# Probabilistic Caching Placement in the Presence of Multiple Eavesdroppers

Authors: Fang Shi; Lisheng Fan; Xin Liu; Zhenyu Na; Yanchen Liu
Journal: Wireless Communications and Mobile Computing
 (2018)

---

## Abstract

The wireless caching has attracted a lot of attention in recent years, since it can reduce the backhaul cost significantly and improve the user-perceived experience. The existing works on the wireless caching and transmission mainly focus on the communication scenarios without eavesdroppers. When the eavesdroppers appear, it is of vital importance to investigate the physical-layer security for the wireless caching aided networks. In this paper, a caching network is studied in the presence of multiple eavesdroppers, which can overhear the secure information transmission. We model the locations of eavesdroppers by a homogeneous Poisson Point Process (PPP), and the eavesdroppers jointly receive and decode contents through the maximum ratio combining (MRC) reception which yields the worst case of wiretap. Moreover, the main performance metric is measured by the average probability of successful transmission, which is the probability of finding and successfully transmitting all the requested files within a radiusR. We study the system secure transmission performance by deriving a single integral result, which is significantly affected by the probability of caching each file. Therefore, we extend to build the optimization problem of the probability of caching each file, in order to optimize the system secure transmission performance. This optimization problem is nonconvex, and we turn to use the genetic algorithm (GA) to solve the problem. Finally, simulation and numerical results are provided to validate the proposed studies.


---

## Body

## 1. Introduction

The arrival of big data era has led to a growing communication business, and the demand for wireless data rates becomes higher and higher. In order to reduce the transmission load and capacity crunch, caching is emerging as an important technology in the next generating wireless networks. The main idea behind caching is to store parts of the popular contents in caching helpers’ memory and leverage the locally stored content to reduce transmission links, thereby reducing the transmission load and speeding up the transmission of requested content. And the different cache strategies have been well studied [1–9]. Specifically in [1], the authors considered a cluster-centric small-cell networks with combined design of cooperative caching and transmission policy and proposed a combined caching scheme to increase the local content diversity. The distributed caching placement has been studied in [2, 3], and in [3], the authors proposed to combine two recent schemes, distributed caching of content in small cells and cooperative transmissions from nearby base stations/BSs to achieve unprecedented content delivery speeds while reducing backhaul cost and delay. The probabilistic caching placement was studied in [4–7]. Departing from the conventional cache hit optimization in cache-enabled wireless networks, the authors in [4] considered an alternative optimization approach for the probabilistic caching placement in stochastic wireless D2D caching networks, proposed the cache-aided throughput, and provided a closed-form approximation of cache-aided throughput. Different from [4], the authors in [5] studied a probabilistic small-cell caching strategy and considered two kinds of network architectures: the small-cell base stations (SBSs) are always active and the SBSs are activated on demand by mobile users (MUs). The authors in [6, 7] proposed to use different optimization strategies to optimize the probabilistic caching placement. In addition, in paper [8], the analysis, design, and optimization of geographic caching were presented; and in paper [9], a hybrid caching scheme was studied which was jointly optimized with the transmission schemes to achieve a fine balance between the signal cooperation gain and the caching diversity gain.The emergence of cache and wireless devices has solved a lot of problems, such as reducing transmission load, traffic, and energy consumption of the backhaul. The existing works on the wireless caching and transmission mainly focus on the communication scenarios without eavesdroppers, for instance, [4–9]. But when the eavesdroppers appear, it is of vital importance to investigate the physical-layer security for the wireless caching aided networks. In recent years, some researchers also have took into account the problem of secure caching, such as [10–13]. Specially, in [10], the problem of secure caching in the presence of an external wiretapper for both centralized and decentralized cache placement was analyzed. In [11], unmanned aerial vehicles assisted secure transmission for scalable videos in hyperdense networks via caching was studied. The authors in [12] studied a cooperative network with caching relays to reduce the transmission links overheard by the eavesdropper. Moreover, a novel hybrid cache placement was proposed to cache the popular contents, and the closed expressions of the secrecy outage probability and average secrecy capacity were obtained. The authors in [13] studied a framework of communication, caching, and computing- (3C-) oriented small-cell networks with interference alignment, in which caching and computing are exploited to simplify the network topology, improve the throughput, reduce the backhaul load, and guarantee the quality of experience of users.The works about the physical-layer security have been studied such as the works in [14–16]. In [14], Wyner proved that the secure communication is feasible without cryptography technology as long as the eavesdropper’s instantaneous channel is worse than the legitimate user’s instantaneous channel. Based on Wyner’s wiretap channel model, the authors in [15] studied the secrecy capacity over Gaussian channel. And the knowledge about the wireless information-theoretic security has been studied in [16]. In addition, the secrecy performance of wireless communication has been studied in [17–19]. Specifically in [17], the impact of cochannel interference and wiretap on the security performance of multiple amplify-and-forward (AF) relaying networks has been studied. In [18], the physical-layer security of a multiantenna transmission system in the presence of Poisson distributed eavesdroppers was analyzed, and the two different cases including the eavesdroppers being colluding and noncolluding were also analyzed in the paper. The relaying techniques for enhancing the physical-layer security have been studied in [19–26].According to the above analysis, the main idea of this paper is to design, analyze, and optimize the probabilistic caching placement based on the security of transmission. Without loss of generality, the locations of relays are modeled by a homogeneous PPP. Moreover, considering the randomness of eavesdroppers’ positions, we also model the locations of eavesdroppers by a homogeneous PPP, and the eavesdroppers jointly receive and decode contents through MRC reception which yields the worst case of wiretap. In addition, the main performance metric is measured by the average probability of successful transmission; the analytical result and analytical lower bound of the average probability of successful transmission are presented in the performance analysis. Due to the nonconvex nature and the complication of the average probability of successful transmission, it is too complicated to get a closed-form solution. Therefore, the GA is used to find the optimal solution instead of deriving a closed-form solution. And in order to better evaluate the proposed caching placement, we use the most popular content (MPC) caching placement as a standard for comparison, where the method of MPC caching placement is to cache the most popular contents in all relays. Finally, the numerical and simulation results are provided to validate the proposed studies.The novelties and main contributions of this paper can be summarized as follows:(i)
Based on the security of transmission, the probabilistic caching placement is designed in the presence of multiple eavesdroppers which follow the homogeneous PPP.
(ii)
The main performance metric is measured by the average probability of successful transmission, and both the analytical result and the analytical lower bound of the average probability of successful transmission are presented. Moreover, GA is used to optimize the average probability of successful transmission to maximize the system performance.
(iii)
The simulation results are provided to demonstrate the studies that the optimized probabilistic caching placement is superior to the MPC caching placement, and the system secure performance can be improved by increasing the transmit power, the cache size, and the intensity of relays but will deteriorate with larger intensity of eavesdroppers.The rest of this paper is organized as follows. In Section2, we introduce the system model and study the probabilistic caching placement and the file transmission. In Section 3, the system performance is analyzed. And the optimization of probabilistic caching placement is presented in Section 4. The numerical and simulation results are provided in Section 5. The conclusions are presented in Section 6.Notations. In this paper, we use P
i
f
i
n
d and P
i
s
u
c to represent the probability of finding the requested file i and the successful probability of transmitting the file i, respectively. Moreover, we use P
¯
s
u
c to represent the average probability of successful transmission and use P
¯
l
o
w
s
u
c to represent the lower bound of average probability of successful transmission.


