I Introduction
Ultradense deployment of small cell base stations (BSs), relay nodes, and distributed antennas is considered as a de facto solution for realizing the significant performance improvements needed to accommodate the overwhelming future mobile traffic demand [1]. Traditional network expansion techniques like cell splitting are often utilized by telecom operators to achieve the expected throughput, which is less efficient and proven not to keep up with the pace of traffic proliferation in the near future. Heterogeneous networks (HetNets) then become a promising and attractive network architecture to alleviate the problem. “HetNets” is a broad term that refers to the coexistence of different networks (e.g., traditional macrocells and small cell networks like femtocells and picocells), each of them constituting a network tier. Due to differences in deployment, BSs in different tiers may have different transmit powers, radio access technologies, fading environments and spatial densities. HetNets are envisioned to change the existing network architectures and have been introduced in the LTEAdvanced standardization [2, 3].
Massive work has been done in HetNets scenario mainly related to cell association scheme [4, 5, 6], cacheenabled networks [7], physical layer security [8], etc. In [4], the pertinent user association algorithms designed for HetNets, massive MIMO networks, mmWave scenarios and energy harvesting networks have been surveyed for the future fifth generation (5G) networks. Bethanabhotla et al. [5] investigated the optimal usercell association problem for massive MIMO HetNets and illustrated how massive MIMO could also provide nontrivial advantages at the system level. The joint downlink cell association and wireless backhaul bandwidth allocation in a twotier HetNet is studied in [6]. In [7], Yang et al. aimed to model and evaluate the performance of the wireless HetNet where the radio access network (RAN) caching and devicetodevice (D2D) caching coexist. The physical layer security of HetNets where the locations of all BSs, mobile users (MUs) and eavesdroppers are modeled as independent homogeneous PPPs in [8].
From the mobile operators point of view, the commercial viability of network densification depends on the underlying capital and operational expenditure [9]. While the former cost may be covered by taking up a high volume of customers, with the rapid rise in the price of energy, and given that BSs are particularly powerhungry, energy efficiency (EE) has become an increasingly crucial factor for the success of dense HetNets [10]. Recently, loads of work [11, 12, 13, 14, 15] has investigated the EE in the 5G network scenarios. In [11], Niu et al. investigated the problem of minimizing the energy consumption via optimizing concurrent transmission scheduling and power control for the mmWave backhauling of small cells densely deployed in HetNets. A selforganized crosslayer optimization for enhancing the EE of the D2D communications without creating harmful impact on other tiers by employing a noncooperative game in a threetier HetNet is proposed in [12]. To jointly optimize the EE and video quality, Wu et al. [13] presented an energyquality aware bandwidth aggregation scheme. In [14], Yang et al. investigated the energyefficient resource allocation problem for downlink heterogeneous OFDMA networks. The mobile edge computing offloading mechanisms are studied in 5G HetNets [15].
Different from most prior work analyzing network performance where the propagation path loss between the BSs and the MUs follows the same powerlaw model, in this paper we consider the coexistence of both nonlineofsight (NLoS) and lineofsight (LoS) transmissions, which frequently occur in urban areas. More specifically, for a randomly selected MU, BSs deployed according to a homogeneous Poisson point process (PPP) are divided into two categories, i.e., NLoS BSs and LoS BSs, depending on the distance between BSs and MUs. It is well known that LoS transmission may occur when the distance between a transmitter and a receiver is small, and NLoS transmission is common in office environments and central business districts. Moreover, as the trend of ultradense network deployment, the distance between a transmitter and a receiver decreases, the probability that a LoS path exists between them increases, thereby causing a transition from NLoS transmission to LoS transmission with a higher probability [16]. In this context, Ding et al. [16] studied the coverage and capacity performance by using a multislop path loss model incorporating probabilistic NLoS and LoS transmissions. The coverage and capacity performance in millimeter wave cellular networks are studied in [17, 18, 19]. In [17], a threestate statistical model for each link was assumed, in which a link can either be in an NLoS, LoS or an outage state. In [18], selfbackhauled millimeter wave cellular networks are characterized assuming a cell association scheme based on the smallest path loss. However, both [17] and [18] assume a noiselimited network, ignoring intercell interference, which may not be very practical since modern wireless networks work in the interferencelimited region. In [19], the coverage probability and capacity were calculated in a millimeter wave cellular network based on the smallest path loss cell association model assuming multipath fading modeled as Nakagami fading, respectively. However, shadowing was ignored in their models, which may not be very practical for an ultradense heterogeneous network.
