# On the Energy-Efficient Deployment for Ultra-Dense Heterogeneous Networks with NLoS and LoS Transmissions

We investigate network performance of ultra-dense heterogeneous networks (HetNets) and study the maximum energy-efficient base station (BS) deployment incorporating probabilistic non-line-of-sight (NLoS) and line-of-sight (LoS) transmissions. First, we develop an analytical framework with the maximum instantaneous received power (MIRP) and the maximum average received power (MARP) association schemes to model the coverage probability and related performance metrics, e.g., the potential throughput (PT) and the energy efficiency (EE). Second, we formulate two optimization problems to achieve the maximum energy-efficient deployment solution with specific service criteria. Simulation results show that there are tradeoffs among the coverage probability, the total power consumption, and the EE. To be specific, 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.

## Authors

• 92 publications
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• ### Joint User Association and BS Switching Scheme for Green Heterogeneous Cellular Network

A dense deployment of on-grid small cell base stations (SBSs) in a heter...
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• ### Base Station Coordination Scheme for Multi-tier Ultra-dense Networks

In this paper, we consider a relative received link power (RRLP)-based c...
12/18/2020 ∙ by Sudarshan Mukherjee, et al. ∙ 0

• ### Base Station Switch-off with Mutual Repulsion in 5G Massive MIMO Networks

When small cells are densely deployed in the fifth generation (5G) cellu...
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• ### Energy Efficient Optimization of Wireless-powered 5G Full Duplex Cellular Networks: A Mean Field Game Approach

This paper studies the power allocation of an ultra-dense cellular netwo...
03/07/2019 ∙ by Xiaohu Ge, et al. ∙ 0

• ### On the Effective Energy Efficiency of Ultra-reliable Networks in the Finite Blocklength Regime

Effective Capacity (EC) indicates the maximum communication rate subject...
01/29/2018 ∙ by Mohammad Shehab, et al. ∙ 0

• ### Self-energy recycling for low-power reliable networks: Half-duplex or Full-duplex?

Self-energy recycling (sER), which allows transmit energy re-utilization...
11/12/2021 ∙ by Dian Echevarría Pérez, et al. ∙ 0

• ### A Lightweight Cell Switching and Traffic Offloading Scheme for Energy Optimization in Ultra-Dense Heterogeneous Networks

One of the major capacity boosters for 5G networks is the deployment of ...
09/16/2021 ∙ by Attai Ibrahim Abubakar, et al. ∙ 0

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## I Introduction

Ultra-dense 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 LTE-Advanced standardization [2, 3].

Massive work has been done in HetNets scenario mainly related to cell association scheme [4, 5, 6], cache-enabled 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 user-cell 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 two-tier 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 device-to-device (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 power-hungry, 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 self-organized cross-layer optimization for enhancing the EE of the D2D communications without creating harmful impact on other tiers by employing a non-cooperative game in a three-tier HetNet is proposed in [12]. To jointly optimize the EE and video quality, Wu et al. [13] presented an energy-quality aware bandwidth aggregation scheme. In [14], Yang et al. investigated the energy-efficient 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 power-law model, in this paper we consider the co-existence of both non-line-of-sight (NLoS) and line-of-sight (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 ultra-dense 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 multi-slop 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 three-state statistical model for each link was assumed, in which a link can either be in an NLoS, LoS or an outage state. In [18], self-backhauled millimeter wave cellular networks are characterized assuming a cell association scheme based on the smallest path loss. However, both [17] and [18] assume a noise-limited network, ignoring inter-cell interference, which may not be very practical since modern wireless networks work in the interference-limited 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 multi-path fading modeled as Nakagami- fading, respectively. However, shadowing was ignored in their models, which may not be very practical for an ultra-dense 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:

1. 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 log-normal shadowing, Rayleigh fading and incorporating probabilistic NLoS and LoS transmissions.

2. 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 energy-efficient BS deployment is obtained.

3. Network design insights: We compare the optimal BS deployment strategies in different network scenarios, i.e., assuming the fixed transmit power, the density-dependent 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 energy-efficient 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) , 111 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 intra-cell interference in the analysis of our paper. However, adjacent BSs which are not serving the connected MU may cause inter-cell 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.

### Ii-a Signal Propagation Model

The long-distance signal attenuation in tier

is modeled by a monotone, non-increasing 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 by

and particularly assume that it follows a log-normal 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 one-parameter distance-based NLoS/LoS transmission probability model is applied. That is,

 pNLk(∥Xk,i∥)+pLk(∥Xk,i∥)=1, (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 piece-wise functions and step functions, respectively. Ding et al. [16] considered to be a linear function and a two-piece 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 BS-MU 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 non-homogeneous 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 density-dependent transmit power is further assumed and analyzed mentioned in [25], i.e., , where is the radius of an equivalent disk-shaped coverage area in the -th tier with an area size of and is the per tier SINR threshold.