## 2. System Model

Figure1 depicts the system model of a wireless caching network, which comprises a cellular base station (BS), a random legitimate user (D), multiple DF relays {
R
k
∣
k
=
1,2
,
…
,
K
} with cache capacity, and multiple eavesdroppers {
E
l
∣
l
=
1,2
,
…
,
L
} which can overhear messages and bring out the issue of information security [27–30]. Without loss of generality, the locations of relays are modeled by the homogeneous PPP Φ
r with intensity λ
r and the eavesdroppers’ positions also follow the homogeneous PPP Φ
e with intensity λ
e.Figure 1
System model.In this system model, we consider BS has no direct link withD and eavesdroppers [31–33], the transmission is performed only via relays [34–36], and all relays can successfully send the files in their local cache to D within radius R. For legitimate D, if the requested file can be found in relays, the nearest relay directly transfers the file to D; otherwise, the file will be transmitted from BS to the nearest base station and then transmitted to D. Moreover, all wireless links are subjected to Rayleigh flat fading channel with a path loss governed by the exponent α
>
2 [37–39].


### 2.1. Cache Placement

We assume that there areN files that have been requested to D, which all have the same size. The case of unequal size will not be considered in this paper, but we can always assume that any file can be divided into blocks of the same size, so the similar analysis also can still be applied. In this paper, the files are characterized by their popularity, namely, the probability that a file is requested by the user. The request probability follows the Zipf distribution, which has been widely used in the literature [1–9]; that is, the request probability of ith file is(1)
f
i
=
i
-
γ
∑
j
=
1
N
j
-
γ
,where γ is the Zipf parameter with the popularity skewness. According to the request probability, we can find that f
1
≥
f
2
≥
⋯
≥
f
N and ∑
i
=
1
N
f
i
=
1.In this paper, we consider each relay has the same cache memory sizeC
R
(
C
R
<
N
) and the unit of storage/size is file. Because relays cannot store all files (K
C
R
<
N), relays need to judiciously choose which files to store. Thus, we apply the probabilistic caching placement to the file’s cache placement and by optimizing the cache placement to prove the system performance.In the probabilistic model, the contents are independently placed in the cache memories of different relays, according to the same distribution. Therefore, if each relay cachesith file with a certain probability q
i
(
0
≤
q
i
≤
1
) independently, we denote by q
=
[
q
1
,
…
,
q
N
] the caching probabilities of file i
∈
[
1
,
N
], and due to the cache storage limit, we have(2)
∑
i
=
1
N
q
i
≤
C
R
.In this paper, in order to alleviate the traffic and decline the transmission links, our goal is to find an optimal local caching strategy to optimize the system performance. Therefore, we only consider the secure transmission in local devices.


