Tradeoff between Delay and Physical Layer
Security in Wireless Networks

arXiv:1802.00194v1 [] 1 Feb 2018

Yi Zhong, Xiaohu Ge, Tao Han, Qiang Li, Jing Zhang

Abstract
Exchange of crucial and confidential information leads to the unprecedented attention on the security
problem in wireless networks. Though the security has been studied in a number of works, the joint
optimization of the physical layer security and the end-to-end delay management, which requires a
meticulous cross-layer design, has seldom been evaluated. In this work, by combining the tools from
stochastic geometry and queueing theory, we analyze the tradeoff between the delay and the security
performance in large wireless networks. We further propose a simple transmission mechanism which
splits a message into two packets and evaluate its effect on the mean delay and the secrecy outage
probability. Our numerical results reveal that the security performance is better for larger path loss
exponent when the density of legitimate nodes is large, and it is reverse when the density is small.
Moreover, it is observed that by introducing the simple mechanism of message split, the security
performance is greatly improved in the backlogged scenario and slightly improved in the dynamic
scenario when the density of legitimate transmitters is large. In summary, this work provides an
understanding and a rule-of-thumb for the practical design of wireless networks where both the delay
and the security are key concerns.

Index Terms
Delay, physical layer security, Poisson point process, queueing theory, stochastic geometry

The authors are with School of Electronic Information and Communications, Huazhong University of Science and Technology,
Wuhan, P. R. China. (e-mail: yzhong@hust.edu.cn). This research was supported by the National Natural Science Foundation
of China (NSFC) grant No. 61701183.
Part of this work was submitted to the 2018 IEEE International Conference on Communications (ICC).

I. I NTRODUCTION
A. Motivations
The wireless access becomes ubiquitous for the civilian and military applications [1], including
the exchange of crucial and confidential information like the banking related data, leading
to unprecedented attention to the security problem for the exchange of information between
nodes in the wireless networks. Comparing with the traditional security methods based on
the cryptographic techniques at the upper layers of the communication protocols in a wireless
network [2], the physical layer security is an information-theoretic based approach that guarantees
the security of wireless links by exploiting the randomness of wireless channels, which makes
it more difficult for the attackers to decipher the delivered messages. From the point of view of
the information theory, the security problem can be modeled by the wiretap channel proposed by
Wyner in his pioneering work [3], where a source sends the information to a legitimate receiver
at the presence of an eavesdropper.
On the other hand, due to the emergence of new types of latency-critical applications such
as the command-and-control of drones, the advanced manufacturing, the vehicular networks [4]
and the tactile Internet, the delay performance becomes an indispensable metric that should be
considered in the wireless transmissions [5]. Though the delay in wireless networks has been
studied by a number of works [6], [7], the joint optimization of the physical layer security
and the end-to-end delay management has seldom been studied in the literature, which requires
a meticulous cross-layer design. The relationship between the physical layer security and the
end-to-end delay is exceedingly complex, which is affected by a number of factors, such as
the random time-varying channel, the irregular deployment of nodes, the wireless transmission
mechanisms and so on.

[

Fig. 1. Cipher system with random arrival messages: Alice attempts to deliver messages to Bob, while keeping it secret from
Eve who intercepts the transmissions.

An intuitive tradeoff between the delay and the physical layer security could be observed
from Figure 1, where a transmitter named Alice attempts to deliver messages to a legitimate
receiver named Bob, while keeping it secret from an eavesdropper named Eve who intercepts
the transmissions. Alice maintains a buffer of infinite capacity to store the incoming packets,
and the time is divided into discrete slots with equal duration. In each time slot, Alice attempts
to transmit its head-of-line packet with a certain probability p if its buffer is not empty. A
failure occurs if Bob cannot decode the packets sent by Alice. When a failure occurs, Alice
retransmits the packet in the next time slot with the same access probability p. The security
constraint is that the probability of a packet delivered by Alice failing to achieve perfect secrecy
from the detrimental eavesdropper Eve is less than certain threshold. Increasing the transmit
power of Alice reduces the delay but makes it more easier for Eve to decode the message,
thus decreasing the security as well. Moreover, in wireless networks with multiple legitimate
links and multiple eavesdroppers, the tradeoff between the delay and the physical layer security
is much more perplexed. For example, increasing the transmit probability p may decrease the
delay for legitimate links and increase the chance for Eve to intercept the messages since the
probability to schedule the packets is increased, but it also increases the interference in wireless
networks which makes it harder for Eve to intercept the messages.
The motivation of this work is to evaluate the tradeoff between the delay and the physical
layer security in wireless networks. Particularly, we model the spatial distribution of legitimate
nodes and eavesdroppers in wireless networks by using the tools from the point process theory.
Through combining the point process theory and the queueing theory, we analyze the tradeoff
between the delay and the physical layer security theoretically. Our work will provide a useful
guideline for the design of secure wireless transmissions while meeting the end-to-end delay
requirements imposed by the mobile applications in wireless networks.
B. Related Works
The work in [8] extends the Wyner’s wiretap channel model [3] to the case of broadcast
channels with confidential messages, which shows that perfect secrecy could be achieved if the
legitimate link has a better channel than the link of eavesdropper. The work in [9] demonstrates
that the secrecy capacity, i.e., the maximum achievable rate of the legitimate link while guaranteeing that the eavesdropper cannot decode the message, is the difference of the maximum
achievable rates between the legitimate link and the eavesdropping link.

Most of the studies on physical layer security in the literature focused on the scenes of a small
number of nodes. For instance, the works in [10], [11] studied the security problem in the case of
a small number of nodes with multiple antennas, and the works in [12]–[14] studied the security
problem in the cooperative transmission and in the multiple access channels. The works in [15],
[16] studied the secure and reliable transmission for the multiple-input multiple-output systems.
Recently, due to the applications of stochastic geometry tools in the wireless networks [17], a
few studies have been carried out to analyze the security problem in large wireless networks.
For example, the works in [18]–[21] studied the connectivity problem in the wireless network
where the legitimate nodes and the eavesdroppers are randomly distributed in space, and the
works in [22]–[24] evaluated the coverage and the capacity with the limitation of security. The
work in [25] studied the information-theoretic secrecy performance in large cellular networks
by modeling the locations of both base stations and mobile users as two independent Poisson
point processes (PPP) [26]–[30]. The work in [31] studies the throughput of large decentralized
wireless networks with physical layer security constraints, and the authors also utilize the tools
from stochastic geometry to model the wireless networks.
Though the relationship between the security and the capacity of large wireless networks
has been well explored in the literature, few works have been done to evaluate the relationship
between the security and the delay in large wireless networks. Part of the reason is that the
analysis of delay in large wireless network is far more difficult than the analysis of coverage
and capacity since the delay is a complicated function of all links and is affected by an unusually
large number of variables such as the network load, the medium access protocol, the path loss,
and so on. In [32], the joint optimization of the physical layer security and the end-to-end delay is
studied in the wireless body area networks. In [33], the resource allocation problem is studied in
the cognitive radio network with delay and security constraints. However, these works considers
few nodes in the network and have not obtained the analytical results for the large wireless
networks. Recently, several tentative works appear to analyze the delay in large scale wireless
networks, such as [34]. The basic idea is to combine the stochastic geometry and the queueing
theory, which results in the challenge of interacting queues problem [35]–[37], i.e., the service
rates of the queues in the networks rely on the status of all queues in the networks. Our previous
works have derived the conditions for the stability of queues [38] and the bounds for the delay
[6] in large wireless networks.

