{D}}_{k,j,e}^{(n)}$
$\displaystyle\times\left[\left(\ln\left(\tilde{\alpha}_{k,e}\right)+m_{e}-\frac{s_{e}\lambda_{k,j,e}}{\phi_{e}}\right)\frac{1-Q\left(\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)}{\mathcal{A}_{e}^{(n)}}\right.$
$\displaystyle\left.+\frac{s_{e}}{\phi_{e}}\frac{1}{(\mathcal{A}_{e}^{(n)})^{2}\sqrt{2\pi}}\exp{\left(-\frac{(\bar{\mathcal{B}}_{k,j,e}^{(n)})^{2}}{2}\right)}\right.$
$\displaystyle\left.+\frac{s_{e}}{\phi_{e}}\frac{\bar{\mathcal{B}}_{k,j,e}^{(n)}\left(1-Q\left(\bar{\mathcal{B}}_{k,j,e}^{(n)}\right)\right)}{(\mathcal{A}_{e}^{(n)})^{2}}\right],$
(69)

where

$\displaystyle\mathcal{A}_{e}^{(n)}$
$\displaystyle=\sqrt{2nK_{1}+\frac{1}{\phi_{e}^{2}}},~{}~{}~{}\bar{\mathcal{B}}_{k,j,e}^{(n)}=\frac{-nK_{2}+\frac{\lambda_{k,j,e}}{\phi_{e}^{2}}}{\mathcal{A}_{e}^{(n)}},$
$\displaystyle\mathcal{C}_{k,j,e}^{(n)}$
$\displaystyle=2nK_{3}+\frac{\lambda_{k,j,e}^{2}}{\phi_{e}^{2}},$
$\displaystyle\bar{\mathcal{D}}_{k,j,e}^{(n)}$
$\displaystyle=\exp{\left(-\frac{1}{2}\left(\mathcal{C}_{k,j,e}^{(n)}-(\bar{\mathcal{B}}_{k,j,e}^{(n)})^{2}\right)\right)}.$
(70)

Finally, $I^{(0)}_{{k,j,e}}$, $I^{(+)}_{{k,j,e}}$, and $I^{(-)}_{{k,j,e}}$
from (-B), (-B), and (-B) are expressed in (27), (1), and (1), respectively,
in Proposition 1.

###  -C Solutions of the basic integrals.

In this section, we shall derive (in closed-form) the four basic integrals
which are encountered multiple times to achieve the approximate asymptotic ASC
and approximate POI. Here we assume that $\mathcal{A}$ and $\mathcal{B}$ are
arbitrary constants and $f_{Q}(t)$ is the standard normal PDF. In this case,
the solution of the following integrals for any $\mathcal{A}$ and
$\mathcal{B}$ are given by

$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{0}\exp\left(-\frac{1}{2}\left(\mathcal{A}t-\mathcal{B}\right)^{2}\right)\mbox{d}t=\frac{Q\left(\mathcal{B}\right)}{\mathcal{A}},$
(71)
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{0}t\exp\left(-\frac{1}{2}\left(\mathcal{A}t-\mathcal{B}\right)^{2}\right)\mbox{d}t$
$\displaystyle=-\frac{1}{\mathcal{A}^{2}\sqrt{2\pi}}\exp{\left(-\frac{\mathcal{B}^{2}}{2}\right)}+\frac{\mathcal{B}Q\left(\mathcal{B}\right)}{\mathcal{A}^{2}},$
(72)
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\exp\left(-\frac{1}{2}\left(\mathcal{A}t-\mathcal{B}\right)^{2}\right)\mbox{d}t=\frac{1-Q\left(\mathcal{B}\right)}{\mathcal{A}},$
(73)
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}t\exp\left(-\frac{1}{2}\left(\mathcal{A}t-\mathcal{B}\right)^{2}\right)\mbox{d}t$
$\displaystyle=\frac{1}{\mathcal{A}^{2}\sqrt{2\pi}}\exp{\left(-\frac{\mathcal{B}^{2}}{2}\right)}+\frac{\mathcal{B}\left(1-Q\left(\mathcal{B}\right)\right)}{\mathcal{A}^{2}}.$
(74)

The proof easily follows by converting the exponential form in all the
integrals into standard normal form by changing the integration variable and
the corresponding limits.

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