### 2.2. File Transmission

When a file request occurs, and there is at least one relay that stored the requested file within the radiusr, the request would be satisfied and the relay would directly transmit the file to D. If there is more than one relay which has the requested file, the file will be transmitted from the nearest one. In the case where the requested file can not be found in relays, the file must be forwarded from core network to D assisted by nearest relay. Because we only consider the secure transmission in local devices, in the following, we will only analyze the local transmission.We assume the channel state information (CSI) is known toD; therefore, when D sends the request, the nearest relay R
k
(
k
∈
K
) which has cached the requested file directly transmits the file to D. According to [18], the received S
N
R at D can be shown as follows:(3)
S
N
R
D
=
ρ
η
d
,where ρ
=
p
r
/
σ
2, p
r is the transmit power at relay, σ
2 is the noise power, η
d
≜
|
h
R
k
,
D
|
2
r
R
k
,
D
-
α is the channel gains for D, h
R
k
,
D denotes the channel parameters of R
k
→
D, and r
R
k
,
D represents the distance from D to the nearest relay R
k.The received SNR at a random eavesdropperE
l is given by(4)
S
N
R
E
l
=
ρ
η
e
l
,where η
e
l
≜
h
R
k
,
E
l
2
r
R
k
,
E
l
-
α, h
R
k
,
E
l denotes the channel parameters of R
k
→
E, and r
R
k
,
E
l represents the distance from R
k to E
l.


## 2.1. Cache Placement

We assume that there areN files that have been requested to D, which all have the same size. The case of unequal size will not be considered in this paper, but we can always assume that any file can be divided into blocks of the same size, so the similar analysis also can still be applied. In this paper, the files are characterized by their popularity, namely, the probability that a file is requested by the user. The request probability follows the Zipf distribution, which has been widely used in the literature [1–9]; that is, the request probability of ith file is(1)
f
i
=
i
-
γ
∑
j
=
1
N
j
-
γ
,where γ is the Zipf parameter with the popularity skewness. According to the request probability, we can find that f
1
≥
f
2
≥
⋯
≥
f
N and ∑
i
=
1
N
f
i
=
1.In this paper, we consider each relay has the same cache memory sizeC
R
(
C
R
<
N
) and the unit of storage/size is file. Because relays cannot store all files (K
C
R
<
N), relays need to judiciously choose which files to store. Thus, we apply the probabilistic caching placement to the file’s cache placement and by optimizing the cache placement to prove the system performance.In the probabilistic model, the contents are independently placed in the cache memories of different relays, according to the same distribution. Therefore, if each relay cachesith file with a certain probability q
i
(
0
≤
q
i
≤
1
) independently, we denote by q
=
[
q
1
,
…
,
q
N
] the caching probabilities of file i
∈
[
1
,
N
], and due to the cache storage limit, we have(2)
∑
i
=
1
N
q
i
≤
C
R
.In this paper, in order to alleviate the traffic and decline the transmission links, our goal is to find an optimal local caching strategy to optimize the system performance. Therefore, we only consider the secure transmission in local devices.


## 2.2. File Transmission

When a file request occurs, and there is at least one relay that stored the requested file within the radiusr, the request would be satisfied and the relay would directly transmit the file to D. If there is more than one relay which has the requested file, the file will be transmitted from the nearest one. In the case where the requested file can not be found in relays, the file must be forwarded from core network to D assisted by nearest relay. Because we only consider the secure transmission in local devices, in the following, we will only analyze the local transmission.We assume the channel state information (CSI) is known toD; therefore, when D sends the request, the nearest relay R
k
(
k
∈
K
) which has cached the requested file directly transmits the file to D. According to [18], the received S
N
R at D can be shown as follows:(3)
S
N
R
D
=
ρ
η
d
,where ρ
=
p
r
/
σ
2, p
r is the transmit power at relay, σ
2 is the noise power, η
d
≜
|
h
R
k
,
D
|
2
r
R
k
,
D
-
α is the channel gains for D, h
R
k
,
D denotes the channel parameters of R
k
→
D, and r
R
k
,
D represents the distance from D to the nearest relay R
k.The received SNR at a random eavesdropperE
l is given by(4)
S
N
R
E
l
=
ρ
η
e
l
,where η
e
l
≜
h
R
k
,
E
l
2
r
R
k
,
E
l
-
α, h
R
k
,
E
l denotes the channel parameters of R
k
→
E, and r
R
k
,
E
l represents the distance from R
k to E
l.


## 3. Performance Analysis

In this section, we will analyze the cache hit probability and the average probability of successful transmission, and the average probability of successful transmission is defined as the main performance metric. Moreover, the analytical result and the analytical lower bound of the average probability of successful transmission are presented in this section.


### 3.1. Cache Hit Probability

In this paper, we define the cache hit probability as a probability that the userD successfully finds the requested file in a given area. From the system model, we know that relays are modeled by a PPP Φ
r with intensity λ
r, so the relays caching the ith file also follow a PPP with density q
i
λ
r. According to the notion of stochastic geometry, in a given area within the radius r, the expected number of relays caching the ith file can be calculated as(5)
E
K
=
q
i
λ
r
π
r
2
.And from [3–7], we find that, for a PPP distribution with density λ, the probability that there are n nodes in an area within the radius r is(6)
F
n
,
r
,
λ
=
π
r
2
λ
n
n
!
e
-
π
r
2
λ
.Therefore, if we assume userD is located at the origin and find the requested file in an area within the radius R, the probability of finding at least one relay caching the ith file within a radius R is(7)
P
i
f
i
n
d
=
1
-
F
0
,
R
,
q
i
λ
r
=
1
-
e
-
π
q
i
λ
r
R
2
.