C. Contributions
In this work, we evaluate the tradeoff between the delay and the security performance in
large wireless networks for both the backlogged scenario and the dynamic scenario. To our best
knowledge, this is the first work to explore the joint optimization of the delay and the security
problem in large wireless networks. By combining the tools from stochastic geometry and queueing theory, we introduce some new analytical approaches and derive the close-formed results for
the mean delay and the secrecy outage probability (i.e., the probability of a transmission failing
to achieve perfect secrecy). We also propose a simple transmission mechanism which splits a
message into two packets and analyze its effect on the delay and security performance.
Our results reveal that the security performance of a large wireless network is better for larger
path loss exponent when the density of legitimate nodes is large, and it is reverse when the
density is small. We also find that under the condition that a certain confidential rate (i.e., the
achievable rate of confidential messages with the constraint of perfect secrecy) is guaranteed,
the delay performance is better in high signal-to-interference ratio (SIR) regime than that in low
SIR regime. By introducing the simple mechanism of message split, it is shown that the delay
performance in the backlogged scenario is improved when the density of legitimate transmitters
is large. Meanwhile, introducing message split greatly improves the security performance in the
backlogged scenario, while slightly improves the security performance in the dynamic scenario
for small density of legitimate nodes.
To sum up, our contributions in this work could be summarized as follows.
•

The joint optimization of the delay and the security problem in large wireless networks,
which is seldom explored in the literature, is evaluated through numerical analysis.

•

New analytical approaches based on the combination of stochastic geometry and queueing
theory are proposed to derive the mean delay and the secrecy outage probability in the
scenario of random traffic.

•

Numerical evaluation based on the theoretical analysis is conducted to gain insight for the
practical design of wireless networks where both the delay and the security are key concerns.

The remaining part of the paper is organized as follows. Section II describes the propagation
model, the traffic model, and the secrecy transmission rate. Section III analyzes the delay in two
different scenarios respectively, i.e., the backlogged scenario and the dynamic scenario. Section
IV proposes and analyzes a simple transmission mechanism, in which a confidential message

Fig. 2. Topology of the network, where the squares denotes the legitimate transmitters, the circles denote the legitimate receivers
and the triangles denote the eavesdroppers.

is divided into two packets to be delivered independently. Section V numerically analyzes the
tradeoff between the security and the delay performance. Finally, Section VI concludes the paper.
II. S YSTEM M ODEL
We consider an ad hoc network consisting of both legitimate nodes and eavesdroppers in
a large two-dimensional space. The locations of the legitimate transmitters are modeled as a
homogeneous PPP Φl with intensity λl , and the locations of the eavesdroppers are modeled
as another independent homogeneous PPP Φe with intensity λe . Each legitimate transmitter
is paired with a legitimate receiver at a fixed distance r0 and a random orientation. Without
loss of generality, we consider a typical legitimate transmitter located at the origin, and the
corresponding legitimate receiver is located at x0 with |x0 | = r0 (see Fig. 2). We consider the
discrete time system where the time is divided into discrete slots with equal duration, and each
transmission attempt occupies exactly one time slot. In practice, the locations of the transmitters
and the receivers are generated once at the beginning and then kept unchanged during all time
slots. Therefore, the topology of the network could be considered as static.

A. Propagation Model
The propagation model in this work consists of two parts, i.e., the path loss and the fading.
We assume that the standard power path loss model is used, and the average received power at
a receiver y from a transmitter x is denoted by
Pr = Pt |x − y|−α,

(1)

where Pt is the transmit power, Pr is the received power, and α is the path loss exponent
(α ≥ 2). The fading of each link is assumed to be Rayleigh block fading with zero mean and
unit variance, and it is independent between different links. The power fading coefficients in
different time slots are independent identically distributed (i.i.d.) and are constant during one
time slot. Note that the channel model in our work is not as comprehensive as that in [39] which
considers the path loss, the Rayleigh fading, the azimuth angle of departure and the log-normal
shadowing. However, the channel model in our work is a common assumption that has been
widely accepted in the literatures, such as [40]. Meanwhile, using the model of the path loss and
the Rayleigh block fading may greatly reduce the complexity of the derivations and make the
analysis tractable, and close-formed results could be obtain to gain intuitive insight. Due to the
fact that most of the existing wireless networks are interference-limited, i.e., the noise is much
smaller compared with the interference, we ignored the thermal noise in our analysis. Without
loss of generality, the transmit power of all legitimate transmitters is normalized.
When all transmitters are active, the SIR at the typical legitimate receiver x0 is
hx0 r0−α
,
(2)
SIRx0 = P
−α
y∈Φl hy |y − x0 |
where hx0 is the fading coefficient between the typical legitimate transmitter at the origin and
the corresponding receiver at x0 , {hy } is the fading coefficients between the transmitter y ∈ Φl
and the typical legitimate receiver x0 .
When all transmitters are active, the SIR at an eavesdropper xe ∈ Φe is
he |xe |−α
,
(3)
SIRxe = P
−α
y∈Φl hy |y − xe |
where he is the fading coefficient between the legitimate transmitter at the origin and the
eavesdropper at xe .
B. Traffic Model
For completeness consideration, we consider two scenarios for the traffic arrival process, i.e.,
the backlogged scenario and the dynamic scenario (see Figure 3).

/HJLWLPDWH7[

/HJLWLPDWH5[

(DYHVGURSSHU

Fig. 3. Two scenarios for the traffic arrival process, i.e., the backlogged scenario and the dynamic scenario.

1) Backlogged scenario: In the backlogged scenario, we assume that the network is backlogged, i.e., the transmitters always have packets to transmit when they are scheduled for
transmission. In this way, a meaningful and practically relevant metric in the backlogged scenario
is the duration to successfully deliver one packet, which is closely related to the number of
retransmissions of a packet. This type of delay, which ignores the queueing delay (i.e., the
waiting time of a packet until it is served), is also defined as local delay [40]–[42]. Without loss
of generality, we call it the delay for simplicity in the backlogged scenario.
2) Dynamic scenario: In the dynamic scenario, we assume that the packets arrive at the
legitimate transmitters as a stochastic process. The packets arrival process at each legitimate
transmitter xi ∈ Φl is assumed to be an independent Bernoulli process of arrival rate ξ per time
slot. By the definition of the Bernoulli process, ξ is the probability of an arrival of a packet at
user xi in any time slot.
We assume that each legitimate transmitter maintains a buffer with infinite capacity to store
the incoming packets. In each time slot, each legitimate transmitter attempts to transmit its
head-of-line packet with probability p if the queue at the transmitter is not empty. As for the
retransmission mechanism, we assume that if a packet fails for transmission in the scheduled
time slot, the packet will be added into the head of the queue of the corresponding legitimate
transmitter and wait to be rescheduled again.

&RQQHFWLRQ
)DLOXUH

6HFUHF\
)DLOXUH

[

W

[
H

Fig. 4. Two failure events for a transmission attempt, i.e., the connection failure and the secrecy failure.

C. Secrecy Transmission Rate
According to Wyners secure encoding scheme [3], [43], a transmitter may choose two rate
thresholds, i.e., the rate threshold of codewords Rt and the rate threshold of confidential messages
Rs . The difference between the two rate thresholds, denoted by Re = Rt − Rs , is the rate cost
of securing the messages against eavesdropper. If the rate of a legitimate link is less than the
threshold Rt , the legitimate receiver cannot decode the message with an arbitrarily small error.
Meanwhile, if the rate of every eavesdropping link is less than the threshold Re , perfect secrecy
could be achieved to guarantee the confidential rate Rs .
With the above two rate thresholds, we define two failure events for a transmission attempt,
i.e., the connection failure and the secrecy failure (see Figure 4).
1) Connection Failure: The connection failure means that a transmission attempt of a packet
sent by a legitimate transmitter cannot be successfully decoded by the corresponding receiver.
The probability of connection failure for a transmission attempt is
Pcf = P{log2 (1 + SIRx0 ) < Rt } = P{SIRx0 < 2Rt − 1}.

