Outage Probability Analysis for D2D-Enabled Heterogeneous Cellular Networks with Exclusion Zone:A Stochastic Geometry Approach

Yulei Wang,Li Feng,★,Shumin Yao,2,Hong Liang,Haoxu Shi and Yuqiang Chen

1School of Computer Science and Engineering,Macau University of Science and Technology,Taipa,Macau,China

2Department of Broadband Communication,Peng Cheng Laboratory,Shenzhen,China

3School of Artificial Intelligence,Dongguan Polytechnic,Dongguan,China

ABSTRACT

Interference management is one of the most important issues in the device-to-device(D2D)-enabled heterogeneous cellular networks(HetCNets)due to the coexistence of massive cellular and D2D devices in which D2D devices reuse the cellular spectrum.To alleviate the interference,an efficient interference management way is to set exclusion zones around the cellular receivers.In this paper,we adopt a stochastic geometry approach to analyze the outage probabilities of cellular and D2D users in the D2D-enabled HetCNets.The main difficulties contain three aspects:1)how to model the location randomness of base stations,cellular and D2D users in practical networks;2) how to capture the randomness and interrelation of cellular and D2D transmissions due to the existence of random exclusion zones;3)how to characterize the different types of interference and their impacts on the outage probabilities of cellular and D2D users.We then run extensive Monte-Carlo simulations which manifest that our theoretical model is very accurate.

KEYWORDS

Device-to-device(D2D)-enabled heterogeneous cellular networks(HetCNets);exclusion zone;stochastic geometry(SG);Matérn hard-core process(MHCP)

1 Introduction

As the number of wireless connected devices grows explosively in the upcoming sixth generation(6G)era[1,2],it can be foreseen that heterogeneous cellular(i.e.,smartphones)and device-to-device(i.e.,wearable devices) devices will densely coexist to extensively collect and frequently exchange information[3]and have widespread application prospects in many fields[4,5].In the device-to-device(D2D)-enabled heterogeneous cellular networks (HetCNets) [6,7],D2D devices reuse the cellular spectrum,which may result in severe interference for the reception of cellular signals at the base stations(BSs).To alleviate the interference,an efficient interference management way is to set exclusion zones around the receivers[8–10].That is,when a BS is receiving desired signals from cellular devices,the exclusion zone,defined as a cycle region,centered at the BS is set,in which the D2D devices are inhibited to perform any transmissions.

1.1 Motivation

Whether the information can be successfully transmitted is an important performance metric for wireless devices.In this paper,we aim to theoretically analyze the outage probabilities (i.e.,unsuccessful transmission probabilities)of cellular and D2D devices in D2D-enabled HetCNets with exclusion zone.However,we are facing the following major difficulties.First,in practical network deployment,cellular and D2D devices are randomly located in the space,while BSs are deployed with strict requirements and restrictions,the two facts affect the outage probabilities greatly.It is difficult to model the location randomness of BSs,cellular and D2D devices in practice.Second,cellular transmissions occur randomly,leading to random exclusion zones around the BS receivers,further leading to random D2D transmissions.It is difficult to capture the randomness and interrelation of cellular and D2D transmissions.Third,different types of cellular and D2D devices perform transmissions concurrently; they mutually interfere with each other.It is difficult to characterize the different types of interference and their impacts on the outage probabilities of cellular and D2D devices.The above three difficulties motivate this study.

1.2 Contributions

Consider a D2D-enabled HetCNet with exclusion-zone,we theoretically analyze the outage probabilities of cellular and D2D devices and make the following novel contributions:

· We adopt a stochastic geometry(SG)approach to solve the abovementioned three difficulties.To address difficulty 1,we use Matérn hard-core process (MHCP) to model the real location distribution of BSs and use homogeneous Poisson point processes (HPPPs) to capture the location randomness of cellular and D2D devices in practical networks.To address difficulty 2,we first model the transmitting cellular devices by a thinned HPPP,and then model the activated D2D devices outside the exclusion zones by Poisson hole process(PHP).To address difficulty 3,we characterize mutual interference among the concurrent cellular and D2D transmissions by approximating MHCP of receiving BSs and PHP of activated D2D devices with PPPs and further estimating the intensities of different types of transmitting cellular and D2D devices.

