# Performance Analysis of Three-Dimensional Clustered Device-to-Device Networks for Internet of Things

Authors: Haejoon Jung; In-Ho Lee
Journal: Wireless Communications and Mobile Computing
 (2017)

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## Abstract

Internet of things (IoT) is a smart technology that connects anything anywhere at any time. Intelligent device-to-device (D2D) communication, in which devices will communicate with each other autonomously without any centralized control, is an integral part of the Internet of Things (IoT) ecosystem. Thus, for D2D applications such as local file sharing or swarm sensing, we study communications between devices in proximity in ultra-dense urban environments, where devices are stacked vertically and dispersed in the horizontal plane. To reflect the spatiotemporal correlation inherently embedded in the D2D communications, we model and analyze clustered D2D networks in three-dimensional (3D) space based on Thomas cluster process (TCP), where the locations of clusters follow Poisson point process, and cluster members (devices) are normally distributed around their cluster centers. We assume that multiple device pairs in the network can share the same frequency band simultaneously. Thus, in the presence of cochannel interference from both the same cluster and the other clusters, we investigate the coverage probability and the area spectral efficiency of the clustered D2D networks in 3D space.


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## Body

## 1. Introduction

Fifth-generation (5G) networks are being developed to support dramatically increasing data traffic with various multimedia applications [1]. As more devices are embraced to connect everything, everywhere, and everyone, networks become dense with unprecedented rise of mobile traffic. In this context, device-to-device (D2D) communication, which relieves the burden of base stations (BSs), is an important feature for various types of mobile networks in the future cellular systems [2–4]. Through the D2D communication, wireless devices can constantly interact to each other as well as with their environments, which is the key 5G enabler for the Internet of Things (IoT) [5–7]. The D2D communications to create, gather, and share information involve various types of devices such as sensors, smartphones, cars, health care gadgets, and home appliances [8].Motivated by such emerging applications of the D2D communications, in this paper, we model and analyze D2D networks in three-dimensional (3D) space based on stochastic geometry [9]. To be specific, we consider 3D multicluster D2D networks, where devices in close proximity form a clustered network architecture. Poisson point process (PPP) is a widely used to analyze various types of networks (e.g., [10, 11]) including D2D networks, for its mathematical tractability. However, it cannot capture the fact that a device typically has multiple proximate devices, any of which is a potential serving device, with correlation in space and time.To overcome this limitation, the authors in [12] develop a more realistic model for two-dimensional (2D) D2D networks, where the devices locations are modeled as a Poisson cluster process, in particular a variant of a Thomas cluster process [9], where the D2D network consists of multiple clusters, and cluster members (devices) are normally distributed around the center of clusters. Different from the widely used uniform spatial distribution assumption with PPP as in [13], the model proposed in [12] reflects the spatiotemporal correlation in the content demand in D2D networks in the IoT environments as indicated in [14, 15]. Using this model, they investigate 2D clustered D2D networks for local information sharing with each cluster [16–18].However, as highlighted in [19–22], a 2D space model assumed in [12] may not be suitable for dense urban environments with high-rise buildings, where both devices and small-cell BSs are distributed over the 3D space. In [12], the coverage probability of wireless networks has been studied for various 2D deployment scenarios without much consideration for the vertical component of node distributions. However, to better model the future wireless environments (especially for the IoT applications) with ultra-dense deployments of devices and BSs, we need to consider the spatial distribution in the vertical space as well as the horizontal plane, as noted in [19–22]. For this reason, we extend the analytic framework of [12] in 2D space (on the horizontal plane) into 3D space. To our knowledge, this is the first study to model 3D D2D networks using TCP.The contributions of this paper are fourfold. First, we derive the probability distributions of distance between two devices that belong to (i) the same cluster and (ii) two different clusters in the 3D space. Second, we provide the exact mathematical expressions of the coverage probability and the area spectral efficiency of the 3D clustered D2D networks. Third, the approximate upper and lower bounds of the coverage probability are obtained, which are useful in the coverage analysis to gain insights into system design guidelines. Moreover, we present numerical and simulation results to validate our analysis and compare the 2D and 3D TCP models with various system parameters.