### Ii-B Cell Association Scheme

Cell association scheme [26] plays a crucial role in network performance determining BS coverage, MU hand-off regulation and even facility deployment of small cells. Conventionally, a typical MU is connected to the BS if and only if

 PdBmk,m>PdBmj,n,j≠k, (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

 ¯¯¯¯¯¯¯¯¯¯¯PdBmk,m>¯¯¯¯¯¯¯¯¯¯PdBmj,n,j≠k, (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

 ¯¯¯¯¯¯¯¯¯¯¯PdBmk,m+△%dBk,m>¯¯¯¯¯¯¯¯¯¯PdBmj,n+△dBj.n,j≠k, (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.

### Ii-C 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

 SINRk(∥Xk,i∥)=PkAUkhUkgUklUk(∥Xk,i∥)[k=1]K∑∑Xk,j∈Φk∖Xk,iPkAUkhUkgUklUk(∥∥Xk,j∥∥)+η, (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]

 T({λk},{Tk},{BUk})=[k=1]K∑λkAkpcondcov,klog2(1+Tk)=[k=1]K∑λkpcov,klog2(1+Tk), (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 per-tier coverage probability. Compared with the area spectral efficiency (ASE), which is defined as

 ASE({λk},{Tk},{BUk})=[k=1]K∑E[λklog2(1+SINRk(∥Xk,i∥))], (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.,

 E({λk},{Tk},{BUk})=T({λk},{Tk},{BUk})[k=1]K∑λk(akPk+bk), (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 bit-error probability and per-MU 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.

### Iii-a 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

 ¯¯¯¯¯¯¯¯¯RLk,i=∥Xk,i∥⋅(BLkgLk)−1/αLk, (11)

respectively. Then Eq. (2) can be written as

 Preck,i=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩PNLk,i=hNLk(¯¯¯¯¯¯¯¯¯RNLk,i)−αNLk,with probability pNLk(∥Xk,i∥)PLk,i=hLk(¯¯¯¯¯¯¯¯¯RLk,i)−αLk,with probability % pLk(∥Xk,i∥). (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

 λNLk(t)=ddtΛNLk([0,t]), (13)

and

 λLk(t)=ddtΛLk([0,t]), (14)

respectively, where

 ΛNLk([0,t])=EgNLk[2πλk∫t(BNLkgNLk)1/αNLkz=0pNLk(z)zdz] (15)

and

 ΛLk([0,t])=EgLk[2πλk∫t(BLkgLk)1/αLkz=0pLk(z)zdz]. (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.

### Iii-B 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

 pMIRPcov({λk},{Tk},{BUk}) =[k=1]K∑∫∞r=0e−TkηrαNLkλNLk(r)[j=1]K∏[LMIRPINLj(TkrαNLk)LMIRPILj(TkrαNLk)]dr (17)

where

 LMIRPINLj(s)=exp[−∫∞y=0λNLj(y)1+yαNLj/sdy], (18)

and

 LMIRPILj(s)=exp[−∫∞y=0λLj(y)1+yαLj/sdy]. (19)
###### Proof:

See Appendix A.

In pursuit of the analytical results of the PT and the EE, the NLoS/LoS coverage probability and per-tier 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

 pMIRPcov,NL({λk},{Tk},{BUk})=[k=1]K∑pMIRPNL,k({λk},{Tk},{BUk}) =[k=1]K∑∫∞r=0e−TkηrαNLkλNLk(r)[j=1]K∏[LMIRPINLj(TkrαNLk)LMIRPILj(TkrαNLk)]dr, (20)

and

 pMIRPcov,L({λk},{Tk},{BUk})=[k=1]K∑pMIRPL,k({λk},{Tk},{BUk}) =[k=1]K∑∫∞t=0e−TkηrαLkλLk(r)[j=1]K∏[LMIRPI% NLj(TkrαLk)LMIRPILj(TkrαLk)]dr, (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 per-tier coverage probability for a typical MU which is covered by the -th tier with the MIRP association scheme is given by

 pMIRPcov,k({λk},{Tk},{BUk}) =∫∞r=0e−TkηrαNLkλNLk(r)[j=1]K∏[LMIRPINLj(TkrαNLk)LMIRP% ILj(TkrαNLk)]dr +∫∞t=0e−TkηrαL% kλLk(r)[j=1]K∏[LMIRPINLj(TkrαLk)LMIRPILj(TkrαLk)]dr. (22)
###### Proof:

This corollary can be derived from Theorem 1 by rearranging the terms in Eq. (17) and thus the proof is omitted here.

