The ever-increasing requirements of Internet-enabled applications and services have exhaustively strained the capacity of conventional cellular networks. One promising technology for augmenting the throughput of the fifth generation (5G) wireless systems is exploiting new spectrum resources, e.g. millimeter wave (mmWave) [2, 3, 4, 5, 6]. Recently, the mmWave band from 30 GHz to 300 GHz has been applied in numerous commercial scenarios to enhance the network capacity, such as local area networking , personal area networking  and fixed-point access links . In contrast to the traditional sub-6 GHz communications, mmWave has two distinguishing properties . One is the sensitivity to blockage effects, which dramatically increases the penetration loss for mmWave signals . As a result, the path loss of non-line-of-sight (NLOS) transmissions is much more severe than that of line-of-sight (LOS) links [12, 13]. The other feature of mmWave networks is the small wavelength, which shortens the size of antenna elements so that large antenna arrays can be employed at devices for enhancing the directional array gain [11, 10]. This property significantly reduces the path loss, inter-cell interferences, noise power and thus improving the system throughput .
Accordingly, several works have paid attention to these two distinctive features when analyzing mmWave networks. The primary article  proposed a directional beamforming model with a simplified path loss pattern to analyze the mmWave communications. Then, authors in  optimized the path loss model by a stochastic blockage scheme. However, the antenna pattern in this work was over-simplified such that it failed to depict the exact properties of a practical antenna, for example, the front-back ratio, beamwidth, and nulls . Then, a realistic antenna pattern was introduced in . To capture the randomness of networks, stochastic geometry has been widely applied in numerous studies [10, 13, 18, 15]. More specifically, the locations of base stations (BSs) follow a Poisson Point Process (PPP). Since mmWave is able to support ultra-high throughput in short-distance communications , a recent work  considered a Poisson Cluster Process (PCP) instead of PPP to evaluate short-range mmWave networks, which obtains a close characterization of the real world.
In addition to expanding the available spectrum range, another significant objective of 5G cellular networks is improving the spectral efficiency . Lately, non-orthogonal multiple access (NOMA) has kindled the attention of academia since it realizes multiple access in the power domain rather than the traditional frequency domain . The main merit of such approach is that NOMA possesses a perfect balance between coverage fairness and universal throughput . In contrast to the conventional orthogonal multiple access (OMA), the successive interference cancellation (SIC) is applied at near NOMA users, which have robust channel conditions [21, 23]. The detailed process is that the receiver with SIC first subtract the partner’s information from the received signal and then decode its own message . Since NOMA users are capable of sharing same frequency resource at the same time, numerous advantages are proposed in recent works, such as improving the edge throughput, decreasing the latency and strengthening the connectivity [25, 26, 27, 28, 29].
Currently, extensive articles related to NOMA have been published [30, 31, 28, 32, 33, 29]. Firstly, the power allocation strategies for NOMA networks were introduced in  to assure the fairness for all users. Then, in a single cell scenario, the physical layer security was studied in , the downlink sum-rate and outage probability were analyzed in , and the uplink NOMA performance with a power back-off method was investigated in . However, the aforementioned articles focus on the noise-limited system and inter-cell interference is ignored for tractability of the analysis. In fact, such interference is an important factor when studying the coverage performance, especially in the sub-6 GHz networks. The authors in  offered a dense multiple cell network with the aid of applying NOMA techniques. Under this model, both uplink and downlink transmissions were evaluated. Regarding the mmWave networks with NOMA, since acquiring the complete channel state information (CSI) is complicated, two recent works [34, 35] focused on a random beamforming method without considering the locations of users. Then, the beamforming strategy and power allocation coefficients were jointly optimized in  and  for maximizing the system throughput. In addition to the channel gain as studied in [36, 37, 35], the distance-dependent path loss is also an important parameter for the received signal power. Therefore, it also affects the power allocation in NOMA. Note that stochastic geometry is able to characterize all communication distances between transceivers by providing a spatial framework. Like mmWave communications, stochastic geometry has also been utilized in NOMA networks [33, 29] to model the locations of primary and secondary NOMA receivers.