## 2. System Model

We consider a D2D network in 3D space, where the devices participating communications exist in clusters by the nature of D2D communications [12]. We assume that each device communicates with other devices in the same cluster, while the devices across clusters do not communicate directly (or, the intercluster communications may use orthogonal channels). As shown in Figure 1, the locations of the devices in 3D space are modeled by a TCP, where the cluster centers follow a homogeneous PPP Φc with density λc. Also, the cluster members (devices) are independent and identically distributed (i.i.d.) according to a symmetric normal distribution with variance σ2 around each cluster center x∈Φc with the density function of the device locations y∈R3 relative to a cluster center as(1)fYy=12π3/2σ3exp⁡-y22σ2,where σ is the scattering parameter.Figure 1
An example illustration of a three-dimensional clustered D2D network based on TCP.The devices in the cluster ofx∈Φc are denoted by Nx, which has two subsets: (i) transmitting devices Ntx and (ii) receiving devices Nrx. Suppose the set of simultaneously transmitting devices in the cluster is Bx⊆Ntx, and its cardinality Bx follows a Poisson distribution with mean λt. In other words, the number of simultaneously active transmitting devices (Dev-Txs) inside each cluster is a Poisson random variable (RV) with mean λt. Therefore, excluding the serving (or desired) Dev-Tx, we assume that the number of interfering devices follows a Poisson distribution with mean (λt-1). As in [12], without loss of generality, we perform analysis based on a typical device in a representative cluster x0∈Φc, where the typical device is regarded as the device receiver of interest. We assume that the typical device is located at the origin.We assume that the serving Dev-Tx is located aty0 inside the cluster x0∈Φc. Thus, the distance between the serving Dev-Tx and the typical device is denoted by r=y0+x0. Hence, with the transmit power of each device denoted by P0, the received power at the typical device is(2)S=P0h0r-α=P0h0x0+y0α,where α is the path-loss exponent and h0 is the power gain of small scale fading channel, which follows exponential distribution with unit mean, as in [12, 19–21]. The typical device suffers from two types of cochannel interference: (i) intracluster interference caused by the simultaneously active Dev-Txs in the same cluster and (ii) intercluster interference caused by the Dev-Txs in the other clusters, which are represented as(3)Iintra=∑y∈Bx0\y0P0hyx0x0+yα,(4)Iinter=∑x∈Φc\x0∑y∈BxP0hyxx+yα,respectively. Consequently, assuming interference-limited networks, the signal-to-interference-ratio (SIR) at the typical device is(5)SIRr=SIintra+Iinter=h0/x0+y0α∑y∈Bx0\y0hyx0/x0+yα+∑x∈Φc\x0∑y∈Bxhyx/x+yα,where P0 is canceled, since we assume the fixed transmit power for all Dev-Txs.


## 3. Distance Distributions

In this section, we derive the probability distributions of the distances from the typical device to intra- and intercluster devices for system performance analysis associated withSIR. We assume that the content of interest for a typical device in a given cluster is available at a device chosen uniformly at random in the cluster, as in [12]. Based on this assumption, we derive the distance distributions from the typical device to the serving Dev-Tx, intra- and intercluster interferers.


### 3.1. Distances between Typical Device and Intracluster Dev-Txs

For the intracluster devices, letDtx0 be the set {Di}i=1:Ntx0 of distances from the typical device to the set of possible Dev-Txs Ntx0 in the cluster x0∈Φc, where di=x0+y is the realization of Di. We note that the index i will be omitted when it is clear from the context. To delve into the distance statistics of D2D links, we first derive the probability distribution function (PDF) of the distance v0=x0 between the cluster center x0 and the typical device at the origin. Then, using this result, the PDF of the separation between the intracluster Dev-Tx and the typical device will be derived.Lemma 1 (probability distribution ofv0=x0).
The PDF ofv0 is given by(6)fv0x=2πx2σ3exp⁡-x22σ2,where x≥0.Proof.
Based on the 3D Gaussian distribution defined in (1), Z=v02/σ2 is the squared sum of three i.i.d. standard (zero mean and unit variance) Gaussian random variables, which corresponds to the PDF as(7)fZz=z2πe-z/2,where z≥0. Therefore, by the change of variables, we can obtain the PDF in (6).Lemma 2 (probability distribution ofD=x0+y).
The PDF of the separation between the typical device and the Dev-Tx in the same cluster is given by(8)fDd=d22πσ3exp⁡-d24σ2,where d≥0.Proof.
The locations of the cluster centerx0 and the Dev-Txs y are i.i.d. random vectors in R3, where the three components follow i.i.d. Gaussian distributions with zero mean and variance of σ2. Suppose Z=x0+y2/2σ2, which is the squared sum of three i.i.d standard Gaussian random variables. Thus, Z follows a chi-squared distribution with 3 degrees of freedom with the PDF:(9)fZz=z2πe-z/2.Therefore, the PDF of D=σ2Z in (8) can be derived by the change of variables (it is noted that the PDF and conditional PDF of D are, resp., obtained by extending the probability distribution analysis in 2D to 3D space).


### 3.2. Conditional Distribution ofD Given x0

The distances of the typical device to the Dev-Txs in the same clusters, which are required to calculateS and Iintra in SIR, are correlated because of the common factor x0. Therefore, conditioning the relative location of the cluster center, x0, to typical device, we can treat the locations of the intracluster devices as i.i.d. RVs, which means that the distances between the typical device and the intracluster devices are i.i.d. To exploit this property, the following lemma gives the conditional distribution of D given x0.Lemma 3 (conditional probability distribution ofD=x0+y given x0).
The conditional PDF ofD for a given x0 is derived as (10)fDd∣v0=d3/2σ2v0e-v02+d2/2σ2I1/2v0dσ2,where d≥0 and I1/2(t)=∑k=0∞1/k!Γt+k+1t/21/2+2k is the modified Bessel function with order 1/2.Proof.
LetZ=x0+y2/σ2. Because y2/σ2 is the squared sum of three i.i.d standard Gaussian RVs, conditioned on v0=x0, Z follows a noncentral chi-square distribution with the PDF:(11)fZz∣v0=12σ2zv021/4e-v02+σ2z/2σ2I1/2v0zσ.Since D=x0+y=σZ, its PDF in (10) can be obtained by the change of variables (it is noted that the PDF and conditional PDF of D are, resp., obtained by extending the probability distribution analysis in 2D to 3D space).