### 3.2. Probability of Successful Transmission

In this paper, we define the probability of successful transmission as the probability of finding and then successfully transmitting the requested file within a radiusR. In order to analyze the probability of successful transmission, we firstly analyze the secrecy capacity which is the difference between the capacities of the legitimate channel C
D and the equivalent wiretap channel C
E. Based on the system model, the secrecy capacity can be expressed as [40–43](8)
C
s
=
log
2
⁡
1
+
S
N
R
D
-
l
o
g
2
1
+
S
N
R
E
+
,where [
x
]
+ returns max
⁡
(
0
,
x
), S
N
R
E
=
ρ
η
e, and η
e
=
∑
E
l
∈
Φ
e
‍
η
e
l is equivalent wiretap channel gain.Therefore, when theith file is requested by D, we use r
i to represent the distance to the nearest relay which has cached ith file. The probability of successful transmission can be shown as follows [44–47]:(9)
P
i
s
u
c
=
P
r
log
2
⁡
1
+
S
N
R
D
-
log
2
⁡
1
+
S
N
R
E
>
R
s
=
P
r
η
d
>
M
+
τ
η
e
=
E
r
i
∫
0
∞
∫
M
+
τ
y
∞
f
η
d
x
f
η
e
y
d
x
d
y
,where R
s is the target secrecy rate, M
=
(
2
R
s
-
1
)
/
ρ, τ
=
2
R
s, and f
η
e and f
η
d are the probability distribution function (PDF) of η
e and η
d, respectively.Because eavesdroppers jointly receive and decode contents with MRC reception, we haveη
e
=
∑
E
l
∈
Φ
e
η
e
l. But since the randomness of eavesdroppers’ positions, the exact closed-form expression for the PDF of η
e is difficult to obtain. However, by using the result from [48] and applying the PDF of η
d as f
η
d
(
x
)
=
r
i
α
e
-
r
i
α
x, we can calculate the successful probability of transmitting ith file P
i
s
u
c as follows:(10)
P
i
s
u
c
=
E
r
i
e
-
r
i
α
M
∫
0
∞
e
-
r
i
α
τ
y
f
η
e
y
d
y
=
E
r
i
e
-
r
i
α
M
L
η
e
s
=
∫
0
∞
e
-
r
i
α
M
L
η
e
s
f
r
i
d
r
i
,where s
=
r
i
α
τ and L
η
e
(
s
) is the Laplace transform of η
e. According to [48], we have(11)
L
η
e
s
=
E
Φ
e
e
-
s
η
e
=
E
Φ
e
exp
⁡
-
s
∑
e
l
∈
Φ
e
η
e
l
=
E
Φ
e
∏
e
l
∈
Φ
e
E
h
e
l
exp
⁡
-
s
h
e
l
2
r
e
l
-
α
=
a
exp
⁡
-
E
h
e
l
∫
0
∞
λ
e
1
-
e
x
p
-
s
h
e
l
2
r
e
l
-
α
2
π
r
e
l
d
r
e
l
=
b
exp
⁡
-
λ
e
π
E
h
e
l
h
e
l
4
/
α
Γ
1
-
2
α
s
2
/
α
,where step (a) holds for the probability generating functional lemma (PGFL) over PPP [48], step (b) holds for the integration formula ∫
0
∞
x
m
exp
⁡
-
β
x
n
d
x
=
Γ
γ
/
(
n
β
γ
), γ
=
(
m
+
1
)
/
n, E
h
e
l
h
e
l
4
/
α can be calculated as(12)
E
h
e
l
h
e
l
4
/
α
=
∫
0
∞
x
2
/
α
f
h
e
l
2
x
d
x
=
Γ
1
+
2
α
,and f
h
e
l
2
x
=
e
-
x is the PDF of wiretap channel gain h
R
k
,
E
l. Therefore, substitute(13)
L
η
e
s
=
exp
⁡
-
β
s
2
/
α
,where β
=
λ
e
π
Γ
1
+
2
/
α
Γ
1
-
2
/
α. In this paper, we assume conditioning on r
i
≤
R as a result of the maximum distance, and the PDF of r
i is given by(14)
f
r
i
=
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
e
-
π
q
i
λ
r
r
i
2
0
≤
r
i
≤
R
0
r
i
>
R
.Substituting (13) and (14) in (10), the successful probability of transmitting the file i is given by(15)
P
i
s
u
c
=
∫
0
∞
exp
⁡
-
r
i
α
M
-
β
τ
2
/
α
r
i
2
×
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
.