· With our model,we theoretically analyze the outage probabilities of cellular and D2D devices,which are formulated as functions of system parameters,including the intensities of transmitting cellular and D2D devices,the minimum distance among BSs,and the radius of exclusion zones around BSs.

· We verify the accuracy of our theoretical model via extensive Monte-Carlo simulations.

The rest paper is organized as follows.Section 2 presents the related work.Section 3 introduces some stochastic geometry preliminaries.Section 4 specifies the system model of a D2D-enabled HetCNet with exclusion zone.Section 5 theoretically analyzes the outage probabilities of cellular and D2D devices with SG approach.Section 6 verifies the accuracy of our theoretical model via extensive Monte-Carlo simulations.Finally,Section 7 concludes the paper.For the ease of reference,Table 1 lists the main notations and their meanings.

Table 1 : Notions and their meanings

2 Related Work

This section presents existing works in terms of performance analysis of D2D-enabled HetCNets by setting exclusion zones or enabling D2D devices to operate in half-/full-duplex mode.

2.1 Set Exclusion Zones

Setting exclusion zones can effectively alleviate the interference among cellular and D2D transmissions.Many existing works have studied the exclusion zones set around transmitters or receivers.

Setexclusionzonesaroundtransmitters.Chu et al.in[11]adopted SG to study energy-harvestingbased D2D communication,where they set guard zones (also called exclusion zones) around D2D transmitters to protect D2D transmissions from interference emitted from the cellular devices.Flint et al.in [12] set guard zones around first-tier transmitters in two-tier heterogeneous networks,where they consider the exclusive relationship among the first-tier transmitters and model the spatial distribution of first-tier transmitters by Poisson hard-core process (PHCP).However,we study the D2D-enabled HetCNets with SG and set exclusion zones around the receivers to protect cellular transmissions from interference by D2D transmissions.

Setexclusionzonesaroundreceivers.Hasan et al.in [8] introduced guard zone around each receiver to balance the interference and spatial reuse,but this study is for wireless ad hoc networks.Tefek et al.in[9]set two types of exclusion zones around primary receivers and secondary transmitters in two-tier cognitive networks,and analyze the transmission capacities and outage probabilities of primary and secondary users with SG approach.Chen et al.in[10]studied decentralized opportunistic access for D2D underlaid cellular networks with SG and impose cellular guard zones around the BSs where no D2D transmitters can lie in,but they do not consider the minimum distance among BSs.Different from the above works,we adopt SG to study the D2D-enabled HetCNets and consider the exclusive relationship among BSs in practical networks.Besides,D2D devices can operate in half-/full-duplex mode optionally.The performance frameworks in the above works are not suitable in our research scenario.

2.2 Operate in Half-/Full-Duplex Mode

Despite the HD mode,each DU can operate in FD mode optionally to further promote to double the spectral efficiency.Some previous works analyze the performance of HD/FD D2D transmissions.

OperateinHDmode.Huang et al.in [13] studied the energy-efficient mode selection for D2D communications in cellular networks,which enable HD D2D users to select approximated modes,and then analyze the success probability and ergodic capacity for both cellular and D2D links using SG.Sun et al.in[14]controled the transmit power for D2D transmitters based on the statistical channelstate information to mitigate interference among D2D and cellular communications in D2D-underlaid cellular networks,and adopt SG to analyze the success probability and the energy efficiency of D2D communications.In contrast,we analyze the D2D transmissions where D2D devices can operate in HD/FD mode optionally,and we employ exclusion zones around the BS receivers to alleviate the interference.

OperateinFDmode.Badri et al.in [15] and [16] studied FD D2D communication in cellular networks,which enable D2D users to optionally work in HD/FD mode to alleviate the interference and guarantee the quality of service (QoS) of the cellular users.However,they do not consider the real deployment of BSs.Different from the above works,our study captures the location randomness of BS,cellular and D2D devices in real networks,the randomness and interrelation of cellular and D2D transmissions as well as the mutual interference and characterize their impacts on the outage probabilities of cellular and D2D transmissions.