### 3.3. Distances to Serving Dev-Tx and Interferers:r, w, and u

Let the distances from the typical device to the serving Dev-Tx and intracluster interferer ber=x0+y0 and w=x0+y, respectively. Their conditional PDFs given that v0=x0 are same as (10). In other words, fR(r∣v0)=fD(r∣v0) and fW(w∣v0)=fD(w∣v0). In addition, conditioned on the distance v=x between one of the other clusters x∈Φc and the typical device, the distances {u=x+y,∀y∈Bx} between the typical device and the intercluster interfering Dev-Txs in x∈Φc are i.i.d., following the conditional PDF fU(u∣v)=fD(u∣v0=v) given in (10). Also, the PDF of v=x is identical to the PDF of v0=x0 defined in (6).


## 3.1. Distances between Typical Device and Intracluster Dev-Txs

For the intracluster devices, letDtx0 be the set {Di}i=1:Ntx0 of distances from the typical device to the set of possible Dev-Txs Ntx0 in the cluster x0∈Φc, where di=x0+y is the realization of Di. We note that the index i will be omitted when it is clear from the context. To delve into the distance statistics of D2D links, we first derive the probability distribution function (PDF) of the distance v0=x0 between the cluster center x0 and the typical device at the origin. Then, using this result, the PDF of the separation between the intracluster Dev-Tx and the typical device will be derived.Lemma 1 (probability distribution ofv0=x0).
The PDF ofv0 is given by(6)fv0x=2πx2σ3exp⁡-x22σ2,where x≥0.Proof.
Based on the 3D Gaussian distribution defined in (1), Z=v02/σ2 is the squared sum of three i.i.d. standard (zero mean and unit variance) Gaussian random variables, which corresponds to the PDF as(7)fZz=z2πe-z/2,where z≥0. Therefore, by the change of variables, we can obtain the PDF in (6).Lemma 2 (probability distribution ofD=x0+y).
The PDF of the separation between the typical device and the Dev-Tx in the same cluster is given by(8)fDd=d22πσ3exp⁡-d24σ2,where d≥0.Proof.
The locations of the cluster centerx0 and the Dev-Txs y are i.i.d. random vectors in R3, where the three components follow i.i.d. Gaussian distributions with zero mean and variance of σ2. Suppose Z=x0+y2/2σ2, which is the squared sum of three i.i.d standard Gaussian random variables. Thus, Z follows a chi-squared distribution with 3 degrees of freedom with the PDF:(9)fZz=z2πe-z/2.Therefore, the PDF of D=σ2Z in (8) can be derived by the change of variables (it is noted that the PDF and conditional PDF of D are, resp., obtained by extending the probability distribution analysis in 2D to 3D space).


## 3.2. Conditional Distribution ofD Given x0

The distances of the typical device to the Dev-Txs in the same clusters, which are required to calculateS and Iintra in SIR, are correlated because of the common factor x0. Therefore, conditioning the relative location of the cluster center, x0, to typical device, we can treat the locations of the intracluster devices as i.i.d. RVs, which means that the distances between the typical device and the intracluster devices are i.i.d. To exploit this property, the following lemma gives the conditional distribution of D given x0.Lemma 3 (conditional probability distribution ofD=x0+y given x0).
The conditional PDF ofD for a given x0 is derived as (10)fDd∣v0=d3/2σ2v0e-v02+d2/2σ2I1/2v0dσ2,where d≥0 and I1/2(t)=∑k=0∞1/k!Γt+k+1t/21/2+2k is the modified Bessel function with order 1/2.Proof.
LetZ=x0+y2/σ2. Because y2/σ2 is the squared sum of three i.i.d standard Gaussian RVs, conditioned on v0=x0, Z follows a noncentral chi-square distribution with the PDF:(11)fZz∣v0=12σ2zv021/4e-v02+σ2z/2σ2I1/2v0zσ.Since D=x0+y=σZ, its PDF in (10) can be obtained by the change of variables (it is noted that the PDF and conditional PDF of D are, resp., obtained by extending the probability distribution analysis in 2D to 3D space).


## 3.3. Distances to Serving Dev-Tx and Interferers:r, w, and u

Let the distances from the typical device to the serving Dev-Tx and intracluster interferer ber=x0+y0 and w=x0+y, respectively. Their conditional PDFs given that v0=x0 are same as (10). In other words, fR(r∣v0)=fD(r∣v0) and fW(w∣v0)=fD(w∣v0). In addition, conditioned on the distance v=x between one of the other clusters x∈Φc and the typical device, the distances {u=x+y,∀y∈Bx} between the typical device and the intercluster interfering Dev-Txs in x∈Φc are i.i.d., following the conditional PDF fU(u∣v)=fD(u∣v0=v) given in (10). Also, the PDF of v=x is identical to the PDF of v0=x0 defined in (6).


## 4. Performance Analysis:Pc and ASE

In this section, we investigate the coverage probability,Pc, and the area spectral efficiency, ASE, of the clustered D2D network. We first find the Laplace transforms of the two interference terms to characterize SIR. Then, we derive the exact expressions of Pc and ASE.