### 3.3. The Average Probability of Successful Transmission

In this paper, we define the average probability of successful transmission as the probability of finding and then successfully transmitting all the requested files within a radiusR. Therefore, based on the above analysis, the average probability of successful transmission is given by(16)
P
¯
s
u
c
=
∑
i
=
1
N
f
i
P
i
f
i
n
d
P
i
s
u
c
=
∑
i
=
1
N
f
i
1
-
e
-
π
q
i
λ
r
R
2
×
∫
0
∞
exp
⁡
-
r
i
α
M
-
β
τ
2
/
α
r
i
2
×
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
,following constraints(17)
∑
i
=
1
N
q
i
≤
C
R
,
q
i
∈
0,1
,
i
∈
1
,
N
.From (16), we can find P
¯
s
u
c is a function of various factors, for example, r
i, α, R
s, q
i, R, λ
r as well as λ
e. For any given r
i, α, R
s, R, λ
r, and λ
e,  P
¯
s
u
c solely depends on the caching probability q
i. But since the complication of P
¯
s
u
c, it is complicated to obtain a closed-form expression for P
¯
s
u
c. Thus, in this subsection, we derive its analytical lower bound. The analytical lower bound presents a conservative estimation of P
¯
s
u
c. If the lower bound is higher than the success threshold, the exact P
¯
s
u
c can be definitely guaranteed. The details about the analytical lower bound are shown as follows. We rewrite (10) as(18)
P
i
s
u
c
=
E
r
i
exp
⁡
-
r
i
α
M
exp
⁡
-
β
τ
2
/
α
r
i
2
=
E
r
i
exp
⁡
-
r
i
α
M
E
r
i
exp
⁡
-
β
τ
2
/
α
r
i
2
.According to Jensen’s inequality, we have(19)
P
i
s
u
c
≥
exp
⁡
-
M
E
r
i
r
i
α
exp
⁡
-
β
τ
2
/
α
E
r
i
r
i
2
.Based on the PDF of r
i in (14),  E
r
i
[
r
i
α
] can be calculated as(20)
E
r
i
r
i
α
=
∫
0
∞
r
i
α
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
=
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
×
Γ
1
+
α
/
2
π
q
i
λ
r
1
+
α
/
2
.
E
r
i
r
i
2 can be calculated as(21)
E
r
i
r
i
2
=
∫
0
∞
r
i
2
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
=
Γ
2
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
.Substituting (20) and (21) in (19), we can obtain the closed-form expression of the lower bound P
i
l
o
w
s
u
c as(22)
P
i
l
o
w
s
u
c
=
exp
⁡
-
M
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
×
Γ
1
+
α
/
2
π
q
i
λ
r
1
+
α
/
2
×
exp
⁡
-
β
τ
α
/
2
Γ
2
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
.Therefore, the closed-form expression of the lower bound P
l
o
w
s
u
c is given by(23)
P
¯
l
o
w
s
u
c
=
∑
i
=
1
N
f
i
P
i
f
i
n
d
P
i
l
o
w
s
u
c
=
∑
i
=
1
N
f
i
1
-
e
-
π
q
i
λ
r
R
2
×
exp
⁡
-
M
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
×
Γ
1
+
α
/
2
π
q
i
λ
r
1
+
α
/
2
×
e
x
p
-
β
τ
2
/
α
Γ
2
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
.


## 3.1. Cache Hit Probability

In this paper, we define the cache hit probability as a probability that the userD successfully finds the requested file in a given area. From the system model, we know that relays are modeled by a PPP Φ
r with intensity λ
r, so the relays caching the ith file also follow a PPP with density q
i
λ
r. According to the notion of stochastic geometry, in a given area within the radius r, the expected number of relays caching the ith file can be calculated as(5)
E
K
=
q
i
λ
r
π
r
2
.And from [3–7], we find that, for a PPP distribution with density λ, the probability that there are n nodes in an area within the radius r is(6)
F
n
,
r
,
λ
=
π
r
2
λ
n
n
!
e
-
π
r
2
λ
.Therefore, if we assume userD is located at the origin and find the requested file in an area within the radius R, the probability of finding at least one relay caching the ith file within a radius R is(7)
P
i
f
i
n
d
=
1
-
F
0
,
R
,
q
i
λ
r
=
1
-
e
-
π
q
i
λ
r
R
2
.


## 3.2. Probability of Successful Transmission

In this paper, we define the probability of successful transmission as the probability of finding and then successfully transmitting the requested file within a radiusR. In order to analyze the probability of successful transmission, we firstly analyze the secrecy capacity which is the difference between the capacities of the legitimate channel C
D and the equivalent wiretap channel C
E. Based on the system model, the secrecy capacity can be expressed as [40–43](8)
C
s
=
log
2
⁡
1
+
S
N
R
D
-
l
o
g
2
1
+
S
N
R
E
+
,where [
x
]
+ returns max
⁡
(
0
,
x
), S
N
R
E
=
ρ
η
e, and η
e
=
∑
E
l
∈
Φ
e
‍
η
e
l is equivalent wiretap channel gain.Therefore, when theith file is requested by D, we use r
i to represent the distance to the nearest relay which has cached ith file. The probability of successful transmission can be shown as follows [44–47]:(9)
P
i
s
u
c
=
P
r
log
2
⁡
1
+
S
N
R
D
-
log
2
⁡
1
+
S
N
R
E
>
R
s
=
P
r
η
d
>
M
+
τ
η
e
=
E
r
i
∫
0
∞
∫
M
+
τ
y
∞
f
η
d
x
f
η
e
y
d
x
d
y
,where R
s is the target secrecy rate, M
=
(
2
R
s
-
1
)
/
ρ, τ
=
2
R
s, and f
η
e and f
η
d are the probability distribution function (PDF) of η
e and η
d, respectively.Because eavesdroppers jointly receive and decode contents with MRC reception, we haveη
e
=
∑
E
l
∈
Φ
e
η
e
l. But since the randomness of eavesdroppers’ positions, the exact closed-form expression for the PDF of η
e is difficult to obtain. However, by using the result from [48] and applying the PDF of η
d as f
η
d
(
x
)
=
r
i
α
e
-
r
i
α
x, we can calculate the successful probability of transmitting ith file P
i
s
u
c as follows:(10)
P
i
s
u
c
=
E
r
i
e
-
r
i
α
M
∫
0
∞
e
-
r
i
α
τ
y
f
η
e
y
d
y
=
E
r
i
e
-
r
i
α
M
L
η
e
s
=
∫
0
∞
e
-
r
i
α
M
L
η
e
s
f
r
i
d
r
i
,where s
=
r
i
α
τ and L
η
e
(
s
) is the Laplace transform of η
e. According to [48], we have(11)
L
η
e
s
=
E
Φ
e
e
-
s
η
e
=
E
Φ
e
exp
⁡
-
s
∑
e
l
∈
Φ
e
η
e
l
=
E
Φ
e
∏
e
l
∈
Φ
e
E
h
e
l
exp
⁡
-
s
h
e
l
2
r
e
l
-
α
=
a
exp
⁡
-
E
h
e
l
∫
0
∞
λ
e
1
-
e
x
p
-
s
h
e
l
2
r
e
l
-
α
2
π
r
e
l
d
r
e
l
=
b
exp
⁡
-
λ
e
π
E
h
e
l
h
e
l
4
/
α
Γ
1
-
2
α
s
2
/
α
,where step (a) holds for the probability generating functional lemma (PGFL) over PPP [48], step (b) holds for the integration formula ∫
0
∞
x
m
exp
⁡
-
β
x
n
d
x
=
Γ
γ
/
(
n
β
γ
), γ
=
(
m
+
1
)
/
n, E
h
e
l
h
e
l
4
/
α can be calculated as(12)
E
h
e
l
h
e
l
4
/
α
=
∫
0
∞
x
2
/
α
f
h
e
l
2
x
d
x
=
Γ
1
+
2
α
,and f
h
e
l
2
x
=
e
-
x is the PDF of wiretap channel gain h
R
k
,
E
l. Therefore, substitute(13)
L
η
e
s
=
exp
⁡
-
β
s
2
/
α
,where β
=
λ
e
π
Γ
1
+
2
/
α
Γ
1
-
2
/
α. In this paper, we assume conditioning on r
i
≤
R as a result of the maximum distance, and the PDF of r
i is given by(14)
f
r
i
=
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
e
-
π
q
i
λ
r
r
i
2
0
≤
r
i
≤
R
0
r
i
>
R
.Substituting (13) and (14) in (10), the successful probability of transmitting the file i is given by(15)
P
i
s
u
c
=
∫
0
∞
exp
⁡
-
r
i
α
M
-
β
τ
2
/
α
r
i
2
×
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
.


