1

Interference Exploitation in D2D-enabled CellularNetworks: A Secrecy Perspective

Chuan Ma, Jiaqi Liu, Xiaohua Tian, Hui Yu, Ying Cui and Xinbing Wang

Abstract—Device-to-device (D2D) communication underlayingcellular networks is a promising technology to improve networkresource utilization. In D2D-enabled cellular networks, interfer-ence generated by D2D communications is usually viewed asan obstacle to cellular communications. However, in this paper,we present a new perspective on the role of D2D interferenceby taking security issues into consideration. We consider alarge-scale D2D-enabled cellular network with eavesdroppersoverhearing cellular communications. Using stochastic geometry,we model such a network and analyze the signal-to-interference-plus-noise ratio (SINR) distributions, connection probabilitiesand secrecy probabilities of both the cellular and D2D links. Wepropose two criteria for guaranteeing performances of securecellular communications, namely the strong and weak perfor-mance guarantee criteria. Based on the obtained analytical resultsof link characteristics, we design optimal D2D link schedulingschemes under these two criteria respectively. Both analyticaland numerical results show that the interference from D2Dcommunications can enhance physical layer security of cellularcommunications and at the same time create extra transmissionopportunities for D2D users.

Index Terms—D2D communication, cellular network, physicallayer security, link scheduling, stochastic geometry.

I. INTRODUCTION

Recently, there has been a rapid increase in the demand oflocal area services and proximity services (ProSe) among thehighly-capable user equipments (UEs) in cellular networks.Consequently, device-to-device (D2D) communication, whichenables direct communication between UEs that are in proxim-ity, has been proposed as a competitive technology componentfor next generation cellular networks. The integration of D2Dcommunication to cellular networks holds the promise ofmany types of advantages [1]: allowing for high-rate low-delaylow-power transmission for proximity services, increasingfrequency reuse factor and network capacity, facilitating newtypes of peer-to-peer services, etc. For these reasons, D2Dcommunication has strongly appealed to both academia [2],[3] and industry [4]–[6].

However, the introduction of D2D communication alsobrings a number of technical challenges, such as devicediscovery, mode selection and intra-cell interference man-agement. Intra-cell interference, referring to the interferencebetween D2D and cellular links that share the same time-frequency resources within a cell, becomes a major issue inD2D-enabled cellular networks. Especially the interferencegenerated by D2D links, if not properly managed, would

The authors are with the Department of Electronic Engineering, ShanghaiJiao Tong University, China, e-mail: {oknewkimi, 13-liujiaqi, xtian, yuhui,cuiying, xwang8}@sjtu.edu.cn.

severely hamper the performance of cellular links in thenetwork. To guarantee reliable cellular communications inD2D-enabled cellular networks, extensive research has beenundertaken on the management of interference generated byD2D communications. To date, most proposed schemes canbe categorized into the following three types.

• Interference avoidance: Orthogonal time-frequency re-source allocation schemes are adopted to avoid interfer-ence from D2D links to cellular links [7].

• Interference coordination: Intelligent power control andlink scheduling schemes are employed to mitigate inter-ference from D2D links to cellular links [8]–[10].

• Interference cancellation: Advanced coding and decodingmethods are used at cellular and/or D2D links to cancelinterfering signals from desired signals [11], [12].

In the above work, the interference generated by D2Dcommunications is purely viewed as an obstacle to cellularcommunications. However, when privacy and security issuesare taken into consideration, such interference may play acompletely different role. We consider a D2D-enabled cellularnetwork with eavesdroppers overhearing cellular communi-cations. According to literature on physical layer security[13], [14], perfect secrecy of cellular communications can beachieved at the physical layer by adopting secrecy codingschemes. References [15] and [16] further introduced coop-erative jammers which transmit jamming signals to eaves-droppers to improve secrecy capacity. In D2D-enabled cellularnetworks with eavesdroppers, D2D transmitters can play therole of cooperative jammers for cellular communications ifthe interference from D2D links to eavesdropping links ismore severe than that to cellular links. In this situation,interference generated by D2D links is not harmful to buthelpful for secure cellular communications. Based on theabove observation, we propose the notion of D2D interferenceexploitation in D2D-enabled cellular networks, which meansthat the interference generated by D2D communications canbe exploited to enhance secure cellular communications and atthe same time create extra transmission opportunities for D2Dusers. Our prior work [17] put forward this idea and studied thesecrecy performance for point-to-point models. In this paper,we extend our analysis to a large-scale D2D-enabled cellularnetwork and investigate the effect of D2D communications onsecrecy performance of the cellular network.

The main contributions of this paper are summarized asfollows.

(1) We model a large-scale D2D-enabled cellular networkin the presence of eavesdroppers via stochastic geometry, and

2

derive the general expressions for signal-to-interference-plus-noise ratio (SINR) distributions of typical cellular links, eaves-dropping links and D2D links, and the connection probabilityand secrecy probability of typical cellular links. To illustratethe use of the general expressions, we further derive accurateclosed-form bounds for interference-limited case.

(2) Based on the obtained analytical results, we first intro-duce the strong performance guarantee criterion for cellularcommunications. Under this criterion, the average number ofreliable and secure cellular links should not be reduced byintroducing D2D links. Then, we analyze the feasible region ofD2D scheduling parameters for satisfying the strong guaranteecriterion, and design optimal D2D link scheduling schemeswithin the feasible region to maximize the numbers of cellularlinks and D2D links respectively.

(3) We also introduce the weak performance guarantee crite-rion for cellular communications. This criterion corresponds toa certain level of performance degradation of cellular links. Weanalyze the feasible region of D2D scheduling parameters aswell as reasonable values of the required minimum connectionand secrecy probability of cellular links, and design optimalD2D link scheduling schemes within the feasible region tomaximize the numbers of cellular links and D2D links re-spectively.

The remaining part of this paper is organized as follows.Section II presents the related work. Section III describesthe system model. Section IV analyzes SINR distributions,connection probabilities and secrecy probabilities. Sections Vand VI investigate optimal D2D link scheduling problems withrespect to two different criteria. Section VII presents numericalresults and section VIII concludes the paper. A summary ofthe notations used in this paper is given in Table I.

II. RELATED WORK

Interference management for D2D communication. Xu etal. [7] proposed a combinatorial auction approach to allo-cate orthogonal resources between cellular and D2D users.Kaufman et al. [8] presented an opportunistic communicationscheme in which the D2D network can communicate as a fullyloaded cellular network. Pei and Liang [9] designed a spectrumsharing protocol that enables D2D users to communicate bi-directionally. Fodor et al. [10] introduced a network codingscheme to integrated D2D-cellular networks such that a higherspectral and energy efficiency can be achieved. Min et al. [11]designed an interference cancellation scheme that exploits aretransmission of the interference from the base station. Ma etal. [12] proposed two superposition coding based cooperativerelaying schemes to exploit the transmission opportunities forD2D users. All the above work viewed the D2D interferenceas an obstacle to cellular communications, and focused on theavoidance, mitigation and cancellation of D2D interference.However, in this paper, we present a new perspective on therole of D2D interference and investigate how to exploit suchinterference.

Physical layer security for wireless networks. Security isan important issue in wireless networks due to the openwireless medium [18], [19]. Physical layer security using an

information-theoretic point of view has attracted considerablerecent attention. Wyner proposed the wire-tap channel modeland the concept of perfect secrecy for point-to-point commu-nication in his pioneering work [13]. Csiszár and Körner [14]extended Wyner’s results to broadcast channels. Goel et al.[15] and Tang et al. [16] showed that cooperative jammingcan improve secrecy capacity of point-to-point systems. Theresearch on physical layer security for large-scale wirelessnetworks focused mainly on connectivity [20], [21], secrecycapacity [22], [23] and capacity scaling laws [24]. In [20],[21], secrecy communication graph and percolation theorywere employed to analyze secure connectivity in large-scalewireless networks. In [22] and [23], the throughput cost ofachieving a certain level of security in interference-limitednetworks was analyzed. In [24], the asymptotic behavior ofsecrecy capacity of ad hoc network was investigated.