### 4.1. Laplace Transform of Intracluster Interference

Conditioned onv0=x0, we first derive the Laplace transform of Iintra as(12)LIintras∣v0=Ee-sIintra=E∏y∈Bx0\y0Ehyx0exp⁡-shyx0x+yα=aEBx0∏y∈Bx0\y011+sy+x0-α=bexp⁡1-λt∫R3sy+x0-α1+sy+x0-αfYydy=cexp⁡1-λt∫0∞sw-α1+sw-αfWw∣v0dw,where (a) follows from the exponentially distributed hx0 with unit mean and (b) follows from the probability generating functional (PGF) of Poisson process of the intracluster interferers with mean (λt-1). Also, (c) follows from w=x0+y.


### 4.2. Laplace Transform of Intercluster Interference

The Laplace transform ofIinter is given by(13)LIinters=Ee-sIinter=EΦc∏x∈Φc\x0EBx∏y∈BxEhyxexp⁡-shyxx+yα=aEΦc∏x∈Φc\x0EBx∏y∈Bx11+sy+x-α=bEΦc∏x∈Φc\x0exp⁡∫0∞-λtsu-α1+su-αfUu∣vdu=cexp⁡4πλc∫0∞κv-1v2dv,where κv=exp⁡∫0∞-λtsu-α/1+su-αfUu∣vdu and (a) follows from the exponentially distributed hx0 with unit mean. Also, (b) and (c) follow from the PGF of Poisson process (with the mean of λc and (λt-1) resp.).


### 4.3. Coverage Probability and Area Spectral Efficiency

Lettingβ denote the SIR threshold for successful decoding at the receiver, which is a function of modulation and coding, the coverage probability is(14)Pc=PSIR>β=ERPSIRR>β∣R=ERPh0>βrαIintra+Iinter∣R=r=EREe-βrαIintra+Iinter∣R=r=∫0∞∫0∞LIinterβrαLIintraβrα∣v0×fRr∣v0fv0v0drdv0.Therefore, letting the area spectral efficiency be defined as the average achievable rate per unit bandwidth per unit area as in [12], the area spectral efficiency is given by(15)ASE=λtλclog2⁡1+βPc,where λtλc is the average density of simultaneously active Dev-Txs of the whole D2D network.


## 4.1. Laplace Transform of Intracluster Interference

Conditioned onv0=x0, we first derive the Laplace transform of Iintra as(12)LIintras∣v0=Ee-sIintra=E∏y∈Bx0\y0Ehyx0exp⁡-shyx0x+yα=aEBx0∏y∈Bx0\y011+sy+x0-α=bexp⁡1-λt∫R3sy+x0-α1+sy+x0-αfYydy=cexp⁡1-λt∫0∞sw-α1+sw-αfWw∣v0dw,where (a) follows from the exponentially distributed hx0 with unit mean and (b) follows from the probability generating functional (PGF) of Poisson process of the intracluster interferers with mean (λt-1). Also, (c) follows from w=x0+y.


## 4.2. Laplace Transform of Intercluster Interference

The Laplace transform ofIinter is given by(13)LIinters=Ee-sIinter=EΦc∏x∈Φc\x0EBx∏y∈BxEhyxexp⁡-shyxx+yα=aEΦc∏x∈Φc\x0EBx∏y∈Bx11+sy+x-α=bEΦc∏x∈Φc\x0exp⁡∫0∞-λtsu-α1+su-αfUu∣vdu=cexp⁡4πλc∫0∞κv-1v2dv,where κv=exp⁡∫0∞-λtsu-α/1+su-αfUu∣vdu and (a) follows from the exponentially distributed hx0 with unit mean. Also, (b) and (c) follow from the PGF of Poisson process (with the mean of λc and (λt-1) resp.).


## 4.3. Coverage Probability and Area Spectral Efficiency

Lettingβ denote the SIR threshold for successful decoding at the receiver, which is a function of modulation and coding, the coverage probability is(14)Pc=PSIR>β=ERPSIRR>β∣R=ERPh0>βrαIintra+Iinter∣R=r=EREe-βrαIintra+Iinter∣R=r=∫0∞∫0∞LIinterβrαLIintraβrα∣v0×fRr∣v0fv0v0drdv0.Therefore, letting the area spectral efficiency be defined as the average achievable rate per unit bandwidth per unit area as in [12], the area spectral efficiency is given by(15)ASE=λtλclog2⁡1+βPc,where λtλc is the average density of simultaneously active Dev-Txs of the whole D2D network.


## 5. Approximate Upper and Lower Bounds ofPc

Because the exact expressions ofPc and ASE are unwieldy, we provide easy-to-compute upper and lower bounds of Pc. In particular, the lower bound is in a closed form, which can be readily evaluated. As stated in Section 2, r and w are correlated because of the common factor x0. For analytical tractability to derive the two approximate bounds, we allow separate deconditioning on r and w as in [12], which implies that r and w are i.i.d. following the PDF in (8).


### 5.1. Upper Bound ofPc

Since the intracluster interferers are significantly closer to the typical device compared to the intercluster Dev-Txs,Iintra is dominant in the denominator of SIR. Thus, we can derive the approximate upper bound of SIR by ignoring Iinter, which corresponds to the upper bound of Pc. By the i.i.d. assumption of r and w, the Laplace transform of Iintra can be approximated as L~Iintra(s)=e1-λt∫0∞sw-α/1+sw-αfW(w)dw, where fW(w) follows the PDF in (8). Thus, the upper bound of Pc is given by(16)Pc~=ERPh0>βrαIintra∣R=r=∫0∞L~IintraβrαfRrdr,where fR(r) follows the PDF in (8).