3 Stochastic Geometry Preliminaries

Stochastic geometry(SG)approach,which provides various powerful tools to model the spatial location distribution of wireless devices and characterize the interference effect,has been widely used in wireless networks[17].Many existing works[15,16,18]adopted homogeneous Poisson point process (HPPP) to model the spatial distribution of the wireless devices,which assumes devices are independently distributed and is the most popular spatial point process owing to its mathematical tractability.However,in practical networks,the transmitters are deployed with strict requirements and restrictions in order to alleviate interference,extend coverage region and reduce deployment costs,and thus an exclusion zone among the locations of the transmitters naturally arises.In this context,hard-core point process(HCPP)[12,19,20],which forbids devices to lie closer than a certain minimum distance has drawn much attention,such as PHCP[12]or MHCP[19,20].According to whether there is a practical exclusive relationship among devices,we assume that the D2D and cellular users follow HPPPs and assume that base stations follow MHCP in our study.Below,we briefly present some terminologies and SG tools involved in this paper.Readers can refer to[21–24]for further details.

Definition 1.(Poisson point process)A spatial point processΦ={xi,i∈N+}⊂Rdwith intensity measure ∧is a Poisson point process(PPP)[18],if the random number of points of Φ for every bounded Borel setB⊂Rdhas a Poisson distribution with mean ∧(B),that is,

where ∧(B)represents the average number of points falling in the given setB.For an HPPP,∧(B)=λ|B|,whereλis the intensity of Φ and represents the average number of points falling in per unit area or volume,|B|is the Lebesgue measure(i.e.,area)of setBin Euclidean space.

Definition 2.(Matérn hard-core process of type I) An MHCP of type I ΦM1is formed from a dependent thinning of an HPPP Φ={xi,i∈N+} ⊂Rdwith intensityλ.First,each pointxi∈Φ is marked if it has a neighbor within distancer.Then,remove all marked points.All the remaining points of Φ form an MHCP of type I ΦM1.Mathematically,ΦM1is described as

ΦM1={xi∈Φ: ∀xj∈Φis not inb(xi,r)}

whereb(xi,r)represents a ball centered atxi∈Φ with radiusr.The intensityλM1of ΦM1is given by

Definition 3.(Matérn hard-core process of type II[25])An MHCP of type II ΦM2is formed from a dependent thinning of an HPPP Φ={xi,i∈N+} ⊂Rdwith intensityλ.First,each pointxi∈Φ is marked independently with a random markMi∈(0,1).Then,a pointxi∈Φ is retained in ΦM2if and only if the ballb(xi,r)does not contain any point of Φ with mark smaller thanMi.Mathematically,ΦM2is described as

The probability that each pointxi∈ Φ is retained in ΦM2can be expressed as PM2=[26].Then,the intensityλM2of ΦM2is given byλM2=λPM2=which can be further written with the intensityλM1of ΦM1asλM2=

Definition 4.(Poisson hole process) Let Φ1={xi,i∈N+} ⊂Rdwith intensityλ1and Φ2={yi,i∈N+} ⊂Rdwith intensityλ2(λ2≫λ1)be two independent PPPs in a given bounded Borel setB⊂Rd.For each pointxi∈Φ1,remove all the pointsyi∈Φ2inb(xi,r).All the removed points of Φ2form the Hole-0 process Φh0[27]with intensityλh0=and the remaining points of Φ2form the Poisson hole process (PHP) ΦPHP(also named as Hole-1 process [27]) with intensity

Definition 5.(Probability generating functional) Let Φ={xi,i∈N+} ⊂Rdbe a spatial point process with intensity measure ∧,for any measurable functionf(x): Rd→[0,1],the probability generating functional(PGFL)of Φ is defined as

wherexi∈Φ represents the orthogonal coordinates of points in Φ.For an inhomogeneous PPP with intensity functionλ(x),the PGFL of Φ can expressed as

For an HPPP with intensityλ,the PGFL of Φ can expressed as

We convert the above integral from orthogonal coordinates to polar coordinates,i.e.,

wherexi=(rsinω,rcosω),ωis the polar angle and follows uniform distribution in[0,2π].

Definition 6.The Laplace transform(LT)Lof random variableXis defined as

wherefX(x)is the probability density function(PDF)ofX.