Stochastic geometry for wireless networks. As a mathemat-ical tool to study random spatial patterns, stochastic geometrycan be used to model and analyze interference, connectivityand coverage in large-scale wireless networks [25]. Recentyears, many tractable models have been proposed for analyzingad hoc [26], [27], cellular [28], [29] and D2D [30]–[32] net-works via stochastic geometry. Specially, in [30]–[32], a D2D-enabled cellular network was modeled by two independentPoisson point processes, and the SINR distributions of bothcellular and D2D links were derived. References [33], [34]studied secrecy performance of ad hoc networks via stochasticgeometry. Different from the above work, we consider a morecomplex scenario that the D2D-enabled cellular network isoverheard by eavesdroppers, and study the effect of D2D com-munications on secure cellular communications. The analysisof such complex scenario is more challenging than that of D2Dnetworks without eavesdroppers [30]–[32] and that of ad hocnetworks with eavesdroppers [33], [34].

III. SYSTEM MODEL

In this section, we elaborate on the network model anddescribe the secrecy coding scheme.

A. Network Model

We consider a hybrid network consisting of cellular links,D2D links and a set of eavesdroppers that overhear the trans-mission of cellular links1 over a large two-dimensional space,as shown in Fig.1. The base stations (BSs) are assumed to bespatially distributed as a homogeneous Poisson point process(PPP) Φb of intensity λb, and an independent collection ofcellular users is assumed to be located according to some in-dependent stationary point process Φc. The downlink scenariois considered for cellular communications, and each cellularuser is assumed to connect to its strongest BS instantaneously,i.e. the BS that offers the highest received SINR. We assumethere is no intra-cell interference between cellular links due toorthogonal multiple access within a cell.

1We do not consider the security requirement for D2D links in the model.Therefore, a transceiver pair is not allowed to use D2D mode if it has somesecurity requirement.

3

Table I: Notations used in the paperNotation Description

Φb Poisson point process of base stationsΦd Poisson point process of D2D linksΦe Poisson point process of eavesdroppersPb Transmission power of base stationsPd Transmission power of D2D usersα Path loss exponent

(δ = 2

α

)Tϕ SINR threshold for connection of cellular linksTϵ SINR threshold for secrecy of cellular linksTσ SINR threshold for connection of D2D links

p(c)con

(Tϕ

)Connection probability of cellular links

p(c)sec (Tϵ) Secrecy probability of cellular links

p(d)con (Tσ) Connection probability of D2D links

ϕ Connection probability requirement for cellular linksϵ Secrecy probability requirement for cellular linksNc Average number of perfect cellular links per unit areaNd Average number of perfect D2D links per unit area

The spatial locations of eavesdroppers in the network aremodeled as a homogeneous PPP Φe of intensity λe. We assumethat each cellular link is exposed to all the eavesdroppers andits secrecy data rate is determined by the most detrimentaleavesdropper, i.e. the eavesdropper with the highest receivedSINR of cellular signal. The locations of D2D transmitters inthe network are arranged according to a homogeneous PPP Φd

of intensity λd, and for a given D2D transmitter, its associatedreceiver is assumed to be located at a fixed distance l awaywith isotropic direction. It is noted that other distributions ofl can be easily incorporated into the framework, and somediscussions are provided in the following section.

The transmission powers are assumed to be Pb at BSs andPd at D2D transmitters. We adopt a unified channel modelthat comprises standard path loss and Rayleigh fading forboth cellular and D2D links: given transmission power Pof transmitter xi, the received power at receiver xj can beexpressed as Ph ∥xi − xj∥−α, where h is the fading factorfollowing an exponential distribution with unit mean, i.e.h ∼ exp (1), and α > 2 is the path loss exponent. In laterparts of this paper we use parameter δ to denote 2

α for brevityof expressions. In addition, the noise power at the receiver isassumed to be additive and constant with value σ2.

B. Secrecy Coding

The cellular links are assumed to be eavesdropped in themodel. To fight against eavesdropping, each cellular trans-mitter (say BS for downlink scenario) adopts secrecy codingscheme, such as Wyner code [13], to encode the data beforetransmission. We assume Wyner code is employed in thispaper and thus two kinds of rates, namely the rate of thetransmitted codewords Rc and the rate of the transmittedmessage Rm (Rc > Rm), need to be determined at the cellulartransmitter. The design of Rc and Rm should consider bothconnection and secrecy of the cellular link:

• Connection. If the rate of the transmitted codewords Rc

is above the capacity of the cellular link, the receivedsignal at the cellular receiver can be decoded with anarbitrarily small error, and thus perfect connection of

Base station

Cellular user

D2D user

Eavesdropper

Figure 1: Network model.

the cellular link can be achieved. Otherwise, connectionoutage occurs in the cellular link.

• Secrecy. The rate redundancy Rc − Rm is to providesecrecy. If the rate redundancy is above the capacityof the most detrimental eavesdropping link, the receivedsignal at any eavesdropper provides no information aboutthe transmitted message, and thus perfect secrecy of thecellular link can be achieved. Otherwise, secrecy outageoccurs in the cellular link.

Perfect cellular transmission implies that both perfect connec-tion and perfect secrecy of the cellular link are achieved. Sincedecoding capability is generally determined by the receivedSINR, the perfect transmission of a cellular link can be definedin terms of received SINRs of both the cellular link and theeavesdropping link:

Definition 1. [Perfect transmission] A cellular transmissionis said to be perfect if SINRc > Tϕ and SINRe < Tϵ,where SINRc, SINRe denote the received SINRs at the cellularreceiver and the most detrimental eavesdropper respectively,and Tϕ, Tϵ represent the corresponding threshold SINR values.

Perfect transmission cannot be always achieved due to time-varying wireless environment. Therefore, in practical networksconstraints on connection probability and secrecy probabilityare usually pre-set to control network performance. Accord-ingly, we have the following definition:

Definition 2. [(ϕ, ϵ)-Perfect transmission] A cellular trans-mission is said to be (ϕ, ϵ)-perfect if P (SINRc > Tϕ) ≥ ϕand P (SINRe < Tϵ) ≥ ϵ, where 0 ≤ ϕ, ϵ ≤ 1 denotethe required minimum connection probability and minimumsecrecy probability respectively.

By the above definitions, perfect transmission is equiv-alent to (1, 1)-perfect transmission. In addition, for (ϕ, ϵ)-perfect transmission, 1− ϕ and 1− ϵ represent the maximumconnection outage probability and maximum secrecy outageprobability respectively.

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IV. ANALYSIS OF SECRECY TRANSMISSION IND2D-ENABLED CELLULAR NETWORKS

In this section, we analyze the SINR distributions of typicallinks in large-scale D2D-enabled cellular networks, and derivethe connection probability and secrecy probability of cellularlinks as well as the connection probability of D2D links. Thederived results are the baseline for designing link schedulingschemes in later sections.

A. Connection of Cellular Links

Without loss of generality, we conduct analysis on a typicalcellular user located at the origin. The distance between theBS located at point x and the typical cellular user is denotedby rx, and the fading factor between BS x and the typicalcellular user is denoted by gx which is i.i.d exponential, i.e.gx ∼ exp (1). Then, the received SINR at the typical cellularuser from BS x can be expressed as

SINRc (x) =Pbgxr

−αx

σ2 + Ic (x), (1)

where

Ic (x) =∑

xi∈Φb\{x}

Pbgi ∥xi∥−α+∑

yi∈Φd

Pdhi ∥yi∥−α (2)

is the cumulative interference from all other BSs that arelocated at xi with fading factor gi and D2D transmitters thatare located at yi with fading factor hi

2.We assume that each cellular user associates with its

strongest BS. Thus a cellular user is connected to the networkwhen its SINR from the strongest BS is above the thresholdTϕ, while it is dropped from the network when SINR isbelow Tϕ. The link that connects the typical cellular userand its strongest BS is referred to as typical cellular link, andthe connection probability of the typical cellular link can bedefined as

p(c)con (Tϕ)△= P

[maxx∈Φb

SINRc (x) > Tϕ

]. (3)

In the following theorem and proposition, we provide an up-per bound on the connection probability and show a sufficientcondition under which the bound can be achieved.

