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Massive Multiuser MIMO in Heterogeneous

Cellular Networks with Full Duplex Small

Cells

Sunila Akbar, Student Member, IEEE, Yansha Deng, Member, IEEE, Arumugam

Nallanathan, Fellow, IEEE, Maged Elkashlan, Member, IEEE, and George K.

Karagiannidi, Fellow, IEEE

Abstract

Full duplex (FD) communication has emerged as an attractive solution for increasing the network

throughput, by allowing downlink (DL) and uplink (UL) transmissions in the same spectrum. However,

only employing FD base stations in heterogeneous cellular networks (HCNs) cause coverage reduction,

due to the DL and UL interferences as well as the residual loop interference. We therefore propose

HCNs with half duplex (HD) massive multiuser multiple-input multiple-output (MIMO) macrocell base

stations (MBSs) to relax the coverage reduction, and FD small cell base stations (SBSs) to improve

spectrum efﬁciency. A tractable framework of the proposed system is presented, which allows to derive

exact and asymptotic expressions for the DL and the UL rate coverage probabilities, and the DL and

the UL area spectral efﬁciencies (ASEs). Monte carlo simulations conﬁrm the accuracy of the analytical

results, and it is revealed that equipping massive number of antennas at MBSs enhances the DL rate

coverage probability, whereas increasing FD SBSs increases the DL and the UL ASEs. The results also

demonstrate that by tuning the UL fractional power control, a desirable performance in both UL and

DL can be achieved.

Index Terms

Heterogeneous cellular networks, massive multiuser MIMO, full duplex, spectral efﬁciency, stochas-

tic geometry.

S. Akbar, Y. Deng, and A. Nallanathan are with Center for Telecommunications Research, King’s College London, London,

UK (e-mail: {sunila.akbar, yansha.deng, arumugam.nallanathan}@kcl.ac.uk).

M. Elkashlan is with Queen Mary University of London, London E1 4NS, UK (e-mail: maged.elkashlan@qmul.ac.uk).

G. K. Karagiannidi is with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki,

54124 Thessaloniki, Greece (e-mail: geokarag@auth.gr).

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I. INTRODUCTION

The emerging ﬁfth-generation (5G) wireless communication system targets higher data rates,

roughly 1000 times of the current fourth-generation (4G) system to support exponential increase

in wireless data transmissions [1]. In order to meet this target, heterogeneous cellular networks

(HCNs) are proposed to boost the network capacity through dense deployment of small cell base

stations (SBSs) [2], and multiuser multiple-input multiple-output (MIMO) with large number of

antennas at the base station (BS) enables ﬁne-grained beamforming towards each mobile user

(MU), which brings ultra high throughput [3].

Recently, increasing research has been conducted on full-duplex (FD) communication, which

allows transmitting and receiving data simultaneously, within the same frequency band [4]. In

theory, FD data transmission is capable of doubling the spectral efﬁciency of half-duplex (HD)

system. However, FD has been previously regarded as hard to be realized in practice due to its

high residual self-interference (SI) problem. Fortunately, the recent advances on SI cancellation,

such as antenna separation schemes [5], beamforming-based techniques [6], and digital circuit

domain schemes [7], have demonstrated the feasibility of FD transmission for short to medium

range wireless communications. For instance, FD transmission can be realized at the access points

through shared or separated antenna conﬁgurations [8]. In terms of antenna usage, the efﬁciency

of the shared antenna conﬁguration is higher than that of the separated one [9]. Besides, the shared

antenna conﬁguration is a promising alternative for separated antenna conﬁguration in short range

communications, where the transmit power is low and the antenna isolation requirement is less

rigorous compared with medium to long range communication [9]. As a promising candidate for

FD technology due to the low transmit power, the SBSs increases network capacity and coverage

[10]. As such, leveraging the use of FD technology at the SBSs and massive multiuser MIMO

at the macrocell base stations (MBSs) provide a potential solution to improve spectral efﬁciency

of HCNs.

A. Related Work

1) Multiuser MIMO in HCNs: [11] presented the coverage probability and area spectral

efﬁciency (ASE) for the downlink (DL) MU in HCNs with multiuser MIMO. It is shown that

for a given total number of transmit antennas, it is preferable to distribute the antennas across

large number of single-antenna BSs rather than small number of multi-antenna BSs. The work

in [10] was extended to [12], which studied the load balancing strategy, which maximizes the

3

coverage probability. In [13], it is shown that massive multiuser MIMO BSs and small cells

BSs operating in time division duplexing (TDD) mode lead to high area throughput, which can

be further improved by installing more BS antennas or deploying more small cells. In [14], the

trade-off between the link reliability and the ASE of HCNs with multiuser MIMO was studied.

2) FD Communication in Cellular Networks: The performance gains brought by FD transmis-

sion in cellular networks have been studied in [15–22]. [15] concludes that making different tiers

operate in different duplex modes in heterogeneous networks enhances the network throughput.

In [16], the ASE was derived for small cell networks with FD, and the self-interference (SI) was

shown to be dominant compared to the aggregate interference. It is shown in [17] that small

cell in-band wireless backhaul has the potential to increase the throughput of massive multiuser

MIMO systems. The authors in [18] investigated the spectrum and energy efﬁciency of the

massive MIMO-enabled FD cellular networks. In [19], the rate coverage probability of a massive

multiuser MIMO-enabled wireless backhaul networks was evaluated, where each SBS can be

conﬁgured with either in-band or out-of-band FD backhaul mode. In [20], the authors studied

the joint in-band backhauling and interference mitigation problem in HCNs, which consists of a

massive multiuser MIMO MBS overlaid with self-backhauled small cells. Furthermore, the work

in [21] proposed in-band α-duplex scheme in multi-cell networks with FD operation in each cell,

which allows a partial overlap between DL and uplink (UL) frequency bands. The results in [21]

demonstrated that the overlap parameter, α, can be optimized to achieve maximum FD gain. In

[22], the cell association problem in multi-tier in-band FD networks was investigated. It is shown

that the proposed decoupled cell association, where MUs can be served by different BS in the

UL and DL transmission, outperforms the coupled cell association in which MUs associate to

the same BS in both DL and UL.

3) Spectral Efﬁciency and Link Reliability: The two important metrics to evaluate the perfor-

mance of HCNs, spectral efﬁciency and outage probability were evaluated in HCNs with wireless

power transfer in [23, 24]. The trade-off between the ASE and the link reliability was discussed

in wireless ad-hoc networks [25–27]. In these networks, increasing the density of transmitters

affects both link reliability and ASE, therefore the trade-off between them is essential to balance

both aspects. The trade-off between the ASE and the coverage probability has been studied in

massive multiuser MIMO HCNs [14] and a mixed multi-cell system composed of FD and HD

small cells [28].

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B. Motivation and Contributions

The aforementioned literature laid a solid foundation for the feasibility of FD communica-

tion in cellular networks. However, operating all the BSs in FD mode is likely to erode the

performance gain of the FD communication since simultaneous DL and UL operation on the

same band brings increased interference, and thus reduced coverage. Inspired by the work in

[15], where making different tiers operate in different duplex modes in heterogeneous networks

enhances the network throughput, we focus on HCNs, where only small cells operate in FD

mode, and the macrocells operate in HD mode. We consider FD deployment at the SBSs, due

to low transmit power and low mobility of the associated MUs [10]. However, we note that

increasing the FD SBSs increases the ASE of the network, they also increase the interference

due to the simultaneous DL and UL transmission on the same band which decreases coverage

[28]. In order to compensate this cost, a simple solution is to employ massive multiuser MIMO at

the BSs. However, due to the facts that: 1) a more powerful SI cancellation scheme is required to

make FD MIMO systems feasible, and 2) the residual interference at each receive chain increase

linearly with the number of antennas [29], we have considered massive multiuser MIMO only

at the MBSs. For the proposed HCNs, massive antennas at MBSs ensure coverage over large

areas, while SBSs act as capacity-drivers [13]. Furthermore, we employ distance-proportional

fractional power control in the UL, which provides coverage improvement to the cell-edge MUs

and efﬁcient utilization of MUs’ battery [30, 31].