I-a Motivation and Contribution
As mentioned earlier, although mmWave obtains a large amount of free spectrum, the unparalleled explosion of Internet-enabled services, especially for augmented reality (AR) and virtual reality (VR) services, will drain off such bandwidth resource. Introducing NOMA to mmWave networks is an ideal way to further improve the spectrum efficiency. In addition, in dense networks with a large number of users, the combination of mmWave communications with NOMA is capable of providing massive connectivity and high system throughput. Therefore, we are interested in the average performance of NOMA-enabled mmWave networks with multiple small cells111The mmWave network mentioned in this paper refer to the multi-cell network with a content-centric nature, e.g., Internet of Things (IoT) networks with central controllers, multi-cell sensor networks with central BSs, and so forth.. With the aid of the PCP as discussed in [38, 13], we proposed a spatial framework to evaluate the effect of communication distances under three general user selection schemes. An actual antenna array pattern  is also applied to enhance the analytical accuracy. The main contributions of this work are as follows:
We consider the coverage performance and system throughput for proposed clustered mmWave networks with NOMA under three distinctive scenarios: 1) Fixed Near User and Random Far User (FNRF) Scheme, where near user is pre-decided and far user is selected randomly from the remaining farther intra-cluster users; and 2) Random Near User and Fixed Far User (RNFF) Scheme, where far user is pre-decided and near user is chosen at random from the rest possible closer NOMA receivers; and 3) Fixed Near User and Fixed Far User (FNFF) Scheme, where both near user and far user are pre-decided.
We characterize the distance distributions for both intra-cluster NOMA users and inter-cluster interfering BSs. With the aid of Rayleigh distribution, we propose a ranked-distance distribution. Based on such distribution, the exact probability density functions (PDFs) of intra/inter-cluster distances under three distance-dependent user selection schemes are deduced.
We derive Laplace transform of interferences to simplify the notation of analysis. Then, different coverage probability and system throughput expressions for three scenarios are figured out based on proposed distance distributions. Specifically, closed-form approximations are derived under a sparse network assumption. It analytically shows that small antenna scale and massive noise power ruin the coverage performance of near user. Moreover, the equation of system rate for traditional OMA is also provided for comparison.
We demonstrate that: 1) the proposed mmWave networks with NOMA achieves higher system throughput than traditional mmWave networks with OMA and NOMA-enabled mmWave networks with the random beamforming; 2) NLOS signals can be ignored in our system due to the severe path loss; 3) when considering the coverage, 73 GHz is the best choice for near user, while 28 GHz is the best for far user; and 5) there is an optimal number of antenna elements to achieve the maximum system rate.
The rest of this paper is organized as follows: In Section II, we introduce our network model, in which the NOMA users follow a PCP and all BSs are located in the center of clusters. In Section III, the distance distributions for intra/inter cluster transceivers are analyzed based on the Rayleigh distribution. In Section IV, we derive novel theoretical expressions for the coverage probability and system throughput. In Section V, Monte Carlo simulations and numerical results are discussed for validating the analysis and offering further insights. In Section VI, our conclusions and future work are proposed.
Ii Network Model
Ii-a Spatial Model
As shown in Fig. 1, we consider the downlink of a clustered mmWave network with NOMA. The locations of all transceivers are modeled with the aid of one typical PCP, which is a tractable variant of Thomas cluster process222Compared with Matern cluster process, Thomas cluster process is more suitable to model the outdoor scenarios as all clusters in such process have no geographical boundary. . Regarding the proposed PCP, it is a two-step point process. Firstly, parent points are distributed following a homogeneous Poisson Point Process (HPPP) with density
. More specifically, every parent point is uniformly distributed in the considered areaand the number of parent points obeys , where is the probability function . Secondly, the offspring points around one parent point at
are independent and identically distributed (i.i.d.) following symmetric normal distributions with varianceand mean zero. These offspring points form a cluster, which can be denoted by . Noted that the parent points are not included in this point process. Therefore, the entire set of points in the PCP can be expressed as follows :
In our spatial model, the locations of BSs and users are modeled by the parents points and the offspring points , respectively. Based on this assumption, the distance from one user at to the central BS at
follows a two-dimensional Gaussian distribution and its probability density function is given by
Due to the content diversity, we assume that the users in each cluster have same requests and they are served by the central BS. In order to satisfy the pairing requirement of NOMA techniques, the number of intra-cluster users is fixed as , namely . All BSs serve one pair of users at each time slot333We study the two-user pairing scenario in this paper. Other pairing schemes for more than two users can be extended from this work.. As a result, there is no mutual interference among all pairs of users in each cluster, but the inter-cluster interference from other BSs still exists. To ensure the generality, a typical BS is randomly chosen to be located at the origin of the considered plane. The corresponding cluster is the typical cluster.