(4)

Let θt = 2Rt − 1 be the SIR threshold for the connection failure event. Due to the retransmission
mechanism, if a transmission attempt of a packet fails, the packet will be added into the head
of the queue and wait to be rescheduled for transmission.

2) Secrecy Failure: The secrecy failure means that a transmission attempt of a packet sent
by a legitimate transmitter is not perfectly secure against the eavesdroppers. The probability of
secrecy failure for a transmission attempt is
Y
Y
P{SIRxe < 2Re − 1}.
P{log2 (1 + SIRxe ) < Re } = 1 −
Psf = 1 −

(5)

xe ∈Φe

xe ∈Φe

Let θe = 2Re − 1 be the SIR threshold for the secrecy failure event. Since a packet may be
retransmitted for several times due to the connection failure, the delivery of a packet is perfectly
secure only when all the transmission attempts are successful.
Therefore, the delay in the backlogged scenario is the number of time slots required by a
legitimate transmitter to successfully deliver a packet to the corresponding legitimate receiver.
On the other hand, the delay in the dynamic scenario is the number of time slots between the
arrival of a packet and the successful delivery of that packet. Meanwhile, the secrecy performance
can be characterized by the secrecy outage probability, which in this paper is defined as the
probability that at least one transmission attempt of a packet is not perfectly secure. Let Nr
be the total number of scheduled transmissions until a packet is successfully decoded at the
legitimate receiver, and Nr is a geometric distributed random variable with success probability
1 − Pcf , i.e., Nr ∼ Geo(1 − Pcf ). Then, the secrecy outage probability with given Nr is
Pso = 1 − (1 − Psf )Nr .

(6)

The relationship between the mean delay and the secrecy outage probability is not straightforward. Intuitively, the equation (6) reveals that increasing the total number of transmissions Nr
raises the mean delay as well as the secrecy outage probability. However, since the probability of
secrecy failure Psf may decrease, the secrecy outage probability may also decrease. Therefore,
the theoretical analysis of the mean delay and the secrecy outage probability is necessary. In
the following discussions, we derive the expressions to reveal the exact relationship between the
mean delay and the secrecy outage probability.
The system model described above is a general mathematical framework that could be applied
to analyze the security and delay performance of general large wireless networks, corresponding
to a number of application scenarios. For example, the proposed mathematical framework could
be applied in the military communications, where the soldiers from one army want to eavesdrop
the confidential messages from the hostile army. In this application scenario, the soldiers from
the eavesdropping army could be considered as eavesdroppers and the soldiers from the other

army could be considered as the legitimate nodes. Another application scenario could be the
prevention of theft in the wireless banking transactions, where the thieves could be considered
as the eavesdroppers and the banking users could be considered as the legitimate nodes. In
these application scenarios, all eavesdroppers try to eavesdrop the confidential messages from
all legitimate users. Thus, the legitimate users should prevent their confidential messages from
being eavesdropped by any one of the eavesdroppers.
III. D ELAY

AND

S ECRECY O UTAGE P ROBABILITY

The delay in the backlogged scenario is exactly the transmission delay caused by retransmission (i.e., the time to successfully transmit a packet that is served), while the delay in the
dynamic scenario consists of two parts, i.e., the queueing delay and the transmission delay. We
analyze the delay in the two different scenarios respectively in the following discussions.
A. Backlogged Scenario
In the backlogged scenario, whenever a transmitter is scheduled to access the channel it
always has packet to transmit. The backlogged assumption is widely used in the analysis of
wireless communication systems. With the backlogged assumption, the obtained interference in
the network is an upper bound for the interference without backlogged assumption. Therefore, the
performance derived with the backlogged assumption will be a useful bound for the performance
of the practical system. With the backlogged assumption, considering the queueing delay is
meaningless since the queueing delay is always infinite. However, a meaningful and practically
relevant metric in the backlogged scenario is the transmission delay. The transmission delay in the
backlogged scenario is exactly the local delay, which has been well explored in the literature,
such as [40] and [42]. Another benefit of introducing the backlogged assumption is that the
theoretical analysis of delay becomes tractable since the interacting between different queues
is avoided. Concise and close-formed results could be obtained by the backlogged assumption,
which further leads to more insights. From [40], we get the following lemma, which gives the
analytical expression for the mean delay of the typical legitimate link.
Theorem 1. The mean delay of the typical legitimate link in the backlogged scenario is
D=


1
exp λl πr02 θtδ C(δ)p(1 − p)δ−1 ,
p

where θt = 2Rt − 1, δ = 2/α and C(δ) = Γ(1 + δ)Γ(1 − δ) = 1/sinc(δ).

(7)

Proof. The proof is ignored in this paper for conciseness since it is similar to the proof in [40,
Lemma 2] and [42, Theorem 3].
The result in Theorem 1 is closed-form and can be easily evaluated, and it shows directly how
the mean delay varies with the transmit probability p. The equation (7) indicates that the mean
delay goes to infinite both when p decreases to zero and when p increases to one since δ < 1
holds. When the transmit probability p decreases to zero, a packet is scheduled for transmission
for very small probability, and when the transmit probability p increases to one, the interference
and the interference correlation may greatly deteriorate the delay performance.
In the following, we derive the secrecy outage probability for the backlogged scenario. Note
that Nr is the total number of transmissions until a packet is successfully decoded at the legitimate
receiver. Then, Theorem 1 gives the analytical expression for the mean times of transmissions
E{Nr } = D in the backlogged scenario. For the typical legitimate link, when the realization of
Φl is given, the connection failure probability is determined by the fading and random access
mechanism, which are independent among different time slots. Thus, the connection failure
Φl
l
probability with given Φl , denoted by PΦ
cf , is the same for all time slots. Let Nr be the total

number of transmissions until a packet is successfully decoded at the legitimate receiver when
l
Φl is given. Then, NrΦl is a geometric distributed random variable with success probability PΦ
cf ,

Φl
l
i.e., NrΦl ∼ Geo(1−PΦ
cf ). In order to guarantee the perfect secrecy, each of the Nr transmission
l ,Φe
attempts should be perfect secure; otherwise, the message will be eavesdropped. Letting PΦ
sf

be the secrecy failure probability with given Φl and Φe , the secrecy outage probability is
Φl 

l ,Φe Nr
)
Pso = 1 − EΦl ,Φe (1 − PΦ
sf
∞
hX
i
Φl
l ,Φe n
)
P{N
=
n}
(1 − PΦ
= 1 − EΦl ,Φe
r
sf

n=1

= 1 − EΦl ,Φe

∞
hX

= 1 − EΦl ,Φe

∞ 
h 1 − P Φl X

(a)

Φl n−1
l ,Φe n
l
) (1 − PΦ
(1 − PΦ
cf )(Pcf )
sf

n=1

= EΦl ,Φe



1−

cf

l
PΦ
cf

i

 Φl n i
l ,Φe
Pcf
1 − PΦ
sf

n=1

Φl ,Φe
Psf
,
Φl ,Φe Φl
l
Pcf
PΦ
cf + Psf

(8)

where (a) holds since NrΦl is a geometric distributed random variable with success probability
1 − Pcf . The last equation holds due to the sum formula of infinite geometric series whose

successive terms have a common ratio. With the above discussion, we could further derive the
secrecy outage probability in the following theorem.
Theorem 2. The secrecy outage probability of the typical link in the backlogged scenario can
be approximated by

exp pλl θλδeC(δ) − 1
e


.
Pso ≃
exp pλl θλδeC(δ) + exp − pλl πr02 θtδ C(δ) − 1

(9)

e

Proof. See Appendix A.