4 System Model

This section specifies the system model of a D2D-enabled heterogeneous cellular network(HetCNet)with exclusion-zone in terms of network deployment,channel model,intensities of transmitting cellular and D2D users.

4.1 Network Deployment

We study a D2D-enabled HetCNet with exclusion-zone,which consists of multiple base stations(BSs),lots of cellular users(CUs)and D2D users(DUs),as shown in Fig.1.

Figure 1 :Overview of a D2D-enabled heterogeneous cellular network(HetCNet)with exclusion-zones around the BSs

In a typical HetCNet,the CUs and DUs are randomly located in the space,we model the locations of CUs and DUs by two independent HPPPs Φc,Φd⊂R2with intensitiesλcandλd,respectively;since any two BSs cannot be arbitrarily close to each other in practical network deployment,we model the location of BSs by an MHCP of typekΦkbwith intensityλMkb(k={1,2}),which is formed by dependent thinning of another HPPP Φ0b⊂R2with intensityλ0b(λc≫λ0b,λd≫λ0b)[19,20].According to the definition of MHCP(i.e.,Definitions 2 and 3),λMbkcan be expressed as

whereRbis the minimum distance among any two BSs.

We assume that Φ0b,Φcand Φdare independent,the locations of BSs,CUs and DUs are independent with each other.We assume that each CU transmits to its geographically nearest BS with a fixed powerPc,where a Voronoi tessellation is formed,as shown in Figs.2a–2c.The mean area of each Voronoi cellSvcan be expressed as[19,21–23]

In order to facilitate the analysis,we approximate the Voronoi cell as a circle with radiusRv[28–30],i.e.,

We assume that DUs utilize the uplink cellular channel to perform D2D transmissions and may choose to operate in either HD or FD mode to transmit to its nearest DU with a fixed powerPd.When adopting FD mode,we assume the imperfect self-interference cancellation at the DU receiver side.Due to spectrum sharing,the D2D transmissions may interfere with the reception of cellular signals at the BS.To manage the interference,exclusion zones around BSs are set.In the exclusion zone centered at each receiving BS with radiusdb,the DUs cannot be activated to perform D2D transmissions.As Figs.1 and 2 show,DUs in the exclusion zones of BSs are non-activated;in contract,DUs outside the exclusion zones are activated.

4.2 Channel Model

We assume that all the wireless signals in D2D and cellular transmissions undergo large-and small-scale channel fading.We characterize the large-scale channel fading by the distance dependent power-law path loss modell=‖x-y‖-α=R-α[31],whereR=‖x-y‖is Euclidean distance between a transmitterxand a receivery,andαis the path-loss exponent which usually satisfies 2＜α＜6 [32].We characterize the small-scale channel fading with Rayleigh fading that is modeled by an independent and identically distributed (i.i.d.) power fading coefficientH(square of the amplitude fading coefficient)[33],which follows exponential distribution with mean 1/μ,i.e.,H～Exp(μ)[18].Besides,we assume that the thermal noise at the receiver is additive white Gaussian noise with zero mean and varianceσ2[34,35].

Figure 2 : (Continued)

Figure 2 :A snapshot of Voronoi tessellation of BSs,CUs and DUs in a 1000 m×1000 m square region,where λc=100 CUs km-2,λd=100 DUs km-2,Rb=100 m,db=50 m.(a).BSs follow an HPPP with λ0b=30 BSs km-2;(b).BSs follow an MHCP of type I with =BSs km-2;(c).BSs follow an MHCP of type II with =19.4 BSs km-2

4.3 Intensity of Transmitting CUs

We assume that all CUs perform ALOHA mechanism to access the channel with probabilityptcto transmit data to its associated BSs[15].Let Φtcdenote the set of transmitting CUs with intensityλtc.According to independent thinning process of HPPP Φc,Φtcis an HPPP andλtccan be expressed as

