Modeling and Analysis of D2D Millimeter-Wave Networks with Poisson Cluster Processes

07/12/2018 ∙ by Wenqiang Yi, et al. ∙ Queen Mary University of London 0

This paper investigates the performance of millimeter wave (mmWave) communications in clustered device-to-device (D2D) networks. The locations of D2D transceivers are modeled as a Poisson Cluster Process (PCP). In each cluster, devices are equipped with multiple antennas, and the active D2D transmitter (D2D-Tx) utilizes mmWave to serve one of the proximate D2D receivers (D2D-Rxs). Specifically, we introduce three user association strategies: 1) Uniformly distributed D2D-Tx model; 2) Nearest D2D-Tx model; 3) Closest line-of-site (LOS) D2D-Tx model. To characterize the performance of the considered scenarios, we derive new analytical expressions for the coverage probability and area spectral efficiency (ASE). Additionally, in order to efficiently illustrating the general trends of our system, a closed-form lower bound for the special case interfered by intra-cluster LOS links is derived. We provide Monte Carlo simulations to corroborate the theoretical results and show that: 1) The coverage probability is mainly affected by the intra-cluster interference with LOS links; 2) There exists an optimum number of simultaneously active D2D-Txs in each cluster for maximizing ASE; and 3) Closest LOS model outperforms the other two scenarios but at the cost of extra system overhead.

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I Introduction

The unparalleled explosion of Internet-enabled mobile devices, applications and services is promoting the development of wireless communication networks. As the spectrum resource is limited in the forth generation (4G) cellular networks, 5G new radio (NR) standard has been considered to be the foundation for the next generation of mobile networks, which works over frequencies both below and above 6 GHz. Apart from extensive researches on sub-6 GHz, such as 2.4 GHz [1] and TV white space [2, 3], millimeter wave (mmWave) has recently received significant attention due to a huge range of free spectrum [4, 5, 6]. Numerous protocols show that mmWave frequencies from 30 GHz to 300 GHz have already been utilized in different commercial networks including local area networking in IEEE 802.11ad [7], personal area networking in IEEE 802.15.3c [8] and fixed-point access links in IEEE 802.16.1 [9]. Comparing to traditional networks in 4G, the first distinguishing feature of mmWave networks is the small wave length, which helps to deploy huge antenna arrays at transceivers for enhancing the array gain [10]. This feature reduces inter-cell interferences, the additional noise power and the frequency-dependent path loss [5]. Another differentiating feature is that mmWave signals are sensitive to blockage effects [10]. Moreover, mmWave signals experience more serious penetration loss than the sub-6 GHz carriers when passing through the blockage [11]. Therefore, the path loss laws for line-of-sight (LOS) links and blockage-dependent non-line-of-sight (NLOS) links are significantly different in mmWave networks [6, 12]. A plenty of practical channel measures demonstrate that the path loss exponent of NLOS is more massive than LOS’s, because the complicated scattering environment contributes to the severe path loss for NLOS links [12, 13, 14].

Accordingly, various articles focus on these two features when modeling mmWave networks. The primary work [15] employed a directional beamforming to fulfill the array gain, but the path loss model was simplified and hence failed to fully reflect mmWave propagation features. Then, site-specific simulation [16] and stochastic blockage model [17, 18, 19] were proposed to investigate the performance of mmWave networks with the impact of the blockage. Stochastic geometry is an effective tool to capture the randomness of the networks [20] and recently it was applied in mmWave networks [10, 21]. More particularly, base station locations were modeled as a Poisson Point Process (PPP) on the plane [21]. Under this model, a framework combining random blockage process and directional antenna beamforming was designed, which shown a close characterization of the reality [10].