### 5.2. Lower Bound ofPc

We first derive lower bounds ofLIintra(s) and LIinter(s) in closed forms. Then, using the two, the lower bound of Pc will be obtained.Lemma 4 (lower bound ofLIintra(s)).
The lower bound on the Laplace transform ofIintra is(17)LIintras≥LIintra∗s=exp⁡1-λt6πσ3s3/α3π/αsin⁡3π/α.Proof.
See AppendixA.Lemma 5 (lower bound ofLIinter(s)).
The lower bound on the Laplace transform ofIinter is given by(18)LIinters≥LIinter∗s=exp⁡-43s3/αλcλt3π2/αsin⁡3π/α.Proof.
See AppendixB.With (17) and (18) along with the independent deconditioning assumption, we can obtain the approximate lower bound of Pc in a closed form as(19)Pc≥∫0∞LIinter∗βrαLIintra∗βrαfRrdr=a∫0∞exp⁡-ρr3r22πσ3exp⁡-r24σ2dr,where ρ=3πβ3/α/αsin⁡3π/αλt-1/6πσ3+4πλcλt/3 and (a) follows from fR(r) following (8). Because ρ≥0 (∵α≥2 and λt≥1), we can obtain the lower bound in (20), where Bi(a)=1/π∫0∞cos⁡t3/3+atdt is the Airy function, the derivative of which is Bi′(a). Moreover, F22 is the generalized hypergeometric function [23].(20)Pc∗=108ρ4/3σ4F221/2,1;1/3,2/3;-1/432ρ2σ6+e-1/864ρ2σ632/3πBi1/4833ρ4/3σ4-1233πρ2/3σ2Bi′1/4833ρ4/3σ4648πρ7/3σ7.


## 5.1. Upper Bound ofPc

Since the intracluster interferers are significantly closer to the typical device compared to the intercluster Dev-Txs,Iintra is dominant in the denominator of SIR. Thus, we can derive the approximate upper bound of SIR by ignoring Iinter, which corresponds to the upper bound of Pc. By the i.i.d. assumption of r and w, the Laplace transform of Iintra can be approximated as L~Iintra(s)=e1-λt∫0∞sw-α/1+sw-αfW(w)dw, where fW(w) follows the PDF in (8). Thus, the upper bound of Pc is given by(16)Pc~=ERPh0>βrαIintra∣R=r=∫0∞L~IintraβrαfRrdr,where fR(r) follows the PDF in (8).


## 5.2. Lower Bound ofPc

We first derive lower bounds ofLIintra(s) and LIinter(s) in closed forms. Then, using the two, the lower bound of Pc will be obtained.Lemma 4 (lower bound ofLIintra(s)).
The lower bound on the Laplace transform ofIintra is(17)LIintras≥LIintra∗s=exp⁡1-λt6πσ3s3/α3π/αsin⁡3π/α.Proof.
See AppendixA.Lemma 5 (lower bound ofLIinter(s)).
The lower bound on the Laplace transform ofIinter is given by(18)LIinters≥LIinter∗s=exp⁡-43s3/αλcλt3π2/αsin⁡3π/α.Proof.
See AppendixB.With (17) and (18) along with the independent deconditioning assumption, we can obtain the approximate lower bound of Pc in a closed form as(19)Pc≥∫0∞LIinter∗βrαLIintra∗βrαfRrdr=a∫0∞exp⁡-ρr3r22πσ3exp⁡-r24σ2dr,where ρ=3πβ3/α/αsin⁡3π/αλt-1/6πσ3+4πλcλt/3 and (a) follows from fR(r) following (8). Because ρ≥0 (∵α≥2 and λt≥1), we can obtain the lower bound in (20), where Bi(a)=1/π∫0∞cos⁡t3/3+atdt is the Airy function, the derivative of which is Bi′(a). Moreover, F22 is the generalized hypergeometric function [23].(20)Pc∗=108ρ4/3σ4F221/2,1;1/3,2/3;-1/432ρ2σ6+e-1/864ρ2σ632/3πBi1/4833ρ4/3σ4-1233πρ2/3σ2Bi′1/4833ρ4/3σ4648πρ7/3σ7.


## 6. Numerical Results

In this section, we present numerical results to validate our analysis and discuss the impacts of system parameters. For simulations, the device locations are randomly drawn from a TCP over 100× 100 × 100 m3 cube. The cluster centers follow PPP with intensity λc, and devices are normally distributed around their cluster centers. Moreover, the number of the Dev-Txs in each cluster follows a Poisson distribution with mean λt. Also, we assume the path-loss exponent α of 4, as in [12, 19, 20]. The simulation results are obtained from 106 random realizations of device distribution (network topology) and Rayleigh fading channel.