The main contribution of this work can be summarized as follows:

•We model the K-tier HCNs with multiuser MIMO MBSs operating in HD mode and SBSs

operating in FD mode. We consider only the DL transmission for MBSs, and both DL and

UL transmissions for SBSs. We characterize the network interference generating from the

distributed FD SBSs and UL MUs for performance evaluation.

•We derive analytical expressions for the DL rate coverage probability, the DL ASE of the

macrocells and small cells, the UL rate coverage probability, and the UL ASE of small cells

to evaluate the link reliability and spectral efﬁciency. To examine the impact of massive

multiuser MIMO antennas at the MBS, we derive easy to compute expressions for the

asymptotic DL and UL rate coverage probabilities, and asymptotic DL and UL ASEs as

the number of antennas at the MBSs grows large.

•Numerical results demonstrate the effectiveness of massive multiuser MIMO at the MBSs

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TABLE I: Frequent Notations

Notation Deﬁnition Notation Deﬁnition

PMTransmit power of MBS PkTransmit power of kth tier SBS

ρkReceiver’s sensitivity at the kth tier SBS UL power control factor

αMPath loss exponent for macrocell αkPath loss exponent for kth tier small cell

βFrequency dependent constant value hRSI ,k

Residual self interfering channel

of a kth tier BS

NoNoise power Γ(.)Gamma function

B(.)[., .]Incomplete beta function 2F1[., .;.;.]Gauss hypergeometric function

and the FD SBSs in enhancing the rate coverage probabilities and the ASEs. Moreover,

we show that the distance-proportional fractional power control can be tuned to achieve

a desirable performance in both DL and UL, where decreasing the power control factor

degrades the UL rate coverage probability, but improves the DL rate coverage probability.

C. Paper Organization and Notations

The rest of the paper is organized as follows. In Section II, we discuss the system model of

HCNs with multiuser MIMO at the MBSs and FD operation at the SBSs. In Section III, we

present the cell association and derive the rate coverage probability and ASE both in the DL

and the UL. In Section IV, we evaluate DL and UL rate coverage probabilities and ASEs for

the massive multiuser MIMO regime. We present the performance comparison of the proposed

HCNs with FD SBSs with the conventional HCNs with HD SBSs in Section V. Finally, numerical

results are discussed in Section VI before the paper is concluded in Section VII.

The notations commonly used throughout the paper are presented in Table I.

II. SY ST EM MO DE L

A. Network Model

We consider K-tier HCNs, where the MBSs and the SBSs are spatially located in R2, following

homogeneous Poisson point process (HPPP), ΦbMand Φbkwith intensity λbMand λbk(k=

2,· · · , K), respectively. We consider massive multiuser MIMO at the MBSs, where each BS

is equipped with Nantennas, serving UMMUs (1< UMN), and operates in HD mode.

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MBS

SBS

DL/UL Signal Interference

Signals

Self Interference

Corresponding

Apx. Interference

Characterization

MU

Fig. 1: Example cells of the proposed HCNs with HD multiuser MIMO MBS and FD SBSs and

the interference characterizations.

Each SBS is equipped with single antenna, and is transmitting and receiving at the same time in

FD mode [32, 33]. All the MUs have single antenna and operate in HD mode. In this work, we

focus on the DL performance of the macrocell without pilot contamination, while the SBSs have

transmissions in the DL and UL simultaneously due to the FD operation. The performance of

the UL transmission of MBSs can be easily analyzed following our analytical framework, where

the scheduled macrocell MUs simultaneously transmit to their serving massive multiuser MBS

per resource block in the UL [34]. Linear receive ﬁlters are then used for UL signal detection.

In HCNs with FD small cells, the DL and UL small cell transmissions occur simultaneously

by reusing the spectrum of the DL macrocell transmissions due to the full frequency reuse in

HCNs [35]. The network is assumed to be fully-loaded, such that each MBS has UMactive

MUs [11, 12] 1, and each SBS serves one active DL MU and one active UL MU in each time

1Here, we limit ourselves to the ideal assumption with the ﬁxed number of UMin each macrocell to ensure equivalent

performance at the macrocell MU in each macrocell as in [36, 37]. We note that in [38], the probability mass function of the

number of users served by a generic BS was derived by approximating the area of a Voronoi cell via a gamma-distributed random

variable. However, the result in [38] cannot be applied in this paper, since the Euclidean plane is not divided into Voronoi cells

based on the considered cell association methods. We highlight that it is an important work to study the case of the ﬂexible

UMfollowing a certain distribution in less-dense scenarios.

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instant [22], as shown in Fig. 1. Accordingly, the intensity of the active DL MUs in HCNs is

λDL

u=UMλbM+

K

P

k=1

λbk, whereas the active UL MUs per tier are modeled by an independent

HPPP Φukwith intensity λUL

uk=λbk. The analysis will be performed at a typical MU, which is

assumed to be at the origin.

B. Channel Model

We model the channel path loss over the distance |x|as β|x|−α, where βis the frequency

dependent constant value and αis the path loss exponent. The channels are modeled as inde-

pendent and identically distributed (i.i.d.) quasi-static Rayleigh fading. We assume TDD mode,

where channel reciprocity can be exploited and allows a BS to estimate its DL channels from

UL pilots sent by the MUs. Therefore, the resulting number of pilots scales linearly with the

number of MUs, and is independent of the number of antennas in contrast to frequency division

duplexing (FDD). We consider time division multiple access (TDMA), where several MUs share

the same channel in different time slots, thus the BS transmit power is independent of the density

of active MUs, and there is no intra-cell interference in each cell. In a snap of time, each MBS

can serve UMMUs and each FD SBS serves one DL and one UL MU per channel.

C. BS and MU Transmit Power Allocation

Each MBS and SBS transmit with ﬁxed power, PMand Pk, respectively. To limit the UL

interference and reduce the overall power consumption of MUs, we employ distance-proportional

fractional power control [30] in the UL, where the MU at a distance dfrom the associated SBS

adjusts its transmit power with, Pu=ρkβ−dαj, to compensate large-scale fading. Note that,

0≤≤1, is the power control factor, and ρkis the receiver sensitivity of the kth tier SBS.

D. Massive multiuser MIMO

Each MBS transmit UMdata streams using linear zero-forcing beamforming (ZFBF) with the

equal transmit power allocation [39], thus the uncorrelated intra-cell interference is suppressed.

We assume sophisticated channel estimation design with sufﬁcient training information that

guarantees perfect CSI [40]. In the training phase, each MU sends a pre-assigned orthogonal

pilot sequence to the MBS, which is perfectly estimated by the MBS without pilot contamination,

therefore the perfect channel state information is available at the BSs and MUs. The non-

pilot contamination assumption is valid when the pre-assigned pilot sequences used in different

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macrocells are orthogonal to each other [41]. The maximum number of MUs per MBS depends

on the dimension of the UL pilot ﬁeld. Accordingly, the number of channel vectors that can

be estimated and for which the DL precoder can be designed is determined. In our model, we

consider ﬁxed number of MUs served by each MBS.

