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

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.


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][3][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][6][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][17][18].

Wireless Communications and Mobile Computing
However, as highlighted in [19][20][21][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][20][21][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.

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 Φ  with density   .Also, the cluster members (devices) are independent and identically distributed (i.i.d.) according to a symmetric normal distribution with variance  2 around each cluster center  ∈ Φ  with the density function of the device locations  ∈ R 3 relative to a cluster center as where  is the scattering parameter.
The devices in the cluster of  ∈ Φ  are denoted by N  , which has two subsets: (i) transmitting devices N   and (ii) receiving devices N   .Suppose the set of simultaneously transmitting devices in the cluster is B  ⊆ N   , and its cardinality |B  | follows a Poisson distribution with mean   .In other words, the number of simultaneously active transmitting devices (Dev-Txs) inside each cluster is a Poisson random variable (RV) with mean   .Therefore, excluding the serving (or desired) Dev-Tx, we assume that the number of interfering devices follows a Poisson distribution with mean (  − 1).As in [12], without loss of generality, we perform analysis based on a typical device in a representative cluster  0 ∈ Φ  , 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 at  0 inside the cluster  0 ∈ Φ  .Thus, the distance between the serving Dev-Tx and the typical device is denoted by  = ‖ 0 +  0 ‖.Hence, with the transmit power of each device denoted by  0 , the received power at the typical device is where  is the path-loss exponent and ℎ 0 is the power gain of small scale fading channel, which follows exponential distribution with unit mean, as in [12,[19][20][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 respectively.Consequently, assuming interference-limited networks, the signal-to-interference-ratio (SIR) at the typical device is where  0 is canceled, since we assume the fixed transmit power for all Dev-Txs.

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 with SIR.
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. in the cluster  0 ∈ Φ  , where   = ‖ 0 + ‖ is the realization of   .We note that the index  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 V 0 = ‖ 0 ‖ between the cluster center  0 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 of
where  ≥ 0. Proof.Based on the 3D Gaussian distribution defined in (1),  = V 2 0 / 2 is the squared sum of three i.i.d.standard (zero mean and unit variance) Gaussian random variables, which corresponds to the PDF as where  ≥ 0. Therefore, by the change of variables, we can obtain the PDF in (6).
Lemma 2 (probability distribution of  = ‖ 0 +‖).The PDF of the separation between the typical device and the Dev-Tx in the same cluster is given by where  ≥ 0.
Proof.The locations of the cluster center  0 and the Dev-Txs  are i.i.d.random vectors in R 3 , where the three components follow i.i.d.Gaussian distributions with zero mean and variance of  2 .Suppose  = ‖ 0 + ‖ 2 /2 2 , which is the squared sum of three i.i.d standard Gaussian random variables.Thus,  follows a chi-squared distribution with 3 degrees of freedom with the PDF: Therefore, the PDF of  =  √ 2 in ( 8) can be derived by the change of variables (it is noted that the PDF and conditional PDF of  are, resp., obtained by extending the probability distribution analysis in 2D to 3D space).

Conditional Distribution of 𝐷
Given ‖ 0 ‖.The distances of the typical device to the Dev-Txs in the same clusters, which are required to calculate  and  intra in SIR, are correlated because of the common factor  0 .Therefore, conditioning the relative location of the cluster center,  0 , 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  given ‖ 0 ‖.
Proof.Let  = ‖ 0 + ‖ 2 / 2 .Because ‖‖ 2 / 2 is the squared sum of three i.i.d standard Gaussian RVs, conditioned on V 0 = ‖ 0 ‖,  follows a noncentral chi-square distribution with the PDF: Since  = ‖ 0 + ‖ =  √ , its PDF in (10) can be obtained by the change of variables (it is noted that the PDF and conditional PDF of  are, resp., obtained by extending the probability distribution analysis in 2D to 3D space).