The secrecy outage probability given by Theorem 2 is closed-form, and the relationship
between the secrecy outage probability and the system parameters can be observed directly from
the equation (9). The equation (9) also verifies the intuition that the secrecy outage probability
is positively correlated with the intensity of the eavesdroppers λe and is negative correlated with
the intensity of the legitimate transmitters λl . Specially, we get Pso → 0 as λe → 0, and Pso → 1
as λe → +∞. The reason is that increasing the intensity of eavesdroppers may increase the
probability that an eavesdropper appears nearby the legitimate links. Meanwhile, increasing the
intensity of legitimate transmitters may increase the interference in the wireless network, thus
preventing the message from being eavesdropped. Note that when either p → 0 or λl → 0 holds,
the secrecy outage probability goes to one, i.e., Pso → 1, indicating that the secrecy outage
event happens with high probability when there are very few active legitimate transmitters. This
observation illustrates that increasing the number of concurrent transmissions in the wireless
networks is helpful to improve the secrecy performance, i.e., increasing the interference in the
wireless networks is helpful from the aspect of the security performance.
B. Dynamic Scenario
In the dynamic scenario, the delay consists of the queueing delay and the transmission delay.
The analysis of the delay in the dynamic scenario requires the combination of the stochastic
geometry and the queueing theory, which induces the interacting queues problem. In order to
bypass these difficulties, we propose an approach to approximate the delay in the dynamic
scenario. We first simplify the interaction between the queues and assume that all interfering
transmitters are active with the same probability q. Then, we derive the connection failure
probability Pcf for the typical legitimate link in each time slot. The service rate of the queue
at the typical legitimate transmitter will be p(1 − Pcf ), which means that the probability for a

packet at the typical user being scheduled and successfully transmitted in a time slot is p(1−Pcf ).
Then, ρ =

ξ
p(1−Pcf )

represents the average proportion of time occupied by a transmitter. When the

queue is empty, the transmitter is inactive, and when the transmitter is occupied, it is active with
probability p due to the random access. Therefore, the active probability of the typical legitimate
ξ
, 1}. Note that Pcf is a function of q. By solving the equation
link is min{pρ, 1} = min{ 1−P
cf
ξ
min{ 1−P
, 1} = q, we obtain a solution q = q ⋆ which could be used to derive the approximated
cf

delay for the wireless network.
Since the transmissions of the typical user in different time slots are affected by the independent
fading and the independent random access, the probability for a transmission attempt to be
scheduled and successful in all time slots is the same, which is p(1 − Pcf ). Therefore, the
queueing system at the typical transmitter is a Geo/G/1 queue, or a discrete-time single server
retrial queue [44], [45]. In the Geo/G/1 queue, the arrival process of the packets is a Bernoulli
process with intensity ξ packets per time slot. The arrival process of the packets is called the
geometric arrival since the probability that a packet arrives in a time slot is ξ, and the number of
time slots between two adjacent arrivals is a geometric random variable. The success probability
is p(1 − Pcf ) and the service times of packets are i.i.d. with geometric distribution. From [44],
we get the mean delay D for the Geo/G/1 queue with given Pcf as

1−ξ

if p(1 − Pcf ) > ξ
p(1−Pcf )−ξ
D=
 ∞
if p(1 − Pcf ) ≤ ξ.

(10)

With the above discussions, we could obtain the mean delay in the dynamic scenario in the

following theorem.
Theorem 3. The mean delay of the typical legitimate link in the dynamic scenario is
1 − 1/ξ
D≃
.
pλl πr02 θtδ C(δ)
1 + W(−ξλ
πr 2 θ δ C(δ))
l

(11)

0 t

Proof. By assuming that all interfering transmitters are active with the same probability q, we
get the connection failure probability as
i
q
+
1
−
q
1 + θt r0α |y|−α
y∈Φl

= 1 − exp − qλl πr02 θtδ C(δ) .

Pcf = 1 − EΦl

hY

ξ
Considering the equation min{ 1−P
, 1} = q, we get

 cf
ξ
 , 1 = q.
min
exp − qλl πr02 θtδ C(δ)

(12)

(13)

Solving the above equation, we get the solution


W(−ξλl πr02 θtδ C(δ))
⋆
q = min −
,1 .
λl πr02 θtδ C(δ)

(14)

where W(z) is the Lambert-W function with the defining equation W(z) exp(W(z)) = z. Then,
the connection failure probability is


ξλl πr02 θtδ C(δ)
ξ
,1− ξ .
Pcf = 1 − ⋆ = min 1 +
q
W(−ξλl πr02 θtδ C(δ))

(15)

Having derived the connection failure probability, the mean delay is
D=

1−ξ
1 − 1/ξ
.
=
pλl πr02 θtδ C(δ)
p(1 − Pcf ) − ξ
1 + W(−ξλ
2
δ
πr θ C(δ))
l

(16)

0 t

Therefore, we get the results in the theorem.
It is hard to interpret the relationship between the mean delay and the system parameters since
the property of the Lambert-W function is not intuitive. However, we could further simplify the
above results for some special cases. Note that when x → 0, the asymptotic expansions of the
Lambert-W function is
W(x) =

∞
X
(−n)n−1
n=1

n!

3
8
= x − x2 + x3 − x4 + ....
2
3

(17)

Therefore, we have W(x) ∼ x when x → 0. When λl r02 θtδ C(δ) → 0, the result given by Theorem
3 can be simplified into
D≃

1 − 1/ξ
1+

pλl πr02 θtδ C(δ)
W(−ξλl πr02 θtδ C(δ))

≃

1 − 1/ξ
1 − p/ξ

(18)

The above result indicates that when either λl → 0, r0 → 0, or θt → 0 holds, the delay
could be approximated into simple form. Then, the relationship between the mean delay D, the
transmit probability p and the arrival rate ξ could be easily obtained.
The secrecy failure event happens in the transmitting process and is not affected by the
queueing process. Thus, the secrecy outage probability in the dynamic scenario is similar to
that in the backlogged scenario since the secrecy outage is only determined by the delivering
process of a packet. The only difference between the secrecy outage probability in the backlogged
scenario and that in the dynamic scenario is that the active probabilities for the two scenarios
are different. In the backlogged scenario, the active probability for the interfering transmitters
is the transmit probability p since all transmitters are backlogged and always have packets to

transmit. However, in the dynamic scenario, the active probability is q ⋆ . Therefore, we get the
following theorem, which gives the secrecy outage probability in the dynamic scenario.
Theorem 4. The secrecy outage probability of the typical link in the dynamic scenario can be
approximated by

exp q⋆ λl θλδeC(δ) − 1
e


,
Pso ≃
exp q⋆ λl θλδeC(δ) + exp − q ⋆ λl πr02 θtδ C(δ) − 1

(19)

e

where q ⋆ is given by

⋆

q = min




W(−ξλl πr02 θtδ C(δ))
−
,1 .
λl πr02 θtδ C(δ)

(20)

Proof. The proof is similar to that of Theorem 2.
The equation (19) shows that Pso ≃ 0 as λe → 0 or θe → ∞, indicating that when the number
of eavesdroppers is very small or when the SIR threshold to eavesdrop the message is very large,
the secrecy outage event hardly ever happens. On the other hand, when λe → ∞ or θe → 0, we
get Pso ≃ 1, indicating that when the eavesdroppers are densely deployed or when the messages
are easily intercepted by the eavesdroppers, the secrecy outage event almost always happens.
Similar to the discussions after Theorem 3, we can also simplify the result given by Theorem
4. Note that W(x) ∼ x as x → 0. When r02 θtδ C(δ) → 0, the active probability given by the
equation (20) can be simplified as
q ⋆ = min{ξ, 1}.