Theorem 1. The connection probability of the typical cellularlink in D2D-enabled cellular networks is bounded from aboveby

p(c)con (Tϕ) ≤ 2πλb

∫ ∞

0

exp(−P−1

b Tϕrαxσ

2 − πr2xTδϕµ)rxdrx,

(4)

where µ = λb

[1 + λd

λb

(Pd

Pb

)δ]sinc−1 δ.

Proof: The probability that the strongest SINR is aboveTϕ equals the probability that at least one SINR is above Tϕ.Therefore, the connection probability of the typical cellular

2To distinguish different links, in this paper we use g ∼ exp (1) , h ∼exp (1) to represent the fading factor for links related to cellular transmitter(say BS) and D2D transmitter respectively. It is noted that there is no essentialdistinction between these two symbols.

link under the strongest-BS association model can be derivedas follows:

p(c)con (Tϕ)△= P

[maxx∈Φb

SINRc (x) > Tϕ

]= P

[ ∪x∈Φb

SINRc (x) > Tϕ

]

= E

[1

( ∪x∈Φb

SINRc (x ) > Tϕ

)](a)

≤ E

[∑x∈Φb

1 (SINRc (x) > Tϕ)

]

= E

[∑x∈Φb

1

(Pbgxr

−αx

σ2 + Ic (x)> Tϕ

)](b)= λb

∫R2

E[1

(Pbgxr

−αx

σ2 + I ′c> Tϕ

)]dx

= λb

∫R2

P[Pbgx r

−αx

σ2 + I ′c> Tϕ

]dx

(c)= λb

∫R2

e−P−1b Tϕr

αxσ2

EI′c

[e−P−1

b Tϕrαx I′

c

]dx

(d)= λb

∫R2

e−P−1b Tϕr

αxσ2

LI′c

(P−1b Tϕr

αx

)dx

= 2πλb

∫ ∞

0

e−P−1b Tϕr

αxσ2

LI′c

(P−1b Tϕr

αx

)rx drx.

(5)

(a) follows from the property of union, and the equalityholds if at most one BS in the network can provide a SINRabove the threshold. In (b), I ′c =

∑xi∈Φb

Pbgi ∥xi∥−α+∑

yi∈ΦdPdhi ∥yi∥−α, which includes the term related to the

tagged BS x and thereby is quite different from Ic (x). Thederivation of (b) follows from the Campbell-Mecke Theorem[35]: E

[∑x∈Φ f (x,Φ \ {x})

]= λ

∫R2 E [f (x,Φ)] dx. (c)

follows from the Rayleigh distribution assumption of channelfading and the independence of noise and interference. In (d),LI′

c(·) denotes the Laplace transform of I ′c. To complete the

proof, we next derive the expression of LI′c(s).

Let I ′c = I ′c−c + I ′c−d, where I ′c−c =∑

xi∈ΦbPbgi ∥xi∥−α

and I ′c−d =∑

yi∈ΦdPdhi ∥yi∥−α denote the interference

from cellular links and D2D links respectively. Then it isstraightforward to get

LI′c(s) = LI′

c−c(s) · LI′

c−d(s) , (6)

since EI′c

[e−sI′

c

]= E

[e−sI′

c−c

]· E[e−sI′

c−d

]. The Laplace

transform of I ′c−c is given by

LI′c−c

(s) = E

[exp

(−s

∑xi∈Φb

Pbgi ∥xi∥−α

)]

= EΦb,g

[ ∏xi∈Φb

exp(−sPbgi ∥xi∥−α

)]

= EΦb

[ ∏xi∈Φb

Eg

[exp

(−sPbgi ∥xi∥−α

)]]

5

(e)= exp

(−λb

∫R2

(1− Eg

[e−sPbgi∥xi∥−α

])dxi

)(f)= exp

(−λb2π

∫ ∞

v=0

v

1 + s−1P−1b vα

dv

)(g)= exp

(−λbπ (sPb)

δΓ (1− δ) Γ (1 + δ)

)(h)= exp

(−πλbP

δb s

δ

sinc δ

). (7)

(e) follows from the probability generating functional (PGFL)of PPP [35]: E

[∏x∈Φ f (x)

]= exp

(−λ∫R2 (1− f (x)) dx

).

(f) follows from the double integral in polar coordinates. In(g), Γ (x) =

∫∞0

tx−1e−t dt is the gamma function and δ = 2α .

(h) is obtained by using the property of gamma function:Γ (1 + x) Γ (1− x) = πx

sinπx = 1sinc x , for 0 < x < 1.

Similarly, we have

LI′c−d

(s) = exp

(−πλdP

δd s

δ

sinc δ

). (8)

By plugging (7) (8) into (6), we get

LI′c(s) = exp

(−πλbP

δb s

δ

sinc δ− πλdP

δd s

δ

sinc δ

). (9)

Therefore,

LI′c

(P−1b Tϕr

αx

)= exp

(−πr2xT

δϕµ), (10)

where µ = λb

[1 + λd

λb

(Pd

Pb

)δ]sinc−1 δ. Then by plugging

(10) into (5), we complete the proof.The following proposition provides a sufficient condition

under which the equality in (4) holds.

Proposition 1. The upper bound on p(c)con (Tϕ) is achieved

when Tϕ > 1 (0 dB).

Proof: The proof follows from [29, Lemma 1]. Accordingto the lemma, at most one BS can provide a SINR greater than1 if Tϕ > 1. Therefore, the equality in step (a) of (5) holdsand the upper bound on p

(c)con (Tϕ) is achieved.

By Proposition 1, when Tϕ > 1, the upper bound onp(c)con (Tϕ) is the exact value of p

(c)con (Tϕ). However, when

Tϕ ≤ 1, there exists a gap between p(c)con (Tϕ) and the

upper bound. In the simulation, we will evaluate the gap bynumerical results and show the tightness of the upper boundwhen Tϕ ≤ 1. For tractability, in the following part of thispaper, we represent the exact value of p

(c)con (Tϕ) by its upper

bound, or equivalently assume Tϕ > 1.Considering a special case that the network is interference-

limited, we get the following corollary.

Corollary 1. In interference-limited D2D-enabled cellularnetworks, the connection probability of the typical cellularlink is bounded from above by

p(c)con (Tϕ) ≤sinc δ[

1 + λd

λb

(Pd

Pb

)δ]T δϕ

, (11)

and the equality holds when Tϕ > 1.

Proof: Following from Theorem 1 with σ2 → 0.

Remark 1. Corollary 1 shows that p(c)con (Tϕ) is inversely

correlated with the density ratio λd/λb and transmission powerratio Pd/Pb. The result is intuitive since given λb and Pb, alarger density and transmission power of D2D communicationsintroduce more severe interference to cellular communications.

B. Secrecy of Cellular Links

The analysis is conducted based on the typical cellular linkthat comprises a typical cellular user located at the originand a typical BS located at x0. For a eavesdropper locatedat z, its distance to the typical BS is denoted by rz , andthe fading factor for this eavesdropping link is denoted bygz, gz ∼ exp (1). Then, the received SINR at eavesdropper zfrom the typical BS can be expressed as

SINRe (z) =Pbgzr

−αz

σ2 + Ie (z), (12)

where

Ie (z) =∑

xi∈Φb\{x0}

Pbgi ∥xi − z∥−α+∑

yi∈Φd

Pdhi ∥yi − z∥−α

(13)is the cumulative interference from all other BSs (except thetypical BS located at x0) that are located at xi with fadingfactor gi and D2D transmitters that are located at yi withfading factor hi.