### Iii-C Coverage Probability with the MARP Association Scheme

With the MARP association scheme, the typical MU is associated with the BS which offers the maximum long-term averaged received power by averaging out the effect of multi-path 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

 pMARPcov({λk},{Tk},{BUk}) =[k=1]K∑pMARPk,NL({λk},{Tk},{BUk})+[k=1]K∑pMARPk,L({λk},{Tk},{BUk}) =[k=1]K∑∫∞r=0e−TkηrαNLkλNLk(r)[j=1]K∏[LMARP1INLj(TkrαNLk)LMARP1ILj(TkrαNLk)] ×e−[j=1]K∑[ΛLj([0,rαNLk/αLj])+ΛNLj([0,rαNLk/αNLj])]dr ×e−[j=1]K∑[ΛLj([0,rαLk/αLj])+ΛNLj([0,rαLk/αNLj])]%dr, (23)

where

 LMARP1INLj(s)=exp[−∫∞y=rαNLk/αNLjλNLj(y)1+yαNLj/sdy], (24)
 LMARP1ILj(s)=exp[−∫∞y=rαNLk/αLjλLj(y)1+yαLj/sdy], (25)
 LMARP2INLj(s)=exp[−∫∞y=rαLk/αNLjλNLj(y)1+yαNLj/sdy], (26)

and

 LMARP2ILj(s)=exp[−∫∞y=rαLk/αLjλLj(y)1+yαLj/sdy]. (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 per-tier 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

 pMARPcov,NL({λk},{Tk},{BUk})=[k=1]K∑pMARPNL,k({λk},{Tk},{BUk}) =[k=1]K∑∫∞r=0e−TkηrαNLkλNLk(r)[j=1]K∏[LMARP1INLj(TkrαNLk)LMARP1ILj(TkrαNLk)] ×e−[j=1]K∑[ΛLj([0,rαNLk/αLj])+ΛNLj([0,rαNLk/αNLj])]dr (28)

and

 pMARPcov,L({λk},{Tk},{BUk})=[k=1]K∑pMARPL,k({λk},{Tk},{BUk}) =[k=1]K∑∫∞r=0e−TkηrαLkλLk(r)[j=1]K∏[LMARP2I% NLj(TkrαLk)LMARP2ILj(TkrαLk)] ×e−[j=1]K∑[ΛLj([0,rαLk/αLj])+ΛNLj([0,rαLk/αNLj])]%dr, (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 per-tier coverage probability for a typical MU which is covered by the -th tier with the MARP association scheme is given by

 pMARPcov,k({λk},{Tk},{BUk})
 =∫∞r=0e−TkηrαNLkλNLk(r)[j=1]K∏[LMARP1INLj(TkrαNLk)LMARP% 1ILj(TkrαNLk)] ×e−[j=1]K∑[ΛLj([0,rαNLk/αLj])+ΛNLj([0,rαNLk/αNLj])]dr +∫∞r=0e−TkηrαL% kλLk(r)[j=1]K∏[LMARP2INLj(TkrαLk)LMARP2% ILj(TkrαLk)] ×e−[j=1]K∑[ΛLj([0,rαLk/αLj])+ΛNLj([0,rαLk/αNLj])]%dr. (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.

See Appendix C.

### Iii-D 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.,

 T({λk},{Tk},{BUk})=[k=1]K∑λkpMARPcov,k({λk},{Tk},{BUk})log2(1+Tk). (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 power-hungry, 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) energy-efficient system design. The improvement in hardware may have achieved its bottleneck due to the limit of Moore’s law, while the energy-efficient system design has a great potential in the future 5G networks. In the following, two energy-efficient optimization problems are proposed trying to obtain insights of the system design.

### Iv-a 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

 OP1:maxλk pMARPcov({λk},{Tk},{BUk}) s. t. C1: [k=1]K∑λk(akPk+bk)⩽Pmax (32) C2: λk⩾0,∀k∈K

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.

### Iv-B 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

 OP2:maxλk E({λk},{Tk},{BUk}) s. t. C1: pMARPcov({λk},{Tk},{BUk})⩾pmincov (33) C2: λk⩾0,∀k∈K

We will show in the simulation results that tradeoff exists between the coverage probability and the EE.

### Iv-C 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 well-suited for tackling the problem considering that the objective function derivative is not available analytically and its accurate evaluation is resource-intensive. 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 energy-efficient deployment solution, in the following, we focus on the problem of finding the optimal BS density in a 2-tier HetNet.

## V Results and Insights

A 2-tier 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.

### V-a 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 density-dependent 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 density-dependent transmit power are illustrated, respectively. By utilizing a density-dependent 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 density-dependent 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.

### V-B 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.