However, the aforementioned models only deploy mmWave into a conventional cellular structure where devices download the information from a base station. In this structure, the path loss is serious due to the long distance between transceivers, while mmWave is capable of supporting high rate with short-range networks [22]. In order to achieve a higher quality cellular network, a key short-distance technology with enormous potential termed device-to-device (D2D) has kindled the interest of academia [23]. To be more specific, D2D networks enable direct links between proximal devices without the aid of cellular networks [24]. When comparing with the traditional architecture in 4G networks, the received power at the intended D2D receiver (D2D-Rx) is typically much higher due to the shorter link distance [25]. With the content centric nature, D2D networks are able to satisfy spatiotemporal correlation in the content demand [26, 27]. In particular, a user downloads popular files from any of the surrounding transmitters rather than a base station [28, 29, 30]. The set of proximate devices is termed a cluster in D2D networks, which corresponds to a hotspot in the heterogeneous cellular networks [31].

The same with mmWave networks, stochastic geometry has also been successfully applied in D2D communications. The primary approach for D2D networks was fixing a D2D transmitter (D2D-Tx) at the origin in a plane and D2D-Rxs were modeled using a PPP [23]. The limitation for this approach is the lack of enough D2D-Txs. As a further development, D2D-Txs were located following a PPP, while D2D-Rxs were modeled as a Poisson Dipole Process (PDP) where every D2D-Tx had a fixed distance to its corresponding D2D-Rx [32, 33, 34, 35]. However, the fixed distance assumption is extremely restrictive. Then the condition was relaxed by assuming that the D2D-Rx was uniformly located within a circle around the serving D2D-Tx [36, 37, 38]. Although the distance is variable, the intended D2D-Rx still fails to choose the serving device from multiple proximate transmitters, which is the fundamental nature of D2D networks [28, 29, 30]. Very recently, a realistic tractable D2D structure [25, 39] was proposed following a Poisson Cluster Process (PCP) 111The PCP model is regarded as a promising method for analytically studying various kinds of networks, such as device-to-device, ad hoc network and sensor networks. However, the shortage of experiments in terms of PCP will motivate our future work., where the intended user had multiple randomly distributed D2D-Txs and each of them had the ability to be the active serving device. However, this work only focuses on sub-6 GHz networks, while more attention should be paid on the performance of mmWave networks under this architecture as it outperforms sub-6 GHz in short-distance communications.

I-a Motivation and Contribution

As discussed above, mmWave communications have been studied in a variety of scenarios, but there is still short of researches on a short-distance communication system. This shortage motivates us to contribute this treatise. Note that the tractable D2D model mentioned in [25] has a perfect short-distance communication architecture. To increase the capacity and signal-to-interference-plus-noise ratio (SINR) coverage of future wireless networks, it is ideal to deploy mmWave into this D2D structure. Different from [25], four main issues are carefully addressed in our paper. Firstly, the propagation environment is replaced by two kinds of path loss laws and nakagami-M fading channels due to blockage sensitivity of mmWave signals. Secondly, we employ a sectorial model for analyzing the antenna beamforming. Thirdly, three different user association strategies are proposed to evaluate our system. Lastly, we compare the performance of various carrier frequencies in terms of SINR coverage probability. On the other side, different from PPP modeled mmWave networks [10], the employment of the PCP results in a unique interference from inter-clusters [25], which is not negligible in D2D networks. The prime contributions of this paper are as follows:

  • We analyze the coverage performance and area spectral efficiency (ASE) for three different scenarios: i) Uniform Distribution Model, where the connected D2D-Tx is uniformly distributed in the same cluster of the typical D2D-Rx; ii) Closest Distribution Model, where the connected D2D-Tx is the nearest transmitter in the same cluster of the typical D2D-Rx; and iii) Closest LOS Model, where the connected D2D-Tx is the closest transmitter with an LOS link in the same cluster of the typical D2D-Rx.

  • We characterize the distribution of distances from the typical D2D-Rx to the serving D2D-Tx and intra/inter-interfering devices. Moreover the exact probability density functions (PDFs) of distances for three scenarios are presented.

  • We work out Laplace transforms of intra/inter-cluster interfering powers, using which different coverage probability expressions for three scenarios are derived. Additionally, a closed-form lower bound for an intra-interfered case is presented. We analytically demonstrate that the coverage probability has a positive correlation with the directivity gain at the typical D2D-Rx, while it has the inverse correlation with the number of interfering D2D-Txs. Finally, ASEs are characterized based on the derived coverage probabilities.