In contrast to prior work, we investigate the HetNets in a more realistic scenario, i.e., NLoS and LoS transmissions in desired signal and interference signal are both considered. Besides, we also explore the optimal BS deployment under the quality of service (QoS) constraint. The main contributions of this paper are summarized as follows:

A unified framework: We propose a unified framework, in which the user association strategies based on the maximum instantaneous received power (MIRP) and the maximum average received power (MARP) can be studied, assuming lognormal shadowing, Rayleigh fading and incorporating probabilistic NLoS and LoS transmissions.

Performance optimization: We formulate two optimization problems under different QoS constraints, i.e., the maximal total power consumption and the minimal coverage probability. Utilizing solutions of the above optimization problems, the maximum energyefficient BS deployment is obtained.

Network design insights: We compare the optimal BS deployment strategies in different network scenarios, i.e., assuming the fixed transmit power, the densitydependent transmit power, with and without considering the static power consumption in BSs. Through our results, the maximum coverage probability with ideal power consumption is superior to that with practical power consumption when the total power constraint is small and inferior to that with practical power consumption when the total power constraint becomes large. Moreover, the maximum EE is a decreasing function with respect to the coverage probability constraint.
The remainder of this paper is organized as follows. Section II introduces the system model, network assumptions, and performance metrics. In section III, the coverage probability, the potential throughput (PT) and the EE of the HetNets are derived with the MIRP and the MARP association schemes, respectively. In Section IV, two optimization problems for energyefficient BS deployment are formulated. In Section V, the analytical results are validated via Monte Carlo simulations. Besides, the insights of BS deployment are studied. Finally, Section VI concludes this paper and discusses possible future work.
Ii System Model
In this paper, a tier HetNet is considered, which consists of macrocells, picocells, femtocells, etc. BSs of each tier are assumed to be spatially distributed on the infinite plane and locations of BSs follow independent homogeneous Poisson point processes (HPPPs) denoted by with a density (aka intensity) , ^{1}^{1}1 means is defined to be another name for ., where denotes the location of BS in the th tier. MUs are deployed according to another independent HPPP denoted by with a density (). BSs belonging to the same tier transmit using the same constant power and sharing the same bandwidth. Besides, within a cell assume that each MU uses orthogonal multiple access method to connect to a serving BS for downlink and uplink transmissions and therefore there is no intracell interference in the analysis of our paper. However, adjacent BSs which are not serving the connected MU may cause intercell interference which is the main focus of this paper. It is further assumed that each MU can possibly associate with a BS belonging to any tier, i.e., open access policy is employed.
Without loss of generality and from the Slivnyak’s Theorem [20], we consider the typical MUwhich is usually assumed to be located at the origin, as the focus of our performance analysis.
Iia Signal Propagation Model
The longdistance signal attenuation in tier
is modeled by a monotone, nonincreasing and continuous path loss function
and decays to zero asymptotically. The fast fading coefficient for the wireless link between a BS and the typical MU is denoted as .are assumed to be random variables which are mutually independent and identically distributed (i.i.d.) and also independent of BS locations
, thus can be denoted as for the sake of simplicity. Similarly, the shadowing is denoted byand particularly assume that it follows a lognormal distribution with zero mean and standard deviation
. Note that the proposed model is general enough to account for various propagation scenarios with fasting fading, shadowing, and different path loss models.To characterize shadowing effect in urban areas which is a unique scenario in our analysis, both NLoS and LoS transmissions are incorporated. That is, if the visual path between a BS and the typical MU is blocked by obstacles like buildings, trees, and even MUs, it is an NLoS transmission. Otherwise it is a LoS transmission. The occurrence of NLoS and LoS transmissions depend on various environmental factors, including geographical structure, distance, and cluster. In this work, a oneparameter distancebased NLoS/LoS transmission probability model is applied. That is,
(1) 
where and denote the probability of the occurrence of NLoS and LoS transmissions, respectively, is the distance between the BS and the typical MU.