We assume that each cellular link is exposed to all theeavesdroppers. According to the secrecy requirement, if thereexists some eavesdropper z such that SINRe (z) is above thethreshold Tϵ, the cellular link is not secure. Therefore, thesecure transmission of a cellular link is determined by its mostdetrimental eavesdropper, and the secrecy probability of thetypical cellular link can be defined as

p(c)sec (Tϵ)△= P

[maxz∈Φe

SINRe (z) < Tϵ

]. (14)

The expression of p(c)sec (Tϵ) is given by the following

theorem.

Theorem 2. The secrecy probability of the typical cellularlink in D2D-enabled cellular networks is

p(c)sec (Tϵ) = exp

(−2πλe

∫ ∞

0

e−P−1b Tϵr

αz σ2−πr2zT

δϵ µrz drz

),

(15)

where µ = λb

[1 + λd

λb

(Pd

Pb

)δ]sinc−1 δ.

Proof: The probability that the most detrimental SINR isbelow Tϵ equals the probability that all the SINRs are belowTϵ. Therefore, the secrecy probability of the typical cellularlink can be derived as follows:

p(c)sec (Tϵ)△= P

[maxz∈Φe

SINRe (z) < Tϵ

]= P

[ ∩z∈Φe

SINRe (z) < Tϵ

]

= EΦe,Φd

[1

( ∩z∈Φe

SINRe (z) < Tϵ

)]

6

(a)= EΦe,Φd

[ ∏z∈Φe

1 (SINRe (z) < Tϵ)

]

= EΦe

[ ∏z∈Φe

EΦd[1 (SINRe (z) < Tϵ | z)]

]

= EΦe

[ ∏z∈Φe

P (SINRe (z) < Tϵ | z)

](b)= EΦe

[ ∏z∈Φe

(1− e−P−1

b Tϵrαz (σ

2+Ie(z)))]

= EΦe

[ ∏z∈Φe

(1− e−P−1

b Tϵrαz σ2

LIe(z)

(P−1b Tϵr

αz

))](c)= exp

(−2πλe

∫ ∞

0

e−P−1b Tϵr

αz σ2

LIe(z)

(P−1b Tϵr

αz

)rz drz

).

(16)

(a) follows from the property of intersection. (b) follows fromthe Rayleigh distribution assumption of channel fading and theindependence of noise and interference. (c) follows from theprobability generating functional (PGFL) of PPP. To completethe proof, we next derive the expression of LIe(z) (s).

First, shift the coordinates so that eavesdropper z is locatedat the origin. It is noted that translations do not change the dis-tribution of PPP [22]. Thus Ie (z) can be replaced by Ie whichequals Ie (z = 0). Then let Ie = Ie−c + Ie−d, where Ie−c =∑

xi∈Φb\{x0} Pbgi ∥xi∥−α and Ie−d =∑

yi∈ΦdPdhi ∥yi∥−α

denote the interference from cellular links and D2D linksrespectively. It is straightforward to get

LIe (s) = LIe−c (s) · LIe−d(s) . (17)

Following the steps similar to (7)− (10) and using Slivnyak’sTheorem of PPP [35], we have

LIe(z)

(P−1b Tϵr

αz

)= exp

(−πr2zT

δϵ µ), (18)

where µ = λb

[1 + λd

λb

(Pd

Pb

)δ]sinc−1 δ. Plug (18) into (16),

then the proof is completed.By letting σ2 → 0 in (15), we get the following result for

interference-limited cases.

Corollary 2. In interference-limited D2D-enabled cellularnetworks, the secrecy probability of the typical cellular linkis

p(c)sec (Tϵ) = exp

− λe sinc δ

λb

[1 + λd

λb

(Pd

Pb

)δ]T δϵ

. (19)

Remark 2. By Corollary 2, p(c)sec (Tϵ) is negatively correlatedwith the eavesdropper intensity and is positively correlatedwith the D2D intensity and power. This is because, a largerpopulation of eavesdroppers can reduce the average distanceof eavesdropping links, while a larger population of D2D userscan generate more interference to eavesdropping links.

C. Connection of D2D Links

We conduct analysis on a typical D2D link that comprisesa typical D2D transmitter and a typical D2D receiver locatedat distance l away with isotropic direction. Assume that thetypical D2D receiver is located at the origin and denote thefading factor for the typical D2D link by h0, h0 ∼ exp (1).Then, the received SINR at the typical D2D receiver can beexpressed as

SINRd =Pdh0l

−α

σ2 + Id, (20)

where

Id =∑

xi∈Φb

Pbgi ∥xi∥−α+

∑yi∈Φd\{y0}

Pdhi ∥yi∥−α (21)

is the cumulative interference from all the base stations that arelocated at xi with fading factor gi and other D2D transmitters(except the typical D2D transmitter located at y0) that arelocated at yi with fading factor hi.

The connection probability of the typical D2D link can bedefined as

p(d)con (Tσ)△= P [SINRd > Tσ] , (22)

where Tσ is the SINR threshold.

Theorem 3. The connection probability of the typical D2Dlink in D2D-enabled cellular networks is

p(d)con (Tσ) = exp(−P−1

d Tσlασ2 − πl2T δ

σν), (23)

where ν = λd

[1 + λb

λd

(Pb

Pd

)δ]sinc−1 δ.

Proof: Based on a fixed distance l, the connection prob-ability can be derived as follows:

p(d)con (Tσ)△= P [SINRd > Tσ]

= P[h0 > P−1

d Tσlα(σ2 + Id

)]= e−P−1

d Tσlασ2

LId

(P−1d Tσl

α). (24)

Following approaches similar to those in previous proofs, wehave

LId

(P−1d Tσl

α)= exp

(−πl2T δ

σν), (25)

where ν = λd

[1 + λb

λd

(Pb

Pd

)δ]sinc−1 δ. Combining (24) and

(25), we obtain the result.

Corollary 3. In interference-limited D2D-enabled cellularnetworks, the connection probability of the typical D2D linkis

p(d)con (Tσ) = exp

(−πl2T δ

σλd

[1 +

λb

λd

(Pb

Pd

)δ]sinc−1 δ

).

(26)

Remark 3. The results of Theorem 3 and Corollary 3can be extended to cases where l is variable. Denote thepdf of l as fl (l), then p

(d)con (Tσ) can be computed as∫∞

0P [SINRd > Tσ | l] · fl (l) dl. For example, assume l is

Rayleigh distributed, fl (l) = 2πλdle−πλdl

2

, then p(d)con (Tσ) =

λd/(T δσν + λd

)for interference-limited cases.

7

D. Performance Guarantee for Cellular TransmissionsBy Corollary 2 and 3, the secrecy probability of cellular

links p(c)sec (Tϵ) and the connection probability of D2D links

p(d)con (Tσ) can be enhanced by increasing λd and/or Pd.

However, by Corollary 1, increasing λd and/or Pd wouldreduce the connection probability of cellular links p

(c)con (Tϕ).

Therefore, the scheduling parameters of D2D links (λd, Pd)should be carefully designed to improve the performanceof D2D communications and meanwhile guarantee a certainperformance level of cellular communications.

We propose the following two performance guarantee cri-teria for cellular communications:

• Strong guarantee criterion: the probability of perfectcellular transmissions3 should not to be reduced by in-troducing D2D communications, i.e.,

p(c)con (Tϕ) p(c)sec (Tϵ) ≥ p(c)(0)con (Tϕ) p

(c)(0)sec (Tϵ) , (27)

where p(c)(0)con (Tϕ) and p

(c)(0)sec (Tϵ) denote the connection

probability and secrecy probability of cellular links in theabsence of D2D links respectively.

• Weak guarantee criterion: (ϕ, ϵ)-perfect cellular trans-mission should be guaranteed after introducing D2Dcommunications, i.e.,

p(c)con (Tϕ) ≥ ϕ and p(c)sec (Tϵ) ≥ ϵ, (28)

where ϕϵ ≤ p(c)(0)con (Tϕ) p

(c)(0)sec (Tϵ).