4.4 Intensity of Transmitting DUs

Recall that we set exclusion zones at the side of BS which is performing reception from transmitting CUs in its Voronoi cell,that is,when no CUs are transmitting in a cell,the BSs may not perform reception,the exclusion zones are not set.Letprbdenote the receiving probability of BSs,which is equal to the probability that there is at least a transmitting CU in a given Voronoi cellSv.According to definition of PPP(i.e.,Definition 1),prbcan be expressed as

where ∧(E[Sv])is the intensity measure of Φtcand can be expressed as

Let Φrbdenote the set of receiving BSs with intensityλrb.Due to the non-availability of any known PGFL for MHCP,for the ease of analysis,we use a PPPwith same intensityλMkbto approximate the MHCPof receiving BSs1The approximated PPP is inhomogeneous with constant positive density.[20,36–38],and the accuracy of such approximation is also validated in[39].According to independent thinning of ΦMbk′,Φrbis a PPP andλrbcan be expressed as

LetΞdbdenote the union of exclusion zones formed by all receiving BSs.Since each receiving BS form an exclusion zone,Ξdbcan be expressed as

whereb(yi,db)is an exclusion zone centered atyi∈Φrbwith radiusdb.

In Ξdb,the DUs cannot be activated to perform D2D transmissions.According to the definition of PHP(i.e.,Definition 4),for the two PPPs of receiving BSs Φrband DUs Φd,the activated DUs outside the exclusion zones(i.e.,Ξdb)naturally form a PHP Φadwith intensityλad,i.e.,

Recall that the DUs can choose to operate in either HD or FD mode[15].We assume that a DU operates in HD and FD with probabilitypHandpF,respectively,such thatpH+pF=1.Due to the non-availability of any known PGFL for PHP,for the ease of analysis,we use a PPP Φa′dwith same intensityλadto approximate the PHP Φadof activated DUs[30,40].According to independent thinning of PPP Φad′,Φad′can be regarded as the union of two independent PPPs ΦHof activated HD DUs with intensityλHand ΦFof activated FD DUs with intensityλF,that is,=ΦH∪ΦF[15].Hence,λH,λFcan be expressed as

We assume that half of HD DUs are transmitters and half of them are receivers[15].Hence,the transmitting HD DUs form a thinned PPPwith intensity=λH/2.Similarly,all FD DUs are transceivers at the same time.The transmitting FD DUs form a thinned PPP ΦtFwith intensityλtF=λF.

5 Outage Probability Analysis

This section theoretically analyzes the outage probabilities of CUs and DUs with stochastic geometry approach.

5.1 Outage Probability of CUs Pc

We first analyze the outage probability of CUs.Consider a cellular transmission from a tagged CUc0to a tagged BSb0in a distanceRc.LetSINRb(Rc,Ib) denote the signal-to-interference-plus-noise ratio(SINR)betweenb0receivedc0’s signal powerSband its suffered interference signal powerIbplus noise powerσ2.Hence,theSINRb(Rc,Ib)atb0can be expressed as

whereσ2is the noise power.In Eq.(11),Sbcan be expressed as

wherePcis the transmission power of CU,Hcis the power fading coefficient betweenc0andb0.

In Eq.(11),Ibcan be expressed as

For the tagged transmitterc0,the transmission is unsuccessful ifSINRb(Rc,Ib)atd0is smaller than a certain SINR thresholdθ2In general,SINR is real ratio value with no unit,SINR threshold θ is given in decibel(dB),the real value(no unit)of which is given by θ=10θ/10.When comparing SINR with θ,it should be in same scale..Let Pcdenote the outage probability ofc0,which is defined as the mean value of P([SINRb(Rc,Ib)]＜θ),i.e.,

wherefRc(rc) is the probability density function (PDF) ofRc[41].Below,we expressfRc(rc) and EIb[P(SINRb(rc,Ib)＜θ|rc)]in sequence.

LetFRc(rc)denote the cumulative distribution function(CDF)ofRc.FRc(rc)can be expressed as

Further,fRc(rc)can be obtained by taking derivative ofFRc(rc)with respect torc,i.e.,

Then,we express EIb[P(SINRb(rc,Ib)＜θ|rc)]as

where EX[AX] is the expectation ofAXwith respect toX.Eq.(a) holds becauseHcfollows an exponential distribution with mean 1/μ,i.e.,Hc～Exp(μ).According to the CDF of an exponential distribution,iffHc(hc)=μe-μhc,P(Hc＜h0)=FHc(h0)=∫-μhcdhc=1-exp(-μh0).Eq.(b) follows from the definition of LT (i.e.,Definition 6) of interferenceIcb,IH b,andIFbevaluated atrespectively.We express them in sequence below.