Regarding the mathematical form of (or ), Blaunstein et al. [21] formulated as a negative exponential function, i.e., , where is a parameter determined by the density and the mean length of the blockages lying in the visual path between BSs and the typical MU. Bai et al. [22] extended Blaunstein’s work by using random shape theory which shows that is not only determined by the mean length but also the mean width of the blockages. [17] and [19] approximated by using piecewise functions and step functions, respectively. Ding et al. [16] considered to be a linear function and a twopiece exponential function, respectively; both are recommended by the 3GPP. It is important to note that the introduction of NLoS and LoS transmissions is essential to model practical networks, where a MU does not necessarily have to connect to the nearest BS. Instead, for many cases, MUs are associated with farther BSs with stronger signal strength.
It should be noted that the occurrence of NLoS and LoS transmissions is assumed to be independent for different BSMU pairs. Though such assumption might not be entirely realistic (e.g., NLoS transmission caused by a large obstacle may be spatially correlated), Bai et al. [22, 19] showed that the impact of the independence assumption on the SINR analysis is negligible.
For a specific tier , note that from the viewpoint of the typical MU, each BS in the infinite plane is either an NLoS BS or a LoS BS to the typical MU. Accordingly, a thinning procedure on points in the PPP is performed to model the distributions of NLoS BSs and LoS BSs, respectively. That is, each BS in will be kept if a BS has an NLoS transmission with the typical MU, thus forming a new point process denoted by . While BSs in form another point process denoted by , representing the set of BSs with LoS path to the typical MU. As a consequence of the independence assumption of NLoS and LoS transmissions mentioned in the last paragraph, and are two independent nonhomogeneous PPPs with intensity functions and , respectively.
Based on assumptions above, the received power of the typical MU from a BS is defined as follows.
Definitation 1. The received power of the typical MU from a BS , i.e., is
(2) 
where and denote the respective path loss for NLoS and LoS transmissions at the reference distance (usually at 1 meter). For simplicity, denote and let , where the superscript used distinguishes NLoS and LoS transmissions and denotes the path loss exponent for NLoS or LoS transmission in the th tier. Recently, [23] and [24] took bounded path loss model and stretched exponential path loss model into consideration, in which several interesting performance trends are found and will be investigated in our future work.
Remark 1. Apart from the fixed transmit power, a densitydependent transmit power is further assumed and analyzed mentioned in [25], i.e., , where is the radius of an equivalent diskshaped coverage area in the th tier with an area size of and is the per tier SINR threshold.
IiB Cell Association Scheme
Cell association scheme [26] plays a crucial role in network performance determining BS coverage, MU handoff regulation and even facility deployment of small cells. Conventionally, a typical MU is connected to the BS if and only if
(3) 
where is the instantaneous received power with dBm unit from the BS and Eq. (3) is known as the MIRP association scheme.
In practical, is usually averaged out in time and frequency domains to cope with fluctuations caused by channel fading. In this text, a typical MU is connected to the BS if and only if
(4) 
where denotes the average received power with dBm unit from the BS and Eq. (4) is known as the MARP association scheme.