The first criterion requires that introducing D2D links to cel-lular networks should not degrade the performance of cellularlinks, while the second criterion can tolerate a certain levelof performance degradation of cellular links. In the followingsections, we explore D2D link scheduling issues under boththe strong and weak guarantee criteria.

V. OPTIMAL D2D LINK SCHEDULING UNDER STRONGGUARANTEE CRITERION FOR CELLULAR LINKS

Based on the derived analytical results in previous section,we design D2D link scheduling schemes, which determine theintensity and power of D2D links, under the strong perfor-mance guarantee criterion for cellular links. For purposes ofmathematical tractability, we consider the interference-limitedscenario where σ2 → 0.

A. Feasible Region of D2D Scheduling Parameters

Let A be a Borel set of R2 with unit measure, i.e. |A| =1. Hence, the intensity measure of the set of perfect cellulartransmissions in A can be defined as

Nc (A)△= E

[∑i∈Φb

1perA (i)

], (29)

where

1perA (i) =

{1, if i ∈ A is perfect transmission,

0, otherwise.(30)

3Here we approximate the probability of perfect cellular transmissions byp(c)con

(Tϕ

)·p(c)sec (Tϵ) using the assumption that the interference at each point

is independent, and its validity is evaluated in numerical results. It is notedthat such assumption is also used in some stochastic geometry literature [33].

The indicator function 1perA (i) allows to count the number of

perfect cellular links in A. Therefore, Nc (A) is a measure ofthe average number of perfect cellular links in A. Due to thestationary of PPPs, Nc (A) is independent of A. Hence, weuse Nc to denote the average number of perfect cellular linksper unit area, and

Nc = λbp(c)con (Tϕ) p

(c)sec (Tϵ) . (31)

By Corollary 1 and 2, we have

Nc = λbax exp (−bx) , (32)

where a = sinc δT δϕ

> 0, b = λe sinc δλbT δ

ϵ> 0, x = 1

1+λdλb

(PdPb

)δ

∈ (0, 1]. From (32) we can see that D2D communications

introduce a factor of λd

λb

(Pd

Pb

)δinto the performance metrics

of cellular communications. By letting λd

λb

(Pd

Pb

)δ= 0, i.e.

x = 1, in (32), we get

N (0)c = λba exp (−b) , (33)

where the superscript (0) indicates the case in the absence ofD2D links in the network.

By comparing (27) and (31), we can see that the strongguarantee criterion (27) is equivalent to

Nc ≥ N (0)c . (34)

The following lemma shows the feasible region of D2Dscheduling parameters under criterion (34).

Lemma 1. The feasible region constrained by the strongguarantee criterion is

Fstr =

{(λd, Pd) :{

λdPδd ≤

(− b

Wp(−be−b)− 1)λbP

δb , if b > 1

λd = Pd = 0, if b ≤ 1

,

(35)

where Wp (·) is the real-valued principal branch of LambertW-function.

Proof: The proof comprises two steps. In the first step,we derive the solution to Nc = N

(0)c , and in the second step,

we further derive the solution to Nc ≥ N(0)c based on the

results obtained in the first step.(1) Evaluate the equation Nc = N

(0)c . By (32) and (33),

Nc = N(0)c is equivalent to

eb(1−x) =1

x. (36)

The derivation of the solution to (36) is as follows:

eb(1−x) =1

x⇒ −bxe−bx = −be−b

(a)⇒ yey = −be−b

(b)⇒ y = Wp

(−be−b

)∪Wm

(−be−b

)(c)⇒ x = −1

bWp

(−be−b

)∪−1

bWm

(−be−b

). (37)

8

(a) employs a change of variable y = −bx. In (b), the solutionof y has two branches due to −be−b ∈

(−1

e , 0). Wp (·) and

Wm (·) are the real-valued principal branch and the otherbranch of Lambert W-function respectively [36]. It is notedthat Wp (·) and Wm (·) are also denoted as W0 (·) and W-1 (·)in some literature. (c) makes an inverse variable change thatx = −1

by.By examining the two branches of the solution in (37), we

find that − 1b Wp

(−be−b

)∈ (0, 1] and −1

b Wm

(−be−b

)∈

[1,∞). Considering 0 < x ≤ 1, we reject the second branchand obtain the final solution to Nc = N

(0)c :

x0 = −1

bWp

(−be−b

). (38)

(2) Evaluate Nc ≥ N(0)c , which is equivalent to

xe−bx ≥ e−b. (39)

Let f (x) = xe−bx (0 < x ≤ 1). Then, (39) is equivalent to

f (x) ≥ f (1) . (40)

Next, we derive the solution to (40). Taking the derivative off (x) with respect to x, we get

f ′ (x) = (1− bx) e−bx. (41)

Three cases are considered according to the value of b.Case 1: b < 1. In this case, f ′ (x) > 0 and thereby f (x)

monotonically increases in 0 < x ≤ 1. Therefore, the solutionto (40) is x = 1, i.e.,

Fstr = {(λd, Pd) : λd = Pd = 0} . (42)

Case 2: b = 1. It is easy to verify that the solution is also(42).

Case 3: b > 1. In this case, f ′ (x) > 0 when x ∈(0, 1

b

)and f ′ (x) < 0 when x ∈

(1b , 1]. Thus f (x) monotonically

increases in 0 < x < 1b , and monotonically decreases in

1b < x ≤ 1. In addition, by the results obtained in step(1), f (x) = f (1) holds at point x0 = − 1

b Wp

(−be−b

)and x1 = − 1

b Wm

(−bx0e

−bx0)= 1, where x0 < x1 = 1.

Therefore, the solution to (40) is x ∈ [x0, 1], i.e.,

Fstr =

{(λd, Pd) : λdP

δd ≤

(− b

Wp (−be−b)− 1

)λbP

δb

}.

(43)Combining the results of case 1-3, we complete the proof.Remark 4. Lemma 1 shows that, under the strong guaranteecriterion, the performance of cellular links is hampered byD2D links if λe ≤ T δ

ϵ

sinc δλb, and is enhanced otherwise. This isbecause, for D2D links, their interfering effect on cellular linksis critical when the eavesdropper intensity is small, while theirjamming effect on eavesdropping links becomes dominantwhen the intensity is large.

B. D2D Link Scheduling Schemes

Based on the derived feasible region of D2D schedulingparameters, we study the D2D link scheduling problems foroptimizing network performance. Note that we focus only onthe case of b > 1, since when b ≤ 1 D2D communicationsshould be blocked by the network.

The first problem is how to obtain the maximum averagenumber of perfect cellular transmissions per unit area, i.e.,

max(λd,Pd)

Nc, s.t. (λd, Pd) ∈ Fstr. (44)

Lemma 2. The optimal solution of problem (44) is

F ′str =

{(λd, Pd) : λdP

δd = (b− 1)λbP

δb

}. (45)

Proof: According to the proof of Lemma 1, f ′ (x) =0 ⇒ x = 1

b , and f ′′ ( 1b

)< 0. Therefore, the optimal solution

is x = 1b .

Lemma 2 shows that there exist a series of (λd, Pd) pairsthat can achieve maximum Nc. Next, we further study whichpair(s) among them can achieve the optimal D2D performance.

We employ the average number of perfect D2D links perunit area as the metric for D2D communications, which isdefined as

Nd = λdp(d)con (Tσ) . (46)

By Corollary 3, we have

Nd = λd exp

(− c

1− xλd

), (47)

where c =πl2T δ

σ

sinc δ > 0. Then, the optimization problem is

P1 :max

(λd,Pd)Nd, s.t. (λd, Pd) ∈ F ′

str. (48)

Theorem 4. The optimal solution of Problem 1 is

S1 =

{(λ∗

d, P∗d ) : λ

∗d =

b− 1

bc, P ∗

d = (bcλb)1δ Pb

}. (49)

Proof: The constraint of P1 is equivalent to x = 1b ,

thereby Nd = λd exp(− bc

b−1λd

), which depends only on

λd. Similar to the proof of Lemma 1, Nd monotonicallyincreases in λd ∈

(0, b−1

bc

), and monotonically decreases in

λd ∈(b−1bc ,∞

). Therefore, the maximum value of Nd is ob-

tained at point λ∗d = b−1

bc . By (45), we have P ∗d = (bcλb)

1δ Pb.