In Eq.(17),the LT ofIbcatb0is given as

Proof:

In the above proof,Eq.(a)can be obtained from the fact thatRiandHiare mutually independent.Eq.(b)follows from the property of exponential distribution,i.e.,Eq.(c)is due to the definition of expectation ofHi.Eq.(d)holds becauseHifollows an exponential distribution with mean 1/μ3In our simulation,we set μ=1,i.e.,Hi～Exp(1).,i.e.,fHi(hi)=.According to the PGFL(i.e.,Definition 5)of PPP Φtc,we can obtain

In Eq.(e),R2is the area in which the interfering CUs locate.Eq.(f)converts the expression from orthogonal coordinates to polar coordinates.Eq.(g) follows by changing the variablei.e.,y=henceybelongs to (0,∞).Eq.(h) can refer to Eq.3.241.4of[42].Eq.(i)follows from the Euler’s reflection formula Γ(x)·Γ(1-x)=,where Γ(x)=x＞0 is the complete gamma function.

For the special caseα=4,we have

In Eq.(17),the LT ofIbHatb0is given as

For the special caseα=4,we have

where(a)follows

In Eq.(17),the LT ofIbFatb0is given as

5.2 Outage Probability of DUs Pd

We next analyze the outage probability of DUs in HD/FD mode.Consider a D2D transmission from a tagged DUdt0to the other tagged DUdr0in a distanceRd.LetSINRd(Rd,Id)denote the SINR betweendr0receiveddt0’s signal powerSdand its suffered interference signal powerIdplus noise powerσ2.Hence,theSINRd(Rd,Id)atdr0can be expressed as

whereσ2is the noise power.In Eq.(21),Sdcan be expressed as

wherePdis the transmission power of DU,Hdis the power fading coefficient betweendt0anddr0.

In Eq.(21),Idcan be expressed as

For the tagged transmitterdt0,the transmission is unsuccessful ifSINRd(Rd,Id) atdr0is smaller than a certain SINR thresholdθ.Let Pddenote the outage probability ofdt0,which is defined as the mean value of P([SINRd(Rd,Id)]＜θ),i.e.,

wherefRd(rd)is the PDF ofRd.Below,we expressfRd(rd)and EId[P(SINRd(rd,Id)＜θ|rd)].

Recall that the DU transmits to its nearest DU.Given the tagged DUdr0in the origin and a nearest distancerd,there is no DU closer thanrd,which means that there is no DU in the diskb(dr0,rd).According to the definition of PPPΦad(i.e.,Definition 1),the PDF thatRdis not smaller thanrd[41]can be derived as

where|Sb|=πr2dis the area ofb(dr0,rd).

LetFRd(rd)denote the CDF ofRd.FRd(rd)can be expressed as

Further,fRd(rd)can be obtained by taking derivative ofFRd(rd)with respect tord,i.e.,

Then,we express EId[P(SINRd(rd,Id)＜θ|rd)]as

In Eq.(28),the LT ofIdcatdr0is given as

If a D2D pair operates in the HD mode:

In Eq.(28),the LT ofIdHatdr0is given as

In Eq.(28),the LT ofIdFatdr0is given as

If a D2D pair operates in the FD mode:

In Eq.(28),the LT ofIdHatdr0is given as

In Eq.(28),the LT ofIdFatdr0is given as

6 Model Evaluation

In this section,we validate the accuracy of our theoretical model via extensive Monte-Carlo simulations and illustrate the outage probabilities of CUs and DUs in the D2D-enabled HetCNet with exclusion-zone.Table 2 shows the parameter settings for each simulation in Figs.3–6,respectively.In Table 2,we use pattern ‘x:y:z’to represent that a parameter takes value fromxtozwith an increasing step ofy,use pattern‘x,y’to represent that a parameter takes valuexandy,respectively.For example,in first row of Table 2,‘-20:10:30’means that parameterθtakes value from-20 to 30 dB with an increasing step of 10 dB,‘-50,-70’means that parameterκtakes value-50 and-70 dB,respectively.In our simulations,we set the simulation region as a circular disk with radius 104m.For each simulation,we run 104iterations to obtain the average value.In all figures,we use labels‘ana’and‘sim’to denote the theoretical and simulation results,respectively.