In this paper, we focus on a typical pair of users from the typical cluster, where the paired User and User represent near user and far user, respectively. To analyze the performance of proposed networks, we introduce three user selection strategies for comparison which are as follows: 1) FNRF Scheme, where User is the -th nearest receiver to the typical BS and User is randomly chosen from the rest farther NOMA users in the typical cluster; 2) RNFF Scheme, where User is the -th nearest receiver to the typical BS and User is randomly chosen from the rest nearer NOMA users in the typical cluster; and 3) FNFF Scheme, where User and User are pre-decided and .
Ii-B Blockage Effects
One remarkable characteristic of mmWave networks is that it is sensitive to be blocked by obstacles. Therefore, line-of-sight (LOS) links have a distinctive path loss law with non-line-of-sight (NLOS) transmissions. Note that each cluster can be visualized as a dense mmWave network due to the small variance of NOMA users. Under this condition, one obstacle may block all receivers behind it, so we adopt the LOS disc to model the blockage effect [10, 41]. This blockage model fits the practical scenarios better than other patterns , especially for most urban scenarios with high buildings. Accordingly, the LOS probability inside the LOS disc with a radius is one, while the NLOS probability outside the disc is one. With the aid of such model, we provide the path loss law of our proposed networks with a distance as follows
where is the intercept and is the path loss exponent. and represent the LOS and NLOS links, respectively. is the unit step function.
Ii-C Uniform Linear Array
The channel model of mmWave is significantly different from the sub-6GHz networks due to the high free-space path loss. We adopt a popular model proposed in , where each BS employs the uniform linear array (ULA) antenna with
elements. However, an omnidirectional antenna pattern is considered at NOMA users for simplifying the analysis. Hence the channel vector of mmWave signals from the BS to Usercan be expressed as
where is a vector and is the number of multi-path. For -th path, is the complex small-scale fading gain and is the spatial angle-of-departure (AoD). Due to the highly directional beamforming and quasi-optical property of mmWave signals. we assume in this paper, then the index can be dropped. For mmWave communications, follows independent Nakagami- fading . The is the transmit array response vector, which is expressed as follows:
where is uniformly distributed over , and is the antenna index. Here, denotes the spacing among antennas, denotes the wavelength, and denotes the physical AoD. In this paper, we consider , namely a critically sampled environment.
Ii-D Analog Beamforming
Another constraint for mmWave networks is the high cost and power consumption for signal processing components. We adopt analog beamforming in this work for achieving a low complexity beamforming design. More particularly, the directions of beams are controlled by phase shifters. We invoke the optimal analog precoding which implies that the BSs try to align the direction of beams with the AoD of channels. Hence high beamforming gains can be obtained. In our system, we assume User is the primary user which requires higher quality of the service than User . Therefore, the main beam direction of the typical BS is towards User . The optimal analog vector for User can be expressed as
Then based on this precoding design, the effective channel gain at User aligning with the optimal analog beamforming is given by
Regarding any other User , the effective channel gain is as follows
where denotes the normalized Fejér kernel with parameter . Note that has a period of two. Therefore, is uniformly distributed over .
Ii-E Signal Model
We assume that in the typical cluster, the typical BS is located at . Then, User located at and User located at are paired and served by the same beam. The distances of them obey . Moreover, the power allocation coefficients satisfy the conditions that and , which is for fairness considerations . In terms of other clusters, the interfering BS located at provides an optimal analog beamforming for User , which is chosen uniformly at random. As a consequence, the received signal is given by
where represents the channel vector from BS at to User and .
We assume that perfect SIC is carried out at User , and hence User first decodes the signal of User with the following signal-to-interference-plus-noise-ratio (SINR)
where . is the noise power normalized by .
If this decoding is successful, User then decodes the signal of itself. Based on (II-E), the SINR of User to decode its own message can be expressed as
Regarding User , it directly decodes its own message by treating the signal of User as the interference. Based on (II-E), the SINR of User is given by
Iii Distance Distributions
In this section, we discuss the distance distribution of NOMA users and BSs, which is the basis for analyzing the performance of our system. To simplify the notation, we first introduce a typical distribution named Rayleigh Distribution in the following part [13, 38].
Under Rayleigh Distribution, the PDF is given by
and the cumulative distribution function (CDF) is as follows
where is the variance parameter as mentioned in (2).