## 3.3. The Average Probability of Successful Transmission

In this paper, we define the average probability of successful transmission as the probability of finding and then successfully transmitting all the requested files within a radiusR. Therefore, based on the above analysis, the average probability of successful transmission is given by(16)
P
¯
s
u
c
=
∑
i
=
1
N
f
i
P
i
f
i
n
d
P
i
s
u
c
=
∑
i
=
1
N
f
i
1
-
e
-
π
q
i
λ
r
R
2
×
∫
0
∞
exp
⁡
-
r
i
α
M
-
β
τ
2
/
α
r
i
2
×
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
,following constraints(17)
∑
i
=
1
N
q
i
≤
C
R
,
q
i
∈
0,1
,
i
∈
1
,
N
.From (16), we can find P
¯
s
u
c is a function of various factors, for example, r
i, α, R
s, q
i, R, λ
r as well as λ
e. For any given r
i, α, R
s, R, λ
r, and λ
e,  P
¯
s
u
c solely depends on the caching probability q
i. But since the complication of P
¯
s
u
c, it is complicated to obtain a closed-form expression for P
¯
s
u
c. Thus, in this subsection, we derive its analytical lower bound. The analytical lower bound presents a conservative estimation of P
¯
s
u
c. If the lower bound is higher than the success threshold, the exact P
¯
s
u
c can be definitely guaranteed. The details about the analytical lower bound are shown as follows. We rewrite (10) as(18)
P
i
s
u
c
=
E
r
i
exp
⁡
-
r
i
α
M
exp
⁡
-
β
τ
2
/
α
r
i
2
=
E
r
i
exp
⁡
-
r
i
α
M
E
r
i
exp
⁡
-
β
τ
2
/
α
r
i
2
.According to Jensen’s inequality, we have(19)
P
i
s
u
c
≥
exp
⁡
-
M
E
r
i
r
i
α
exp
⁡
-
β
τ
2
/
α
E
r
i
r
i
2
.Based on the PDF of r
i in (14),  E
r
i
[
r
i
α
] can be calculated as(20)
E
r
i
r
i
α
=
∫
0
∞
r
i
α
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
=
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
×
Γ
1
+
α
/
2
π
q
i
λ
r
1
+
α
/
2
.
E
r
i
r
i
2 can be calculated as(21)
E
r
i
r
i
2
=
∫
0
∞
r
i
2
2
π
q
i
λ
r
r
i
1
-
e
-
π
q
i
λ
r
R
2
exp
⁡
-
π
q
i
λ
r
r
i
2
d
r
i
=
Γ
2
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
.Substituting (20) and (21) in (19), we can obtain the closed-form expression of the lower bound P
i
l
o
w
s
u
c as(22)
P
i
l
o
w
s
u
c
=
exp
⁡
-
M
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
×
Γ
1
+
α
/
2
π
q
i
λ
r
1
+
α
/
2
×
exp
⁡
-
β
τ
α
/
2
Γ
2
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
.Therefore, the closed-form expression of the lower bound P
l
o
w
s
u
c is given by(23)
P
¯
l
o
w
s
u
c
=
∑
i
=
1
N
f
i
P
i
f
i
n
d
P
i
l
o
w
s
u
c
=
∑
i
=
1
N
f
i
1
-
e
-
π
q
i
λ
r
R
2
×
exp
⁡
-
M
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
×
Γ
1
+
α
/
2
π
q
i
λ
r
1
+
α
/
2
×
e
x
p
-
β
τ
2
/
α
Γ
2
π
q
i
λ
r
1
-
e
-
π
q
i
λ
r
R
2
.


