capacity- and trust-aware bs cooperation in non-uniform...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2729346, IEEETransactions on Vehicular Technology

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Capacity- and Trust-aware BS Cooperation inNon-uniform HetNets: Spectral Efficiency and

Optimal BS DensityHuici Wu, Student Member, IEEE, Ning Zhang, Member, IEEE, Xiaofeng Tao, Senior Member, IEEE,

Zhiqing Wei, Member, IEEE, and Xuemin (Sherman) Shen, Fellow, IEEE

Abstract—This paper studies base station (BS) cooperation innon-uniform heterogeneous networks (HetNets). Considering thelimited capacity at BS and existence of untrusted small cellbase stations (SBSs) in practical scenarios, a novel capacity-and trust-aware BS cooperation strategy is proposed. The BScooperation is performed in a user centric manner, based on theaverage received signal strength at users, the capacity of BSs, andthe trustworthiness of SBSs. Furthermore, with the proposedBS cooperation, the statistics of aggregate information-signalstrength and interference strength are theoretically analyzed,based on stochastic geometry. Then, expressions for spectralefficiency (SE) and area spectral efficiency (ASE) are analyticallyderived. In addition, to study the impact of the SBS density on theSE and ASE, the optimal densities of normal SBSs to maximizethe SE and ASE are proved to exist and obtained. Finally,simulations and numerical evaluations validate the theoreticalanalysis and reveal that (i) with the awareness of BS capacityand trustworthiness, optimal cooperative thresholds achieving themaximum SE and ASE exist, which decrease with high path-lossexponent; and (ii) in high path-loss fading environment with highexistence probability of untrusted SBSs, more normal SBSs arerequired to be deployed to achieve the maximum SE and ASEperformance.

Index Terms—BS cooperation, optimal BS density, HetNets,stochastic geometry

I. INTRODUCTION

With proliferation of connected wireless devices (e.g., smartphones, tablets, connected vehicles [1], etc.) and emergenceof diverse wireless services (e.g., video streaming, social

Manuscript received November 17, 2016; accepted July 13, 2017. Thiswork was supported in part by the National Natural Science Foundation forDistinguished Young Scholars of China under Grant 61325006, in part by theNational Natural Science Foundation of China (No. 61231009, No. 61601055),in part by the 111 Project of China under the Grant B16006, and in part by theNatural Sciences and Engineering Research Council of Canada. The associateeditor coordinating the review of this paper and approving it for publicationwas Honggang Wang. Corresponding author: Xiaofeng Tao

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]

H. Wu and X. Tao are with the National Engineering Lab for MobileNetwork Technologies, Beijing University of Posts and Telecommunications,Beijing 100876, China (e-mail: {dailywu, taoxf}@bupt.edu.cn).

Z. Wei is with the Wireless Technology Innovation (WTI) Institute, BeijingUniversity of Posts and Telecommunications, Beijing 100876, China (e-mail:[email protected]).

N. Zhang is with the Department of Computing Science at Texas A&MUniversity-Corpus Christi (e-mail: [email protected]).

X. Shen is with the Department of Electrical and Computer Engineer-ing, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail:[email protected]).

Macro Cell

Micro Cell

Pico Cell

Femto Cell

Fig. 1. An illustration of heterogeneous networks with untrusted SBSs. Themacro BSs are overlaid by three tiers of low-power BSs and APs in the samegeographic area. Untrusted SBSs may exist in the proximity of normal SBSs.

networking, etc.), mobile data will dramatically increase. Itis predicted that the overall mobile data traffic generatedby these devices will grow to 30.6 exabytes per month by2020, which is an eightfold increase over 2015 [2]. Nextgeneration (5G) wireless networks expect to accommodatesuch a mobile data surge, through three main technical as-pects: spectrum extension, spectrum efficiency enhancement,and network densification [3]–[6]. As the dominant theme in5G, network densification is to densely deploy diverse smallcell base stations (SBSs) coexisting with conventional macrobase stations (MBSs), giving rise to heterogeneous networks(HetNets).

The HetNet architecture, making mobile terminals closer toaccess points (APs) or BSs, can not only improve networkcoverage and capacity, but also provide higher spectrum effi-ciency. However, it also poses many technical challenges, suchas resource/interference management, mobility management,security, etc. The resulting interference, including inter-tierinterference and intra-tier interference, is one of the toptechnical challenges that severely deteriorate the performanceof HetNets. To cope with the severe interference in HetNets,Coordinated Multi Point (CoMP) transmission, either in theform of multi-node joint transmission (JT) [7]–[9] or coordi-nated scheduling and coordinated beamforming (CS/CB) [10],[11], has been acknowledged as a promising technique in coor-



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dinating small cell BSs and macro BSs to mitigate interferenceand further improve the performance and efficiency of thenetwork.

A. Related Work and Motivation

In literatures, joint transmission with multiple nodes acrosstiers in HetNets has been investigated either by removing thedominant interferers [12], [13] or transforming the interferersto joint transmitters [14]–[19]. The performance analysis ofjoint transmission is significant for providing theoretical guid-ance and insights to practical operation of HetNets. Stochasticgeometry has been widely applied in modeling the distributionof wireless nodes in HetNets, where the BSs in differenttiers are modeled as independent homogeneous Poisson PointProcesses (PPPs). The existing works have also developedtractable results to characterize the statistics of interferenceand signal-to-interference-and-noise ratio (SINR) in HetNetswith various BS cooperation strategies.

The SINR distribution with noncoherent joint-transmission(NC-JT) BS cooperation was studied in a homogeneous(single-tier) wireless cellular network [14]. A cooperative BSjoins to transmit the user’s data only if the instantaneousreceived-signal-strength (RSS) on the corresponding link isabove a given threshold. The same cooperation strategy wasalso investigated in a K-tier HetNet in [15], where spectral ef-ficiency (SE) for the typical user was derived. [16] investigatedthe performance of JT and CB with quantized and delayedchannel state information (CSI). An adaptive multi-modetransmission scheme was proposed, which switches betweenJT and CB to maximize the data rate. In [17], a location-awareBS cooperation scheme was proposed in a two-tier HetNetto enhance the performance of the users suffering from highcross-tier interference. Outage probability, achievable data rateand load per BS were analyzed. A multi-cell cooperationscheme was proposed to improve the coverage performanceof cell-edge users in our prior work [18], where the coverageprobability of a randomly located user was derived. A generalBS cooperation across tiers was proposed and analyzed in[19]. K tiers of BSs are sorted in an increasing order bylong-term average RSS and a typical user connects to the nstrongest BSs. General expressions for coverage probability,diversity gain and power gain are derived. Following theircontributions, Gaurav studied spatio-temporal cooperation inHetNets [12], where the cooperative transmission was extend-ed to simultaneous multi-BS cooperation and retransmissionat a cooperative BS. Two cooperation strategies includingjoint transmission and BS silencing were investigated andexpressions for coverage probabilities were provided. Note thatthe studies on BS cooperation in the aforementioned literaturesare assumed with light-loaded BSs, which does not considerthe impact of the practical BS load on the BS cooperation andthe system performance.