Distances to
Serving Dev-Tx and Interferers: , , and .Let the distances from the typical device to the serving Dev-Tx and intracluster interferer be  = ‖ 0 +  0 ‖ and  = ‖ 0 + ‖, respectively.Their conditional PDFs given that V 0 = ‖ 0 ‖ are same as (10).In other words, In addition, conditioned on the distance V = ‖‖ between one of the other clusters  ∈ Φ  and the typical device, the distances { = ‖ + ‖, ∀ ∈ B  } between the typical device and the intercluster interfering Dev-Txs in  ∈ Φ  are i.i.d., following the conditional PDF (10).Also, the PDF of V = ‖‖ is identical to the PDF of V 0 = ‖ 0 ‖ defined in (6).

Performance Analysis: P c and ASE
In this section, we investigate the coverage probability, P c , 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 P c and ASE.

Laplace Transform of Intracluster
Interference.Conditioned on V 0 = ‖ 0 ‖, we first derive the Laplace transform of  intra as where () follows from the exponentially distributed ℎ  0 with unit mean and () follows from the probability generating functional (PGF) of Poisson process of the intracluster interferers with mean (  − 1).Also, () follows from  = ‖ 0 + ‖.

Laplace Transform of Intercluster Interference.
The Laplace transform of  inter is given by where ) and () follows from the exponentially distributed ℎ  0 with unit mean.Also, () and () follow from the PGF of Poisson process (with the mean of   and (  − 1) resp.).

Coverage Probability and Area Spectral Efficiency.
Letting  denote the SIR threshold for successful decoding at the receiver, which is a function of modulation and coding, the coverage probability is 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 where     is the average density of simultaneously active Dev-Txs of the whole D2D network.

Approximate Upper and Lower Bounds of P c
Because the exact expressions of P c and ASE are unwieldy, we provide easy-to-compute upper and lower bounds of P c .In particular, the lower bound is in a closed form, which can be readily evaluated.As stated in Section 2,  and  are correlated because of the common factor  0 .For analytical tractability to derive the two approximate bounds, we allow separate deconditioning on  and  as in [12], which implies that  and  are i.i.d.following the PDF in (8).

Upper
, where   () follows the PDF in (8).Thus, the upper bound of P c is given by where   () follows the PDF in (8).

Lower
Proof.See Appendix B.

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 m 3 cube.The cluster centers follow PPP with intensity   , 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   .Also, we assume the path-loss exponent  of 4, as in [12,19,20].The simulation results are obtained from 10 6 random realizations of device distribution (network topology) and Rayleigh fading channel.

Impacts of System Parameters. Figures 2(a) and 2(b) show
how the coverage probability P c varies, as the average number of simultaneously active Dev-Txs   increases, with   = 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 P c * 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 P c derived in the previous section are validated.Specifically, comparing the two figures, when   is large, the actual P c is closer to the lower bound P c * compared to the upper bound Pc , as in Figure 2(a), because the large   results in the higher intercluster interference  inter , which is ignored in the Pc .On the other hand, for small   , the gap between the exact P c and its upper bound Pc is significantly smaller compared to the difference from its lower bound P c * , as in Figure 2(b), since the intracluster interference  intra is dominant relative to the intercluster interference  inter .In either case, the exact P c curve is always bounded by Pc and P c * .In Figures 3 and 4, we observe the impacts of  and   on exact P c , 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   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 , which means the larger spatial scattering of the devices from the cluster center, results in the lower P c .This can be attributed to the increased impact of  inter , while the P c curves only with  intra do not change as indicated by the dashed curves in the figure.The P c curves only with  intra stay the same regardless of   , because the variations of the serving and interfering Dev-Txs cancel each other.We can observe the same trend in Figure 4: as   increases, P c decreases because of the increased intercluster interference  inter .On the other hand, the coverage probability P c only with the intracluster interference  intra does not vary under the variation in the cluster density   .
Figures 5 and 6 show the exact area spectral efficiency ASE, numerically obtained by (15), versus the average number of simultaneously active Dev-Txs   .In the figure, the horizontal axis indicates   , while the vertical axis is ASE.Also, the solid, dashed, and dash-dotted lines represent the theoretical results with different system parameters (  , , and ), while the circle and triangle markers represent the corresponding simulation results.Lastly, the optimal   in each graph is indicated by the "x"-marker.As shown in Figures 2, 3, and 4, we can observe the great correlation between the simulation and theoretical results.If comparing the curves with  = 0.3 and 0.5 with the same   in Figure 5, ASE increases, as  decreases, which is expected from the result of P c in Figure 3.Moreover, for the fixed  = 0.5, the curves with   = 0.03 show significantly higher ASE compared to the curve with   = 0.01, because of the greater multiplication factor in (15).In Figure 6, the higher  makes ASE increase for small   , while the curves with higher  decrease more rapidly compared to the curves with smaller , as   increases.One of the most important design aspects is the optimal   to maximize the ASE, which determines network operation and wireless resource allocation.In Figure 5,   ≈ 2 gives the best ASE for all the three curves, which indicates its low sensitivity to  and   .On the other hand, if we increase the SIR threshold   as in Figure 6, the optimal value of   decreases (4.25, 2, 1 for  = −5, 0, 5 dB, resp.), because lower  can accommodate more simultaneous users (devices).