(21)

Note that if ξ > 1, the arrival rate is larger than the transmit probability, thus, the queues in the
network will be unstable. A more relevant case is ξ < 1, which is a necessary condition for the
queues to be stable. If both r02 θtδ C(δ) → 0 and ξ < 1 hold, the active probability will be
q ⋆ = ξ.
Plugging q ⋆ = ξ into (19), we get the secrecy outage probability when r02 θtδ C(δ) → 0 as

exp ξλl θλδeC(δ) − 1
e


Pso ≃
exp ξλl θλδeC(δ) + exp − ξλl πr02 θtδ C(δ) − 1
e

λe
exp ξλl θδ C(δ) − 1
e

≃
exp ξλl θλδeC(δ)
e


λe
= 1 − exp −
.
ξλl θeδ C(δ)

(22)

(23)

The relationship between the secrecy outage probability and the system parameters, such as the
intensity of eavesdroppers λe , the intensity of legitimate transmitters λl , the SIR threshold θe
and so on, could be observed from the equation (23) directly.
IV. P ERFORMANCE

OF

M ESSAGE S PLIT

In this section, we propose and analyze a simple transmission mechanism, in which a confidential message is divided into two packets to be delivered independently. A confidential message
is successfully decoded at the legitimate receiver only when both of the two packets from
the message are successfully decoded at the receiver. Meanwhile, the perfect security could
be guaranteed if at least one of the two packets from the same message cannot be decoded
by an eavesdropper. This mechanism has a direct effect on the delay and may also affect the
security performance. Therefore, in this section, we explore this simple transmission mechanism
to provide a guidance to tradeoff the delay and the security of the wireless network.
A. Backlogged Scenario
Since one message is split into two packets, the size of each packet is halved. When fixing
the duration of the time slots, the requirement for the rate of the confidential message could be
reduced from Rs to Rs′ = Rs /2. The threshold for the rate of the eavesdropping link is also
reduced from Re to Re′ = Re /2. Then, the rate threshold of codewords becomes Rt′ = Rs′ + Re′ =
(Rs + Re )/2.
Note that the mean delay is doubled since both of the two packets should be successful. By
the similar derivations as Theorem 1, we get the following corollary which gives the mean delay
in the case of message split.
Corollary 1. The mean delay of the typical legitimate link in the backlogged scenario with
message split is
D=
′

where θt′ = 2Rt − 1.


2
exp λl πr02 θt′δ C(δ)p(1 − p)δ−1 ,
p

(24)

Comparing with the result given by (7), we observe that the mean delay is doubled and θt
is replaced by θt′ . However, since the effect of the two changes are in the opposite direction,
whether the mean delay is increased or not is still uncertain, which depends on the parameters
such as the transmit probability p, the intensity of legitimate transmitters λl , and so on.

In order to intercept the confidential message, both of the two packets from the same message
should be successfully decoded at the eavesdropper. Therefore, by the similar derivations as that
in Theorem 2, we get
Corollary 2. The secrecy outage probability of the typical link in the backlogged scenario with
message split can be approximated by

2

exp pλl θλ′δeC(δ) − 1
e


,
Pso ≃
exp pλl θλ′δeC(δ) + exp − pλl πr02 θt′δ C(δ) − 1

(25)

e

where

θe′

=2

R′e

− 1.

The secrecy outage probability in the backlogged scenario with message split is closed-form.
The result also reveals that Pso → 0 as λe → 0, and Pso → 1 as λe → +∞. Though whether
the message split decreases the secrecy outage probability is still not intuitive from Corollary 2,
the numerical analysis could be conducted to evaluate the effect of message split.
B. Dynamic Scenario
The analysis of the dynamic scenario is different from that of the backlogged scenario when the
message split is introduced in the transmission mechanism. The arrival process of the messages
is still a Bernoulli process of arrival rate ξ. Since each message is split into two packets which
are transmitted independently, at least two time slots are needed to successfully deliver one
message. However, the queueing system is still a Geo/G/1 queue because the number of time
slots between two adjacent arrivals is still a geometric random variable. When each message
is divided into two packets, the service time of each message is the summation of the service
time of the two packets. Since the service time of each packet is a geometric random variable,
the total service time for each message is the summation of two geometric distributed random
variables. The analysis of this queueing system is rather difficult because the service rate is
not a well know distribution. Thus, we use the formula for the mean delay obtained from the
M/M/1 queueing system to approximate the mean delay in the dynamic scenario with message
split. With these discussions, we get the following corollary, which gives the mean delay in the
dynamic scenario with message split.

Corollary 3. The mean delay of the typical legitimate link in the dynamic scenario with message
split is
D≃−

1
pξλl πr02 θt′δ C(δ)
2W(−ξλl πr02 θt′δ C(δ))

+ξ

,

(26)

′

where θt′ = 2Rt − 1.
Proof. By approximating the queueing system with the M/M/1 model, the mean delay can be
approximated as
D≃

1
,
µ−ξ

(27)

where ξ is the arrival rate of the messages, and µ is the service rate. The mean delay in the
dynamic scenario with message split could be obtained by the service rate of the Geo/G/1 queue.
Similar to the derivations in Theorem 3, the connection failure probability of a packet is


ξλl πr02 θt′δ C(δ)
,1− ξ ,
(28)
Pcf = min 1 +
W(−ξλl πr02 θt′δ C(δ))
′

where θt′ = 2Rt − 1. The service time for each packet is a geometric random variable with
success probability p(1 − Pcf ), and the mean service time for each packet is
mean service time for each message is
p(1−Pcf )
.
2

2
,
p(1−Pcf )

1
.
p(1−Pcf )

Thus, the

and the mean service rate for each message is

Plugging into the equation (27), we get the mean delay as
D ≃ −

1
pξλl πr02 θt′δ C(δ)
2W(−ξλl πr02 θt′δ C(δ))

+ξ

.

(29)

Therefore, we get the result in the corollary.
By the expansion of the Lambert-W function given by the equation (17), we get the mean
delay when λl πr02 θt′δ C(δ) → 0 as
D≃

1
p
2(1+ξλl πr02 θt′δ C(δ))

−ξ

≃

2
.
p − 2ξ

(30)

Furthermore, when p = 1 and ξ → 0, the mean delay D is exactly two time slots, which is
the case where the transmission attempts of a packet is scheduled with probability one and is
always successful when it is scheduled.
To eavesdrop the confidential message at an eavesdropper in the dynamic scenario with
message split, both of the two packets from the same message should be successfully decoded
at the eavesdropper. By the similar derivations as that in Theorem 4, we get

Corollary 4. The secrecy outage probability of the typical link in the dynamic scenario with
message split can be approximated by

2

exp q⋆ λl θλ′δe C(δ) − 1
e


Pso ≃
,
exp q⋆ λl θλ′δe C(δ) + exp − q ⋆ λl πr02 θt′δ C(δ) − 1

(31)

e

where q ⋆ is given by

⋆

q = min
′




W(−ξλl πr02 θt′δ C(δ))
,1 ,
−
λl πr02 θt′δ C(δ)