On the other hand, although BS cooperation can providebetter coverage and capacity performance in HetNets, dis-tributing the information signal of mobile users to multiplecooperative BSs may cause information leakage due to thedevelopment of large-scale, distributed, ever more open and

diverse architecture of HetNets [20]. Since SBSs can bedeployed by users, the adversaries can easily deploy theirSBSs, making the network more vulnerable to eavesdroppingand attacks. Moreover, these SBSs are usually deployed incluster manner, like in stadium, mall and etc, which makesthe information security problem more severe since the illegalaccess points such as pseudo BSs are much easier to maskthemselves and act as regular BSs among the clusters, asdepicted in Fig. 1. These illegal nodes can lunch differentattacks such as man-in-the-middle attack (MITM) [21], denial-of-service (DoS) attack [22], replay attack and eavesdropping,etc. These attacks may cause disable or information leakage ofwireless transmissions. For example, in the man-in-the-middleattack (also known as fire brigade attacks), the attacker accessto the communication channel between two end terminalsand control over the end-users confidential message. DoSattacks make the network resources unavailable to the endusers and the replay attacks cause ambiguous informationtransmission. Therefore, the trustworthiness and security statusof BSs and access points should be considered into the BScooperation to guarantee the information security in wirelesscommunications.

To sum up, the main factors for BS cooperation in practicalsystems: i) the limited capacity of BSs, and ii) the trustwor-thiness of SBSs, are rarely considered in existing literatures.Specifically, the BSs were usually considered to be capableof serving any number of users during cooperation, which isnot reasonable in practice due to the densely deployment ofSBSs in HetNets and the scarcity of frequency and spatialresources. Thus, the BS capacity, i.e., the maximum number ofmobile terminals that can be associated with a BS, is limitedin practice. In addition, the SBSs are not always trusted inpractical deployment since the SBSs could be compromisedor easily deployed by malicious users with illegal purpose.Moreover, to make a efficient attack, especially eavesdroppingattack, the untrusted SBSs are always deployed around thelegitimate transmitter. Therefore, it is of great importance toaddress those issues to harvest the benefits of BS cooperationin practical systems.

B. Contribution and OrganizationIn this paper, considering the limited capacity and trustwor-

thiness of BSs, we propose a capacity- and trust-aware BScooperation strategy in the non-uniform HetNets. Two tiersof HetNet including trusted MBSs and partly trusted SBSsis studied, where Gaussian Poisson Process (GPP) is appliedto model the cluster property of SBSs and the existence ofthe untrusted SBSs. All the BSs are equipped with multipleantennas and can simultaneously serve multiple users in atime-frequency channel. The BSs participate in cooperationin a user-centric manner and the mobile users can adaptivelyupdate the cooperation BS cluster according to their servicerequirement and actual network status, including the trustwor-thiness of BSs and the traffic load of BSs. In a nutshell, thecontribution of this paper is summarized as follows.

• We propose a capacity- and trust-aware BS cooperationscheme performing in a user centric way, whereby d-ifferent BSs are selected to cooperatively serve users,



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based on the received signal power from potential BSsat users, the capacity of BSs, and the trustworthiness ofSBSs. Furthermore, we theoretically analyzed the spectralefficiency (SE) and area spectral efficiency (ASE) withstochastic geometry.

• The optimal SBS densities that maximize the SE and ASEare proved to exist and obtained. The analyses show thatthe optimal SBS density maximizing the ASE is denserthan that maximizing the SE.

• Extensive simulation results are provided to validate thetheoretical analysis and to show the impact of the systemparameters on the SE and ASE performance. The resultsindicate that i) with the awareness of BS capacity andtrustworthiness, there exist optimal cooperative thresholdsfor the determination of cooperative BSs to achievethe best SE and ASE performance, and lower optimalcooperative thresholds are needed with higher path-lossexponent; ii) more normal SBSs can be deployed in highpath-loss fading environments to achieve the best SEand ASE performance; and iii) larger distance repulsionbetween BSs can provide better SE and ASE performancewhen the area for candidate cooperative is too largeor too small. Moreover, small distance repulsion canprovide better performance with the optimal cooperativethreshold.

The reminder of this paper is organized as follows. Thesystem model, the proposed BS cooperation strategy andmathematical preliminaries are presented in Section II. InSection III, expressions of SE and ASE for the proposedBS cooperation strategy are analytically derived. Then, theexistence of optimal small BS densities maximizing the SE andASE are proved and the optimal SBS densities are provided inSection IV. Simulation results are given in Section V. Finally,Section VI concludes this paper.

Notation: Lower case characters are used for vectors. De-tailed notations are summarized as in Table I.

TABLE INOTATIONS SUMMARY

Symbol DescriptionΦ1 (λ1) ∈ R2 PPP model of MBSs with density λ1

Φg (λg) ∈ R2 GPP model of all SBSs (normal and untrustedSBS) with density λg

Φ2 (λ2) ∈ R2 PPP model of normal SBSs with density λ2

P tx1 , P tx

2 Transmit power of MBS and SBSN1, N2 Number of antennas at MBS and SBSK1, K2 Capacity of MBS and SBSq Existence probability of untrusted SBSsα Path-loss exponentΓ (x, y), γ (x, y) Upper and lower incomplete gamma functionB (o, r) Circular area centered at o and with radius r1 (∗) Indicator function with value 1 if the event ∗

is true, and 0, otherwise.

II. SYSTEM MODEL AND MATHEMATICAL PRELIMINARY

In this section, the system model, including spatial modelof wireless nodes and the proposed BS cooperation strategy,will be firstly described. Then, the definition of GPP will bepresented.

A. System model and the proposed BS cooperation strategy

Downlink transmission in a two-tier HetNet is considered,where BSs are equiped with multiple antennas. Each MBS andSBS is equipped with N1 and N2 antennas, and serves at mostK1 (K1 6 N1) and K2 (K2 6 N2) users at a resource block,respectively. Single omni-antenna is equipped at mobile users.The transmit power at MBS and SBS are fixed as P tx

1 andP tx2 , respectively. In this paper, we mainly focus on the BS

cooperation in a reference channel.

2D

Active macro BS

Inactive macro BS

Normal active SBS

Inactive SBS

Mobile user

Untrusted SBS

Fig. 2. Two-tier HetNets with untrusted SBSs.

1) Distribution of BS and mobile users: The distributionof MBSs are modeled as a two-dimensional homogeneousPPP Φ1 ∈ R2 with spatial density λ1. Consider the intra-tier repulsion among BSs, a MBS is not retained if there areany other MBSs located around it. Specifically, if any otherMBSs are located in a circle area centered at the MBS withradius D1, i.e., located inside B (MBS,D1), the MBS will beremoved, as depicted in Fig. 2. Then, for an arbitrary MBS,it retains in the process with probability

υ1 =1− e−πD2

1λ1

πD21λ1

(1)

and all the retained MBSs can be approximated as a reformu-lated PPP Φ

′

1 ∈ R2 with thinning density λ′

1 = λ1υ1 [23],[24].

Normal SBSs and untrusted SBSs are considered. Thenormal SBSs are distributed according to a PPP Φ2 ∈ R2

with density λ2. To model the existence of untrusted SBSs,we assume that an untrusted SBS exists near a normal SBSwith probability q and the distance between untrusted SBSand normal SBS follows some certain distribution. As aresult, the distribution of all SBSs, including normal SBSsand untrusted SBSs, are formulated as a Gaussian-PoissonPoint Process Φg ∈ R2, with spatial density λ2 of the parentprocess Φ2 ∈ R2. In the GPP model, one or two pointsare included in a cluster. If there is only one point, thenthe point is considered to be normal SBS. If there are twopoints (referred to as a couple), the point at the position ofparent PPP is considered to be normal SBS while the otheris untrusted SBS. The untrusted SBS can eavesdrop or attack



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the confidential messages transmitted by its couple or interruptthe transmission.