Relaxing the constraint of Problem 1 to the strong guaranteecriterion (27), we have

P2 :max

(λd,Pd)Nd, s.t. (λd, Pd) ∈ Fstr. (50)

Theorem 5. The optimal solution of Problem 2 is

S2 =

{(λ∗

d, P∗d ) : λ

∗d =

b+Wp

(−be−b

)bc

,

P ∗d =

(− bcλb

Wp (−be−b)

) 1δ

Pb

}. (51)

Proof: The constraint of P2 is equivalent to x ∈ [x0, 1].Then, Nd = λd exp

(− c

1−xλd

)≤ λd exp

(− c

1−x0λd

), and

the equality holds iff x = x0. Similar to the proof of Lemma1, the maximum value of the upper bound of Nd is obtained

at point λ∗d = 1−x0

c . By x = x0, P ∗d =

(cλb

x0

) 1δ

Pb.

9

Theorem 4 and 5 give two D2D link scheduling schemes,S1 and S2, under the strong performance guarantee criterionfor cellular links. For schemes S1,

N (1)c =

λba

be−1, (52)

N(1)d =

b− 1

bce−1, (53)

and for schemes S2,

N (2)c = λbae

−b, (54)

N(2)d =

b+Wp

(−be−b

)bc

e−1. (55)

By observing (52)− (55), we can get that

N (1)c > N (2)

c = N (0)c , (56)

N(1)d < N

(2)d . (57)

The results show that schemes S1 can achieve the optimal per-formance for cellular communications, while S2 can providea higher performance level for D2D communications.

VI. OPTIMAL D2D LINK SCHEDULING UNDER WEAKGUARANTEE CRITERION FOR CELLULAR LINKS

In this section, we study the D2D link scheduling problemsunder the weak performance guarantee criterion for cellularlinks. Two D2D link scheduling schemes are designed forinterference-limited networks.

A. Feasible Region of D2D Scheduling Parameters

The weak guarantee criterion (28) requires the minimumconnection probability ϕ and secrecy probability ϵ for cellularlinks. The following lemma gives the feasible region of D2Dscheduling parameters under this criterion.

Lemma 3. The feasible region constrained by the weakguarantee criterion is

Fweak =

{(λd, Pd) :

(b

ln 1ϵ

− 1

)λbP

δb ≤ λdP

δd

≤(a

ϕ− 1

)λbP

δb

}. (58)

Proof: By p(c)con (Tϕ) ≥ ϕ and (11), we have x ≥ 1

aϕ,where x = 1

1+λdλb

(PdPb

)δ is defined in (32). By p(c)sec (Tϵ) ≥ ϵ

and (19), we have x ≤ 1b ln

1ϵ . Therefore, 1

aϕ ≤ x ≤ 1b ln

1ϵ .

Remark 5. By Lemma 1 and 3, Fstr exists only for b > 1,while Fweak exists for any b. This is due to the fact that,comparing to the strong guarantee criterion (27), the weakguarantee criterion (28) relaxes the secrecy constraint andhence provide extra transmission opportunities for D2D users.

Before investigating the D2D link scheduling problems, weshould determine the feasible values of ϕ and ϵ. A reasonablerange of (ϕ, ϵ) is given by

R =

{(ϕ, ϵ) : 0 ≤ ϕ ≤ a, e−b ≤ ϵ ≤ 1,

ϕϵ ≤ ae−b,1

ϕln

1

ϵ≥ b

a

}, (59)

where the first two conditions correspond to the fact that 0 ≤p(c)con (Tϕ) ≤ a and e−b ≤ p

(c)sec (Tϵ) ≤ 1, the third condition

corresponds to the definition of weak guarantee criterion thatϕϵ ≤ p

(c)(0)con (Tϕ) p

(c)(0)sec (Tϵ), and the last one corresponds to

the implied condition in Lemma 3 that 1aϕ ≤ 1

b ln1ϵ . In the

following analysis, we assume (ϕ, ϵ) ∈ R.

B. D2D Link Scheduling Schemes

We first investigate how to obtain the maximum value ofNc, i.e.,

max(λd,Pd)

Nc, s.t. (λd, Pd) ∈ Fweak. (60)

Lemma 4. The optimal solution of problem (60) is

F ′weak = {(λd, Pd) :λdP

δd =

(b

ln 1ϵ

− 1)λbP

δb , if b ≤ 1 or b > 1, ϵ > e−1

λdPδd =

(aϕ − 1

)λbP

δb , if b > 1, ϕ > a

b

λdPδd = (b− 1)λbP

δb , if b > 1, ϵ ≤ e−1, ϕ ≤ a

b

} .

(61)

Proof: Consider two cases: b ≤ 1 and b > 1.(1) b ≤ 1. In this case, Nc monotonically increases in 1

aϕ ≤x ≤ 1

b ln1ϵ . Therefore, the solution of (60) is x = 1

b ln1ϵ .

(2) b > 1. Three cases are considered according to thevalues of 1

aϕ and 1b ln

1ϵ . Case 1: 1

b ln1ϵ < 1

b , i.e., ϵ > e−1. Inthis case, Nc monotonically increases in 1

aϕ ≤ x ≤ 1b ln

1ϵ .

Therefore, the solution is x = 1b ln

1ϵ . Case 2: 1

aϕ > 1b ,

i.e., ϕ > ab . In this case, Nc monotonically decreases in

1aϕ ≤ x ≤ 1

b ln1ϵ . Therefore, the solution is x = 1

aϕ. Case3: 1

aϕ ≤ 1b ≤ 1

b ln1ϵ , i.e., ϵ ≤ e−1, ϕ ≤ a

b . In this case, Nc

monotonically increases in 1aϕ < x < 1

b and monotonicallydecreases in 1

b < x ≤ 1b ln

1ϵ . Therefore, the solution is x = 1

b .Combining the above results, we complete the proof.Lemma 4 shows that there exist a series of (λd, Pd) pairs

that can achieve maximum Nc. Next, we further study whichpair(s) among them can achieve maximum Nd.

P3 :max

(λd,Pd)Nd, s.t. (λd, Pd) ∈ F ′

weak. (62)

Theorem 6. The optimal solution of Problem 3 is

S3 = {(λ∗d, P

∗d ) :

λ∗d=

b−ln 1ϵ

bc , P∗d =

(bc

ln 1ϵ

λb

) 1δPb, if b≤1 or b>1, ϵ>e−1

λ∗d=

a−ϕac , P∗

d =( acϕ λb)

1δ Pb, if b>1, ϕ> a

b

λ∗d=

b−1bc , P∗

d =(bcλb)1δ Pb, if b>1, ϵ≤e−1, ϕ≤ a

b

} . (63)

Proof: Similar to the proof of Theorem 4, the maximumvalue of Nd is obtained at point λ∗

d = 1−xc , where x is the

solution of problem (60) given in Lemma 5.Relaxing the constraint of Problem 3 to the weak guarantee

criterion (28), we have

P4 :max

(λd,Pd)Nd, s.t. (λd, Pd) ∈ Fweak. (64)

10

Table II: A summary of the proposed D2D link scheduling schemesλd Pd Nc Nd

S1b−1bc

(bcλb)1δ Pb

λbab

e−1 b−1bc

e−1

S2

b+Wp

(−be−b

)bc

(− bcλb

Wp(−be−b)

) 1δ

Pb λbae−b

b+Wp

(−be−b

)bc

e−1

S3

b−ln 1

ϵbc

, if case 1a−ϕac

, if case 2b−1bc

, if case 3

(bc

ln 1ϵ

λb

) 1δ

Pb, if case 1(acϕλb

) 1δPb, if case 2

(bcλb)1δ Pb, if case 3

λbaϵb

ln 1ϵ, if case 1

λbϕ exp(− b

aϕ), if case 2

λbab

e−1, if case 3

b−ln 1

ϵbc

e−1, if case 1a−ϕac

e−1, if case 2b−1bc

e−1, if case 3

S4a−ϕac

(acϕλb

) 1δPb λbϕ exp

(− b

aϕ)

a−ϕac

e−1

In the table, (1) a = sinc δTδϕ

, b = λe sinc δλbT

δϵ

, c =πl2Tδ

σsinc δ

;

(2) case 1: b ≤ 1 or b > 1, ϵ > e−1, case 2: b > 1, ϕ > ab

, case 3: b > 1, ϵ ≤ e−1, ϕ ≤ ab

.