  • We show that: 1) The closest LOS model achieves the best performance among three scenarios regarding the coverage probability; 2) Our model is an interference-limited system due to the content centric nature of D2D communications. In addition, the proposed model is mainly interfered by the intra-devices with LOS links; 3) There is an optimal number of active D2D-Txs in a cluster for achieving the maximum ASE; and 4) Large antenna scale for high frequency has limited impact on SINR coverage in our system. 38 GHz is the best carrier frequency for high SINR regions and 28 GHz is the best for low SINR regions.

I-B Organization

The paper is organized as follows. In Section II, considering the blockage and antenna beamforming, the clustered device-to-device mmWave communication networks are modeled in a PCP. In Section III, we derive distribution expressions of distances from the typical D2D-Rx to the serving D2D-Tx and interfering devices. In Section IV, three different distribution scenarios for the serving D2D-Tx are discussed. Coverage probability and ASE algorithms are figured out in this part. In Section V, the numerical results are presented for analyzing and verifying. In Section VI, we propose our conclusion.

Ii System Model

In this section, we present our system model for appraising the performance of the clustered D2D mmWave communication networks. The paper will focus on downlink coverage probability and ASE. The crucial modeling details are discussed below.

Ii-a Spatial Distribution

In this treatise, we adopt one of the typical PCP processes, which is a variant of the Thomas cluster process [39]. More particularly, the devices are located in a group of clusters following a PCP, in which the parent point process follows a PPP with a density , and the offspring point processes with one parent are conditionally independent [40]. In our system, the centers of clusters contribute to the parent points

, and the devices are offspring points. In each cluster, we assume that all devices, which are independent and identically distributed (i.i.d.), follow a symmetric normal distribution around the cluster center with mean zero and variance

. As a result, the location of a device in reference to a cluster center is

(1)

For the tractability of analysis, we assumed that the number of devices in every cluster is same with N in case one wants to allow all transceivers to communicate simultaneously in a special application [25]. Half of the devices are possible transmitters denoted by , and the rest are possible receivers denoted by (

). Each transmitter is capable of supporting one receiver at the same time in our model. Although the number of transceivers is fixed, the quantity of simultaneously active transmitters are different across the clusters, which is assumed to be a poisson distribution variable with mean

denoted by , where . All these devices are the source of interference except the corresponding transmitter that serving the typical user. Thus, the D2D model is shown in Fig. 1(a).

(a) Graphical illustration of spatial distributions for proposed D2D mmWave networks with the aid of a Poisson Cluster Process.
(b) Illustration of the stochastic blockage model and beamforming for mmWave networks.
Fig. 1: The system model of clustered D2D mmWave networks

Without loss of generality, we randomly choose one device as a typical user that is included in the typical cluster. Moreover, the typical user is assumed to be located in the origin of a plane. The center of the typical cluster is and the transmitters in the typical cluster are denoted by . In the proposed network, the performance of the connection is mainly decided by the distance between the typical user and its corresponding transmitter, we provide three different distributions of the corresponding transmitter in the typical cluster for analyzing: 1) Uniform Distribution Model: the corresponding transmitter is uniformly distributed in a set of transmitters in the typical cluster; 2) Closest Distribution Model: the corresponding transmitter is the closest transmitter in the typical cluster; and 3) Closest LOS Model: the corresponding transmitter is the closest transmitter with an LOS link in the typical cluster. Apart from the corresponding transmitter, the rest simultaneously transmitting devices are the source of intra-cluster interference in the typical cluster, so the intra-interfering devices are modeled by a poisson distribution with mean . However, the active inter-transmitters in every inter-cluster, which contribute to inter-cluster interference, are still poisson distributed with mean as we assumed above.