(32)

′

θt′ = 2Rt − 1, and θe′ = 2Re − 1.
Specially, when λl πr02 θt′δ C(δ) → 0, using the expansion of the Lambert-W function given by
(17), we obtain
q ⋆ = min{ξ, 1},

(33)

When the queue is stable, the arrival rate of messages must be smaller than one, i.e., ξ < 1.
Therefore, we have
q ⋆ = ξ.
Then, the secrecy outage probability given by Corollary 4 could be simplified as


2
exp ξλl θλ′δeC(δ) − 1
e


Pso ≃
exp ξλl θλ′δeC(δ) + exp − ξλl πr02 θt′δ C(δ) − 1
e
2


λe
.
≃
1 − exp −
ξλl θe′δ C(δ)

(34)

(35)

V. N UMERICAL E VALUATION
In this section, we evaluate the tradeoff between the mean delay and the secrecy outage
probability through numerical analysis to gain insight. Without special explanation, the numerical
parameters are set as follows in default. The intensities of the legitimate transmitters and the
eavesdroppers are set as λl = 0.05 and λe = 0.01, respectively, and the desired link distance is
r0 = 1. The path loss exponent is α = 4, the transmit probability is p = 0.8, and the packet
arrival rate is ξ = 0.1. The rate threshold of codewords is Rt = 3 bits/Hz, and that of confidential
messages is Re = 1 bit/Hz.
Figure 5 shows the mean delay and the secrecy outage probability as functions of the transmit
probability p in the backlogged scenario. As p increases, both the mean delay and the secrecy
outage probability first decrease then increase. When p starts to increase, the mean delay

12

0.8
α=2.5
α=3
α=4

11

α=2.5
α=3
α=4

0.7

0.6

9

Secrecy outage probability

Mean delay in number of time slots

10

8
7
6
5

0.5

0.4

0.3

4
0.2
3
2

0

0.2
0.4
0.6
0.8
Transmit probability p

0.1

1

0

0.2
0.4
0.6
0.8
Transmit probability p

1

Fig. 5. The mean delay and the secrecy outage probability as functions of the transmit probability p in the backlogged scenario.

41

36

0.7
α=2.5
α=3
α=4

α=2.5
α=3
α=4

31
Secrecy outage probability

Mean delay in number of time slots

0.6

26

21

16

0.5

0.4

0.3

11
0.2
6

1
0.01 0.04 0.07 0.1 0.13 0.16 0.19
Intensity of legitimate transmitters λ
l

0.1
0.01 0.04 0.07 0.1 0.13 0.16 0.19
Intensity of legitimate transmitters λ
l

Fig. 6. The mean delay and the secrecy outage probability as functions of the intensity of legitimate transmitters λl in the
backlogged scenario.

0.6
=2.5
=3
=4

Secrecy outage probability

0.5

0.4

0.3

0.2

0.1

0
100

101
Logarithm of mean delay in number of time slots ln(D)

102

Fig. 7. Tradeoff between the mean delay and the secrecy outage probability when λl increases from 0.01 to 0.09 in the
backlogged scenario.

decreases since the successful delivery of a packet is limited by the transmit probability, while the
secrecy outage probability decreases since the interference for the eavesdropping links increases.
When p continues to grow, the mean delay increases since the interference increases, while the
secrecy outage probability increases since the probability to eavesdrop the message is increased
with large p. Figure 6 shows the mean delay and the secrecy outage probability as functions of λl
in the backlogged scenario. When λl increases, the mean delay increases and the secrecy outage
probability decreases since the interference in the network increases. However, when λl continues
to grow, the secrecy outage probability increases since the distances of the eavesdropping links are
decreased. Figure 7 plots the tradeoff between the mean delay and the secrecy outage probability
when increasing λl in the backlogged scenario. Though the value of λl is not marked in Fig.
7, we can observe from Theorem 1 that λl is linear positive correlated to ln(D) which is the
value of the horizontal ordinate. Note that when the path loss exponent α increases, the secrecy
outage probability decreases for large λl and increases for small λl , indicating that the security
performance is much better when the signal attenuates quickly (i.e., when α is large) for large
λl , and it is reverse for small λl .

1

4
θ =3

θ =3

t

t

θ =10

θ =10

0.9

t

t

θ =15

3.5

θ =15

t

t

Secrecy outage probability

Mean delay in number of time slots

0.8

3

2.5

2

0.7

0.6

0.5

0.4

0.3
1.5
0.2

1

0

0.05
0.1
0.15
0.2
Intensity of legitimate transmitters λ
l

0.1

0

0.05
0.1
0.15
0.2
Intensity of legitimate transmitters λ
l

Fig. 8. The mean delay and the secrecy outage probability as functions of the intensity of legitimate transmitters λl in the
dynamic scenario.

Figure 8 shows the mean delay and the secrecy outage probability as functions of the intensity
of legitimate transmitters λl with different θt in the dynamic scenario. When varying θt , we
keep the rate threshold of confidential messages Rs to be the same (i.e., Rs = 1 bits/Hz). Thus,
the minimal confidential rate Rs = 1 is guaranteed. Figure 8 reveals that although the same
confidential rate is guaranteed, the mean delay is larger and the secrecy outage probability is
smaller in the high SIR regime than those in the low SIR regime.
Figure 9 and Figure 10 show the comparison of the mean delay and the secrecy outage
probability for different scenarios with and without message split. The arrival rate is ξ = 0.2.
Figure 9 reveals that in the backlogged scenario, the mean delay with message split is larger
than that without message split when λl is small, while it is reverse for large λl , indicating
that in the backlogged scenario, message split is beneficial for the delay performance only for
large λl . Figure 9 also reveals that in the dynamic scenario, the delay performance without
message split is always better than that with message split. Figure 10 shows that in the backlogged scenario, introducing message split greatly decreases the secrecy outage probability and

25
Backlogged scenario
Dynamic scenario
Backlogged scenario with message split
Dynamic scenario with message split
Mean delay in number of time slots

20

15

10

5

0

0

0.02

0.04

0.06
0.08
0.1
0.12
0.14
Intensity of legitimate transmitters λ

0.16

0.18

0.2

l

Fig. 9. Comparison of the mean delay in different scenarios with and without message split.

1
Backlogged scenario
Dynamic scenario
Backlogged scenario with message split
Dynamic scenario with message split

0.9

Secrecy outage probability

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

0.05
0.1
0.15
Intensity of legitimate transmitters λl

0.2

Fig. 10. Comparison of the secrecy outage probability in different scenarios with and without message split.

improves the security performance, while in the dynamic scenario, introducing message split
is only beneficial when λl is large. Accordingly, in the actual application scenario such as the
military communications, the message split scheme could be introduced to improve the security
performance at the expense of increasing the delay when the density of the soldiers in the
communications is not very small.
VI. C ONCLUSIONS
In this work, we evaluate the tradeoff between the delay and the security performance in large
wireless networks for both the backlogged scenario and the dynamic scenario. By combining
the tools from stochastic geometry and queueing theory, we derive the close-formed results for
the mean delay and the secrecy outage probability. We also analyzed the effect of a simple
transmission mechanism which splits a message into two packets.
Our results reveal that the security performance of the wireless network is better when the
path loss exponent is larger. We also observe that under the condition that a certain confidential
rate is guaranteed, the delay performance is better in high SIR regime than that in low SIR
regime. Moreover, by introducing the simple mechanism of message split, it is shown that the
delay performance in the backlogged scenario is improved when the density of the legitimate
transmitters is large. Meanwhile, introducing message split greatly improves the security performance in the backlogged scenario, while slightly improves the security performance in the
dynamic scenario when the density of the legitimate transmitters is large.
A PPENDIX A
P ROOF

OF

T HEOREM 2

Let Φk ⊂ Φl be the set of transmitters that are active in time slot k, and let Ik be the interference
at the typical legitimate receiver located at x0 in time slot k. Then, the mathematical expression
for Ik is
Ik =

X

y∈Φl

hy |y − x0 |−α 1(y ∈ Φk ).