Considering the inter-tier repulsion between MBSs and SB-Ss, those SBSs with cluster center located inside the circulararea of any active MBSs will be removed. We define thecircular area of MBSs as interference protection area (dottedcircle in Fig. 2) and the radius of the protection area is D2.Mathematically, an arbitrary SBS cluster retains if the distancebetween the cluster center and its nearest active MBS is greaterthan D2. Also, we define the minimum intra-tier distancerepulsion between any SBSs as D3. Thus an SBS clusterretains with probability υ2 = e−πD2

2λ′1 1−e−πD2

3λ2

πD23λ2

. All theretained SBS clusters can be approximated as a re-formed GPPΦ

′

g ∈ R2 with parent process Φ′

2 ∈ R2 of density λ′

2 = λ2υ2.Mobile users are distributed according to another indepen-

dent homogeneous PPP Φu ∈ R2 with density λu. Withoutloss of generality, a reference user, viewed as a typical user,is located at the origin of the coordinate plane [25].

2) The proposed capacity- and trust-aware BS cooperationstrategy: Perfect CSI1 is assumed to be well known at BSs,e.g., through pilot training. Both path-loss fading l (d) = d−α

and block Rayleigh fading with unit mean are consideredas signal propagation, where d is the distance between thetransmit-receive pair and α (α > 2) is path-loss exponent.Moreover, Zero-forcing (ZF) is adopted at the BSs to servemultiple users simultaneously, with low complexity and fairlygood performance [26], [27]. BS cooperation is performedin a user centric way, whereby different BSs across tiers areselected to cooperatively serve the mobile users, based on theaverage received signal power of the corresponding link, thecapacity of BSs, and the trustworthiness of SBSs. Specifically,since all the MBSs are trusted, an MBS participates in thecooperation if

• the MBS is not full-loaded, and• the average RSS at mobile users from the MBS is greater

than a predetermined cooperative threshold τ1,For the SBSs, the untrusted SBSs are not allowed to

participate in cooperation and a normal SBSs will joint totransmit the users’ data if

• the normal SBSs are in security state, i.e., i) only one SBSin a cluster, or ii) two SBS in a cluster but the untrustedSBS is located outside the normal SBS’s security area,where the average RSS at the untrusted SBS (from itscouple) is below a security threshold τ3.

• the normal SBS is not full-loaded, and• the average RSS at mobile users from the normal SBS is

greater than a predetermined cooperative threshold τ2.Note that when the load of a BS exceeds its capacity, the

BS will be congested and serving requests sending by mobileusers may be blocked. Moreover, the cooperation conditionof the average RSS not only guarantee the effectiveness ofthe signal propagation, but also restricts the geographical areafor the cooperative BSs and hence reduces the implementationoverhead of BS selection.

1We assume perfect CSI in this paper to investigate the fundamental insightsinto the BS cooperation in HetNet. Imperfect CSI will be studied as our futurework.

Remark: Location information of untrusted BS is assumedto be known in this work. When the untrusted SBSs areactive points, such as an active relay or a pseudo BS sendingscam signals, the average RSS can be evaluated at the normalSBS in a cluster by collecting the statistical information. Forexample, statistical analyzing the signaling characteristics andaccounting the area where the pseudo-base station frequentlyupdate their Location Area Code (LAC) can determine theapproximated location of the pseudo-base station. Then, real-time monitoring can position the pseudo-base station. In sucha way, the average RSS can be evaluated. When the untrustedSBSs are silent points, the channel state information can alsobe acquired based on the observation that the untrusted SBSwill use a local oscillator to perform base-band conversionof the signal from normal SBS, and a known impairment ofoscillators is that part of the generated sinusoidal tone back-propagates to the antenna ports and leaks out [28]. This signalcan be detected by the SBSs in the cluster and used for averageRSS estimation purposes.

B. Gaussian-Poisson Point Process

The GPP is a Poisson cluster process with homogeneousindependent clusters. One point or two points are containedin a cluster with probability 1 − q and q, respectively. Ifa cluster has one point, it is located at the center of thecluster. If a cluster has two points, then one of the pointsis located at the center of the cluster and the other one israndomly distributed around the center with random distance.Mathematically, the GPP can be expressed as Φg =

∪y∈Φ2

Φy ,

where Φ2 = {y1, y2, ...} is the parent process of the GPPand {Φy, y ∈ Φ2} denotes the cluster centered at y. Morespecifically, {Φy} can be expressed as:

Φy =

{{y} , 1− q

{y, y + zy} , q (2)

where {zy} ∈ R2 are identically and independently distributedwith Gamma distributed inter-point distance ∥zx∥ as in [29].The probability density function (pdf ) of ∥zx∥ is denoted as∥zx∥ ∼ Gamma (κ, θ) with parameter κ and θ . Thus, thedensity of the GPP is λg = λ2 (1 + q).

III. SE AND ASE ANALYSIS

In this section, SE and ASE are used as the performancemetrics to evaluate the proposed BS cooperation strategy.The SE is defined as the average data rate per frequencybandwidth achievable at a randomly located user and the ASE,average data per frequency bandwidth in a unit area, is definedto measure the average throughput of the HetNet. In thefollowing, the distribution of BS load is firstly analyzed. Then,the received SINR at the typical user is presented, and the SEand ASE are analyzed, based on the statistics of aggregateinformation-signal strength and interference strength.

A. Statistics of BS load

Denote by ρ1,k1 the probability that k1 6 K1 users areassociated with an arbitrary MBS located at x within a



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resource block, then with BS cooperation strategy for MBSs,ρ1,k1 is expressed as

ρ1,k1=

Prob {k1 mobile users with RSS > τ1} , k1<K1∞∑

k=K1

Prob {k mobile users with RSS > τ1} , k1=K1

=

Prob {k1 mobile users inside B (x, χ1)} , k1<K1∞∑

k=K1

Prob {k mobile users inside B (x, χ1)} , k1=K1

(a)=

(πλuχ

21)

k1 exp(−πλuχ21)

k1!, k1 < K1

∞∑k=K1

(πλuχ21)

kexp(−πλuχ

21)

k! , k1 = K1

=

(πλuχ21)


k1!, k1 < K1

γ(K1,πλuχ21)

K1!, k1 = K1

(3)

where (a) follows the PPP of mobile users. χ1 =(

P tx1

τ1

)1/αis

the radius for cooperative MBSs and γ (x, y) =∫ y

0tx−1e−tdt

is lower incomplete gamma function. Similarly, denote byρ2,k2 the probability that k2 6 K2 users are associated withan arbitraty normal SBS located at y within a time slot. Forsimplicity, we denote E1 as the event that only one normalSBS is in a cluster, E2 as the event that two SBSs are in acluster, and E3 as the event that the untrusted SBSs is out ofthe security area. Then the probabilities that events E1, E2

and E3 occur are respectively given as follows.

Prob {E1} = 1− q, Prob {E2} = q (4)

Prob {E3} = Prob {Received RSS at the couple 6 τ3}

= Prob{P tx2 ∥zy∥−α

< τ3

}=

Γ(κ, χ3

θ

)Γ (κ)

, η,(5)

where χ3 =(

P tx2

τ3

) 1α

is the minimum radius for the securityarea. Γ (x, y) = 1 − γ (x, y) is the upper incomplete gammafunction. As a result, when a normal SBS is not in security s-tate, the normal SBS is zero-loaded with probability q (1− η).Otherwise, a normal SBS is in security state with probability(1− q + qη). And the load distribution ρ2,ky

2of an secure SBS

can be expressed as:

ρ2,k2=

Prob {k2 mobile users inside B (y, χ2)} , k2 < K2∞∑

k=K2

Prob {k mobile users inside B (y, χ2)}, k2=K2

=

(πλuχ22)


k2!, k2 < K2

γ(K2,πλuχ22)

K2!, k2 = K2

(6)

where χ2 =(

P tx2

τ2

)1/αis the minimum radius of the area for

cooperative SBSs.