Ii-B LOS and NLOS Links

In our system, all transmitters are capable of establishing an LOS or NLOS link to communicate with the typical user when employing mmWave. We assume that the network system is a stochastic blockage model with rectangle Boolean scheme (see Fig. 1(b)), so the probability function of LOS will follow , where is determined by the average size and density of blockages, is the distance between the transmitter and the typical user. In addition, the average LOS distance is  [18]. The probability of an LOS link is assumed to be independent with other links. Although LOS probabilities for different links are not independent in reality, ignoring such correlation will cause negligible loss of accuracy in terms of SINR coverage [18] and demonstration will be offered in Section V. Moreover, various path loss are used to model LOS and NLOS links.

(2)

where , are LOS and NLOS path loss exponents respectively. is the intercept of LOS links and is that of NLOS links.

Ii-C Directional Beamforming

We deploy antenna arrays at all transceivers to accomplish directional beamforming as mentioned in [10]. The antenna pattern is assumed to be a sectorial model and the total directivity gain of links will be , where and are antenna gains at transmitters and receivers, respectively. In each antenna, () is the main lobe beamwidth, then and denote the directivity gain of main lobe and back lobe. Note that angles of arrival and angles of departure for all LOS and NLOS links are independently and uniformly distributed in the range , so random directivity gains of interferences have patterns with the value and probability , where . See Table. I.

i 1 2 3 4
TABLE I: Probability and Value of

For different carrier frequencies, the antenna array should be changeable since higher frequencies allow manufacturing more antenna elements for compensating the possible higher path-loss. Under this condition, when analyzing various carrier frequencies, we will change and into and , respectively, where respects the number of antennas assembled at devices.

Ii-D Channel Model

Assuming the corresponding transmitter is located at to the center of the typical cluster, the distance between the typical user and the corresponding transmitter is (, ). The received power of the typical link is given by

(3)

where is the transmitting power of each device, is the small fading term for link and independent Nakagami fading. As a result,

follows a normalized Gamma random variable. The Nakagami fading parameters are

and for LOS and NLOS links, respectively. and are assumed to be positive integers for simplicity [10].

In our model, the interferences have two sources. One is intra-cluster interference from the typical cluster, and the other is inter-cluster interference from other clusters. The distance between the typical user and the transmitter in the typical cluster is (, ), and that from the typical user to the transmitter in other clusters is (, ). The two kinds of different interference power are expressed as follows

(4)
(5)

As a result, the SINR at the typical user is given by

(6)

where is the thermal noise power normalized by . The power of transmitter can be canceled in SINR. Without any loss of generality, we assume .

Iii Distribution of Distances

We will discuss the distribution of the distances between the typical user and other devices in this section. Before that, we introduce two different distributions as mentioned in [25] below in order to simplify the notation.

Rayleigh Distribution: the probability density function (PDF) is defined as

(7)

where is the scale parameter of Rayleigh distribution.

Rician Distribution: the PDF is defined as

(8)

where is the scale parameter of Rician distribution and is the first kind Modified Bessel Function with zero order.

Iii-a Distribution in Uniform Distribution Model

In this part, the distribution of distances in uniform distribution model will be characterized. We will start the demonstration with the typical cluster and then other clusters.

Iii-A1 Distance Distribution in Typical Cluster

Assuming the set of distances between the typical user and the possible transmitters in the typical cluster is denoted by (). is the realization of and (, ). Since and are Gaussian Random Variables (i.i.d.) with variance, is a Gaussian Random Variable with variance so that can be approximated by a PDF of . However, is conditional on the distance because the transceivers are i.i.d around the cluster center in our system model. Therefore the exact PDF is shown as below [25, Lemma 1]:

(9)

In typical cluster, since elements of are i.i.d and the corresponding transmitter is selected uniformly at random, all distributions of distances including the corresponding transmitter and intra-interfering devices will follow Rician distribution in (9). The results are shown formally as below.

The distance of the typical link: the distance between the typical user and its corresponding transmitter is assumed to be (). As mentioned above, the PDF of typical link distance is .