### 6.1. Impacts of System Parameters

Figures2(a) and 2(b) show how the coverage probability Pc varies, as the average number of simultaneously active Dev-Txs λt increases, with λc=0.3 and 0.05, respectively. In the figures, the circles indicate the simulation results, while the solid line represents the theoretical results obtained numerically using (14). Moreover, the dash-dotted and dashed curves correspond to the upper and lower bounds Pc~ and Pc∗ in (16) and (20), respectively. In both figures, the simulation results show the excellent agreements with the theoretical results, which verifies our analysis. Moreover, the approximate upper and lower bounds of Pc derived in the previous section are validated. Specifically, comparing the two figures, when λc is large, the actual Pc is closer to the lower bound Pc∗ compared to the upper bound Pc~, as in Figure 2(a), because the large λc results in the higher intercluster interference Iinter, which is ignored in the Pc~. On the other hand, for small λc, the gap between the exact Pc and its upper bound Pc~ is significantly smaller compared to the difference from its lower bound Pc∗, as in Figure 2(b), since the intracluster interference Iintra is dominant relative to the intercluster interference Iinter. In either case, the exact Pc curve is always bounded by Pc~ and Pc∗.Figure 2
P
c versus λt: comparison with the upper and lower bounds.
(a)
P
c with λc=0.3 clusters/m3, σ=0.3, and β=0 dB
(b)
P
c with λc=0.5 clusters/m3, σ=0.3, and β=0 dBIn Figures3 and 4, we observe the impacts of σ and λc on exact Pc, numerically obtained by (14), respectively. In the figures, we consider three scenarios in the presence of (i) only intracluster interference, (ii) only intercluster interference, and (iii) both intra- and intercluster interferences, which correspond to the dashed, dash-dotted, and solid lines, respectively. Moreover, the triangles and circles indicate the corresponding simulation results. In both figures, when λt grows, the intracluster interference, indicated by the dashed line, is dominant compared to the intercluster interference, which is indicated by the dash-dotted line. Also, in Figure 3, the larger σ, which means the larger spatial scattering of the devices from the cluster center, results in the lower Pc. This can be attributed to the increased impact of Iinter, while the Pc curves only with Iintra do not change as indicated by the dashed curves in the figure. The Pc curves only with Iintra stay the same regardless of λt, because the variations of the serving and interfering Dev-Txs cancel each other. We can observe the same trend in Figure 4: as λc increases, Pc decreases because of the increased intercluster interference Iinter. On the other hand, the coverage probability Pc only with the intracluster interference Iintra does not vary under the variation in the cluster density λc.Figure 3
P
c versus λt with β=0 dB, λc=0.03 clusters/m3, and σ={0.3,0.5}.Figure 4
P
c versus λt with β=0 dB, σ=0.5, and λc={0.01,0.03} clusters/m3.Figures5 and 6 show the exact area spectral efficiency ASE, numerically obtained by (15), versus the average number of simultaneously active Dev-Txs λt. In the figure, the horizontal axis indicates λt, while the vertical axis is ASE. Also, the solid, dashed, and dash-dotted lines represent the theoretical results with different system parameters (λc, σ, and β), while the circle and triangle markers represent the corresponding simulation results. Lastly, the optimal λt in each graph is indicated by the “x”-marker.Figure 5
A
S
E versus λt; variations in σ and λc with β=0 dB.Figure 6
A
S
E versus λt with σ=0.5, λc=0.03 clusters/m3, and β={-5,0,5} dB.As shown in Figures2, 3, and 4, we can observe the great correlation between the simulation and theoretical results. If comparing the curves with σ=0.3 and 0.5 with the same λc in Figure 5, ASE increases, as σ decreases, which is expected from the result of Pc in Figure 3. Moreover, for the fixed σ=0.5, the curves with λc=0.03 show significantly higher ASE compared to the curve with λc=0.01, because of the greater multiplication factor in (15). In Figure 6, the higher β makes ASE increase for small λt, while the curves with higher β decrease more rapidly compared to the curves with smaller β, as λt increases. One of the most important design aspects is the optimal λt to maximize the ASE, which determines network operation and wireless resource allocation. In Figure 5, λt≈2 gives the best ASE for all the three curves, which indicates its low sensitivity to σ and λc. On the other hand, if we increase the SIR threshold β as in Figure 6, the optimal value of λt decreases (4.25, 2, 1 for β=-5, 0, 5 dB, resp.), because lower β can accommodate more simultaneous users (devices).