B. Received Signal-to-Interference-plus-Noise Ratio

With the proposed BS cooperation, mobile users not onlyreceive information signals from cooperative serving BSs, butalso suffer from inter-tier and intra-tier interference from allthe other BSs except the serving BSs. The resulting linksfrom cooperative BSs and interfering BSs to the mobile users

are multiple-user multiple-input single-output (MU-MISO)channels. Noncoherent joint transmission2 is employed at thecooperative BSs as in [14], [15] and equal power allocation isadopted. The received SINR at the typical user is thus givenas

γ =Sm + Ss

Im + Is + σ2(7)

where σ2 is white noise power and

Sm =∑

x∈Φ′1∩B(0,χ1)

1 (kx1 < K1)P tx1

kx1 + 1∥x∥−α

hx (8)

is the aggregate information-signal strength from cooperativeMBSs. 1 (kx1 < K1) is indicator function defined such that1 (kx1 < K1) = 1 if the MBS located at x is not full-loaded, and 1 (kx1 < K1) = 0, otherwise. Ss is the aggregateinformation-signal strength from the cooperative SBSs.

Ss =∑

y∈Φ′2∩B(0,χ2)

1 (ky2 < K2)1 (E1 or (E2&E3))P tx2

ky2 + 1∥y∥−α

hy

(9)where 1 (E1 or (E2&E3)) is indicator function indicatingthat the normal SBSs located at y is in security state. Forthe aggregate interference Im, since the interfering MBSsinclude MBSs inside B (o, χ1) and outside B (o, χ1), Im canbe expressed as:

Im = Im,in + Im,out

=∑

x′∈Φ′1∩B(0,χ1)

1

(kx

′

1 = K1

) P tx1

kx′

1

∥∥∥x′∥∥∥−α

hx′

+∑

x′∈Φ′1\B(0,χ1)

1

(kx

′

1 > 0) P tx

1

kx′

1

∥∥∥x′∥∥∥−α

hx′

(10)

where Im,in is the aggregate interference caused by MBSsinside B (o, χ1) and Im,out is the aggregate interference causedby MBSs outside B (o, χ1). Similarly, the interfering SBSsinclude SBSs inside B (o, χ2) and outside B (o, χ2) and theaggregate interference Is can be expressed as:

Is = Is,in + Is,out

=∑

y′∈Φ′2∩B(0,χ2)

1

(ky

′

2 = K2

)1 (E2&E3)

P tx2

ky′

2

∥y∥−αhy′+

1

(0<ky

′

2 <K2

)1 (E2&!E3)

P tx2

ky′

2

∥∥∥y′∥∥∥−α

hy′

+

∑y′∈Φ

′2\B(0,χ2)

1

(ky

′

2 > 0

)P tx2

ky′

2

∥∥∥y′∥∥∥−α

hy′

(11)where !E3 is the opposite event of E3, i.e., the event thatthe untrusted SBSs is inside the security area. Notice that theuntrusted SBSs will not actively transmit signals to any mobileusers, and thus they will not interfere with any communicationlinks.

From the system model, ZF precoding is adopted at BSs.Thus, if a MBS x cooperates to serve the typical user, the

2As stated in [14], noncoherent cooperation outperforms its coherentcounterpart due to less stringent synchronization and CSI requirements



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small-scale fading hx is gamma distributed with parametersN1 − kx1 and 1, i.e., hx ∼ Gamma (N1 − kx1 , 1) [26], [30]. Ifa MBS x

′does not cooperate to serve the typical user, hx′

is approximated as gamma distribution with parameters kx′

1

and 1, i.e., hx′ ∼ Gamma(kx

′

1 , 1)

[27], [31]. Similarly, hyand hy′ are gamma distributions, hy ∼ Gamma (N2 − ky2 , 1),

hy′ ∼ Gamma(ky

′

2 , 1

).

C. Spectral Efficiency

Adopting appropriate adaptive modulation and coding, thespectral efficiency achievable at the typical user is

Rc = E[ln

(1 +

Sm + Ss

Im + Is + σ2

)](nats/s/Hz) (12)

where the expectation is taken average over the statistics ofinformation-signal strength and interference strength, whichconsiders the effect of BS capacity and trustworthiness ofSBSs. Denote by ρ1 = [ρ1,0, ρ1,1, ..., ρ1,K1 ] and ρ2 =[ρ2,0, ρ2,1, ..., ρ2,K2 ] the vectors of the BS load at MBSs andSBSs, respectively. Then, the SE is analytically derived as inthe following theorem.

Theorem 1. With the awareness of BS capacity and trust-worthiness, the spectral efficiency with the proposed BS co-operation strategy in the MU-MISO non-uniform HetNets isapproximated as

Rc ≈∞∫0

exp {−λ1u1 (t)− λ2u2 (t)}

× {1− exp {−λ1w1 (t)− λ2w2 (t)}}e−σ2t

tdt

(13)

where

ui (t) = πυiai

χ2i ρi,Ki

(1−Z

(Ki,

tτiKi

))+(tP tx

i

)2/αψ (Ki, ρi, tτi)

, (14)

wi (t) = πυiaiχ2i

Ki−1∑ki=0

ρi,ki

(1−Z

(Ni − ki,

tτiki + 1

))(15)

with

ψ (a,b, x) =2

α

a∑k=1

bk

∞∫x−1

(1− (kv)

k

(kv + 1)k

)v

2α−1dv, (16)

Z (b, x) =2x−b

αb+ 2×2F1

(b, b+

2

α; b+ 1 +

2

α;− 1

x

). (17)

and a1 = 1, a2 = 1− q + qη.

Proof: Please refer to the detailed proof in Appendix A.

Although Eq. (13) is in an integral form, ψ (a,b, x) andZ (b, x) can be easily calculated by Matlab, and therefore Rc

can be numerically evaluated by applying Monte Carlo integra-tion methods. Compared with existing literatures, expression(13) reflects BS load and security state of small BSs.

D. Area Spectral Efficiency

The ASE is applied as a metric to measure the networkthroughput in a unit area. The average mobile users associatedwith a MBS and a normal SBS are given as

K1 =

K1∑k1=1

ρ1,kk1,K2 = (1− q + qη)

K2∑k2=1

ρ2,kk2 (18)

For a sufficient large area S, the average network throughputper unit area, i.e., ASE, with the proposed BS cooperationstrategy is

ASE =

(Sλ′

1K1 + Sλ′

2K2

)Rc

S=(λ

′

1K1 + λ′

2K2

)Rc

(19)for which the unit is nats/s/Hz/m2.

From (13) and (19), the path-loss exponent α, the existenceprobability of untrusted SBSs (q), the density of trusted BSs(λ1 and λ2), and the distance repulsion among BSs have greatimpact on the SE and ASE performance.

Since the normal SBSs reuse the spectrum resources ofMBSs, the density of normal SBSs (i.e., λ2) has great impacton the SE and ASE performance. When λ2 is below acertain value, the enhancement of information-signal strengthis greater than the increase of interference strength with denserSBSs. This is caused by the fact that the propagation ofinformation signals transform from non-line-of-sight (NLOS)to line-of-sight (LOS) with the denser of SBSs, while the prop-agation of interference signals are still dominated by NLOS.With increasing λ2, the increase of interference strength willdominate the increase of information-signal strength. As a re-sult, the system performance will be firstly improved and thendeteriorated over λ2. To exploit the impact of the deploymentof normal SBSs, we study the optimization problem for thedensity of normal SBSs in the following section.