Distances of intra-cluster interfering links: The distance from intra-cluster interfering device to the typical user is (). Utilizing the same method discussed in (9), the PDF of this case is .

Iii-A2 Distance Distribution in Other Clusters

In other clusters, the set of distances between the typical user and the possible transmitter is denoted by (). is the realization of and (, ). has the same distribution with that in the typical cluster, and the only difference is that is conditional on the distance (). Evidently, the PDF of distances from the typical user to simultaneous transmitters in other clusters is as follows [25, Lemma 2]

(10)

The set of distances of inter-cluster interfering links between the typical user and simultaneously transmitting devices in other clusters is denoted by (). It is conditioned on the distance . As the inter-interfering device is selected at random, has the same distribution with . The distances of inter-cluster interfering link will follow .

Remark 1.

As the corresponding transmitter is located in the typical cluster, there is no difference among three scenarios in terms of distances distribution in inter-clusters. Therefore, the distance of inter-cluster interfering links in other two scenarios are same with uniform distribution model, and we will omit this in the following discussion.

Iii-B Distribution in Closest Distribution Model

In this part, the distribution of distances in closest distribution model will be discussed. Unlike the uniform distribution model above, we assume that the corresponding transmitter in the typical cluster is the nearest one with (, ). In this model, the distribution of distance for the closest link is shown below.

The distance of the closest typical link: The distance from the nearest transmitter to the typical user is conditioned on the distance and the PDF is easy to be deduced from [25, Lemma 3]. We choose the st-closest content available strategy here and the equation is shown below

(11)

where , and .

As the corresponding transmitter is the closest one, the rest distances () from the typical user to intra-interfering transmitters are larger than . They are conditioned on the distance and . The PDF is illustrated below.

Distances of intra-interfering links: The set of distances between the typical user and the rest intra-interfering devices is conditioned on the distance and the closest distance , it is shown below [25, Lemma 4]

(12)

where .

Iii-C Distribution in Closest LOS Model

Different with closest distribution model, we focus on the nearest device with an LOS link in closest LOS model. The set of transmitters with LOS links in the typical cluster is denoted by . On the other hand, the set of NLOS is . And the corresponding transmitter is the nearest one with an LOS link, which has the distance (, ). Note that the probability function of LOS will follow , the distribution of distance between the transmitter with an LOS link and the typical user in the typical cluster is

(13)

Under this condition, the distance for the closest LOS link is distributed as below.

Lemma 1.

(The distance of the closest typical LOS link): The distance of the nearest transmitter with an LOS link is conditioned on the distance and the PDF is

(14)
Proof:

We randomly choose one device in the typical cluster to be the corresponding transmitter. It has a distance to the typical user. Note that the typical cluster has transmitters, so there are transmitters located beyond the circle with the radius of . With the aid of (13), the distance distribution of the closest typical LOS link is derived as above. ∎

As we discussed above, the rest distances with LOS links () from the typical user to simultaneously transmitting devices in the typical cluster must be larger than . They are conditioned on the typical link distance and the distance . The distribution of rest distances is expressed as below.

Lemma 2.

(Distances of intra-interfering LOS links): The distance of the rest typical LOS links are conditioned on the distance and the closest distance with an LOS link , it is

(15)
Proof:

As the locations of LOS transmitters, which follows Rician Distribution, are independent of NLOS devices, (12) also exists in this case. Obviously, if the distance is less than , the probability should be 0. ∎

Different with LOS links, the distance of devices in the typical cluster with NLOS links are distributed as that in uniform distribution model.

Iv Coverage Probability and Area Spatial Efficiency Analysis

In this section, we focus on the coverage probability and ASE in different scenarios depending on the distances distribution.

Iv-a Uniform Distribution Model

In this model, like various strategies have been proposed in recent articles, for example, Uniform Content Availability in D2D networks [25] and RNRF Selection Scheme in NOMA networks [20], the typical user will choose the corresponding transmitter randomly in the typical cluster. This strategy offers a fair opportunity for each device to access the content in the cluster. The benefit of this user association scheme is that networks do not need the additional knowledge of instantaneous channel state information (CSI) which is not available on some networks due to the poor performance of the basic equipment. To make tractable calculation, we first introduce Laplace Transform to figure out the expected value of interference. Then the coverage probability will be derived using Laplace Transform of Interference.