Iii-a Distribution in FNRF Scheme
Under FNRF scheme, we start the analysis of intra-cluster distances from the typical BS to all NOMA users, and then inter-cluster distances from other BSs to the considered NOMA user.
Iii-A1 Distance Distribution of Near User
In the typical cluster, we assume that the distances between NOMA users and the typical BS form a set which can be denoted by . The realization of is defined as , where . Note that
is i.i.d. as a Gaussian random variable with. If the considered NOMA user is selected at random, we are able to drop the index from since every follows the same distribution. Under this condition, is a Gaussian random variable with variance , so the PDF of distance is as follows 
Compared with the aforementioned randomly choosing case, we are more interested in the ordered distance distribution due to the fact that User is always closer to the typical BS than User . Accordingly, we assume that is the distance rank parameter. In other words, the first nearest NOMA user is located at , the second nearest one is located at , and so forth. Assuming the -th closest NOMA user at has a distance to the typical BS, with the aid of the -th order statistic in , the PDF of distance in the typical cluster is given by
Based on the discussion in (III-A1), it is effortless to derive the PDF of near user distance under the FNRF strategy.
Iii-A2 Distance Distribution of Far User
In contrast to near user, far user in the FNRF scheme is randomly chosen from the rest farther NOMA users in the typical cluster. Assuming the possible User is located at with a distance , the distribution of distance is expressed in the following lemma.
The randomly selected far user in the FNRF scheme at has a distance to the typical BS and , so the conditional PDF of distance is given by
When , the probability is zero as far user is defined to be located farther than near user with a distance . Under the other condition , the possible User follows Rayleigh distribution over the rang . Therefore, such distance distribution can be summarized in Lemma 1. ∎
Iii-A3 Distance Distribution of Interfering BSs
The distance distribution of interfering BSs can be deduced from probability generating functional of PPP . The detailed deriving procedure is provided in the next section.
Since the typical pair of users are located in the typical cluster, the distance distribution of interfering BSs is same for all considered user selection strategies and thus we omit the analysis of such distribution in the other scheme.
Iii-B Distribution in RNFF Scheme
Under the RNFF scheme, we focus on the distribution of intra-cluster distances. Both near user and far user have different distributions with those in the FNRF scheme. We first analyze the far user and then the near user.
Iii-B1 Distance Distribution of Far User
The location of considered far user is assumed to be with a distance . Since far user becomes the -th nearest intra-cluster NOMA user, the distribution of distance can be expressed in the following part.
The considered far user under RNFF scheme is the -th closest NOMA receiver located at with a distance and . Therefore the PDF of distance is given by
The proof procedure is similar to Corollary 1, but with the different condition that . ∎
Iii-B2 Distance Distribution of near User
Near user under the RNFF scheme is randomly selected from the remaining closer NOMA users in the typical cluster. We assume the considered near user is located at with a distance . Under this condition, the distance distribution of such near user can be calculated in the following lemma.
The randomly chosen User under RNFF scheme at has a distance to the typical BS, so the PDF of distance is expressed as follows
The proof is similar to Lemma 1 and thus we skip it here. ∎
Iii-C Distribution in FNFF Scheme
Iv Performance Evaluation
In this section, we characterize the coverage performance and system throughput of three different user selection strategies depending on the distributions of intra/inter-cluster distances.
Iv-a FNRF Scheme
The FNRF scheme is suitable for the condition that the primary user (User ) is pre-decided. To enhance the generality, User can be any user in the typical cluster. On the other side, far user (User ) is selected at random from the rest farther NOMA users to provide a fair selection law. All possible far users have the equal opportunity to be the paired one. Moreover, such random selection strategy do not require the instantaneous CSI of User . To make the tractable analysis, we first deduce the Laplace Transform of Interferences in the following part.
Iv-A1 Laplace Transform of Interferences
We only concentrate on the Laplace transform of inter-cluster interferences because there is no interfering device located in the typical cluster. Moreover, the expression is suitable for all user selection strategies due to the fact mentioned in Remark 1.
The inter-cluster interferences are provided from all BSs except the typical BS, then a closed-form approximation for the Laplace transform of such interferences is given by
is Gauss hypergeometric function. over denotes the Gauss-Chebyshev node and . The parameter has a function to balance the complexity and accuracy . Only if the , the equality is established.
See Appendix A. ∎
For most mmWave carrier frequencies, the path loss exponent of LOS communications equals two, namely , which has been proved by several actual channel measures [45, 46, 47]. In terms of the NLOS interferences, numerous papers [10, 48] have indicated that NLOS signals are weak enough to be ignored in mmWave communications. Therefore, we propose the first special case blew to simplify the calculation.