## 4. Optimization of Probabilistic Caching Placement

From the performance analysis, it can be seen that the caching parameterq
i affects the system secure performance significantly. Therefore, in this section, the optimization of probabilistic caching placement is to find the optimal caching probability q
i
∗
(
i
∈
[
1
,
N
]
). But due to the nonconvex nature and the complication of P
¯
s
u
c, it is too complicated to get a closed-form solution of q
i
∗. Based on the above considerations, we utilize the GA to find the optimal solution of q
i
∗ instead of deriving a closed-form solution. The details about the optimization of genetic algorithm are shown in Algorithm 1.Algorithm 1:Optimization of probabilistic caching placement.
Input: input parameters N, λ
r, λ
e, p
r, p
r
m
i
n, p
r
m
a
x
Output: output the optimal caching probability q
∗
=
[
q
1
∗
,
…
,
q
N
∗
]
and the average probability of success transmissionP
-
s
u
c
∗
(1) Initialize P
-
s
u
c
=
z
e
r
o
s
(
0
)
(2)
j
=
1
(3)
for  p
r
=
p
r
m
i
n
:
p
r
m
a
x  do
(4)
[
x
,
f
v
a
l
]
=
g
a
_
m
a
i
n
(
N
,
λ
r
,
λ
e
,
p
r
)
(5)
q
∗
(
j
,
:
)
=
x
(
e
n
d
-
3
)
(6)
P
-
s
u
c
∗
(
j
)
=
-
f
v
a
l
(7)
j
=
j
+
1
(8)
end for
(9)
function  g
a
_
m
a
i
n
N
,
λ
r
,
λ
e
,
p
r
(10) ObjectiveFunction=@
g
a
_
f
i
t
n
e
s
s
(11)
n
v
a
r
s
=
N
+
3
(12)  Initialize L
B and U
B
(13)  ConstraintFunction=@
g
a
_
c
o
n
s
t
r
a
i
n
t
(14)
[
q
i
∗
,
f
v
a
l
]
=
g
a
(
O
b
j
e
c
t
i
v
e
F
u
n
c
t
i
o
n
,
n
v
a
r
s,[
],[
],[
],[
],
L
B
,
U
B
,
… ,C
o
n
s
t
r
a
i
n
t
F
u
n
c
t
i
o
n
,
o
p
t
i
o
n
s
)
(15)
return  [
q
i
∗
,
f
v
a
l
]
(16)
end function
(17)
function  g
a
_
f
i
t
n
e
s
s
x
(18)
q
i
∗
=
x
(
1
:
e
n
d
-
3
)
(19)
p
r
=
x
(
e
n
d
)
(20)  calculate f
i, P
i
f
i
n
d, P
i
s
u
c
(21)
P
-
s
u
c
∗
=
∑
i
=
1
N
f
i
P
i
f
i
n
d
P
i
s
u
c
(22)
y
=
-
P
-
s
u
c
∗
(23)
return  y
(24) end functionNotation.
N denotes the number of total files, λ
r denotes the intensity of relays, λ
e denotes the intensity of eavesdroppers, p
r denotes the transmit power at relay, and p
r
m
i
n and p
r
m
a
x represent the minimum of transmit power and the maximum of transmit power, respectively. In addition, q
∗ represents the optimal caching probability of the ith file, P
¯
s
u
c
∗ represents the average probability of all files successful transmission, and LB and UB represent the lower bound and upper bound of variables, respectively.In Algorithm1, the function g
a
_
m
a
i
n (lines (9)–(16)) is the calling function of GA. The main intension of the function g
a
_
m
a
i
n is to define the number of variables (line (11)), initialize lower bound and upper bound (line (12)), and call the fitness function and constraint function of GA to return the optimal q
i
∗ and the minimum f
v
a
l (line (14)). The fitness function of GA is presented from lines (17) to (24). The main ideas of the fitness function are to take one input vector x, where x has as many elements as number of variables, then compute the value of the function, and return that scalar value in its one return argument y. It is worth noting that all variables consist of the caching probability of N files, the intensity of relays λ
r, the intensity of eavesdroppers λ
e, and the transmit power of relays p
r, so the length of x is equal to N
+
3, where x is the vector of all variables. But because there are only N files, we can get that the length of q should be equal to l
e
n
g
t
h
(
x
)
-
3. Moreover, because the function of GA is to find the minimum value, we define argument y as the negative of P
¯
s
u
c
∗. Similarly, the GA function assumes the constraint function will take one input x, where x has as many elements as number of variables in the problem. Furthermore, the constraint function computes the values of all the inequality and equality constraints and designs two vectors c and c
e
q, respectively, where c
=
sum
⁡
(
x
(
1
:
e
n
d
-
3
)
)
-
C
R and c
e
q
=. The details about the algorithm optimization and the associated analysis can be found in the literature, such as the works [49–52].