Table 2 : Parameters settings for simulations

6.1 Outage Probabilities vs.SINR Threshold in Different BSs Distributions

Here,we verify the accuracy of outage probabilities of CUs Pcand DUs Pdas the SINR thresholdθvaries from-20 to 30 dB,under different BSs distributions,i.e.,MHCP of types I and II (called‘type I’and ‘type II’process for short),which is also compared with PPP.From Fig.3,we have the following observations:

· Given a specific distribution of BSs,both Pcand Pdincrease asθincreases.It is because the increase ofθraises the difficulty of decoding a signal from a CU or DU,respectively.

· Givenθ,Pd(PPP)＞Pd(type II)＞Pd(type I)for DUs,respectively;in contrast,Pc(type I)＞Pc(type II) ＞Pc(PPP) for CUs.The reasons are as follows.In different distributions of BSs,For DUs,the larger the intensity of BSs,the more the exclusion zones of BSs,the smaller the intensity of activated DUs,the larger the average transmission distance from a DU to its nearest DU,the larger the outage probability of DUs;For CUs,the larger the intensity of BSs,the smaller the average area of each Voronoi cell,the smaller the average transmission distance from CU to BS,the smaller the outage probability of CUs.

· Givenθand a specific distribution of BSs,Pd(FD) ＞Pd(HD).It is because FD DUs suffer more self-interference than HD DUs due to imperfect self-interference.Besides,Pd＞Pc.It is because the transmission power of DUs is lower than that of CUs.

We take the example that BSs follow the type II distribution to verify the accuracy of outage probabilities of CUs Pcand DUs Pdand show the new insights below.

Figure 3 :Pc and Pd vs.θ when BSs follows PPP,MHCP of type I and type II

6.2 Outage Probabilities vs.SINR Threshold in Type II Process

Here,we verify the accuracy of outage probabilities of CUs Pcand DUs Pdas the SINR thresholdθvaries from-20 to 30 dB,under different settings,i.e.,self-interference cancellation factorκ=-50,-70 dB and noise powerσ2=-50,-100 dBm.From Figs.4a and 4b,we have the following observations:

· Givenκandσ2,Pcand Pdincreases asθincreases.The reason is similar with that in Fig.3 and omitted.

· Givenθ,the larger theκ,the larger the Pd(FD);Pd(HD)and Pcremains almost unchanged,as shown in Fig.4a.Hence,a largerκmeans that the FD DU suffers more self-interference4Note that the real ratio value of self-interference cancellation factor κ (in dB[15])is given by κ=10κ/10 (no unit[16]).A larger κ means larger self-interference at the FD DU side.,and the outage probability of FD DU also increases.However,the outage probabilities of CU and HD DU are not affected.

· Givenθ,the larger theσ2,the larger the Pcand Pd,as shown in Fig.4b.It is because a largerσ2results in smaller SINRs received at CUs and DUs.With smaller SINRs,we have larger Pcand Pd,respectively.

6.3 Outage Probabilities vs.Minimum Distance of BSs in Type II Process

Here,we verify the accuracy of outage probabilities of CUs Pcand DUs Pdas minimum distance of BSsRbvaries 50 to 300 m,under different settings,i.e.,transmission probability of CUsptc=0.2,0.8,probability of HD DUspH=0.2,0.8.From Figs.5a and 5b,we have the following observations:

· GivenptcandpH,asRbincreases,Pddecreases while Pcincreases.The reasons are as follows.For DUs,asRbincreases,the intensity of BSs decreases; the intensity of activated DUs increases,the average transmission distance from DU to DU decreases,and the outage probability of DU decreases.For CUs,asRbincreases,the intensity of BSs decreases; the average transmission distance from CU to BS increases,and the outage probability of CU increases.

· GivenRb,the largerptc,the larger the Pcand Pd,as shown in Fig.5a.It is because more transmissions from CUs to BSs may bring more mutual interference to the transmissions of CUs and DUs,respectively,and outage probabilities of CUs and DUs also increase.