(36)

With given Φl , the connection failure probability is

l
PΦ
= P hx0 r0−α < θt Ik | Φl
cf



(a)
= 1 − E exp − θt r0α Ik | Φl



 X
α
−α
θt r0 hy |y − x0 | 1(y ∈ Φk ) | Φl
= 1 − E exp −
y∈Φl

= 1−

Y

y∈Φl

h
i


pE exp − θt r0α hy |y − x0 |−α | Φl + 1 − p


p
+
1
−
p
1 + θt r0α |y − x0 |−α
y∈Φl

Y
p
(c)
+
1
−
p
,
= 1−
α
−α
1
+
θ
r
|y|
t
0
y∈Φ
(b)

= 1−

Y

(37)

l

where (a) and (b) follow because the fading coefficients {hy } are i.i.d. exponential distributed
random variables with unit mean, (c) holds due to the stationarity of the PPP.
With given Φl and Φe , the secrecy failure probability is
Y
l ,Φe
P{SIRxe < θe | Φl , Φe }
=
1
−
PΦ
sf
xe ∈Φe

= 1−

Y

xe ∈Φe

= 1−


P he |xe |−α < θe Ik | Φl , Φe

Y 


1 − E exp(−θe Ik |xe |α ) | Φl , Φe

xe ∈Φe

= 1−

X
Y 


hy |y − xe |−α 1(y ∈ Φk )) | Φl , Φe
1 − E exp(−θe |xe |α
y∈Φl

xe ∈Φe

Y 
Y 

E exp(−θe |xe |α hy |y − xe |−α 1(y ∈ Φk )) | Φl , Φe
1−
= 1−
xe ∈Φe

y∈Φl

Y
Y 
1−
= 1−
xe ∈Φe

y∈Φl


p
+
1
−
p
.
1 + θe |xe |α |y − xe |−α

From equation (8), we get the secrecy outage probability of the typical link as


l ,Φe
PΦ
sf
Pso = EΦl ,Φe
Φl ,Φe 
l
1 − PΦ
cf 1 − Psf


Y
Y 

p
+1−p
1−
= EΦl ,Φe 1 −
α
−α
1
+
θ
e |xe | |y − xe |
y∈Φl
xe ∈Φe



Y

p
× 1− 1−
+1−p
1 + θt r0α |y|−α
y∈Φl
 
Y
Y 
 −1
p
.
+1−p
1−
×
α |y − x |−α
1
+
θ
|x
|
e
e
e
y∈Φ
x ∈Φ
e

e

l

(38)

(39)

To derive the exact secrecy outage probability is difficult, and we propose to approximate the
secrecy outage probability. Using the Jensen’s inequality to approximate the secrecy outage
probability, i.e., EX (f (X)) ≃ fX (E[X]) for a random variable X and a function f (·), we get

 Y 

Y

p
+1−p
1−
Pso ≃ 1 − EΦl ,Φe
α |y − x |−α
1
+
θ
|x
|
e
e
e
y∈Φl
xe ∈Φe


Y


p
× 1 − 1 − EΦl
+1−p
α
−α
1
+
θ
t r0 |y|
y∈Φl

 Y 
Y
 −1
p
+1−p
1−
. (40)
× EΦl ,Φe
α
−α
1
+
θ
e |xe | |y − xe |
y∈Φ
x ∈Φ
e

e

l

Using the probability generating functional (PGFL) of the PPP, we have



 
Y
Z 

p
p
= exp − λl
+1−p
1−
+ 1 − p dy
EΦl
1 + θt r0α |y|−α
1 + θt r0α |y|−α
R2
y∈Φl

= exp − pλl πr02 θtδ C(δ) .
(41)
Due to the stationarity of the PPP and the independence between Φl and Φe , we have
 Y 

Y

p
1−
EΦl ,Φe
+1−p
1 + θe |xe |α |y − xe |−α
y∈Φl
xe ∈Φe
Y
 Y 


p
1 − EΦl
= EΦe
+1−p
1 + θe |xe |α |y|−α
y∈Φl
xe ∈Φe

 Y 

2 δ
1 − exp − pλl π|xe | θe C(δ)
= EΦe
xe ∈Φe



= exp − λe


= exp −

Z

R2

 
exp − pλl π|x|2 θeδ C(δ) dx


λe
.
pλl θeδ C(δ)

(42)

Plugging into the equation (40), we have
Pso


1 − exp − pλl θλδeC(δ)
e

≃
1 − exp − pλl θλδeC(δ) + exp − pλl πr02 θtδ C(δ) −
e

exp pλl θλδeC(δ) − 1
e


.
=
exp pλl θλδeC(δ) + exp − pλl πr02 θtδ C(δ) − 1

λe
pλl θeδ C(δ)


(43)

e

Therefore, we get the results in the theorem.

R EFERENCES
[1] X. Ge, S. Tu, G. Mao, C.-X. Wang, and T. Han, “5G ultra-dense cellular networks,” IEEE Wireless Commun., vol. 23,
no. 1, pp. 72–79, Mar. 2016.