IV. OPTIMIZATION OF SBS DENSITY

Inter-tier and intra-tier interferences in HetNets deterioratesystem performance severely. MBSs, well-planed by operators,are always assumed to be deployed with fixed density. Whilethe low-power SBSs are deployed by operators and costumersaccording to connectivity and traffic demands. Appropriatelyincreasing the deployment of SBSs can improve networkcapacity and coverage for both mobile users and the overallnetwork. However, these performance may degrade with theincreasing of SBS density due to the sharp increase of inter-ference3. Hence, in this section, we prove the existence of theoptimal λ2 and provide expressions for the optimal densities.

Differentiating (13) with respect to λ2, we have

dRc

dλ2=

∞∫0

f (t, λ2)e−σ2t

tdt (20)

3As for fixed transmit power at SBSs, the interference increases sharplyand the system performance, like coverage and network capacity, deteriorates.We focus on the SBS density optimization in this paper and will study theinfluence of SBS density with the varying of transmit power in the future.



7

where f (t, λ2) is shown as

f (t, λ2)

=d

dλ2

(exp {−λ1u1 (t)− λ2u2 (t)}×{1− exp {−λ1w1 (t)− λ2w2 (t)}}

)= exp {−λ1w1 (t)} (u2 (t) + w2 (t))

× exp {−λ2 (u2 (t) + w2 (t))− λ1u1 (t)}−u2 (t) exp {−λ2u2 (t)− λ1u1 (t)}

(21)

Defining the function g (z) = ze−λ2z−λ1u1(t), then f (t, λ2)can be re-expressed as

f (t, λ2) = exp {−λ1w1 (t)} g (u2 (t) + w2 (t))− g (u2 (t))(22)

To prove the existence of the optimal normal SBS density, weprove

1) dRc

dλ2= 0 has a unique solution λ∗2, and

2) dRc

dλ2> 0 when λ2 < λ∗2 and dRc

dλ2< 0 when λ2 > λ∗2.

In the following, we provide the theorem and correspondingproof for the existence of optimal λ2 that maximizes thespectral efficiency.

Theorem 2. With the awareness of BS capacity and trust-worthiness, optimal normal SBS density exists such that thespectral efficiency with the proposed BS cooperation strategyis maximized. In addition, the optimal λ2 is the solution of thefollowing equation.

∞∫0

{exp [−λ1w1 (t)] g (u2 (t) + w2 (t))

−g (u2 (t))} e−σ2t

t dt = 0

(23)

Proof: With the definition of g (z), it satisfies g′(z) > 0

when z < 1λ2

, and g′(z) 6 0 when z > 1

λ2, which is a

unimodal function with a unique maximum value at z∗ = 1λ2

.With variable 0 < a 6 1 and a positive y, there always existan z∗ = ay

eλ2y−asatisfying the following inequality.

• When z < z∗, ag(z+y)g(z) > 1, i.e., ag (z + y)− g (z) > 0

• When z > z∗, ag(z+y)g(z) < 1, i.e., ag (z + y)− g (z) < 0

An illustration of function g (z) is depicted as in Fig. 3.Since ui (t) and wi (t) are increasing functions of t and

0 < exp {−λ1w1 (t)} 6 1, there always exists u∗i (t∗) and

w∗i (t

∗) with t∗, satisfying the following inequality.• When t < t∗, f (t, λ2) > 0• When t > t∗, f (t, λ2) < 0

Thus, the integral function (20) starts with a positive value,which is increasing when the upper limit of integral is belowt∗ and decreasing when the upper limit of integral is abovet∗. Function g (z) states that z∗ decreases with increasing ofλ2. Since ui (t) and wi (t) are increasing function of t, t∗

decreases with λ2. As a result, dRc

dλ2is a decreasing function

of λ2. On the other hand, g (u2 (t) + w2 (t)) and g (u2 (t))state that tn ∝ c

λ2with a certain n. Thus, lim t∗ → ∞ with

limλ2 → 0, and dRc

dλ2> 0 hold. Moreover, since

limλ2→∞

exp {−λ1w1 (t)} g (u2 (t) + w2 (t))

g (u2 (t))= 0, (24)

z

g(z)

y2

y1 g(z)

ag(z+y2)

ag(z+y1)

z1*

z1*

Fig. 3. An illustration of Function g (z)

we have limλ2→∞

f (t, λ2) < 0, ∀t and

limλ2→∞

dRc

dλ2= lim

λ2→∞

∞∫0

f (t, λ2)e−σ2t

tdt < 0 (25)

holds.Finally, sum up the above discussions as in Table II4. It is

TABLE IITREND OF VARIABLES AND FUNCTIONS

λ2 t∗ f (t, λ2)dRcdλ2

→ 0 → ∞ \ > 0↑ ↓ \ ↓→ ∞ → 0 < 0 < 0

concluded that the unique optimal value for λ2 exists and canbe obtained by solving the equation dRc

dλ2= 0.

Similarly, we prove the existence of the optimal λ2 thatmaximizes the ASE with the proposed BS cooperation scheme.

Corollary 1. In the MU-MISO non-uniform HetNets with theproposed BS cooperation strategy, there exists optimal normalSBS density maximizing the ASE performance, and the optimalvalue of the SBS density can be obtained by setting (26) tozero.

Proof: Differentiating (19) with respect to λ2, we have

dASE

dλ2= K2υ2Rc +

(λ

′

1K1 + λ′

2K2

) dRc

dλ2. (26)

Since 0 6 K2υ2Rc 6 +∞, combining (25) and Table II, wecan conclude that the ASE is an upper unimodal function andthe unique optimal SBS density can be obtained by setting(26) to zero.

Combining Eq. (20) and (26), since K2υ2Rc > 0 andλ

′

1K1 + λ′

2K2 > 0, the optimal normal SBS density maxi-mizing the ASE is denser than that maximizing the SE.

4↑ means increasing and ↓ means decreasing.



8

V. SIMULATION RESULTS

In this section, simulation results and numerical results areprovided to validate the theoretical analyses and provide someinsights in the MU-MISO HetNets with the proposed BScooperation scheme. The detailed system parameters are givenin Table. III unless otherwise specified. The simulation resultsare obtained through 104 times Monte Carlo simulations in acircular area with radius 10km. For each spatial realization,the MBSs and SBSs are randomly distributed according tohomogeneous PPPs and are deactivated with the distancerepulsion between MBSs and SBSs. An SINR sample isobtained by generating independent Rayleigh random vectorsfor the fading channels.

TABLE IIIPARAMETERS USED IN THE SIMULATIONS

Parameters ValueDensity of MBS, the parent process of SBSand mobile user. (λ1, λ2 and λu)

1π5002

, 5π5002

, 5π5002

Number of antennas at MBS and SBS N1 = 8, N2 = 4Capacity at MBS and SBS K1 = 8, K2 = 4Transmit power at MBS and SBS P tx

1 = 37dBm,P tx2 = 30dBm

Parameters for Gamma distribution of inter-point distance

κ = 25, θ = 2

Thermal noise σ2 (10MHz bandwidth) σ2 = −104dBm

Figs. 4-8 show the comparison of the analytical results andthe simulation results. The gap between the simulation resultsand the analytical results is due to the PPP approximation ofthe retained MBSs and SBSs. However, the gap is quite small,which verifies the accuracy of the PPP approximation and thederived theoretical expressions. Furthermore, Fig. 4-6 showthe SE and ASE with respect to the cooperative thresholdsτ1 and τ2 with different distance repulsions. Firstly, differentcooperative threshold τ2 and path-loss exponent α are analyzedin Fig. 4. It can be seen that optimal cooperative thresholdsalways exist such that the maximum SE and maximum ASEcan be achieved. The reason mainly lies in the fact that withthe limited capacity of BSs, severe congestion will occur whenthe cooperative thresholds are lower than some certain values.In such cases, the average probabilities for a mobile userto get served, i.e., Prob (kx1 < K1) and Prob (ky2 < K2), arealmost zero and the SE will be near zero. With the increasingof cooperative thresholds, there will be less congestion andProb (kx1 < K1) and Prob (ky2 < K2) increases. However, thenumber of BSs that can provide services for a mobile userdecreases as cooperative thresholds rise. As a result, therewill exist optimal cooperative thresholds balancing the servingprobability and number of cooperative BSs, which determinethe best SE and ASE performance. Moreover, low path-lossexponent results in high optimal cooperative threshold τ1.