Aided by cell range expansion (CRE), which is realized by MUs adding a positive cell range expansion bias (CREB) to the received power from BSs in different tiers, more MUs can be offloaded to small cells. That is, if a MU is associated with the BS if and only if
(5) 
where and is the CREB with dB unit in the th and th tier. With proper CREB chosen, the coverage of BSs in some tiers is artificially expanded, allowing MUs more flexible to be associated with BSs which may not provide the strongest received power, thus balancing traffic load to achieve spatial efficiency. However, CRE causes severe interference to small cell MU which impair the QoS of small cell users and thus almost blank subframes (ABS) coordination is needed between macrocell BSs and small cell BSs. However, the analysis of CRE plus ABS is challenging because (i) the association scheme is not only determined by the received power but also the current resource allocation strategy, and (ii) ignoring ABS while using CRE can impair the coverage performance. For simplicity, CRE and ABS are not going to be considered in this paper, which are left as our future work.
IiC Performance Metrics
To evaluate the network performance, the following three metrics, i.e., the coverage probability, the PT and the EE, are focused on.
The coverage probability is the probability that the received SINR is greater than a given threshold, i,e, , where is defined as follows
(6) 
where is the Palm point process [27] representing the set of interfering BSs in the th tier and denotes the noise power at the MU side, which is assumed to be the additive white Gaussian noise (AWGN).
The PT is defined as follows [28, 24]
(7) 
where the network is fully loaded due to the assumption that , is the association probability that the typical MU is connected to the th tier, is the conditional association coverage probability and is the pertier coverage probability. Compared with the area spectral efficiency (ASE), which is defined as
(8) 
the PT implicitly assumes a fixed rate transmission from all BSs in the network, and has a unit of , while the ASE assumes full buffers but it allows each link to adapt its rate to the optimal value for a given SINR, thus avoiding outages at low SINR and the wasting of rate at high SINR [24]. In other words, the PT is a more realistic performance metric and the ASE upper bounds the PT. In our analysis, the PT is chosen as our performance metric.
The EE is defined as the ratio between the PT and the total energy consumption of the network, i.e.,
(9) 
where the coefficient accounts for power consumption that scales with the average radiated power, and the term models the static power consumed by signal processing, battery backup and cooling [29]. Other performance metrics, such as the biterror probability and perMU data rate, can be found using the coverage probability (SINR distribution) following the methods mentioned in [30].
Iii Performance Analysis
In this section, we derive expressions for the considered performance metrics and study the effect of densification on these metrics. It is started by introducing the network transformation and then presenting the analytical expressions with the MIRP and MARP association schemes in the following subsections.
Iiia Network Transformation
Before presenting our main analytical results, firstly the network transformation is introduced, which aims to unify the analysis and to reduce the complexity as well.
Using the manipulation in [31, 32], we define
(10) 
and
(11) 
respectively. Then Eq. (2) can be written as
(12) 
By adopting the Equivalence Theorem in [31], it is concluded that the distance (or ) from a scaled point process for NLoS BSs (or LoS BSs), which still remains a PPP denoted by (or ). are mutually independent with each other, and the intensity measures and intensities are provided in Lemma 1 as below.
Lemma 1. The intensity measure and intensity of can be formulated as
(13) 
and
(14) 
respectively, where
(15) 
and
(16) 
Proof:
The proof can be referred to [31, Appendix A] and thus omitted here. Aided by the network transformation and stochastic geometry tool, the coverage probability, the PT and the EE will be derived in the following.
IiiB Coverage Probability with the MIRP Association Scheme
With the MIRP association scheme, the typical MU is associated with the BS which offers the maximum instantaneous received power as shown in Eq. (3). Using this cell association scheme and considering Lemma 1, the general results of coverage probability in the tier HetNets is given by Theorem 1.
Theorem 1. When , the coverage probability for a typical MU with the MIRP association scheme can be derived as
(17) 
where
(18) 
and
(19) 
Proof:
See Appendix A.
In pursuit of the analytical results of the PT and the EE, the NLoS/LoS coverage probability and pertier coverage probability are presented in the following two corollaries.
Corollary 1. When , the coverage probability for a typical MU which is served by NLoS BSs and LoS BSs with the MIRP association scheme are given by
(20) 
and
(21) 
respectively.
Proof:
This corollary can be derived from Theorem 1 by rearranging the terms in Eq. (17) and thus the proof is omitted here.