### 6.2. Comparison between 2D and 3D TCP Models

In this section, we compare the performance of the 3D clustered D2D networks with the 2D clustered networks studied in [12]. For the comparison, we set the same cluster density per unit space λc (clusters/m2 and clusters/m3 in 2D and 3D spaces, resp.). Figures 7 and 8 show the coverage probability Pc versus the average number of simultaneously active Dev-Txs λt in the 2D and 3D spaces. In the figures, the lined curves represent the theoretical results, while the markers indicate the simulation results. As shown in the figure, we observe that the analytical and simulation results are consistent with each other both for the 2D and 3D cases. For the same parameter set, Pc of the 2D TCP is higher compared to Pc of the 3D TCP, which is consistent with the results assuming uniform node distribution following PPP in [20]. This can be explained by more number of interferers inside volumes with the same radius from the typical device in 3D space compared to 2D space even with the same cluster density λc per unit space. From the figure, the gap between the 2D and 3D curves grows for the larger σ and λc. Furthermore, compared to the 3D space results, the Pc performances in the 2D space are less sensitive to the change in σ and λc as observed in Figures 7 and 8, respectively.Figure 7
Comparison of 2D and 3D withβ=0 dB, λc=0.03, and σ={0.4,0.5}.Figure 8
Comparison of 2D and 3D withβ=0 dB, σ=0.5, and λc={0.01,0.03}.Furthermore, Figure9 displays the area spectral efficiency ASE versus λt graphs of the 2D TCP under the change in β using the same parameters as the 3D TCP case shown in Figure 6: σ=0.5, λc=0.03 clusters/m2, and β={-5,0,5} dB. Overall, the ASE in the 2D TCP is greater compared to the 3D case, because of the higher coverage probability Pc as indicated in Figures 7 and 8. That is, if we use the 2D TCP model for ultra-dense urban environments where the devices exist in 3D space, which will be common in the future wireless networks, both the coverage probability and the area spectral efficiency are overestimated. When comparing the impact of β, we can observe the similar trend in the 2D and 3D models that the higher β gives the higher ASE with small λt. However, while the curves with β=0 dB and 5 dB cross over at around λt=6 in Figure 6, the two curves in Figure 9 keep the measurable gap. Interestingly, the optimal λt to maximize the ASE shows the similar (but not exactly the same) trend to that seen in Figure 6. Thus, in a nutshell, the 2D TCP model can be used to estimate the optimal number of simultaneously active Dev-Txs in the 3D clustered networks; however both Pc and ASE estimated based on the 2D model are significantly overestimated compared to the actual performances of the 3D clustered D2D networks.Figure 9
A
S
E versus λt in two-dimensional (2D) space with σ=0.5, λc=0.03 clusters/m2, and β={-5,0,5} dB.


## 6.1. Impacts of System Parameters

Figures2(a) and 2(b) show how the coverage probability Pc varies, as the average number of simultaneously active Dev-Txs λt increases, with λc=0.3 and 0.05, respectively. In the figures, the circles indicate the simulation results, while the solid line represents the theoretical results obtained numerically using (14). Moreover, the dash-dotted and dashed curves correspond to the upper and lower bounds Pc~ and Pc∗ in (16) and (20), respectively. In both figures, the simulation results show the excellent agreements with the theoretical results, which verifies our analysis. Moreover, the approximate upper and lower bounds of Pc derived in the previous section are validated. Specifically, comparing the two figures, when λc is large, the actual Pc is closer to the lower bound Pc∗ compared to the upper bound Pc~, as in Figure 2(a), because the large λc results in the higher intercluster interference Iinter, which is ignored in the Pc~. On the other hand, for small λc, the gap between the exact Pc and its upper bound Pc~ is significantly smaller compared to the difference from its lower bound Pc∗, as in Figure 2(b), since the intracluster interference Iintra is dominant relative to the intercluster interference Iinter. In either case, the exact Pc curve is always bounded by Pc~ and Pc∗.Figure 2
P
c versus λt: comparison with the upper and lower bounds.
(a)
P
c with λc=0.3 clusters/m3, σ=0.3, and β=0 dB
(b)
P
c with λc=0.5 clusters/m3, σ=0.3, and β=0 dBIn Figures3 and 4, we observe the impacts of σ and λc on exact Pc, numerically obtained by (14), respectively. In the figures, we consider three scenarios in the presence of (i) only intracluster interference, (ii) only intercluster interference, and (iii) both intra- and intercluster interferences, which correspond to the dashed, dash-dotted, and solid lines, respectively. Moreover, the triangles and circles indicate the corresponding simulation results. In both figures, when λt grows, the intracluster interference, indicated by the dashed line, is dominant compared to the intercluster interference, which is indicated by the dash-dotted line. Also, in Figure 3, the larger σ, which means the larger spatial scattering of the devices from the cluster center, results in the lower Pc. This can be attributed to the increased impact of Iinter, while the Pc curves only with Iintra do not change as indicated by the dashed curves in the figure. The Pc curves only with Iintra stay the same regardless of λt, because the variations of the serving and interfering Dev-Txs cancel each other. We can observe the same trend in Figure 4: as λc increases, Pc decreases because of the increased intercluster interference Iinter. On the other hand, the coverage probability Pc only with the intracluster interference Iintra does not vary under the variation in the cluster density λc.Figure 3
P
c versus λt with β=0 dB, λc=0.03 clusters/m3, and σ={0.3,0.5}.Figure 4
P
c versus λt with β=0 dB, σ=0.5, and λc={0.01,0.03} clusters/m3.Figures5 and 6 show the exact area spectral efficiency ASE, numerically obtained by (15), versus the average number of simultaneously active Dev-Txs λt. In the figure, the horizontal axis indicates λt, while the vertical axis is ASE. Also, the solid, dashed, and dash-dotted lines represent the theoretical results with different system parameters (λc, σ, and β), while the circle and triangle markers represent the corresponding simulation results. Lastly, the optimal λt in each graph is indicated by the “x”-marker.Figure 5
A
S
E versus λt; variations in σ and λc with β=0 dB.Figure 6
A
S
E versus λt with σ=0.5, λc=0.03 clusters/m3, and β={-5,0,5} dB.As shown in Figures2, 3, and 4, we can observe the great correlation between the simulation and theoretical results. If comparing the curves with σ=0.3 and 0.5 with the same λc in Figure 5, ASE increases, as σ decreases, which is expected from the result of Pc in Figure 3. Moreover, for the fixed σ=0.5, the curves with λc=0.03 show significantly higher ASE compared to the curve with λc=0.01, because of the greater multiplication factor in (15). In Figure 6, the higher β makes ASE increase for small λt, while the curves with higher β decrease more rapidly compared to the curves with smaller β, as λt increases. One of the most important design aspects is the optimal λt to maximize the ASE, which determines network operation and wireless resource allocation. In Figure 5, λt≈2 gives the best ASE for all the three curves, which indicates its low sensitivity to σ and λc. On the other hand, if we increase the SIR threshold β as in Figure 6, the optimal value of λt decreases (4.25, 2, 1 for β=-5, 0, 5 dB, resp.), because lower β can accommodate more simultaneous users (devices).