Fig. 5 and Fig. 6 show the effect of distance repulsions D1,D2 and D3. Form these two figures, we can see that whenthe cooperative thresholds τ1 and τ2 are small, the BSs arealmost full-loaded with probability 1 and thus the BSs can notparticipate in cooperation but they interferer with the typicaluser. Therefore, a larger distance repulsion between BSs candeactivate more BSs and provide better SE performance butdeteriorate the overall network performance, i.e., the ASE.

−120 −100 −80 −60 −40 −20 00

0.5

1

1.5

2

2.5

3

τ1 (dBm)

SE

(bi

ts/s

/Hz)

−150 −100 −50 00

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−5

τ1 (dBm)

AS

E (

bits

/s/H

z/m

2 )

α=4.28,τ2=−80dBm (sim.)

α=4.28,τ2=−80dBm (ana.)

α=4.28,τ2=−60dBm (sim.)

α=4.28,τ2=−60dBm (ana.)

α=3.75,τ2=−80dBm (sim.)

α=3.75,τ2=−80dBm (ana.)

α=3.75,τ2=−60dBm (sim.)

α=3.75,τ2=−60dBm (ana.)

Fig. 4. SE and ASE with respect to cooperative threshold τ1 for differentpath-loss exponents and cooperative thresholdτ2 (q = 0.5, D1 = 400m,D2 = 100m, D3 = 20m, τ3 = −45dBm).

−120 −100 −80 −60 −40 −20 01

1.5

2

2.5

τ1(dBm)

SE

(bi

ts/s

/Hz)

−120 −100 −80 −60 −40 −20 0

1.8

2

2.2

2.4

2.6

2.8

3

3.2x 10

−5

τ1(dBm)

AS

E (

bits

/s/H

z/m

2 )

D1=200 (ana.)

D1=200 (sim.)

D1=600 (ana.)

D1=600 (sim.)

−30 −20 −10

1.71

1.72

1.73

Fig. 5. SE and ASE with respect to τ1 (α = 4.28, q = 0.5, D2 = 100m,D3 = 20m, τ2 = −80dBm, τ3 = −45dBm).

With the increasing of τ1 and τ2, the loads at BSs are relaxedand more BSs can participate in cooperation to further improvethe system performance. As a result, reducing the distancerepulsion D1 and D2 can retain more BSs and provide betterSE and ASE performance. However, with the increasing ofτ1, the number of MBSs as well as the MBS load are reducedand finally no MBS will cooperate to serve the typical user. Insuch a scenario, a larger distance repulsion D1 can make theMBS more sparse and thus more SBSs can be retained to havethe opportunity to participate in cooperation. Besides, we cansee from Fig. 6 that the intra-tier distance repulsion betweenSBSs has little influence on the SE and ASE performance.

The existence of untrusted SBSs greatly influences theselection of cooperative SBSs and thus affects the systemperformance. Fig. 7 shows the SE and ASE with respect to theexistence probability of untrusted SBSs q for different securitythreshold τ3. The average probability that the untrusted SBSsis out of the security area is η ≈ 1 when τ3 = −35dBm



9

−120 −100 −80 −60 −40 −20 00.5

1

1.5

2

2.5

3

τ2 (dBm)

SE

(bi

ts/s

/Hz)

−120 −100 −80 −60 −40 −20 00.5

1

1.5

2

2.5

3x 10

−5

τ2 (dBm)

AS

E (

bits

/s/H

z/m

2 )

D2=100m, D

3=50m

D2=100m, D

3=10m

D2=300m, D

3=50m

D2=300m, D

3=10m

Fig. 6. SE and ASE with respect to τ2 (α = 4.28, q = 0.5, D1 = 400mτ1 = −80dBm, τ3 = −45dBm).

0 0.2 0.4 0.6 0.8 11.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

q

SE

(bi

ts/s

/Hz)

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−5

q

AS

E (

bits

/s/H

z/m

2 )

τ3=−35dBm (sim.)

τ3=−35dBm (ana.)

τ3=−45dBm (sim.)

τ3=−45dBm (ana.)

τ3=−65dBm (sim.)

τ3=−65dBm (ana.)

Fig. 7. SE and ASE with respect to q (α = 4.28, D1 = 400m, D2 = 100m,D3 = 20m, τ1 = −80dBm, τ2 = −80dBm).

and η ≈ 0 when τ3 = −65dBm. η ≈ 1 means that all thenormal SBSs are in security state and can provide secure datatransmission for mobile users. In such a case, the existenceprobability of untrusted SBSs does not influence the SE andASE performance. η ≈ 0 means that all the normal SBSsin a cluster containing two SBSs are not in security stateand cannot cooperate to transmit user’s data. In such a case,the average number normal SBSs that can participate in thecooperation decreases over q and the SE and ASE performancedeteriorate with the existence probability of untrusted SBSs.

Fig. 8 shows the impact of the density of normal SBSs. Itcan be seen that the SE and ASE are unimodal functions withλ2, which numerically validates Theorem. 2 and Corollary1. Moreover, it can also be seen that the optimal λ2 witha high path-loss exponent is greater than that with a smallpath-loss exponent, and a larger path-loss exponent provideshigher SE and ASE. This result indicates that, to achieve thebest SE and ASE performance, more normal SBSs should bedeployed in high path-loss fading environments. Also, better

100

102

104

0

1

2

3

4

5

6

λ2 (× 1/(π5002))

SE

(bi

ts/s

/Hz)

100

102

104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−3

λ2 (× 1/(π5002))

AS

E (

bits

/s/H

z/m

2 )

α=4.28 (sim.)α=4.28 (ana.)α=3.75 (sim.)α=3.75 (ana.)

Fig. 8. SE and ASE with respect to λ2 (q = 0.5, τ1 = −65dBm, τ2 =−65dBm, τ3 = −35dBm).

system performance can be achieved in high path-loss fadingenvironment.

Fig. 9 and Fig. 10 show the optimal normal SBS density andthe corresponding SE and ASE with respect to the cooperativethreshold τ2 for different path-loss exponent α and existenceprobability of untrusted SBSs q, respectively. From Fig. 9, itcan be seen that the optimal λ2 increases with the cooperativethreshold τ2, which is caused by the fact that denser SBSsare needed to achieve the maximum SE and ASE when thearea for cooperative BSs shrinks. Moreover, higher optimalSBS density is needed with larger q and the optimal λ2maximizing ASE is greater than that maximizing SE with thesame cooperative thresholds. Fig. 10 shows that there is anoptimal cooperative threshold τ2 such that the SE and ASEare maximized. In addition, a lower cooperative thresholdis desired in wireless environments with higher path-lossfading, which is the same to the results obtained in Fig. 4. Inaddition, the maximum SE and ASE do not change with theexistence probability of untrusted SBSs because the existenceof untrusted SBSs only influences the density of the normalSBSs in secure status. As a result, to achieve the same densityof secure normal SBSs, more normal SBSs are needed withhigh existence probability of untrusted SBSs.