· GivenRb,the largerpH,the smaller the Pcand Pd(HD),while Pd(FD)keeps almost unchanged,as shown in Fig.5b.It is because more HD D2D transmissions result in less FD D2D transmissions,which decrease the interference to the transmissions of CUs and HD DUs,respectively.For FD D2D transmission,the self-interference is dominated among the aggregate interference,hence the outage probability of FD DU is not affected.

Figure 5 :Pc and Pd vs.db when(a)ptc=0.2,0.8;and(b)pH=0.2,0.8

6.4 Outage Probabilities vs.Exclusion Zone of BSs in Type II Process

Here,we verify the accuracy of outage probabilities of CUs Pcand DUs Pdas the exclusion zone of BSsdbvaries from 0 to 150 m,under different settings,i.e.,path-loss exponentα=3,4,and transmission power of DUsPd=1,5 dBm.From Figs.6a and 6b,we have the following observations:

· GivenαandPd,asdbincreases,Pcdecreases while Pdincreases.The reasons are as follows.Asdbincreases,the intensity of activated DUs decreases,the average D2D transmission distance increases,and the outage probability of DU increases; meanwhile,less D2D transmissions may bring less mutual interference to the cellular transmissions,hence the probability of CU decreases.

· Givendb,the larger theα,the smaller the Pcand Pd(HD),the larger Pd(FD),as shown in Fig.6a.It is because a largerαmeans larger power reduction of signals as they propagate through space.For CUs and HD DUs,the interference signals decay more than desired signal for their larger transmission distance.The SINRs received at the CU and HD DU are higher,while Pcand Pd(HD)are lower.For FD DUs,the desired signal decays more than self-interference signal for its larger transmission distance.The SINRs received at the FD DU and FD DU are lower,while Pd(FD)are higher.

· Givendb,the largerPd,the smaller the Pdwhile the larger Pc,as shown in Fig.6b.It is because the larger Pd,the larger the received desired signal power at the DU,hence the smaller Pd; in contrast,the largerPd,the larger the received undesired interference power from DUs at the BS,hence the larger Pc.

Figure 6 :Pc and Pd vs.db when(a)α=3,4;(b)Pd=1,5 dBm

7 Conclusion

Heterogeneous cellular and D2D devices will densely coexist to collect and exchange information and hence have wide application prospects in many fields.To mitigate the interference among the concurrent cellular and D2D transmissions,exclusion zones are set around BS receivers.This paper develops a theoretical model to analyze the outage probabilities of cellular and D2D users in D2Denabled HetCNets with exclusion zone.It adopts a stochastic geometry approach to model the location randomness of BSs,cellular and D2D devices.Moreover,it captures the randomness and interrelation between cellular and D2D transmissions and characterizes the complex mutual interference among randomly located cellular and D2D devices.Extensive Monte-Carlo simulation results verify that the theoretical model is very accurate.

Acknowledgement: The authors would like to thank the editor and anonymous reviewers for their valuable suggestions and insightful comments,which have greatly improved the overall quality of this paper.

Funding Statement: This work is funded in part by the Science and Technology Development Fund,Macau SAR (Grant Nos.0093/2022/A2,0076/2022/A2 and 0008/2022/AGJ),in part by the National Nature Science Foundation of China (Grant No.61872452),in part by Special fund for Dongguan’s Rural Revitalization Strategy in 2021(Grant No.20211800400102),in part by Dongguan Special Commissioner Project(Grant No.20211800500182),in part by Guangdong-Dongguan Joint Fund for Basic and Applied Research of Guangdong Province (Grant No.2020A1515110162),in part by University Special Fund of Guangdong Provincial Department of Education (Grant No.2022ZDZX1073).

Author Contributions: The authors confirm contribution to the paper as follows: Yulei Wang: Conceptualization,Methodology,Formal analysis,Software,Validation,Writing–original draft,Writing–review & editing.Li Feng: Conceptualization,Methodology,Writing–review & editing,Supervision,Project administration,Funding acquisition.Shumin Yao: Conceptualization,Methodology,Validation,Writing–review & editing.Hong Liang: Validation,Writing–review & editing.Haoxu Shi:Validation,Writing–review&editing.Yuqiang Chen:Validation,Writing–review&editing.All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials:The data underlying the results presented in the study are available within the article.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.