Corollary 2. When , the pertier coverage probability for a typical MU which is covered by the th tier with the MIRP association scheme is given by
(22) 
Proof:
This corollary can be derived from Theorem 1 by rearranging the terms in Eq. (17) and thus the proof is omitted here.
IiiC Coverage Probability with the MARP Association Scheme
With the MARP association scheme, the typical MU is associated with the BS which offers the maximum longterm averaged received power by averaging out the effect of multipath fading . With this cell association scheme, the primary results of coverage probability is given by Theorem 2.
Theorem 2. The coverage probability for a typical MU with the MARP association scheme is
(23) 
where
(24) 
(25) 
(26) 
and
(27) 
Proof:
See Appendix B.
Remark 2. Note that different from Theorem 1, Theorem 2 can be applied to scenarios without the assumption of a particular range of SINR threshold , e.g., .
Similar to the study for Theorem 1, we provide two corollaries, i.e., the NLoS/LoS coverage probability and the pertier coverage probability, as follows.
Corollary 3. The coverage probability for a typical MU which is served by NLoS BSs and LoS BSs with the MARP association scheme are given by
(28) 
and
(29) 
respectively.
Proof:
This corollary can be derived from Theorem 2 by rearranging the terms in Eq. (23) and thus the proof is omitted here.
Corollary 4. The pertier coverage probability for a typical MU which is covered by the th tier with the MARP association scheme is given by
(30) 
Proof:
This corollary can be derived from Theorem 2 by rearranging the terms in Eq. (23) and thus the proof is omitted here.
Intuitively, the coverage probability with the MIRP association scheme is higher than that with the MARP association scheme. However, it can be proved mathematically which is summarized in the following corollary.
Corollary 5. In the studied tier HetNet, the coverage probability with the MIRP association scheme is higher than that with the MARP association scheme, where the gap is determined by the intensity and the intensity measure.
Proof:
See Appendix C.
IiiD The PT and the EE
As the results with the MIRP and the MARP association schemes are some kind of similar and the MARP association scheme is more practical in the real network, we take the MARP association scheme as an example to evaluate the PT and the EE in the following. The PT with the MARP association scheme can be directly obtained from the coverage probability expressions using Eq. (7), i.e.,
(31) 
While the PT with the MIRP association scheme is similar except for replacing by .
The EE can be derived by using Eq. (9) and we will only provide expressions for it when necessary.
Iv Performance Optimization and Tradeoff
As mentioned, from the mobile operators’ point of view, the commercial viability of network densification depends on the underlying capital and operational expenditure [9]. While the former cost may be covered by taking up a high volume of customers, with the rapid rise in the price of energy, and given that BSs are particularly powerhungry, EE has become an increasingly crucial factor for the success of dense HetNets [10]. There are two main approaches to enhance the energy consumption of cellular networks: 1) improvement in hardware and 2) energyefficient system design. The improvement in hardware may have achieved its bottleneck due to the limit of Moore’s law, while the energyefficient system design has a great potential in the future 5G networks. In the following, two energyefficient optimization problems are proposed trying to obtain insights of the system design.
Iva Optimizing coverage probability with the maximum total power consumption constraint
To pursue a further study on coverage performance, we formulate a theoretical framework which determines the optimal BS density to maximize the coverage probability while guaranteeing that the total area power consumption is lower than a given expected value as follows
s. t.  (32)  
where and are defined in Eq. (9). Note that assumes the MARP association scheme, while the optimization problem with the MIRP association scheme is similar to and omitted here for brevity.
IvB Optimizing the EE under the minimum coverage probability constraint
In this subsection, another framework are formulated which determines the optimal BS density to maximize the EE while guaranteeing QoS of the network, i.e., the coverage probability is higher than a given expected value as follows
s. t.  (33)  
We will show in the simulation results that tradeoff exists between the coverage probability and the EE.