E. Self-Interference Cancellation for FD Small Cells

The SBS in FD mode receives self-interference from its transmitted signal, and performs SI

cancellation to combat it. Since, the amount of SI depends on the transmit power of the SBS,

we deﬁne residual SI power after SI cancellation as [42, 43]

PRSI (Pk) = Pk|hRSI ,k |2,(1)

where hRSI ,k is the residual self interfering channel of a kth tier BS, and hRSI ,k is character-

ized according to the cancellation algorithms. For instance, using digital-domain cancellation

algorithms, hRSI ,k can be modeled as hRSI ,k =hS,k −ˆ

hS,k, where hS,k and ˆ

hS,k are the self-

interfering channel and its estimate channel, respectively [42, 44]. In [45], hRSI ,k is regarded as

a constant value with |hRSI ,k|2=σ2

efor the estimation error variance σ2

e. However, modeling

hRSI ,k is still challenging for other cancellation techniques, such as analog-domain schemes,

propagation-domain schemes, and combined schemes of different domains. The parametrization

of the self-IC capability in (1) can make the analysis more generic. Therefore, in our analysis,

we consider, a constant value for hRSI ,k 2, given as

|hRSI ,k |2= 10LdB,k/10 ,(2)

where LdB,k is the ratio between the residual self-interference after interference cancellation and

the transmit power at the kth tier BS as deﬁned in [15].

F. Cell Association

To obtain the strongest received signal, we consider the maximum received power cell asso-

ciation rule in the DL transmission, where the DL MU connects to the BS, which provides the

2The analysis can be easily extended to the case of random hRSI ,k. For instance, once the probability density function

(PDF) of hRSI ,k is available for a certain self-IC algorithm, we can average the analytical results derived in the paper over the

distribution of hRSI ,k.

9

maximum long-term average received power [46]. The average received power at a typical DL

MU connected to the MBS m(m∈ΦbM) is expressed as

Pr,M=Ga

PM

UM

βXm,uM−αM,(3)

where the array gain Gaof zero-forcing beamforming (ZFBF) transmission is N−UM+ 1 [47].

The average received power at a DL MU that is connected to the kth tier SBS bk(bk∈Φbk),

is expressed as

Pr,bk=PkβXbk,uk−αk.(4)

We remind that for the UL transmission, the MUs can only associate with the FD SBSs.

Considering that the HD UL MU associated to the nearest BS can maximize the UL signal-

to-interference-plus-noise ratio (SINR) [22], we consider the nearest BS cell association in the

UL.

Based on the cell association model, the set of interfering MUs and FD SBSs may correlate.

However, to maintain model tractability, we assume that the set of interfering MUs is independent

of the set of interfering FD SBSs as in [16, 48].

G. SINR Models

1) DL SINR of a Macrocell MU: The SINR for a typical DL macrocell MU uM

0located at

the origin is given as

SINRDL

M=

PM

UMβgo,uM

0Xo,uM

0

−αM

IM,uM

0+IS,uM

0+Ius

ul,uM

0

| {z }

IuM

0

+N0

,(5)

where go,uM

0∼Γ (N−UM+ 1,1) is the small-scale fading channel power gain between the

typical DL macrocell MU and its serving MBS, and Xo,uM

0is the distance between the typical

DL macrocell MU and its serving MBS. In (5), IM,uM

0,IS,uM

0, and Ius

ul,uM

0are the interferences

from the other MBSs, the SBSs, and the UL small cell MUs given as

IM,uM

0=X

x∈ΦM

b\o

PM

UM

hx,uM

0βXx,uM

0

−αM,(6)

IS,uM

0=

K

X

j=2 X

y∈Φj

b

Pjhy,uM

oβXy,uM

0

−αj,(7)

10

and

Ius

ul,uM

0=

K

X

j=2 X

z∈Φj

u

ρjβ−|Rz,bz|αjhz,uM

0βXz,uM

0

−αj,(8)

respectively. In (6), (7), and (8), hx,uM

0∼Γ (UM,1),hy,uM

0∼exp (1), and hz,uM

0∼exp (1)

denote the small-scale fading channel power gains from the MBSs to the typical DL macrocell

MU, from the SBSs to the typical DL macrocell MU, and from the UL small cell MUs to

the typical DL macrocell MU, respectively, and their corresponding distances are denoted as

Xx,uM

0,Xy,uM

0, and Xz,uM

0, respectively. In (8), ρjβ−|Rz,bz|αjis the transmit power of the

UL MU at a distance of |Rz,bz|from its serving SBS, where ρjis the receiver sensitivity at the

SBS of the jth tier and is the power control factor.

2) DL SINR of a Small Cell MU: The SINR for a typical DL small cell MU uk

0located at

the origin can be written as

SINRDL

k=

Pkgo,uk

0βXo,uk

0

−αk

IM,uk

0+IS,uk

0+Ius

ul,uk

0+N0

,(9)

where go,uk

0is the small-scale fading channel power gain between the typical DL small cell MU

and its serving SBS, and Xo,uk

0is the distance between the typical DL small cell MU and its

serving SBS. In (9), IM,uk

0,IS,uk

0, and Ius

ul,uk

0are the interference from the other MBSs, the SBSs,

and the UL small cell MUs, which are given as

IM,uk

0=X

x∈ΦM

b

PM

UM

hx,uk

0βXx,uk

0

−αM,(10)

IS,uk

0=

K

X

j=2 X

y∈Φj

b\o

Pjhy,uk

0βXy,uk

0

−αj,(11)

and

Ius

ul,uk

0=

K

X

j=2 X

z∈Φj

u

ρjβ−|Rz,bz|αjhz,uk

0βXz,uk

0

−αj,(12)

respectively. In (10), hx,uk

0∼Γ (UM,1) denotes the small-scale fading channel power gain

between the MBSs and the typical DL small cell MU. The distances between the typical DL

small cell MU and the MBSs, the other small cell MU, and the UL small cell MU are denoted

as Xx,uk

0,Xy,uk

0, and Xz,uk

0, respectively.

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3) UL SINR of a Small Cell MU: The UL SINR for a typical SBS bk

0located at the origin

can be written as

SINRUL

k=

ρkgo,bk

0βXo,bk

0

αk(−1)

PRSI (Pk) + IM,bk

0+IS,bk

0+Ius

ul,bk

0+N0

,(13)

where ρkis the receiver sensitivity of the serving SBS, go,bk

0∼exp (1) is the small-scale fading

channel power gain between the typical UL small cell MU and its serving SBS, Xo,bk

0is the

corresponding distance, and PRSI (Pk)is the residual SI power after performing cancellation

given in (1). In (13), IM,bk

0,IS,bk

0, and Ius

ul,bk

0are the interference from the MBSs, the other SBSs,

and the other UL small cell MUs given as

IM,bk

0=X

x∈ΦM

b

PM

UM

hx,bk

0βXx,bk

0

−αM,(14)

IS,bk

0=

K

X

j=2 X

y∈Φj

b\bk

0

Pjhy,bk

0βXy,bk

0

−αj,(15)

and

Ius

ul,bk

0=

K

X

j=2 X

z∈Φj

u\o

ρjβ−|Rz,bz|αjhz,bk

0βXz,bk

0

−αj,(16)

respectively. In (14), hx,bk

0∼Γ (UM,1) denotes the small-scale fading channel power gain

between the typical UL small cell MU and the MBSs. The distances between the typical SBS

and the MBSs, the other SBSs, and the other UL small cell MUs are denoted as Xx,bk

0,Xy,bk

0,

and Xz,bk

0, respectively.