Iv-A1 Laplace Transform of Interference

We first derive analytical expressions and approximations on the Laplace transform of intra-cluster interference. As in the real-life world, the number of active D2D pairs is far less than the number of possible transceivers in most clusters. For example, assuming that people in a library form a cluster, the scale of devices sharing study materials simultaneously is much smaller compared with the number of customers because the majority of tasks in a library should be reading and self-study rather than transmitting information. Therefore, we add the assumption in the following content for the tractability of analysis.

Lemma 3.

(Laplace transform of interference in the typical cluster): When , the th conditional Laplace transform of interference in the typical cluster is

(16)

where

(17)
(18)
Proof:

See Appendix A. ∎

Remark 2.

Conditioning on the certain and , and represent the probabilities of intra-cluster interferences from LOS and NLOS links, respectively. Additionally, it is easy to infer that (17) is a monotonic decreasing function with , which means in short-range clustered networks the probability of intra-cluster LOS interference will be high.

Note that the assumption is applicable when the number of simultaneously transmitting devices per cluster is much smaller than the cluster size. As analyzed in the sequel, the assumption is also the regime where the networks will be optimized in terms of ASE. Therefore, the simpler expression will be used as a proxy of exact expression for the analytical equations and approximations.

Assumption 1.

Although distribution of distance from intra-transmitters to the typical user is conditioned on the distance , the analysis is essentially simplified by ignoring this condition. We assume the distance between the typical user and the corresponding transmitter follows as we discussed in the distribution of uniform distribution model.

This assumption is under the consideration that regarding the distribution of intra-cluster devices, the conditioning on the distance is weak enough to be ignored [25]. It is treated as a tight approximation for the following calculation so that more insights can be directly obtained from the analytical results.

Corollary 1.

(Approximation): Under the Assumption 1, the Laplace transform of interference in the typical cluster is approximated as below

(19)

After the analysis of interference in the typical cluster, we now state the analytical result for the Laplace transform of inter-cluster interference.

Lemma 4.

(Laplace transform of interference in other clusters): When , the th conditional Laplace transform of interference in inter-clusters is

(20)
Proof:

See Appendix B. ∎

Remark 3.

Note that the inter-interfering devices are distributed uniformly at random, which means that the above expression is applicable for three proposed scenarios.

Iv-A2 Coverage Probability

We set a pre-decided threshold of SINR to analyze the performance. The SINR that exceeds contributes to the coverage probability. It is expressed as

(21)

where . and are probabilities of the typical LOS link and NLOS link respectively. With the aid of Laplace transform, the tight upper bound expression for is shown in the Theorem 1.

Theorem 1.

Using the Laplace transform of intra-cluster interference Lemma 3 and inter-cluster interference Lemma 4, we figure out the tight upper bound for the coverage probability under uniform distribution model. It is given by

(22)

where

(23)
(24)

and . , .

Proof:

See Appendix C. ∎

Remark 4.

The results derived in Theorem 1 show that the coverage probability of uniform distribution model is independent of the cluster size .

Corollary 2.

(Approximation of coverage probability): Under Assumption 1 the tight upper bound for the probability of coverage in uniform distribution model is

(25)

where

(26)
(27)
Proof:

The proof is same with Theorem 1, except that it is no conditioned with distance . As a result, we use (19) instead of (16). ∎

Assumption 2.

We assume a special case with small and large . As discussed in [25], the coverage probability of the typical user is mainly affected by the intra-cluster interference under such condition, and thus the received interfering signals from inter-transmitters are ignored. Additionally, NLOS signals and noise are negligible in our system due to the nature of mmWave networks [10] and content centric property of D2D architecture [25], respectively. In a word, under this assumption, we only consider the intra-cluster interference and all links are regarded as LOS.