## 6.2. Comparison between 2D and 3D TCP Models

In this section, we compare the performance of the 3D clustered D2D networks with the 2D clustered networks studied in [12]. For the comparison, we set the same cluster density per unit space λc (clusters/m2 and clusters/m3 in 2D and 3D spaces, resp.). Figures 7 and 8 show the coverage probability Pc versus the average number of simultaneously active Dev-Txs λt in the 2D and 3D spaces. In the figures, the lined curves represent the theoretical results, while the markers indicate the simulation results. As shown in the figure, we observe that the analytical and simulation results are consistent with each other both for the 2D and 3D cases. For the same parameter set, Pc of the 2D TCP is higher compared to Pc of the 3D TCP, which is consistent with the results assuming uniform node distribution following PPP in [20]. This can be explained by more number of interferers inside volumes with the same radius from the typical device in 3D space compared to 2D space even with the same cluster density λc per unit space. From the figure, the gap between the 2D and 3D curves grows for the larger σ and λc. Furthermore, compared to the 3D space results, the Pc performances in the 2D space are less sensitive to the change in σ and λc as observed in Figures 7 and 8, respectively.Figure 7
Comparison of 2D and 3D withβ=0 dB, λc=0.03, and σ={0.4,0.5}.Figure 8
Comparison of 2D and 3D withβ=0 dB, σ=0.5, and λc={0.01,0.03}.Furthermore, Figure9 displays the area spectral efficiency ASE versus λt graphs of the 2D TCP under the change in β using the same parameters as the 3D TCP case shown in Figure 6: σ=0.5, λc=0.03 clusters/m2, and β={-5,0,5} dB. Overall, the ASE in the 2D TCP is greater compared to the 3D case, because of the higher coverage probability Pc as indicated in Figures 7 and 8. That is, if we use the 2D TCP model for ultra-dense urban environments where the devices exist in 3D space, which will be common in the future wireless networks, both the coverage probability and the area spectral efficiency are overestimated. When comparing the impact of β, we can observe the similar trend in the 2D and 3D models that the higher β gives the higher ASE with small λt. However, while the curves with β=0 dB and 5 dB cross over at around λt=6 in Figure 6, the two curves in Figure 9 keep the measurable gap. Interestingly, the optimal λt to maximize the ASE shows the similar (but not exactly the same) trend to that seen in Figure 6. Thus, in a nutshell, the 2D TCP model can be used to estimate the optimal number of simultaneously active Dev-Txs in the 3D clustered networks; however both Pc and ASE estimated based on the 2D model are significantly overestimated compared to the actual performances of the 3D clustered D2D networks.Figure 9
A
S
E versus λt in two-dimensional (2D) space with σ=0.5, λc=0.03 clusters/m2, and β={-5,0,5} dB.


## 7. Conclusion

In this paper, we have studied clustered D2D networks in 3D space modeled by TCP for dense urban environments, where devices are distributed over the 3D space. Using stochastic geometry, we have analyzedPc and ASE of the D2D network in the presence of cochannel interference from both the same cluster and the other clusters. We have derived the exact mathematical expressions of Pc and ASE, which were verified with the simulation results. Moreover, the approximate upper and lower bounds on Pc have been derived, which provide design insights. Both the numerical and simulation results indicate that Pc in 3D space is significantly lower compared to 2D space for the same cluster density λc per unit space because of the more interferers within a certain distance. In addition, compared to the 3D space, the 2D TCP model is less sensitive to the system parameters such as the spatial scattering of the devices σ and the cluster density λc. Comparing the two models, we can conclude that the optimal numbers of simultaneously active devices λt to maximize ASE can be similar in the 2D and 3D models. However, it is not appropriate to use the 2D TCP to estimate Pc and ASE of the D2D networks following the 3D TCP especially for the large σ and λc.The study in this paper provides guidelines on how to operate D2D networks in the presence of cochannel interference among devices, which are distributed in clusters in 3D space. The most significant aspect is how much simultaneous traffic to accommodate using the same channel. Through analysis and simulation, we have shown that there exist an optimal number of the simultaneously active D2D links to maximizeASE, and the optimum is smaller in the 3D D2D networks compared to the 2D D2D networks. Based on this result, one can determine the number of the cochannel D2D pairs to allow communicating in each cluster at the same time, which impacts the higher layer design such as wireless resource allocation for given cluster density λc, spatial scattering of devices σ, and quality of service (QoS) requirement characterized by β.



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*Source: 9628565-2017-11-13.xml*