H. Interference Characterization

Characterizing the interference in proposed HCNs is the key challenge in evaluating the system

performance. The reason is the difﬁculty to obtain exact characteristics of the interference from

the UL small cell MUs to the DL macrocell MU, Ius

ul,uM

0in (8), the interference from the UL

small cell MUs to the DL small cell MU, Ius

ul,uk

0in (12), the interference from the MBSs in the

DL to the SBSs in the UL, IM,bk

0in (14), and the interference from the SBSs in the DL to the

SBSs in the UL, IS,bk

0in (15). We characterize the interferences as shown in Fig. 1 using similar

approximation as in [15]. For instance, to characterize Ius

ul,uM

0, we consider a DL macrocell MU

located at a, its serving MBS located at b, a FD SBS located at c, and its associated UL MU

at c+N(c), where N(c)is the relative location of small cell MU to its serving SBS at cin

12

the UL. Generally, the distance between DL macrocell MU aand FD SBS cis greater than

the distance between c+N(c)and c, i.e., ka−ck kN(c)k. Therefore, we assume that the

distance between macrocell MU at aand UL small cell MU at c+N(c)can be approximated

as the distance between a macrocell MU at aand the SBS at c. Likewise, we characterize the

interferences Ius

ul,uk

0,IM,bk

0, and IS,bk

0.

III. PERFORMANCE EVALUATION

Adjusting the number of antennas at MBSs with multiuser MIMO and the FD SBS densities

will affect both the link reliability and ASE of the HCNs. Accordingly, we analyze the perfor-

mance of the DL and the UL transmission of the HCNs in terms of rate coverage probability

and ASE. Since a typical MU can associate with at most one tier, the performance of each tier

as well as per tier association probability determine the overall performance of HCNs in the DL

and the UL as per the law of total probability. To facilitate the analysis, we ﬁrst present the per

tier association probability.

A. DL Cell Association

The probability that a typical MU is associated with the MBS is given as in [46], where the

transmit powers of the MBSs and SBSs are given in (3) and (4), respectively.

ΛM=2πλbMZ∞

0

rexp n−πλbMr2−π

K

X

j=2

λbjPj

ΨPM2/αj

r2αM/αjodr,(17)

where

Ψ =N−UM+ 1

UM

.(18)

Similarly, the probability that a typical MU is associated with the kth tier SBS is given as in

[46]

ΛDL

k=2πλbkZ∞

0

rexp n−π

K

X

j=2

λbj(Pjrαk/Pk)2/αj−πλbMPMΨ

Pk2/αM

r2αk/αModr,(19)

where Ψis given in (18).

13

B. UL Cell Association

In the UL transmission, a typical MUs can only associate with the nearest FD SBS. The

probability that a typical MU is associated with the kth tier SBS is given as [46]

ΛUL

k= 2πλbkZ∞

0

rexpn−

K

X

j=2

πλjr2αk/αjodr.(20)

C. DL Rate Coverage Probability

In this section, we derive the DL rate coverage probability of a typical MU in K-tier HCNs.

The DL rate coverage probability of a random MU in the K-tier HCNs is given by

CDL(RDL) = ΛMCM(RDL) +

K

X

k=2

ΛDL

kCDL

k(RDL),(21)

where ΛMand ΛDL

kare given in (17) and (19), respectively, CM(RDL)is the DL rate coverage

probability between a typical MU and its serving MBS, and CDL

k(RDL)is the DL rate coverage

probability between a typical MU and its serving SBS.

In (21), the DL rate coverage probability between a typical MU and its serving MBS is deﬁned

as

CM(RDL) =E

Xo,uM

0

Pr hSINRDL

MXo,uM

0≥γDLXo,uM

0i,(22)

where SINRDL

Mis given in (5) and γDL is given as

γDL = eRDL −1.(23)

In (23), RDL is the DL rate threshold. Similarly, the DL rate coverage probability of typical MU

at a distance Xo,uk

0from its associated SBS in the kth tier is deﬁned as

CDL

k(RDL) =E

Xo,uk

0

Pr hSINRDL

kXo,uk

0≥γDLXo,uk

0i,(24)

where SINRDL

kand γDL are given in (9) and (23), respectively.

Theorem 1. The DL rate coverage probability of a typical MU associated with the MBS is

derived as

CM(RDL) = 2πλbM

ΛMZ∞

0

x

N−UM

X

n=0

(xαM)n

n!(−1)nXn!

n

Q

l=1

ml(l!)ml

κ(x)

n

Y

l=1 ψ(l)(xαM)mldx, (25)

14

where

κ(x) = exp (−γDLUMqN0

PMβ−ζγDLUMq

PMβ−πλbMx2−π

K

X

j=2

λbkPj

ΨPM2

αjx

2αM

αj),(26)

Pis over all n-tuples of non-negative integers (m1, ..., mn)that satisfy the constraint

n

P

l=1

l . ml=

n,Ψis given in (18),ΛMis given by (17),γDL is given in (23), and ζ(.),ψ(1)(.), and ψ(l)(.)

are given as

ζ(s) =2πλbM

UM

X

ν=1 UM

νPM

UM

βν

sν−sPM

UMβ−ν+2

αM

αM

B−sPM

UMβx−αMν−2

αM

,1−UM+

K

X

j=2

2πλbj(sPjβ

DM

j(x)2−αj

αj−22F1αj−2

αj

,1; 2 −2

αj

;−sPjβ(DM

j(x))−αj+Z∞

0Zr2

0

1

1+(sρjβ(1−))−1u−αj/2rαj

π

K

X

j=2

λbje

−π

K

P

j=2

λbjuλIΦUL

bj

(r)du rdr),

(27)

ψ(1)(q) = −γDLUMN0

PMβ−2πλbMUMγDL x2−αM

αM−22F1αM−2

αM

, UM+ 1; 2 −2

αM

;−γDLqx−αM−

K

X

j=2

2πλbj(

γDLUMPj

PM

DM

j(x)2−αj

αj−22F1αj−2

αj

,2; 2 −2

αj

;−γDLUMPjq

PM

(DM

j(x))−αj−Z∞

0Zr2

0PMβu−αj/2rαj

γDLUMρ

1 + PMβu−αj/2rαj

γDLUMρ−2

q−2π

K

X

j=2

λbje

−π

K

P

j=2

λbjuλIΦUL

bj

(r)du rdr),

(28)

and

ψ(l)(q) =2πλbM

(UM+l−1)!

(UM−1)! (−γDL)

2

αM(q)−l+2

αM

αM

B(−γDLqx−αM)l−2

αM

,1−UM−l+

K

X

j=2

2πλbj(l!

−γDLUMPj

PM2

αM(q)−l+2

αM

αM

B−

γDLUMPjq

PM(DM

j(x))−αMl−2

αM

,−l−Z∞

0Zr2

0PMβu−αj/2rαj

γDLUMρ

l

Y

i=2

(−i)PMβu−αj/2rαj

γDLUMρ+q−(l+1) π

K

X

j=2

λbje

−π

K

P

j=2

λbjuλIΦUL

bj

(r)du rdr).

(29)

In (27),(28) and (29),λIΦUL

bj

(r),γDL, and λIΦUL

bj

(r)are given in (A.8),(23), and (A.8) respec-

tively, and DM

j(x)is the distance between the closest interferring BS of the jth tier and the

typical MMU given as

DM

j(x) = Pj

ΨPM1

αjx

αM

αj,(30)

where Ψis given in (18).

Proof. See Appendix A.