This assumption is a common scene in our real world. For example, people watching the football game sitting around a screen in a bar constitute a cluster. They are able to use the mobile devices to share the short video of the game’s highlights by D2D with mmWave. In this case, the distance between the persons in this cluster is short, which means is small. Additionally, there is a plenty of devices sharing content simultaneously during the peak-time, so is large. Therefore, Assumption 2 is a reasonable simplification for analysis our system.

In this assumption, we have ignored the inter-cluster interference so that it seems like only a certain cluster is taken into account, which is no different from traditional D2D networks. However, in fact, the typical user is randomly chosen across all clusters in our system, which means each cluster has the fair probability to be the typical cluster. This feature ensures that the analysis under Assumption 2 is still able to reflect the whole picture of clustered D2D mmWave networks that have the significantly different clustered structure with traditional D2D communications. Moreover, since we utilize mmWave as our carrier frequency, the propagation environment and antenna pattern are totally different from the traditional ones as well.

With the condition of Assumption 2, the Laplace transform of interference in the typical cluster is shown as follows.

Lemma 5.

(Laplace transform of interference in the typical cluster): When , the th conditional Laplace transform of interference in the typical cluster under Assumption 2 is

(28)

where .

Proof:

Under the Assumption 2, there is no noise and only LOS interference in the typical cluster. We should remove all the useless parts from (16). ∎

Corollary 3.

(Approximation): Combining with the Assumption 1, the approximate Laplace transform of interference in the typical cluster is

(29)
Proof:

The proof procedure is similar as (19) and hence is skipped here. ∎

Theorem 2.

With the aid of Laplace transform of intra/inter interference discussed above, we derive the tight upper bound for the coverage probability for uniform distribution model under Assumption 2. It follows

(30)

where

Proof:

Same with Theorem 1, but there is only LOS intra-cluster interference existed. ∎

The different for various carrier frequencies can be canceled from (30), which means under this assumption, the SINR coverage probability for various carrier frequencies has no relationship with the scale of antenna arrays.

Corollary 4.

(Approximation of coverage probability): under Assumption 1 and Assumption 2 the tight upper bound for the probability of coverage is shown as below.

(31)

where

Proof:

Same with Corollary 2, but only LOS intra-cluster interferences exist. ∎

Corollary 5.

(Closed-form lower bound): Under Assumption 1 and Assumption 2, Theorem 2 has a lower bound, which is shown as below.

(32)

where , .

Proof:

See Appendix D. ∎

Remark 5.

We observe that the coverage probability has a positive correlation with typical directivity gain . On the contrary, it has the inverse correlation with the number of simultaneously transmitting devices and modified expectation gain of antenna . Significantly, the value of should be larger than 2, otherwise, the will be infinite.

Iv-B Closest Distribution Model

Closest distribution model allows the typical user to access with the nearest transmitter in the typical cluster as we mentioned above. When we exploit the limited user s CSI which only contains the location information of devices, the nearest transmitter can be regarded as the energy enhancing device in the typical cluster as described in NNNF Selection Scheme of NOMA networks [20]. In order to analyze the performance of this model, we will use Laplace transform to calculate the SINR coverage in the following part.

Iv-B1 Laplace Transform of Interference

The same with uniform distribution model, we first derive the analytical expressions and approximations for intra-cluster interference with the Laplace transform.

Lemma 6.

(Laplace transform of interference in the typical cluster): In this model, the typical link is the distance from the typical user to its nearest intra-transmitter. When , the th conditional Laplace transform of interference in the typical cluster is given by

(33)
Proof:

The proof process is same with (16), but the distribution of simultaneous transmitters in the typical user is different. In this case, they follows (12) and the range is . ∎

Iv-B2 Coverage Probability

We use the same method as discussed in the uniform distribution model, the coverage probability is as follows

(34)
Theorem 3.

Using Laplace transform of intra/inter-cluster interference discussed above in Lemma 6 and Lemma 4, we figure out the tight upper bound for the coverage probability in closest distribution model.

(35)

where

(36)