IvC Optimal deployment solution
As NLoS and LoS transmissions are incorporated into our model, the coverage probability is not a monotonically increasing function with respect to BS density like the cases in [33, 29, 10, 34] anymore. Besides, the coverage probability function is not convex with respect to , either. Therefore, the optimization problem under consideration should be tackled numerically. Exhaustive search algorithms are wellsuited for tackling the problem considering that the objective function derivative is not available analytically and its accurate evaluation is resourceintensive. Brent’s algorithm [35]
and heuristic downhill simplex method
[36] can be utilized to obtain the solutions of and in exponential time. To gain an analytical insight into the effect of different operational settings on the maximum energyefficient deployment solution, in the following, we focus on the problem of finding the optimal BS density in a 2tier HetNet.V Results and Insights
A 2tier HetNet is considered in our analysis. Macrocell BSs are in Tier 1 and small cell BSs are in Tier 2. We assume that , , , , , , , , , , , , , , [19, 18, 37, 31, 38, 39] unless stated otherwise.
Va Validation of the Analytical Results of Coverage Probability with Monte Carlo Simulations
If fixing , the analytical and simulation results of and the analytical results of configured with are plotted in Fig. 1 and Fig. 2, respectively. As can be observed from Fig. 1, the analytical results match the simulation results well, which validate the accuracy of our theoretical analysis. In Fig. 2, aided by the utilization of a densitydependent BS transmit power, the coverage probability improves a lot as increases.
,,
Fig. 3 and Fig. 4 illustrate the coverage probability vs. the ratio of and , i.e., with the MIRP association scheme and the MIRP association scheme when (or ) is fixed. It is found that in Fig. 3, there is always a coverage peak when is low, medium and high, i.e., (or 0.3725 with the MIRP), (or 0.7868 with the MIRP), (or 0.7476 with the MIRP), which indicates that there exists an optimal when implementing the network design if is fixed. And in Fig. 4, the optimal exists as well. However, compared with Fig. 3, when the fixed value of is sparse, the coverage probability firstly increases and then reaches a peak. Finally it decreases to a certain value. When the fixed value of becomes larger, the coverage probability saturates. Based on the above observations, dense deployment of small cell BSs and macrocell BSs will lead to a better coverage probability. However, there is no need to deploy an infinite number of BSs in a finite area. When approaches infinite if remains fixed, and vice versa, the coverage probability becomes much worse. In contrast, when goes to zero if is fixed, and vice versa, the coverage probability saturates to a certain value.
To have a full picture of the coverage probability with respect to and , present two 3D figures are presented in Fig. 5 and Fig. 6. In Fig. 5, we compare the MIRP and MARP association schemes based on the fixed transmit power. It is found that the coverage probability with the MIRP association scheme is always greater than that with the MARP association scheme as with former association scheme BSs can provide the maximum power all the time even though it is not practical in the real networks. In Fig. 6, coverage probability based on the fixed transmit power and densitydependent transmit power are illustrated, respectively. By utilizing a densitydependent transmit power, the coverage probability improves compared with the HetNets using a fixed transmit power. Besides, it is noted that the coverage probability using a densitydependent transmit power fluctuates with BS density as illustrated in Fig. 6 as well as in Fig. 2. It is because the imperfect power control used in Remark 1 which only depends on BS densities and an approximate equivalent coverage area, the 3D coverage probability appears more unique than that using a fixed transmit power.
VB The PT and the EE
In this subsection, two typical energy consumption scenarios are considered, i.e., practical power consumption and ideal power consumption, denoted by S1 and S2. Recall that the definition of the EE in Eq. (9) have parameters and , thus we define S1 as the HetNets which are configured with [29] and S2 configured with , respectively. Note that S2 accounts for the HetNets with perfect power amplifier and ignoring the static power consumed by signal processing, battery backup, and cooling, etc. In other words, in S2 only radiated power is considered. It is observed that has a greater impact on the PT than in Fig. 7. However, a larger can not always provide a better EE as illustrated in Fig. 8 and Fig. 9. Therefore, there should exist a tradeoff among coverage probability, the PT and the EE, which is revealed in the following subsection.
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