15

Theorem 2. The DL rate coverage probability of a typical MU associated with the kth tier SBS

is derived as

CDL

k(RDL)

=2πλbk

ΛDL

kZ∞

0

xexp (−γDLxαkN0

Pkβ−Ξ(x)−π

K

X

j=2

λbjPjxαk

Pk2

αj−πλbMPMΨ

Pk2

αMx

2αk

αM)dx,

(31)

where Ψ,ΛDL

k,γDL, and Ξ(x)are given in (18),(19), and (23), and (32), respectively. In (31),

Ξ(.)is given as

Ξ(x) =2πλbM

UM

X

ν=1 UM

νγDLxαkPM

PkUMν−γDLxαkPM

PkUM−ν+2

αM

αM

B−γDLxαkPM

PkUM(Dk

M(x))−αMν−2

αM

,1−UM

+

K

X

j=2

2πλbj(γDLxαkPj

PkDk

j(x)2−αj

(αj−2) 2F1αj−2

αj

,1; 2 −2

αj

;−γDLPj

Pk

(Dk

j(x))αk−αj+Z∞

0Zr2

0

1

1+(γDLxαk

Pkβρjβ(1−))−1u−αj/2rαjπ

K

X

j=2

λbje

−π

K

P

j=2

λbjuλIΦUL

bj

(r)du rdr),

(32)

where λIΦUL

bj

(r)and γDL are given in (A.8) and (23), respectively. In (32),Dk

M(x)is the distance

between the closest interfering MBS and the typical small cell MU, and Dk

j(x)is the distance

between the closest interfering BS in the jth tier and the typical small cell MU, given as

Dk

M(x) = ΨPM

Pk1

αMx

αk

αM,(33)

and

Dk

j(x) = Pj

Pk1

αjx

αk

αj.(34)

In (33),Ψis given in (18).

Proof. The proof follows analogous steps to Theorem 1.

D. DL Area Spectral Efﬁciency

The DL ASE measures the capacity of HCNs in the DL deﬁned by [11, 49, 50]. In this section,

we deﬁne the DL ASE of the proposed model as

ASEDL =λbMUMCM(RDL) ln(1 + γDL)

| {z }

ASEDL

MB S

+

K

X

k=2

λbkCDL

k(RDL) ln(1 + γDL)

| {z }

ASEDL

SBS

,(35)

16

where CM(RDL),CDL

k(RDL), and γDL are given in (25), (31), and (23), respectively.

In the following, we present the UL performance metrics which reﬂect the effect of the self-IC,

the density of SBSs, the transmit power of SBSs, and the power control on the UL performance

in the HCNs. We characterize the UL performance in terms of the UL rate coverage probability

and the UL ASE.

E. UL Rate Coverage Probability

In this section, we derive the UL rate coverage probability using

CUL(RUL ) =

K

X

k=2

ΛUL

kCUL

k(RUL),(36)

where ΛUL

kis given in (20), and CUL

kis the UL rate coverage probability between a typical MU

and its serving SBS deﬁned as

CUL

k(RUL) =E

Xo,bk

0

Pr hSINRUL

kXo,bk

0≥γULXo,bk

0i,(37)

where SINRUL

kis given in (13) and γUL is given as

γUL = eRUL −1,(38)

and RUL is the UL rate threshold.

Theorem 3. The UL rate coverage probability of a typical MU associated with the kth tier SBS

is derived as

CUL

k(RUL) = 2πλbk

ΛUL

kZ∞

0

xexp n−γULxαk(N0+Pk|hRS I ,k|2)

(ρkβ−dαj)β−Υ(x)−

K

X

j=2

πλbj(Pj/Pk)2/αj

x2αk/αjodx, (39)

where ΛUL

k,γUL, and |hRS I ,k|2are given in (20),(38) and (2), respectively. In (39),Υ(.)is given

as

Υ(x) =2πλbM

UM

X

ν=1 UM

ν γUL xαkPM

ρkxαkβ−UMαMΓν−2

αMΓUM−ν+2

αM

Γ (UM)+

K

X

j=2

2πλbj(γULPjxαk

ρkxαkβ−

x2−αj

(αj−2)2F11,1−2

αj

; 2 −2

αj

;−γULPj

ρkxαkβ−x(αk−αj)+Z∞

0Zr2

0

1

1 + γULx(1−)αk

ρkρjβ−1u−αj/2rαj

π

K

X

j=2

λbje

−π

K

P

j=2

λbjudu rdr).

(40)

Proof. The proof follows analogous steps to Theorem 1.

17

F. UL Area Spectral Efﬁciency

In this section, we derive the UL ASE in the K-tier HCNs. The UL ASE measures the capacity

of HCNs in the UL, given as

ASEUL =

K

X

k=2

λbkCUL

k(RUL) ln(1 + γUL),(41)

where CUL

k(RUL)and γUL are given in (39) and (38), respectively.

IV. ASYMPTOTIC PERFORMANCE EVALUATION: MASSI VE M ULTIUSER MIMO REGIME

In this section, we analyze the asymptotic performance of K-tier HCNs in which MBSs

are equipped with massive multiuser MIMO antennas. The large number of antennas focusses

energy into ever smaller regions of space to bring huge improvements in throughput and energy

efﬁciency. We refer to the massive multiuser MIMO regime as the case where 1< UMN.

A. SINR Models

1) DL SINR of a Macrocell MU: With massive multiuser MIMO at the MBSs, the SINR for

a typical DL MMU deﬁned in (5) can be simpliﬁed to

SINRDL

MmM=

PMβXo,uM

0

−αM

IMmM,uM

0+IS,uM

0+Ius

ul,uM

0

|{z }

IuM

omM

+N0

,(42)

where the massive multiuser MIMO gain, N−UM+ 1, and the impact of equal power allocation

per backhaul stream (i.e., the denominator of MBS’s transmit power PM

UM) have already been incor-

porated in (45). In (42), IMmM,uM

0=Px∈ΦM

b\o

PM

UMβhx,uM

oXx,uM

0

−αM(a)

≈Px∈ΦM

b\oPMβXx,uM

0

−αM,

the approximation in (a) results due to the fact that with the large number of UM, i.e., (1 <

UMN), the small scale channel fading vanish by the channel hardening effect as in [19]. In

(42), IS,uM

0and Ius

ul,uM

0are given in (7) and (8), respectively.

2) DL SINR of a Small Cell MU: For the massive multiuser MIMO, the DL SINR for a

typical small cell MU deﬁned in (9) can be given as

SINRDL

kmM=

Pkgo,uk

0βXo,uk

0

−αk

IMmM,uk

0+IS,uk

0+Ius

ul,uk

0+N0

,(43)

where IMmM,uk

0≈Px∈ΦM

b\oPMβXx,uk

0

−αM, i.e., there is no short-term fading factor due to

channel hardening effect. In (43), IS,uk

0and Ius

ul,uk

0are given in (11) and (12), respectively.

18

3) UL SINR of a Small Cell MU: For the massive multiuser MIMO case, the UL SINR for

a typical SBS given in (13) can be written as

SINRUL

kmM=

ρkgo,bk

0βXo,bk

0

αk(−1)

PRSI (Pk) + IMmM,bk

0+IS,bk

0+Ius

ul,bk

0+N0

,(44)

where IMmM,bk

0≈Px∈ΦM

b\oPMβXx,bk

0

−αM, i.e., there is no short-term fading factor due to

channel hardening effect. In (44), PRSI (Pk),IS,bk

0, and Ius

ul,bk

0are given in (1), (11) and (12),

respectively.