## 5. Numerical and Simulation Results

In this section, the numerical and simulation results are presented to verify the system secure performance in the presence of multiple eavesdroppers and illustrate the effect of key system parameters. In addition, the system performances are compared with the traditional MPC caching placement. Without loss of generality, the secrecy data rateR
s is set to 0.1 bps/Hz, and the noise power is set to one.As shown in Figure2, this figure depicts the effect of the number of files N on the average probability of successful transmission, where p
r
=
30dB, C
R
=
5, α
=
2.1, γ
=
0.5, R
=
100, λ
r
=
4
×
1
0
-
3, and λ
e
=
1
×
1
0
-
5. From this figure, we can see that the average probability of successful transmission decreases as N increases. And when the number of files N is equal to C
R, the average probability of successful transmission of MPC caching placement is equal to the analytical result of probabilistic caching placement and the analytical lower bound of probabilistic caching placement. However, when N is larger than C
R, the performance of probabilistic caching placement is better than MPC caching placement, and with increasing N, MPC caching placement deteriorates more rapidly than the probabilistic caching placement. The reason is that the MPC caching placement combines all signals to exploit the signal cooperation gain, but the proposed probabilistic caching placement achieves the balance between the signal cooperation gain and the caching diversity gain.Figure 2
Effect ofN on the average probability of successful transmission.Figure3 shows the effect of the cache size of relay C
R on the average probability of successful transmission, where N
=
12, p
r
=
30dB, γ
=
0.5, R
=
100, α
=
2.1, λ
r
=
4
×
1
0
-
3, and λ
e
=
1
×
1
0
-
5. As observed from the figure, the average probability of successful transmission becomes better as C
R increases, and it is also obvious that the performance of probabilistic caching placement is always higher than MPC caching placement. Moreover, from the picture, we can see that as C
R increases, the analytical result of probabilistic caching placement and the analytical lower bound of probabilistic caching placement are quite closer. And the average probability of successful transmission of probabilistic caching placement and MPC caching placement becomes closer as the value of C
R increases.Figure 3
Effect ofC
R on the average probability of successful transmission.Figure4 shows the effect of the intensity of relays λ
r and the intensity of eavesdroppers λ
e on the average probability of successful transmission, where N
=
10, p
r
=
30dB, C
R
=
5, α
=
2.1, γ
=
0.5, and R
=
100. As observed from the figure, the average probability of successful transmission becomes better as λ
r increases, and the average probability of successful transmission with λ
e
=
1
×
1
0
-
5 is higher than that with λ
e
=
1
×
1
0
-
4. Thus, we can find when the value of λ
e increases, the average probability of successful transmission will decrease. In addition, from the figure, we also can find that the analytical result and analytical lower bound of probabilistic caching placement are quite closer with increasing λ
r. And when λ
e
=
1
×
1
0
-
4, the difference between the analytical result and analytical lower bound is more obvious than λ
e
=
1
×
1
0
-
5. Moreover, when λ
e
=
1
×
1
0
-
5, the analytical result and analytical lower bound of probabilistic caching placement are both higher than MPC caching placement. However when λ
e
=
1
×
1
0
-
4, the analytical lower bound of probabilistic caching placement is lower than MPC. But we also can find the difference between the analytical result and the analytical lower bound of probabilistic caching placement becomes quite closer with increasing λ
r. It is worth noting that the performance without considering security is superior to the performance of considering secure transmission, but in the actual situation, eavesdroppers exist, and we cannot just consider the performance of main channel, and we also need to consider the eavesdropper’s channel.Figure 4
Impact ofλ
r and λ
e on the average probability of successful transmission.Figure5 shows the effect of the transmit power at relay p
r and the path loss α on the average probability of successful transmission, where N
=
10, C
R
=
5, γ
=
0.5, R
=
100, λ
r
=
4
×
1
0
-
3, and λ
e
=
1
×
1
0
-
5. From this figure, we can find that the average probability of successful transmission increases as p
r increases. Moreover, the analytical result and the analytical lower bound of probabilistic caching placement is quite closer with increasing p
r. In addition, the analytical result of probabilistic caching placement is always higher than MPC caching placement. And for the probabilistic caching placement, with α
=
2.1, the associated average probability of successful transmission is better than that with α
=
3, so we can obtain that the average probability of successful transmission deteriorates with larger α. Furthermore, when α
=
3 and p
r
≤
12dB, the average probability of successful transmission of probabilistic caching placement is almost equal to MPC caching placement. But when p
r
≥
12dB, the average probability of successful transmission of probabilistic caching placement is always higher than MPC caching placement. The reason is that increasing the value of transmit power p
r can exploit the signal cooperation gain and the caching diversity gain, but the MPC caching placement only can utilize the signal cooperation gain, and the probabilistic caching placement can exploit both the signal cooperation gain and the caching diversity gain.Figure 5
Impact ofp
r and α on the average probability of successful transmission.


## 6. Conclusions

In this paper, we designed, analyzed, and optimized the probabilistic caching placement in the presence of multiple eavesdroppers. And the average probability of successful transmission was defined as the main performance metric, which is the probability of finding and then successfully transmitting all the requested files within a radiusR. Moreover, the analytical result and the analytical lower bound of average probability of successful transmission were both presented. But due to the nonconvex nature and the complication of average probability of successful transmission, the GA was used to find the optimal solution instead of deriving a closed-form solution. Finally, simulation results were provided to support the studies that the proposed probabilistic caching placement is superior to the MPC caching placement. In addition, the system secure performance can be improved by increasing the value of p
r, C
R, and λ
r but will deteriorate with larger N and λ
e.



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*Source: 2104162-2018-05-08.xml*