[2] C. B. Sankaran, “Network access security in next-generation 3GPP systems: A tutorial,” IEEE Commun. Mag., vol. 47,
no. 2, pp. 84–91, Feb. 2009.
[3] A. D. Wyner, “The wire-tap channel,” The Bell System Technical Journal, vol. 54, no. 8, pp. 1355–1387, Oct. 1975.
[4] X. Ge, Z. Li, and S. Li, “5G software defined vehicular networks,” IEEE Commun. Mag., vol. 55, no. 7, pp. 87–93, 2017.
[5] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE
J. on Selected Areas in Commun., vol. 32, no. 6, pp. 1065–1082, Jun. 2014.
[6] Y. Zhong, T. Q. S. Quek, and X. Ge, “Heterogeneous cellular networks with spatio-temporal traffic: delay analysis and
scheduling,” IEEE J. on Selected Areas in Commun., vol. 35, no. 6, pp. 1373–1386, Jun. 2017.
[7] Y. Zhong, M. Haenggi, F.-C. Zheng, W. Zhang, T. Q. Quek, and W. Nie, “Towards a tractable delay analysis in ultradense
networks,” IEEE Commun. Mag., accepted to appear, 2017.
[8] I. Csiszar and J. Korner, “Broadcast channels with confidential messages,” IEEE Trans. on Inform. Theory, vol. 24, no. 3,
pp. 339–348, May 1978.
[9] S. Leung-Yan-Cheong and M. Hellman, “The Gaussian wire-tap channel,” IEEE Trans. on Inform. Theory, vol. 24, no. 4,
pp. 451–456, Jul. 1978.
[10] S. Shafiee and S. Ulukus, “Achievable rates in Gaussian MISO channels with secrecy constraints,” in IEEE Int. Symp. on
Inform. Theory, Jun. 2007.
[11] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas I: the MISOME wiretap channel,” IEEE Trans.
on Inform. Theory, vol. 56, no. 7, pp. 3088–3104, Jul. 2010.
[12] L. Lai and H. E. Gamal, “The relay-eavesdropper channel: cooperation for secrecy,” IEEE Trans. on Inform. Theory,
vol. 54, no. 9, pp. 4005–4019, Sept. 2008.
[13] Y. Liang and H. V. Poor, “Multiple-access channels with confidential messages,” IEEE Trans. on Inform. Theory, vol. 54,
no. 3, pp. 976–1002, Mar. 2008.
[14] E. Tekin and A. Yener, “The general Gaussian multiple-access and two-way wiretap channels: achievable rates and
cooperative jamming,” IEEE Trans. on Inform. Theory, vol. 54, no. 6, pp. 2735–2751, Jun. 2008.
[15] Y. Wu, R. Schober, D. W. K. Ng, C. Xiao, and G. Caire, “Secure massive MIMO transmission with an active eavesdropper,”
IEEE Trans. on Inform. Theory, vol. 62, no. 7, pp. 3880–3900, Jul. 2016.
[16] Y. Wu, J. B. Wang, J. Wang, R. Schober, and C. Xiao, “Secure transmission with large numbers of antennas and finite
alphabet inputs,” IEEE Trans. on Commun., accepted to appear, 2017.
[17] M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the
analysis and design of wireless networks,” IEEE J. on Selected Areas in Commun., vol. 27, no. 7, pp. 1029–1046, Sept.
2009.
[18] M. Haenggi, “The secrecy graph and some of its properties,” in 2008 IEEE Int. Symp. on Inform. Theory, Jul. 2008.
[19] S. Goel, V. Aggarwal, A. Yener, and A. R. Calderbank, “The effect of eavesdroppers on network connectivity: a secrecy
graph approach,” IEEE Trans. on Inform. Forensics and Security, vol. 6, no. 3, pp. 712–724, Sept. 2011.
[20] P. C. Pinto and M. Z. Win, “Continuum percolation in the intrinsically secure communications graph,” in 2010 Int. Symp.
on Inform. Theory and Its Applications, Oct. 2010, pp. 349–354.
[21] X. Zhou, R. K. Ganti, and J. G. Andrews, “Secure wireless network connectivity with multi-antenna transmission,” IEEE
Trans. on Wireless Commun., vol. 10, no. 2, pp. 425–430, Feb. 2011.
[22] A. Sarkar and M. Haenggi, “Secrecy coverage,” in 2010 Conference Record of the Forty Fourth Asilomar Conference on
Signals, Systems and Computers, Nov. 2010, pp. 42–46.
[23] O. O. Koyluoglu, C. E. Koksal, and H. E. Gamal, “On secrecy capacity scaling in wireless networks,” IEEE Trans. on
Inform. Theory, vol. 58, no. 5, pp. 3000–3015, May 2012.

[24] Y. Liang, H. V. Poor, and L. Ying, “Secrecy throughput of MANETs under passive and active attacks,” IEEE Trans. on
Inform. Theory, vol. 57, no. 10, pp. 6692–6702, Oct. 2011.
[25] H. Wang, X. Zhou, and M. C. Reed, “Physical layer security in cellular networks: a stochastic geometry approach,” IEEE
Trans. on Wireless Commun., vol. 12, no. 6, pp. 2776–2787, Jun. 2013.
[26] M. Haenggi, Stochastic geometry for wireless networks.

Cambridge University Press, 2012.

[27] F. Baccelli, P. Mühlethaler, and B. Blaszczyszyn, “Stochastic analysis of spatial and opportunistic ALOHA,” IEEE J. on
Selected Areas in Commun., vol. 27, no. 7, pp. 1105–1119, Sept. 2009.
[28] J. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Trans. on
Commun., no. 99, pp. 1–13, Nov. 2010.
[29] J. Wen, M. Sheng, X. Wang, J. Li, and H. Sun, “On the Capacity of Downlink Multi-Hop Heterogeneous Cellular Networks,”
IEEE Trans. on Wireless Commun., vol. 13, no. 8, pp. 4092–4103, Aug 2014.
[30] M. Sheng, J. Wen, J. Li, B. Liang, and X. Wang, “Performance Analysis of Heterogeneous Cellular Networks With HARQ
Under Correlated Interference,” IEEE Trans. on Wireless Commun., vol. 16, no. 12, pp. 8377–8389, Dec 2017.
[31] X. Zhou, R. K. Ganti, J. G. Andrews, and A. Hjorungnes, “On the throughput cost of physical layer security in decentralized
wireless networks,” IEEE Trans. on Wireless Commun., vol. 10, no. 8, pp. 2764–2775, Aug. 2011.
[32] H. Moosavi and F. M. Bui, “Delay-aware optimization of physical layer security in multi-hop wireless body area networks,”
IEEE Trans. on Inform. Forensics and Security, vol. 11, no. 9, pp. 1928–1939, Sept. 2016.
[33] X. Zhu, B. Yang, C. Chen, L. Xue, X. Guan, and F. Wu, “Cross-layer scheduling for OFDMA-based cognitive radio
systems with delay and security constraints,” IEEE Trans. on Vehic. Tech., vol. 64, no. 12, pp. 5919–5934, Dec. 2015.
[34] B. Blaszczyszyn, M. Jovanovic, and M. K. Karray, “Performance laws of large heterogeneous cellular networks,” in IEEE
13th Inter. Symp. on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), 2015.
[35] R. R. Rao and A. Ephremides, “On the stability of interacting queues in a multiple-access system,” IEEE Trans. on Inform.
Theory, vol. 34, no. 5, pp. 918–930, Sept. 1988.
[36] W. Luo and A. Ephremides, “Stability of N interacting queues in random-access systems,” IEEE Trans. on Inform. Theory,
vol. 45, no. 5, pp. 1579–1587, Jul. 1999.
[37] A. Ephremides and R.-Z. Zhu, “Delay analysis of interacting queues with an approximate model,” IEEE Trans. on Commun.,
vol. 35, no. 2, pp. 194–201, Jan. 1987.
[38] Y. Zhong, M. Haenggi, T. Q. S. Quek, and W. Zhang, “On the stability of static Poisson networks under random access,”
IEEE Trans. on Commun., vol. 64, no. 7, pp. 2985–2998, Jul. 2016.
[39] W. Feng, Y. Wang, D. Lin, N. Ge, J. Lu, and S. Li, “When mmWave Communications Meet Network Densification: A
Scalable Interference Coordination Perspective,” IEEE J. on Selected Areas in Commun., vol. 35, no. 7, pp. 1459–1471,
July 2017.
[40] M. Haenggi, “The local delay in Poisson networks,” IEEE Trans. on Inform. Theory, vol. 59, no. 3, pp. 1788–1802, Mar.
2013.
[41] F. Baccelli and B. Blaszczyszyn, “A new phase transitions for local delays in MANETs,” in Proceedings IEEE INFOCOM,
Mar. 2010.
[42] Y. Zhong, W. Zhang, and M. Haenggi, “Managing interference correlation through random medium access,” IEEE Trans.
on Wireless Commun., vol. 13, no. 2, pp. 928–941, Jan. 2014.
[43] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “On the throughput of secure hybrid-ARQ protocols for Gaussian blockfading channels,” IEEE Trans. on Inform. Theory, vol. 55, no. 4, pp. 1575–1591, Apr. 2009.
[44] I. Atencia and P. Moreno, “A discrete-time Geo/G/1 retrial queue with general retrial times,” Queueing Systems, vol. 48,
no. 1-2, pp. 5–21, 2004.

[45] G. Falin, “A survey of retrial queues,” Queueing systems, vol. 7, no. 2, pp. 127–167, Jun. 1990.