B. Asymptotic DL Rate Coverage Probability

In this analysis, we use the following formula for the DL rate coverage probability of the

macrocell with massive multiuser MIMO [51]

CM(RDL)mM=E

Xo,uM

0

ESINRDL

MhPr hSINRDL

MXo,uM

0≥γDL

MmMXo,uM

0i,(45)

where

γDL

MmM=γDL

Ψ,(46)

and γDL and Ψare given in (23) and (18), respectively. The DL and UL rate coverage probabilities

deﬁnitions for the small cells in the massive multiuser MIMO case will be the same as in the

multiuser MIMO case, which are deﬁned as (24), and (37), respectively.

We present the asymptotic DL rate coverage probability of a typical MU associated with the

MBS and the kth tier SBS in Theorem 4and Theorem 5, respectively.

Theorem 4. For the massive multiuser MIMO regime, the DL rate coverage probability of a

typical MU associated with the MBS is derived as

CM(RDL)mM=2πλbM

ΛMZ∞

0

x1

2−1

πZ∞

0

Imhexp −χ1(x, w) −πλbMχ2(x, w) −

K

X

j=2

2πλbj

{χ3(x, w) + χ4(x, w)} − πλbMx2−π

K

X

j=2

λbkPj

ΨPM2/αj

r2αM/αjidw

wdx,

(47)

19

where

χ1(x, w) =jw PMβ

γDL

MmMxαM−No!,(48)

χ2(x, w) =

Γ1−2

αM+2

αMΓu−2

αM,−jwPMβ

xαM

(−jwPMβ)2

αM

−x2,(49)

χ3(x, w) = Pjαj/2(−jw)β(xαM

ΨPM)2/αj−1

αj−2!2F11,1−2

αj

; 2 −2

αj

;jwΨPM

xαM,(50)

χ4(x, w) = Z∞

0Zr2

0

1

1+(−jwρjβ(1−))−1u−αj/2rαjπ

K

X

j=2

λbje−π

K

P

j=2

λbjuλIΦUL

bj

(r)du rdr,

(51)

and ΛM,γDL

MmM,Ψ, and λIΦUL

bj

(r)are given in (17),(46),(18), and (A.8), respectively.

Proof. See Appendix B.

Theorem 5. For the massive multiuser MIMO regime, the DL rate coverage probability of a

typical MU associated with the kth tier SBS is derived as

CDL

k(RDL)mM=2πλbk

ΛDL

kZ∞

0

x"1

2−1

πZ∞

0

Imexp njwNo−πλbM$1(x, w) −

K

X

j=2

2πλbj

{$2(x, w) + $3(x, w)} − π

K

X

j=2

λbjPjxαk

Pk2

αj−πλbMPMΨ

Pk2

αMx

2αk

αMo

1 + jwPkβ

γDLxαk−1#dw

wdx, (52)

where

$1(x, w) =

Γ1−2

αM+2

αMΓu−2

αM,−jwPMβ

(DM

k(x))αM

(−jwPMβ)2

αM

−(DM

k(x))2,(53)

$2(x, w) = (Pj

Pk)2/αj(−jw)βPk(xαk)2/αj−1

αj−2!2F11,1−2

αj

; 2 −2

αj

;jwβPj

xαk(Dk

j(x))αk−αj,

(54)

$3(x, w) = Z∞

0Zr2

0

1

1+(−jwρjβ(1−))−1u−αj/2rαjπ

K

X

j=2

λbje−π

K

P

j=2

λbjuλIΦUL

bj

(r)du rdr,

(55)

and ΛDL

k,γDL,Ψ,DM

k(x), and λIΦUL

bj

(r)are given in (19),(23),(18),(33), and (A.8), respectively.

20

Proof. The proof follows analogous steps to Theorem 4.

C. Asymptotic UL Rate Coverage Probability

We present the asymptotic UL rate coverage probability of a typical MU associated with the

kth tier SBS in Theorem 6.

Theorem 6. For the massive multiuser MIMO regime, the UL rate coverage probability of a

typical MU associated with the kth tier SBS is derived as

CUL

k(RUL)mM=2πλbk

ΛUL

kZ∞

0

x"1

2−1

πZ∞

0

Imexp njwNo−πλbMϑ1(x, w) −

K

X

j=2

2πλbj{ϑ2(x, w)

+ϑ3(x, w)} −

K

X

j=2

πλbjx

2αk

αjPj

Pk2

αjo1 + jwρkxαkβ−β

γULxαk−1#dw

wdx,

(56)

where

ϑ1(x, w) =(−jwPMβ)2

αMΓ1−2

αM,(57)

ϑ2(x, w) =(−jwPjβ)x2−αj

(αj−2)2F11,1−2

αj

; 2 −2

αj

;jwβPj

xαkx(αk−αj),(58)

ϑ3(x, w) = Z∞

0Zr2

0

1

1 + −jwρjβ1−−1u−αj/2rαjπ

K

X

j=2

λbje−π

K

P

j=2

λbjudu rdr, (59)

and ΛUL

kand γUL are given in (20) and (38), respectively.

Proof. The proof follows analogous steps to Theorem 4.

V. PERFORMANCE COMPARISON WITH THE CONVENTIONAL HD HCNS

In order to compare the performance of the proposed HCNs with FD SBSs with that of the

conventional HD HCNs with HD SBSs, we deﬁne the total ASE of a random MU in HCNs as

ASE =ASEDL +ASEUL,(60)

where ASEDL and ASEUL are given in (35) and (41), respectively. Furthermore, we deﬁne the

ASE of FD small cell MUs as

ASESBS =ASEDL

SBS +ASEUL,(61)

where ASEDL

SBS and ASEUL are given in (35) and (41), respectively.

21

TABLE II: Parameter Values unless speciﬁed

Parameter Value Parameter Value Parameter Value Parameter Value

λbM5×10−5µ20 PM40 dBm P233 dBm

αM3.5α24ρ2−40 dBm 0.9

RDL 0.5 RUL 0.5N0−100 dBm PkhLI 0

6 6.5 7 7.5 8 8.5 9 9.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

N

Exact Simu.

Asymp. Simu.

Exact Analysis

Asymp. Analysis

Rate Coverage Probability

DL Macrocell

DL Small cell

UL Small cell

(a) Rate coverage probability versus small N(multiuser MIMO).

100 200 300 400 500 600 700 800 900 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

N

Exact Simu.

Asymp. Simu.

Exact Analysis

Asymp. Analysis

Rate Coverage Probability

DL Macrocell

DL Small cell

UL Small cell

(b) Rate coverage probability versus large N(massive multiuser

MIMO).

Fig. 2: Rate coverage probability versus the number of MBS antennas with UM= 5.

VI. NUMERICAL RE SU LTS

In this section, we investigate the system performance in the DL and the UL in terms of the

rate coverage probability and the ASE of HCNs with multiuser MIMO antennas at the MBSs

and FD operation at the SBSs. We compare the performance of HCNs with multiuser MIMO

at MBSs and FD at SBSs with that of massive multiuser MIMO at MBSs and FD at SBSs. We

plot the DL rate coverage probability, the DL ASE, the UL rate coverage probability, and the

UL ASE using (21), (35), (36), and (41), respectively. We validate the accuracy of the derived

expressions for a two-tier HCNs with network radius An=π(1000)2km2consisting of HD

macrocells with density λbMand FD small cells with density λb2, via Monte Carlo simulations.

The interference approximations in Section II-H are not made in the simulation. The simulation

is repeated and averaged over 10,000 iterations. The results presented in the ﬁgures of this

22

section validate the accuracy of our approach to characterize the interferences and show that the

assumptions made have a minor effect on the accuracy of the proposed analytical model. Unless

speciﬁed, the parameter values used in this section are listed in Table II.