VI. CONCLUSION

In this paper, a capacity- and trust-aware BS cooperationstrategy has been proposed for the MU-MISO non-uniformHetNets. Closed-form expressions for spectral efficiency andarea spectral efficiency have been analytically obtained withthe proposed cooperation scheme. Furthermore, optimal den-sities of normal SBSs maximizing the SE and ASE have beenproved to exist and derived. The proposed BS cooperationstrategy can be employed in practical systems, where capacityof BSs are limited and not all SBSs are trusted. Simulationresults have validated the theoretical analysis and demon-strated that there also exist optimal cooperative thresholdsfor determination of cooperative BSs to maximize the SEand ASE. Moreover, to achieve the maximum SE and ASE



10

−100 −80 −60 −40 −200

2000

4000

6000

8000

10000

12000

τ2 (dBm)

Opt

imal

λ2 m

axim

izin

g S

E (

× 1/

(π50

02 ))

−100 −80 −60 −40 −200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

4

τ2 (dBm)

Opt

imal

λ2 m

axim

izin

g A

SE

(×

1/(π

5002 ))

α=4.28, q=0.2α=4.28, q=0.8α=3.75, q=0.2α=3.75, q=0.8

α=4.28, q=0.2α=4.28, q=0.8α=3.75, q=0.2α=3.75, q=0.8

Fig. 9. Optimal density of untrusted small BSs with respect to τ2 (D1 =400m, D2 = 100m, D3 = 20m, τ1 = −60dBm, τ3 = −45dBm).

−100 −80 −60 −40 −201

1.5

2

2.5

3

3.5

4

4.5

5

5.5

τ2 (dBm)

Max

imum

SE

(bi

ts/s

/Hz)

−100 −80 −60 −40 −200

0.5

1

1.5

2

2.5x 10

−3

τ2 (dBm)

Max

imum

AS

E (

bits

/s/H

z/m

2 )

α=4.28, q=0.2α=4.28, q=0.8α=3.75, q=0.2α=3.75, q=0.8

α=4.28, q=0.2α=4.28, q=0.8α=3.75, q=0.2α=3.75, q=0.8

Fig. 10. The corresponding maximum SE and ASE.

performance, lower cooperative thresholds are required andmore normal SBS are needed to deployed in the high path-loss fading environments. Besides, for the scenario with a highprobability of the existence of untrusted SBSs, more normalSBSs are required to achieve the best SE and ASE.

For the future work, we will investigate BS cooperation,when reputation based scheme [32] is adopted to monitor andmanage the trustworthiness of SBSs. Interested readers canrefer to [33], [34] for a general overview of the reputationbased system. In addition, cooperation among SBSs withenergy harvesting will also be studied, where the dynamicenergy status of SBSs should be considered when performingcooperation.

APPENDIX APROOF OF THEOREM 1

Before we analyze the spectral efficiency, a useful lemmais presented as follow.

Lemma 1. Let x1, ..., xN , y1, ..., yM be arbitrary non-

negative random variables. Setting x =N∑

n=1xn and y =

M∑m=1

ym, then

E[ln

(1 +

x

1 + y

)]=

∞∫0

Ly (t)− Ly+x (t)

te−tdt (27)

where LA (t) = E[e−tA

]is the Laplace function of random

variable A.

Proof: Detailed proof can be referred to in [35].

With Lemma 1, the SE of a typical user is

Rc(c)=

∞∫0

LIm+Is (t) (1− LSm+Ss (t))

te−σ2tdt

(d)≈

∞∫0

LIm (t)LIs (t) (1− LSm (t)LSs (t))

te−σ2tdt

(28)

where (c) holds due to the independence between informationpower and interference power and (d) is the result of theindependence between the retained MBSs and SBSs in theapproximated PPPs. Related with the Laplace functional ofpoint processes, the Laplace transform of variable Sm can becalculated as

LSm (t) = ESm

[e−tSm

]= EΦ

′1,hx

∏x∈Φ

′1∩B(0,χ1)

e−1(kx

1<K1)tPtx1

kx1+1

∥x∥−αhx

(e)= exp

−2πλ′

1

χ1∫0

(1− Eh

[e−1(k1<K1)t

Ptx1

kx1+1

r−αh])

rdr

(f)=exp

−2πλ′

1

χ1∫0

(1−ρ1,K1−

K1−1∑k1=0

ρ1,k1Eh

[e−t

Ptx1

k1+1 r−αh

])rdr

(29)

where (e) follows the probability generating functional(PGFL) of PPP [25] and (f) follows the distribution of k1and index 1 (k1 < K1). Since h ∼ Gamma (N1 − k1, 1),

Eh

[e−

tPtx1

k1+1 r−αh

]is derived as

Eh

[e−

tPtx1

k1+1 r−αh

]=

(1 +

tP tx1

k1 + 1r−α

)−(N1−k1)



11

Then, LSm (t) can be re-written as

LSm (t) = exp

−2πλ′

1

(1− ρ1,K1)

χ1∫0

rdr−

K1−1∑k1=0

χ1∫0

ρ1,k1(1 + t

P tx1

k1+1r−α)(N1−k1)

rdr

(g)=exp

−πλ′

1

[(1− ρ1,K1)χ

21 −

K1−1∑k1=0

ρ1,k1

(tP tx

1

k1 + 1

)2/α

×

(tτ1

k1+1

)−2/α∫0

uα2 (N1−k1)(

1 + uα2

)(N1−k1)du

(30)

(h)= exp

−πλ′

1

[(1− ρ1,K1)χ

21 −

K1−1∑k1=0

2ρ1,k1(tPtx1 )

2/α

(k1 + 1)2/α

α

×

k1+1tτ1∫0

v2α+(N1−k1)−1

(1 + v)(N1−k1)

dv

(i)=exp

{−πλ

′

1χ21

K1−1∑k1=0

ρ1,k1

(1−Z

(N1 − k1,

tτ1k1 + 1

))}(31)

where (g) and (h) are obtained by replacing

u =(

tP tx1

k1+1

)−2/α

r2 and v = uα2 , respectively. (i) is derived

by applying Eq. (3.194/2) in [36].Similarly, with (9), the Laplace transform of Ss is

LSs (t) = ESs

[e−tSs

]

= exp

−2πλ

′

2

χ2∫0

[1−

Eh

[e−1(k2<K2)1(E1 or (E2&E3))t

Ptx2

k2+1 r−αh

]]rdr

(j)= exp

−2πλ

′

2

χ2∫0

[1− (ρ2,K2 + (1− ρ2,K2) q (1− η))

−a2K2−1∑k2=0

ρ2,k2Eh

[e−t

Ptx2

k2+1 r−αh

]rdr

]

(k)=exp

−2πλ

′

2a2×

K2−1∑k2=0

ρ2,k2

χ2∫0

1− 1(1 + t

P tx2

k2+1r−α)(N2−k2)

rdr

=exp

{−πλ

′

2χ22a2

K2−1∑k2=0

ρ2,k2

(1−Z

(N2 − k2,

tτ2k2 + 1

))}(32)

where (j) holds due to the fact that a normal SB-S does cooperate to transmit users’ data with proba-bility ρ2,K2 + (1− ρ2,K2) q (1− η) and the corresponding

Eh

[e−t

Ptx2

k2+1 r−αh

]= 1. a2 = 1 − q + qη is the probability

that the normal SBS in a cluster is in security state. (k)is the result from the Gamma distribution of h, i.e., hy ∼Gamma (N2 − ky2 , 1).