Special Case 1: When deriving the Laplace transform of interference, we ignore all NLOS interferences due to the negligible impact on the final performance and is assumed to be .
Under special case 1, the tight approximation for Laplace transform of inter-cluster interferences in Lemma 4 can be simplified as follows
Iv-A2 Coverage Probability for Near User
We introduce two SINR thresholds and for User and User , respectively. These thresholds should satisfy the condition to ensure the success of NOMA protocols . Since near user has the SIC procedure, the decoding for User will be success only when . If this condition is satisfied, the coverage probability for near user is the percentage of the received SINR that excess . Therefore, the coverage probability for User under the FNRF scheme can be defined as follows
With the aid of Laplace transform of interferences as discussed in Lemma 4, the expression for coverage probability is shown in the following theorem.
With different value of thresholds and , the coverage probability for User can be divided into two cases. Firstly, for Range 1 : , the expression under the FNRF scheme is given by
On the other hand, for Range 2 : , the coverage probability is changed to
See Appendix B. ∎
It is obvious that if the realistic scenario fits the condition , the coverage probability for and will share the same expression.
Under special case 1, a simpler expression than Theorem 1 is given by
where means using to replace .
In the reality, the coverage radius of the macro BS is always larger than , which means the majority of BSs communicate with the considered user via NLOS links. Note that the received power from NLOS signals is negligible. We propose the second special case.
Special Case 2: In a sparse network, the density of BSs is small enough to ensure that the majority of BSs utilize NLOS links to provide the inter-cluster interferences. Together with the fact that the impact of NLOS signals is tiny, we ignore all inter-cluster interferences and the coverage probability from NLOS links, namely, and . Moreover, we keep assuming as discussed in special case 1.
As NOMA users are randomly distributed in the typical cluster, each of them has an opportunity to communicate with the typical BS through an NLOS link. To ensure the considered number of intra-cluster users is fixed as , we should take every LOS and NLOS NOMA receivers into account when calculating the coverage. Therefore does not indicate that we only consider NOMA receivers with LOS links. It actually means the received SINR at all NOMA users with NLOS links fails to surpass the required threshold.
Under special case 2, the closed-form coverage probability for near user is at the top of next page.
In (34), , , and .
The coverage probability for all users under special case 2 is independent with since such density is only contained in .
With the aid of Corollary 3, we are able to conclude that the coverage probability for near user is a monotonic increasing function with , while it has a negative correlation with and its corresponding threshold. Moreover, for , has a positive correlation with and for , increases with the rise of . These insights can be figured out from (3), which can be rewritten as follows:
where for the range , , while for the range , .
Iv-A3 Coverage Probability for Far User
In contrast to the near user, the coverage probability for User at only depends on . However, as the directional beamforming of the typical BS is aligned towards User , the effective channel gain for User fits (II-D) rather than (7). Note that far user is randomly selected from the farther intra-cluster NOMA receivers. We define the coverage probability for far user as follows
As discussed in Lemma 2 and Laplace transform of interferences, we obtain the coverage probability expression for User in the following theorem.
Under the FNRF scheme, the coverage probability for User at with a distance is given by
See Appendix C. ∎
Under special case 1, the simpler expression than Theorem 2 is shown as follows
The proof procedure is similar to Corollary 2 and thus we omit it here. ∎
Under special case 2, in a sparse network, the closed-form coverage probability for far user is given by
and and .
The coverage probability for far user has the same features with near user as mentioned in Remark 5. The only difference is the relationship to . Corollary 5 demonstrates that the value of is decided by which is fluctuant with the increase of . Such monotonic increasing relation with for near user will not exist in the far user scenario.
Iv-B RNFF Scheme
Comparing with the FNRF scheme, the RNFF strategy focuses on a certain far user which requires continuous services. In this scheme, User is the -th nearest user to the typical BS and User is randomly selected from the rest closer intra-cluster NOMA receivers.
Iv-B1 Coverage Probability for Near User
Under the RNFF scheme, the coverage probability for User with the thresholds and is defined as follows.
As the distance distribution of near user is dependent on the distance of far user , the coverage probability can be expressed in the following part.
Same with FNRF scheme, the coverage probability of near user in the RNFF scheme can be divided into two ranges and and it is given at the top of next page.