A. Impact of number of multiuser MIMO/massive multiuser MIMO antennas at the MBS on the

DL and UL Rate Coverage Probability

Fig. 5(a) and Fig. 5(b) compare the DL and the UL rate coverage probability with the multiuser

MIMO at the MBS to that with massive multiuser MIMO at the MBS as a function of the number

of antennas at the MBS. We see that the asymptotic rate coverage probability of small cell MU

closely matches the exact rate coverage probability in DL and UL both for small and large N.

This observation can be attributed to the fact that changing Nat the interfering MBS does not

change the distributions of short term fading factors hx,uk

0in (10) and hx,bk

0in (14), for the exact

case, which are ignored in the asymptotic case due to channel hardening effect. As expected, the

DL rate coverage probabilities of the macrocell MU and the small cell MU in massive multiuser

MIMO case is higher than those in multiuser MIMO case due to the large antenna array gain.

However, the UL rate coverage probability of MU remains constant with increasing Nfor both

small and large Ndue to that: 1) the UL MU can only associate with the SBSs, and 2) the

interferences from NMBS antennas do not add coherently such that for the same total transmit

power, the interference level from a MBS to an UL MU is the same, regardless of the number

of Nunder i.i.d. Rayleigh fading channels.

B. Impact of number of SBSs density on the DL and UL rate coverage probability

Fig. 3 compares the DL and the UL rate coverage probability with massive multiuser MIMO at

the MBS as a function of the ratio between the SBSs density to the MBSs density (µ=λb2/λbM).

The increase in λb2improves the DL rate coverage probabilities of macrocell MU and small

cell MU. This is according to the the fact that increasing λb2decreases the distance between

the typical small cell MU and the serving SBS. Thus, the MUs transmit with less power due to

distance-proportional fractional power control, which in turn reduces the UL interference for the

macrocell MU and the small cell MU. However, increasing λb2decreases the UL rate coverage

probability due to the increased interference from larger number of SBSs.

23

10 15 20 25 30 35 40 45 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Rate Coverage Probability, N=128,

UM=2

µ

Simu. DL HCNs

Simu. DL Small cell

Simu. DL Macrocell

Simu. UL Small cell

Asymp. Analysis

Fig. 3: Rate coverage probability versus the

ratio between SBSs density to MBSs density.

N

100 200 300 400 500 600 700 800 900 1000

×10

-4

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Area Spectral Efficiency (nats/sec/Hz.m )

2

FD HCNs

HD Small Cells

FD Small Cells

HD HCNs

Asymp. Analysis

Fig. 4: ASE versus the number of MBS anten-

nas with UM= 5.

C. Performance comparison of the proposed HCNs with the conventional HCNs

Fig. 4 compares the ASE of the proposed HCNs with FD SBSs to that of the conventional

HCNs with HD SBSs. We plot the ASE of a random MU in the proposed HCNs using (60),

and that in the conventional HCNs using ASEDL in (35) with no UL interference from the MUs,

i.e., Ius

ul,uM

0= 0 in (5) and Ius

ul,uk

0= 0 in (9). We plot the ASE of a small cell MU of the

proposed HCNs using (61), and that of conventional HCNs using ASEDL

SBS in (35) with no UL

interference from the MUs, i.e., Ius

ul,uk

0= 0 in (9). The ASE of the proposed HCNs is observed

to be higher than that of the conventional HCNs. This suggests the ASE improvement brought

by simultaneous transmission in DL and UL due to FD SBSs which dominates the resulting

additional interferences. With the increase in the number of antennas at the MBSs, the ASE of

the HCNs increases due to the increase in the rate coverage probability with larger Nas shown

in Fig. 2b. Moreover, similar trends are observed for the small cell tier with improved ASE than

that of the HCNs.

D. Impact of SBS density with different number of MBS antennas on the DL Performance

Fig. 5(a) and Fig. 5(b) examine the trade-off between the DL ASE and the DL rate coverage

probability versus the ratio between density of SBSs to density of MBS (µ=λb2/λbM)and

the number of MBS antennas both for multiuser MIMO and massive multiuser MIMO at the

24

0.5 0.52 0.54 0.56 0.58 0.6 0.62

0

1

2

3

4

5

6

7

8x 10−4

Rate Coverage Probability

Simu. N=4,

DL HCNs

DL Small Cell

µ=10

µ=50

DL Macrocell

µ=50 µ=10

µ=10

µ=50

Exact Analysis

UM=2

Simu. N=6,

UM=2

Area Spectral Efficiency (nats/sec/Hz.m

2

)

(a) ASE versus Rate Coverage Probability, small N(multiuser

MIMO).

0.55 0.6 0.65 0.7 0.75 0.8 0.85

1

2

3

4

5

6

7

8

9

10 x 10−4

Rate Coverage Probability

Area Spectral Efficiency (nats/sec/Hz.m

DL HCNs

DL Small Cell

µ=10

µ=50

µ=50

µ=10

µ=10

DL Macrocell

µ=50

Simu. N=64,

Asymp. Analysis

UM=5

Simu. N=128,

UM=5

2

)

(b) ASE versus Rate Coverage Probability, large N(massive

multiuser MIMO).

Fig. 5: The tradeoff between the ASE and the rate coverage probability for various number of

MBS antennas with PkhLI = 0.

MBSs. In Fig. 5(a) and Fig. 5(b), we consider = 0 and the transmit power at the MU Puis

taken as 23 dBm. Clearly, the DL ASE and rate coverage probability with massive multiuser

MIMO at the MBS are higher as compared to those with multiuser MIMO at the MBSs due

to sharp beamforming. The massive number of antennas at MBSs brings the higher DL rate

coverage probability and ASE. The DL ASE and the rate coverage probability of macrocell

MU decreases with increasing the SBSs density due to the increased interference from SBSs.

However, increasing the SBSs density λbkincreases the DL ASE, but decreases the DL rate

coverage probability of small cell MU. With the increase in the SBSs density, the number of DL

small cell transmissions and the aggregate interference from small cells increase, which results

in a trade-off between DL ASE and rate coverage probability for the small cell MUs. We have

shown that the UL interference can be reduced by employing UL power control in the UL,

which improves the DL rate coverage probability in Fig. 3.

E. Impact of SBS density with different MBS and SBS transmit powers on the DL and UL

Performance

Fig. 6 plots the DL and UL ASE and rate coverage probability as a function of the transmit

powers at the MBSs and SBSs. In Fig. 6, we consider = 0 and the transmit power at the MU

25

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−3

Rate Coverage Probability

M=40 dBm, P =30 dBm

P

µ=10

µ=50

µ=50

µ=10

µ=10

µ=50

µ=50

DL HCNs

DL Small Cell

DL Macrocell

UL Small Cell

2

M=40 dBm, P =37 dBm

P2

M=46 dBm, P =30 dBm

P2

Asymp. Analysis

µ=10

Area Spectral Efficiency (nats/sec/Hz.m

2

)

Fig. 6: The tradeoff between the ASE and the rate coverage probability for various MBS and

SBS transmit powers with N= 128 and UM= 5.

Puis taken as 23 dBm. Increasing the MBS transmit power increases the DL ASE and the rate

coverage probability of all tiers, which is due to the increase of SINRDL

Min (5), and the reduced

distance between the typical small cell MU and the associated SBS. Moreover, we observe the

decrease in the UL ASE and the UL rate coverage probability with the increase in PMand

Pk, which is due to the increased cross-tier and co-tier interferences as can be seen from (13).