Theorem 7. The optimal solution of Problem 4 is

S4 =

{(λ∗

d, P∗d ) : λ

∗d =

a− ϕ

ac, P ∗

d =

(ac

ϕλb

) 1δ

Pb

}. (65)

Proof: The constraint of P4 is equivalent tox ∈

[1aϕ,

1b ln

1ϵ

]. Then, Nd = λd exp

(− c

1−xλd

)≤

λd exp(− c

1− 1aϕ

λd

), and the equality holds iff x = 1

aϕ.Therefore, the maximum value of Nd is obtained at point

λ∗d =

1− 1aϕ

c . By x = 1aϕ, we have P ∗

d =(

acϕ λb

) 1δ

Pb.Theorem 6 and 7 give two D2D link scheduling schemes,

S3 and S4, under the weak performance guarantee criterionfor cellular links. For schemes S3,

N (3)c =

λbaϵb ln 1

ϵ , if b ≤ 1 or b > 1, ϵ > e−1

λbϕ exp(− b

aϕ), if b > 1, ϕ > a

bλbab e−1, if b > 1, ϵ ≤ e−1, ϕ ≤ a

b

,

(66)

N(3)d =

b−ln 1

ϵ

bc e−1, if b ≤ 1 or b > 1, ϵ > e−1

a−ϕac e−1, if b > 1, ϕ > a

bb−1bc e−1, if b > 1, ϵ ≤ e−1, ϕ ≤ a

b

, (67)

and for schemes S4,

N (4)c = λbϕ exp

(− b

aϕ

), (68)

N(4)d =

a− ϕ

ace−1. (69)

By observing (66)− (69), we can get that

N (3)c ≥ N (4)

c , (70)

N(3)d ≤ N

(4)d . (71)

The results show that schemes S3 can achieve the optimal per-formance for cellular communications, while S4 can providea higher performance level for D2D communications.Remark 6. Nc, Nd of schemes S3, S4 depend on specificvalues of ϕ and ϵ. Thus it is difficult to compare Nc, Nd ofS1, S2 with those of S3, S4 by analytical results. A summaryof the proposed D2D link scheduling schemes is shown inTable II. In the simulation, we provide numerical illustrationfor performance comparison of these schemes.

VII. SIMULATION RESULTS

In this section, we first present some simulation results tovalidate the proposed model and analytical results, and thenprovide some numerical results on performances of D2D linkscheduling schemes.

A. Validation of Analytical Results

As all the link scheduling schemes are designed based onthe probabilities derived in section IV, in this part we validatethe analytical results of these probabilities by simulations.The simulations employ PPP model with main simulationparameters α = 3, l = 0.05, Tϕ = 0.25, Tϵ = 0.5, Tσ =0.5, λb = 2, λe = 6, and the results are shown in Fig.2.Fig.2(a) shows p(c)con, p

(c)sec, p

(d)con versus λd. As can be observed

from the curves, the analytical results of the probabilities thatare derived in Corollary 1-3, are in quite good agreementwith corresponding simulation results. This fact confirms thatour proposed framework closely matches the practical D2D-enabled cellular network with eavesdroppers. p(c)per, which de-notes the probability of perfect cellular transmissions (see (27)and footnote 3), is another critical metrics for the network. Inthe analysis, we approximate p

(c)per by p

(c)con · p(c)sec. However, in

fact, p(c)per < p(c)con · p(c)sec, since the interference at each point is

not independent in practical networks. Fig.2(b) plots p(c)per and

p(c)con ·p(c)sec in different scenarios. As expected, when Pb is small

and λd is large, the difference between p(c)per and p

(c)con · p(c)sec is

large. It is because small transmission power of cellular links,which can be regarded correspondingly as large transmissionpower of D2D links, and large intensity of D2D links lead tostronger correlation of interference generated by D2D links.This result implies that the approximation is more precisewith larger Pb and smaller λd. Considering that in practicalnetworks, the transmission power of base stations is usuallymuch larger than that of D2D terminals, and the intensity ofD2D links can be properly controlled by the network, theapproximation error of p(c)per is not large.

B. Performance Evaluation of Proposed Schemes

In this part we provide some numerical results to evaluatethe performances of the proposed link scheduling schemes.Fig.3 shows Nc, Nd versus λd and Pd under the strong

11

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

d

Pro

bab

ilit

y (

%)

pccon

(simulation)

pcsec

(simulation)

pdcon

(simulation)

pcper

(simulation)

pccon

(analysis)

pcsec

(analysis)

pdcon

(analysis)

pcper

(analysis)

(a)

1 2 3 4 5 6 7 8 9 1010

15

20

25

30

35

40

d

pc p

er (

%)

Pb=1 (simulation)

Pb=5 (simulation)

Pb=10 (simulation)

Pb=1 (analysis)

Pb=5 (analysis)

Pb=10 (analysis)

(b)

Figure 2: Validation of analytical results of p(c)con, p(c)sec, p

(d)con and p

(c)per.

5

10

15

20

0

5

10

0

0.2

0.4

0.6

0.8

d

Pd

Nc

Nc(1) N

c(2)

Fstr

Fstr'

(a)5

10

15

20

0

5

10

0

2

4

6

8

dP

d

Nd

Fstr'

Fstr

Nd(1)

Nd(2)

(b)

Figure 3: Nc, Nd under the strong guarantee criterion.

guarantee criterion. In the figure, the feasible region F ′str and

the boundary of the feasible region Fstr are shown by themarked curves, and the optimal λd and Pd derived in schemeS1 and S2 are shown by the marked points. Fig.3(a) shows thatwhen (λd, Pd) is below F ′

str, Nc increases as λd, Pd increase;however, when (λd, Pd) is above F ′

str, Nc decreases as λd, Pd

increase. This result reflects that when λd, Pd are small, theeffect of interference from D2D links to eavesdropping links isdominant, but when λd, Pd are large, the effect of interferenceto cellular links becomes dominant. As we can observe fromFig.3, N (1)

c is the maximum value of Nc within F ′str and N

(2)d

is the maximum value of Nd within Fstr, which matches theanalysis in section V. Furthermore, the figure shows that thetransmission power and intensity of D2D links of scheme S1

are smaller than those of scheme S2 respectively. This suggeststhat larger transmission power and intensity of D2D links aredesirable for improving D2D performances.

Fig.4 shows Nc, Nd of scheme S1 and S2 versus λb withdifferent λe. We can see that for both the schemes, as λb

increases, Nc increases while Nd reduces. In addition, for boththe schemes, Nc is smaller when λe is larger, since a largerpopulation of eavesdroppers lowers the probability of secrecycellular transmissions. However, for both the schemes, Nd islarger when λe is larger. This is because when the intensity ofeavesdroppers increases, larger transmission power and inten-sity of D2D links are required to guarantee the performanceof cellular transmissions, which creates more transmissionopportunities for D2D links. Furthermore, comparing betweenthe curves in the figure, we can see that when λb is larger andλe is smaller, the performance of scheme S1 approaches thatof scheme S2.