Since the aggregate interference generated by MBSs insideB (o, χ1) and outside B (o, χ1) are independent, the Laplacetransform of interference power Im can be calculated as

LIm (t) = EIm,in

[e−tIm,in

]EIm,out

[e−tIm,out

](33)

where EIm,in

[e−tIm,in

]is derived as

EIm,in

[e−tIm,in

]= EΦ

′1,hx

′

e−t∑

x′∈Φ

′1∩B(0,χ1)

1

(kx

′

1 =K1

)Ptx1

kx′

1

∥∥∥x′∥∥∥−αhx′

= exp

−2πλ

′

1

χ1∫0

[1−

((ρ1,K1Eh

[e−t

Ptx1

K1r−αh

]+ 1− ρ1,K1

))rdr

]

(l)= exp

−2πλ′

1

χ1∫0

ρ1,K1 −ρ1,K1(

1 + tP tx

1

K1r−α

)K1

rdr

= exp

{−πλ

′

1χ21ρ1,K1

(1−Z

(K1,

tτ1K1

))},

(34)and EIm,out

[e−tIm,out

]is

EIm,out

[e−tIm,out

]= EΦ

′1,hx

′

e−t∑

x′∈Φ

′1\B(0,χ1)

1

(kx

′

1 >0

)Ptx1

kx′

1

∥∥∥x′∥∥∥−αhx′

=exp

−2πλ′

1

∞∫χ1

(1−ρ1,0−

K1∑k1=1

ρ1,k1Eh′

[e−t

Ptx1k1

r−αh′])

rdr

(l)=exp

−2πλ′

1

∞∫χ1

K1∑k1=1

ρ1,k1

1− 1(1+t

P tx1

k1r−α

)k1

rdr

= exp

{−πλ

′

1

(tP tx

1

)2/αψ (K1, ρ1,∗, tτ1)

}.

(35)(l) is the result from hx′ ∼ Gamma

(kx

′

1 , 1)

.Similarly, the Laplace transform of Is can be obtained as

LIs (t) = EIs,in

[e−tIs,in

]EIs,out

[e−tIs,out

], where

EIs,in

[e−tIs,in

]

= exp

−2πλ

′

2

χ2∫0

[1− (1− q + qη)

K2−1∑k2=1

ρ2,k2 − q (1− η)

− (1− q + qη) ρ2,K2Eh

[e−t

Ptx2

K2r−αh

]rdr

= exp

{−πλ

′

2χ22 (1− q + qη) ρ2,K2

(1−Z

(K2,

tτ2K2

))}(36)



12

and

EIs,out

[e−tIs,out

]=exp

−2πλ′

2

∞∫χ2

1− Eh′

e−1(ky

′

2 >0

)tPtx2

ky′

2

r−αh′ rdr

= exp

{−πλ

′

2(1− q + qη)(tP tx

2

)2/αψ (K2, ρ2,∗, tτ2)

}(37)

Substituting (31), (32) and (34)-(37) into (28), (13) forspectral efficiency is proved.

ACKNOWLEDGMENT

The authors would like to thank the China ScholarshipCouncil for supporting Huici Wu the international visiting atUniversity of Waterloo, Waterloo, Canada.

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13

Huici Wu received her B.S. degree in InformationEngineering School from Communication Universityof China, Beijing, China, in 2013. She is currentlypursuing Ph.D. degree with the communications andinformation systems at Beijing University of Post-s and Telecommunications, Beijing, China. FromSeptember 2016 to August 2017, she is visitingthe Broadband Communications Research (BBCR)Group, Department of Electrical and Computer En-gineering, University of Waterloo, Waterloo, ON,Canada. Her research interests are in the area of

wireless communications and networks, with current emphasis on the co-operation and physical layer security in heterogeneous networks.

Ning Zhang (M’15) received the Ph.D degree fromUniversity of Waterloo in 2015. He is now anassistant professor in the Department of ComputingScience at Texas A&M University-Corpus Christi.Before that, he was a postdoctoral research fellowat BBCR lab in University of Waterloo. He wasthe co-recipient of the Best Paper Award at IEEEGLOBECOM 2014 and IEEE WCSP 2015. Hiscurrent research interests include next generationwireless networks, software defined networking, ve-hicular networks, and physical layer security.

Xiaofeng Tao (SM’13) received the B.S. degree inelectrical engineering from Xian Jiaotong University,Xian, China, in 1993 and the M.S.E.E. and Ph.D.degrees in telecommunication engineering from theBeijing University of Posts and Telecommunications(BUPT), Beijing, China, in 1999 and 2002, respec-tively.

He was a Visiting Professor with Stanford Uni-versity, Stanford, CA, USA, from 2010 to 2011; theChief Architect of the Chinese National FuTUREFourth-Generation (4G) TDD working group from

2003 to 2006; and established the 4G TDD CoMP trial network in 2006. Heis currently a Professor with the BUPT and a Fellow of the Institution ofEngineering and Technology. He is the inventor or coinventor of 50 patentsand the author or coauthor of 120 papers in 4G and beyond 4G.

Zhiqing Wei (S’13-M’15) received the B.S. andPh.D. degrees from Beijing University of Posts andTelecommunications (BUPT) in 2010 and 2015 re-spectively. Now he is a lecture at BUPT. His re-search interests are network information theory andoptimization of wireless networks such as cognitiveradio networks, robotic networks, etc.

Xuemin (Sherman) Shen (M’97-SM’02-F’09) re-ceived the B.Sc.(1982) degree from Dalian MaritimeUniversity (China) and the M.Sc. (1987) and Ph.D.degrees (1990) from Rutgers University, New Jersey(USA), all in electrical engineering. He is a Univer-sity Professor and the Associate Chair for GraduateStudies, Department of Electrical and Computer En-gineering, University of Waterloo, Canada. Dr. Shensresearch focuses on resource management, wirelessnetwork security, social networks, smart grid, andvehicular ad hoc and sensor networks. He was an

elected member of IEEE ComSoc Board of Governor, and the Chair of Dis-tinguished Lecturers Selection Committee. Dr. Shen served as the TechnicalProgram Committee Chair/Co-Chair for IEEE Globecom16, Infocom14, IEEEVTC10 Fall, and Globecom07, the Symposia Chair for IEEE ICC10, theTutorial Chair for IEEE VTC11 Spring and IEEE ICC08, the General Co-Chair for ACM Mobihoc15, Chinacom07 and QShine06, the Chair for IEEECommunications Society Technical Committee on Wireless Communications,and P2P Communications and Networking. He also serves/served as theEditor-in-Chief for IEEE Internet of Things Journal, IEEE Network, Peer-to-Peer Networking and Application, and IET Communications; a FoundingArea Editor for IEEE Transactions on Wireless Communications; an AssociateEditor for IEEE Transactions on Vehicular Technology, Computer Networks,and ACM/Wireless Networks, etc.; and the Guest Editor for IEEE JSAC,IEEE Wireless Communications, IEEE Communications Magazine, and ACMMobile Networks and Applications, etc. Dr. Shen received the ExcellentGraduate Supervision Award in 2006, and the Premiers Research ExcellenceAward (PREA) in 2003 from the Province of Ontario, Canada. Dr. Shen isa registered Professional Engineer of Ontario, Canada, an IEEE Fellow, anEngineering Institute of Canada Fellow, a Canadian Academy of EngineeringFellow, a Royal Society of Canada Fellow, and a Distinguished Lecturer ofIEEE Vehicular Technology Society and Communications Society.

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