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.
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 with density . Also, the cluster members (devices) are independent and identically distributed (i.i.d.) according to a symmetric normal distribution with variance around each cluster center with the density function of the device locations relative to a cluster center aswhere is the scattering parameter.
The devices in the cluster of are denoted by , which has two subsets: (i) transmitting devices and (ii) receiving devices . Suppose the set of simultaneously transmitting devices in the cluster is , and its cardinality 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 . As in [12], without loss of generality, we perform analysis based on a typical device in a representative cluster , 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 inside the cluster . Thus, the distance between the serving Dev-Tx and the typical device is denoted by . Hence, with the transmit power of each device denoted by , the received power at the typical device iswhere is the path-loss exponent and 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 asrespectively. Consequently, assuming interference-limited networks, the signal-to-interference-ratio () at the typical device iswhere 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 with . 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, let be the set of distances from the typical device to the set of possible Dev-Txs in the cluster , where 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 between the cluster center 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 ). The PDF of is given bywhere .
Proof. Based on the 3D Gaussian distribution defined in (1), is the squared sum of three i.i.d. standard (zero mean and unit variance) Gaussian random variables, which corresponds to the PDF aswhere . Therefore, by the change of variables, we can obtain the PDF in (6).
Lemma 2 (probability distribution of ). The PDF of the separation between the typical device and the Dev-Tx in the same cluster is given bywhere .
Proof. The locations of the cluster center and the Dev-Txs are i.i.d. random vectors in , where the three components follow i.i.d. Gaussian distributions with zero mean and variance of . Suppose , 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 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).
3.2. Conditional Distribution of Given
The distances of the typical device to the Dev-Txs in the same clusters, which are required to calculate and in , are correlated because of the common factor . Therefore, conditioning the relative location of the cluster center, , 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 .
Lemma 3 (conditional probability distribution of given ). The conditional PDF of for a given is derived as where and is the modified Bessel function with order .
Proof. Let . Because is the squared sum of three i.i.d standard Gaussian RVs, conditioned on , follows a noncentral chi-square distribution with the PDF:Since , 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).
3.3. Distances to Serving Dev-Tx and Interferers: , , and
Let the distances from the typical device to the serving Dev-Tx and intracluster interferer be and , respectively. Their conditional PDFs given that are same as (10). In other words, and . In addition, conditioned on the distance between one of the other clusters and the typical device, the distances between the typical device and the intercluster interfering Dev-Txs in are i.i.d., following the conditional PDF given in (10). Also, the PDF of is identical to the PDF of defined in (6).
4. Performance Analysis: and
In this section, we investigate the coverage probability, , and the area spectral efficiency, , of the clustered D2D network. We first find the Laplace transforms of the two interference terms to characterize . Then, we derive the exact expressions of and .
4.1. Laplace Transform of Intracluster Interference
Conditioned on , we first derive the Laplace transform of aswhere follows from the exponentially distributed with unit mean and follows from the probability generating functional (PGF) of Poisson process of the intracluster interferers with mean . Also, follows from .
4.2. Laplace Transform of Intercluster Interference
The Laplace transform of is given bywhere and follows from the exponentially distributed with unit mean. Also, and follow from the PGF of Poisson process (with the mean of and resp.).
4.3. Coverage Probability and Area Spectral Efficiency
Letting denote the threshold for successful decoding at the receiver, which is a function of modulation and coding, the coverage probability isTherefore, 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 bywhere is the average density of simultaneously active Dev-Txs of the whole D2D network.
5. Approximate Upper and Lower Bounds of
Because the exact expressions of and are unwieldy, we provide easy-to-compute upper and lower bounds of . 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 . 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).
5.1. Upper Bound of
Since the intracluster interferers are significantly closer to the typical device compared to the intercluster Dev-Txs, is dominant in the denominator of . Thus, we can derive the approximate upper bound of by ignoring , which corresponds to the upper bound of . By the i.i.d. assumption of and , the Laplace transform of can be approximated as , where follows the PDF in (8). Thus, the upper bound of is given bywhere follows the PDF in (8).
5.2. Lower Bound of
We first derive lower bounds of and in closed forms. Then, using the two, the lower bound of will be obtained.
Lemma 4 (lower bound of ). The lower bound on the Laplace transform of is
Proof. See Appendix A.
Lemma 5 (lower bound of ). The lower bound on the Laplace transform of is given by
Proof. See Appendix B.
With (17) and (18) along with the independent deconditioning assumption, we can obtain the approximate lower bound of in a closed form aswhere and follows from following (8). Because ( and ), we can obtain the lower bound in (20), where is the Airy function, the derivative of which is . Moreover, is the generalized hypergeometric function [23].
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 , 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 , as in [12, 19, 20]. The simulation results are obtained from random realizations of device distribution (network topology) and Rayleigh fading channel.
6.1. Impacts of System Parameters
Figures 2(a) and 2(b) show how the coverage probability varies, as the average number of simultaneously active Dev-Txs increases, with and , 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 and 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 derived in the previous section are validated. Specifically, comparing the two figures, when is large, the actual is closer to the lower bound compared to the upper bound , as in Figure 2(a), because the large results in the higher intercluster interference , which is ignored in the . On the other hand, for small , the gap between the exact and its upper bound is significantly smaller compared to the difference from its lower bound , as in Figure 2(b), since the intracluster interference is dominant relative to the intercluster interference . In either case, the exact curve is always bounded by and .
(a) with clusters/m3, , and dB
(b) with clusters/m3, , and dB
In Figures 3 and 4, we observe the impacts of and on exact , 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 . This can be attributed to the increased impact of , while the curves only with do not change as indicated by the dashed curves in the figure. The curves only with 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, decreases because of the increased intercluster interference . On the other hand, the coverage probability only with the intracluster interference does not vary under the variation in the cluster density .
Figures 5 and 6 show the exact area spectral efficiency , 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 . 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 and with the same in Figure 5, increases, as decreases, which is expected from the result of in Figure 3. Moreover, for the fixed , the curves with show significantly higher compared to the curve with , because of the greater multiplication factor in (15). In Figure 6, the higher makes 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 , which determines network operation and wireless resource allocation. In Figure 5, gives the best for all the three curves, which indicates its low sensitivity to and . On the other hand, if we increase the threshold as in Figure 6, the optimal value of decreases (4.25, 2, 1 for , 0, 5 dB, resp.), because lower 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 (clusters/m2 and clusters/m3 in 2D and 3D spaces, resp.). Figures 7 and 8 show the coverage probability 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, of the 2D TCP is higher compared to 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 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 versus graphs of the 2D TCP under the change in using the same parameters as the 3D TCP case shown in Figure 6: , clusters/m2, and dB. Overall, the in the 2D TCP is greater compared to the 3D case, because of the higher coverage probability 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 with small . However, while the curves with dB and 5 dB cross over at around in Figure 6, the two curves in Figure 9 keep the measurable gap. Interestingly, the optimal to maximize the 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 and estimated based on the 2D model are significantly overestimated compared to the actual performances of the 3D clustered D2D networks.
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 analyzed and 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 and , which were verified with the simulation results. Moreover, the approximate upper and lower bounds on have been derived, which provide design insights. Both the numerical and simulation results indicate that 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 can be similar in the 2D and 3D models. However, it is not appropriate to use the 2D TCP to estimate and 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 , 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 .
Appendix
A. Proof of Lemma 4
which corresponds to in (17). follows from Jensen’s inequality, and follows from Holder’s inequality.
B. Proof of Lemma 5
which is in (18). follows from the Taylor expansion of an exponential function, and is based on the property of the PDF in (10) that .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2016R1D1A1B03930060).