Fig.5 compares Nc, Nd of the proposed schemes S1 − S4,where we consider three cases of ϕ, ϵ for scheme S3,S4 inthe numerical results. By Fig.5, in case 1, Nc of schemeS3,S4 are smaller than that of scheme S1,S2 respectively,while Nd of scheme S3,S4 are larger than that of schemeS1,S2 respectively; in case 2 and 3, Nc, Nd of scheme S3

equal or approximately equal those of scheme S1 respectively,

12

1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

1

2

3

4

5

6

7

8

b

Nc,

Nd

e = 5

e = 10

Nd(2)

Nd(1)

Nc(1) N

c(2)

Figure 4: Nc, Nd of schemes S1,S2.

while Nc, Nd of scheme S4 are larger than those of scheme S2

respectively. This result suggests that by adjusting parameter ϕand ϵ, different performance levels of cellular and D2D linkscan be achieved. In addition, it is noted that in the simulation bis above 1. For the scenario b ≤ 1, the strong guarantee criteriablocks D2D communications, but the weak guarantee criteriaadmits some D2D links to the network and scheme S3,S4 areoptimal link scheduling schemes. Therefore, scheme S3,S4 areapplicable to a broader range of system parameters, comparingwith scheme S1,S2.

VIII. CONCLUSIONS

In this paper, we focus on a large-scale D2D-enabled cel-lular network in which cellular communications are overheardby eavesdroppers, and propose a framework for modelingsuch a network via stochastic geometry. We derive the ex-pressions for SINR distributions and connection and secrecyprobabilities of cellular and D2D links, based on which wefurther design optimal D2D link scheduling schemes underboth strong and weak performance guarantee criteria forcellular communications. By investigating both analytical andnumerical results, we find out that the interference from D2Dcommunications can be exploited to enhance physical layersecurity of cellular communications and meanwhile createextra transmission opportunities for D2D users. This studyprovides a new perspective on the role of the interferencegenerated by D2D communications.

The main limitation of current model is that the mode(cellular mode or D2D mode) of each user is preset. A majorarea of future work is to study the scenario where each user canchange its communication mode. Another possible extensionof this work is to consider eavesdropper collusion in thenetwork, and design D2D link scheduling schemes that arerobust to colluding eavesdropping.

ACKNOWLEDGMENT

This work is partially supported by NSF China (No.61325012, 61271219, 61202373, 61221001, 61428205); ChinaMinistry of Education Doctor Program (No.20130073110025);

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

e

Nc

Nc(3) (case 2)

Nc(4) (case 2)

Nc(3) (case 3)

Nc(4) (case 3)

Nc(1)

Nc(2)

Nc(3) (case 1)

Nc(4) (case 1)

(a)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 101

2

3

4

5

6

7

8

9

10

11

12

e

Nd

Nd(3) (case 2)

Nd(4) (case 2)

Nd(3) (case 3)

Nd(4) (case 3)

Nd(1)

Nd(2)

Nd(3) (case 1)

Nd(4) (case 1)

(b)

Figure 5: Performance comparison of schemes S1 − S4. Forscheme S3,S4, three cases of ϕ, ϵ are considered. Case 1:ϕ = e−b, ϵ = 0.5, Case 2: ϕ = 1.5

b , Case 3: ϕ = a2b , ϵ = 0.2.

Shanghai Basic Research Key Project (No.11JC1405100,13510711300, 12JC1405200); Shanghai International Coop-eration Project (No. 13510711300); Open Research Fundof National Mobile Communications Research Laboratory,Southeast University (No. 2014D07); SRF for ROCS by SEM.

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Chuan Ma received the B.E. degree in Communica-tion Engineering from Sichuan University, China, in2009 and the M.E. degree in Information and Com-munication Engineering from Zhejiang University,China, in 2012. He is currently a Ph.D. student atShanghai Jiao Tong University, China. His researchinterests are in communication theory, device-to-device communication, heterogeneous networks andsoftware-defined radio.

Jiaqi Liu received the B.E. degree in electronicengineering from Shanghai Jiao Tong University,China, in 2014, where she is currently workingtoward the Ph.D. degree in the Department of Elec-tronic Engineering. Her research interests include so-cial computing, multi-antenna system and software-defined radio.

14

Xiaohua Tian (S’07-M’14) received his B.E.and M.E. degrees in communication engineeringfrom Northwestern Polytechnical University, Xi’an,China, in 2003 and 2006, respectively. He receivedthe Ph.D. degree in the Department of Electrical andComputer Engineering (ECE), Illinois Institute ofTechnology (IIT), Chicago, in Dec. 2010. Since June2013, he has been with Department of ElectronicEngineering of Shanghai Jiao Tong University as anAssistant Professor with the title of SMC-B scholar.He serves as an editorial board member on the

computer science subsection of the journal SpringerPlus, and the guest editorof International Journal of Sensor Networks. He also serves as the technicalprogram committee (TPC) member for IEEE INFOCOM 2014-2015, bestdemo/poster award committee member of IEEE INFOCOM 2014, TPC co-chair for IEEE ICCC 2014 International workshop on Internet of Things 2014,TPC Co-chair for the 9th International Conference on Wireless Algorithms,Systems and Applications (WASA 2014) on Next Generation NetworkingSymposium, local management chair for IEEE ICCC 2014, TPC memberfor IEEE GLOBECOM 2011-2015 on Wireless Networking, Cognitive RadioNetworks and Ad Hoc and Sensor Networks Symposium, TPC memberfor IEEE ICC 2013 and 2015 on Ad Hoc and Sensor Networks and NextGeneration Networking Symposium, respectively.

Hui Yu received the B.S. degree from the Depart-ment of Electrical Engineering, Tong Ji Universityin 1992, and the M.S. degree from the Depart-ment of Electronic Engineering, Shanghai Jiao TongUniversity in 1997. He is currently a senior engi-neer with the Department of Electronic Engineering,Shanghai Jiao Tong University. His research interestsinclude mobile communications, software radio andcognitive radio, channel coding and modulation forwireless communications.

Ying Cui (S’08-M’12) received her B.E. degree inElectronic and Information Engineering from Xi’anJiao Tong University, China, in 2007 and her Ph.D.degree in Electronic and Computer Engineering fromthe Hong Kong University of Science and Tech-nology (HKUST), Hong Kong, in 2011. From Jan.2011 to Jul. 2011, she was a Visiting Assistantin Research in the Department of Electrical Engi-neering at Yale University, US. From Mar. 2012to Jun. 2012, she was a Visiting Scholar in theDepartment of Electronic Engineering at Macquarie

University, Australia. From Jun. 2012 to Jun. 2013, she was a PostdoctoralResearch Associate in the Department of Electrical and Computer Engineeringat Northeastern University, US. From Jul. 2013 to Dec. 2014, she was aPostdoctoral Research Associate in the Department of Electrical Engineeringand Computer Science at Massachusetts Institute of Technology (MIT), US.Since Jan. 2015, she has been an Associate Professor in the Department ofElectronic Engineering at Shanghai Jiao Tong University, China. Her currentresearch interests include delay-sensitive cross-layer control and optimization,future Internet architecture and network coding. She was selected into China’s1000Plan Program for Young Talents in 2013.

Xinbing Wang (SM’12) received the B.S. degree(with hons.) in automation from Shanghai Jiao TongUniversity, Shanghai, China, in 1998, the M.S.degree in computer science and technology fromTsinghua University, Beijing, China, in 2001, andthe Ph.D. degree with a major in electrical andcomputer engineering and minor in mathematicsfrom North Carolina State University, Raleigh, in2006. Currently, he is a professor with the Depart-ment of Electronic Engineering, Shanghai Jiao TongUniversity. His research interests include resource

allocation and management in mobile and wireless networks, TCP asymptoticsanalysis, wireless capacity, cross-layer call admission control, asymptoticsanalysis of hybrid systems, and congestion control over wireless ad hoc andsensor networks. Dr. Wang has been a member of the Technical ProgramCommittees of several conferences including ACM MobiCom 2012, ACMMobiHoc 2012, IEEE INFOCOM 2009-2013.