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   (clusters/m 2 and clusters/m 3 in 2D and 3D spaces, resp.).Figures 7 and 8 show the coverage probability P c versus the average number of simultaneously active Dev-Txs   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, P c of the 2D TCP is higher compared to P c 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   per unit space.From the figure, the gap between the 2D and 3D curves grows for the larger  and   .Furthermore, compared to the 3D space results, the P c performances in the 2D space are less sensitive to the change in  and   as observed in Figures 7  and 8, respectively.Furthermore, Figure 9 displays the area spectral efficiency ASE versus   graphs of the 2D TCP under the change in  using the same parameters as the 3D TCP case shown in Figure 6:  = 0.5,   = 0.03 clusters/m 2 , and  = {−5, 0, 5} dB.Overall, the ASE in the 2D TCP is greater compared to the 3D case, because of the higher coverage probability P c 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 , we can observe the similar trend in the 2D and 3D models that the higher  gives the higher ASE with small   .However, while the curves with  = 0 dB and 5 dB cross over at around   = 6 in Figure 6, the two curves in Figure 9 keep the measurable gap.Interestingly, the optimal   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 P c and ASE estimated based on the 2D model are significantly overestimated compared to the actual performances of the 3D clustered D2D networks.

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 analyzed P c 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 P c and ASE, which were verified with the simulation results.Moreover, the approximate upper and lower bounds on P c have been derived, which provide design insights.Both the numerical and simulation results indicate that P c in 3D space is significantly lower compared to 2D space for the same cluster density   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  and the cluster density   .
Comparing the two models, we can conclude that the optimal numbers of simultaneously active devices   to maximize ASE can be similar in the 2D and 3D models.However, it is not appropriate to use the 2D TCP to estimate P c and ASE of the D2D networks following the 3D TCP especially for the large  and   .
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 maximize ASE, 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   , spatial scattering of devices , and quality of service (QoS) requirement characterized by .

Figure 1 :
Figure 1: An example illustration of a three-dimensional clustered D2D network based on TCP.
Dev-Txs.For the intracluster devices, let D  0  be the set {  } =1:|N  0  | of distances from the typical device to the set of possible Dev-Txs N  0

Figure 2 :Figure 3 :
Figure 2: P c versus   : comparison with the upper and lower bounds.
Bound of P c .Since the intracluster interferers are significantly closer to the typical device compared to the intercluster Dev-Txs,  intra is dominant in the denominator of SIR.Thus, we can derive the approximate upper bound of SIR by ignoring  inter , which corresponds to the upper bound of P c .By the i.i.d.assumption of  and , the Laplace transform of  intra can be approximated as L intra Lemma 4 (lower bound of L  intra ()).The lower bound on the Laplace transform of   is Bound of P c .We first derive lower bounds of L  intra () and L  inter () in closed forms.Then, using the two, the lower bound of P c will be obtained.