Furthermore, we observe that the increase in the SBS density increases the UL rate coverage

probability in contrast to the decreased DL rate coverage probability for small cell MU as shown

in Fig. 5a and Fig. 5b, which is due to the decreased distance between the UL small cell MU

and the serving SBS. It can thus be concluded that the SBS density and the BS transmit power

of each tier can be tuned to achieve joint DL and UL performance gains with FD SBSs.

F. Impact of SI cancellation capability with different SBS transmit power on the DL and UL

Performance

Fig. 7 examines the impact of the SI cancellation capability LdB on the DL and UL rate

coverage probabilities. As expected, increasing LdB decreases the UL rate coverage probability

of the small cell MU. Moreover, increasing the SBS transmit power decreases the UL rate

coverage probability of the small cell MU, due to the increased self interference. However,

26

-120 -115 -110 -105 -100 -95 -90 -85 -80 -75 -70

Rate Coverage Probability

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

LdB

=30 dBm

P

2=37 dBm

P

2

Simu.

Simu.

Asymp. Analysis

DL HCNs

UL Small Cell

Fig. 7: Rate coverage probability versus SI can-

cellation capability for various SBSs transmit

powers with N= 128 and UM= 5.

ρ

-60 -50 -40 -30 -20 -10 0

Rate Coverage Probability

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

=0.9

Simu. with UL-PC,

ε

ε

Simu. without UL-PC

=1

Asymp. Analysis

Simu. with UL-PC,

DL Macrocell

UL Small cell

DL Small cell

Fig. 8: Rate coverage probability versus SBSs

receivers sensitivity for various SBSs power

control factors with N= 128 and UM= 5.

increasing the SBS transmit power increases the DL rate coverage probability of a random MU,

due to the increase of SINRDL

kin (44).

G. Impact of receiver sensitivity at the SBS with different power control factors

Fig. 8 plots the DL and the UL rate coverage probability versus the receiver’s sensitivity at

SBSs ρ2for various power control factors . Increasing ρ2increases the UL rate coverage

probability, and degrades the DL rate coverage probability. This is due to the reason that

decreasing the the SBS receiver sensitivity (i.e., an increase in ρ2) increases the transmit power

required at each MU to perform channel inversion towards serving SBS, which in turn increases

the useful signal power at the its associated SBS and the interference at the other BSs and

MUs. Similarly, higher power control factor improves the UL performance, but degrades the

DL performance. These results demonstrate that ρ2and can be optimized for joint DL and

UL performance gain. We also compare the DL and UL performance of HCNs with UL power

control to that without UL power control when the MUs transmit power is Pu= 23 dBm. The

UL rate coverage probability in HCNs without UL power control is shown to be very small due

to the increased inter-cell interference from the UL MUs.

27

VII. CONCLUSION

In this paper, we have presented a tractable model for massive multiuser MIMO-enabled HCNs

with FD small cells. Relying on stochastic geometry, we have derived the analytical expressions

for DL rate coverage probability and ASE for macrocell and small cells, and UL rate coverage

probability and ASE for small cells. We have also presented the asymptotic expressions as the

number of antennas at MBS goes to inﬁnity. Numerical results demonstrated the beneﬁts brought

by massive multiuser MIMO in achieving high rate coverage probability and the beneﬁts brought

by of FD SBSs in achieving high ASE. It is shown that the SBSs density and the number of

antennas at the MBSs can be used as design parameters to target optimal DL ASE and DL

rate coverage probability. The results also demonstrate that, to achieve similar performance in

the DL and the UL, UL power control should be employed. With the advancements of massive

multiuser MIMO and SI cancellation in FD, the proposed HCNs will prove to be a promising

candidate for 5G systems.

APPENDIX A

PROOF OF THEOREM 1

From (22), the rate coverage probability of the macrocell tier is given by deconditioning over

Xo,uM

0as

CM(RDL) = Z∞

0

Pr SINRDL

M||Xo,uM

0|=x≥γDLXo,uM

0f|Xo,M|(x) dx(A.1)

where f|Xo,M|(x)is the PDF of the distance between a typical MU and its serving MBS given

by [46] as follows

f|Xo,M|(x) =2πλbM

ΛM

xexp −πλbMx2−π

K

X

j=2

λbjPj

ΨPM2/αj

x2αM/αj,(A.2)

Substituting SINRDL

M(x) from (5) into (A.1) and simplifying we obtain

Pr PM

UMβgo,u0Mx−αM

IuM

0+N0

> τ!(a)

=Z∞

0

e−τUMxαM(γDL +N0)

PMβ

N−UM

X

n=0 τUMxαM(γDL+N0)

PMβn

n!dPr(IuM

0≤γDL),

(b)

=

N−UM

X

n=0

(xαM)n

n!(−1)!

dne−τUMxαMN0

PMβLIuM

0τUMq

PMβ

dqnq=xαM

,(A.3)

28

where (a) follows from go,uM

0∼Γ (N−UM+ 1,1) and (b) follows from some some mathematical

manipulations. In (A.3), LIuM

0

is the Laplace transform of the PDF of IuM

0given as

LIuM

0

(s) = LIM,uM

0

(s)LIS,uM

0

(s)LIuS

ul,uM

0

(s),(A.4)

where LIM,uM

0

(s),LIS,uM

0

(s), and LIuS

ul,uM

0

(s)are the Laplace transform of the PDF of IM,uM

0,

IS,uM

0, and IuS

ul,uM

0, respectively. In (A.4), LIM,uM

0

(s)is derived as

LIM,uM

0

(s) =Eexp −sX

m∈ΦM

b\0

PM

UM

hm,uM

0β|Xm,uM

0|−αM

(a)

= exp −2πλbM

UM

X

ν=1 UM

νZ∞

xPM

UMβνsν(r−αM)ν

1 + sPM

UMβr−αMUMrdr,(A.5)

where (a) is obtained by using generating functional of PPP [52], hm,uM

0∼Γ (UM,1) , and using

Binomial expansion. Likewise, LIS,uM

0

(s)is evaluated as

LIS,uM

0

(s) = exp −

K

X

j=2

2πλbjZ∞

DM

j(x)sPjβr−αj

1 + sPjβr−αjrdr,(A.6)

where DM

j(x)is the distance between a typical MU and the closest interfering BS in the jth tier

given in (30). In (A.4), LIuS

ul,uM

0

(s)is evaluated as

LIus

ul,uM

0

(s)(a)

= exp −

K

X

j=2

2πλbjZ∞

0

(1 −exp(−πλbj

AUL

j

r2))ERsρjβ1−Rαjr−αj

1 + sρjβ1−Rαjr−αjrdr,

(A.7)

where (a) follows from the probability generating functional of a PPP and the fact that the UL

interference ﬁeld is a non-homogeneous PPP with distance dependent density function given as

λIΦUL

bj

(r) = λbj(1 −exp(−πλbj

AUL

j

r2)) (A.8)

where (AUL

j=λbj/

K

P

i=2

λbi)is the repulsion parameter as in [53]. In (A.8), the integral has a

lower limit of zero as the nearest UL MU of FD SBS can be arbitrarily close to the typical

macrocell MU. Using the PDF of serving link distances given in (C.1), we derive LIus

ul,uM

0

(s).

Plugging (A.5), (A.6) and (A.7) into (A.4), after some manipulations, LIuM

0

(s)is derived as

LIuM

0

(s) = e−ζ(s),(A.9)

where ζ(s)is given by (27). Substituting (A.9) into (A.3), simplifying using the Faa di Bruno’s

formula, and ﬁnally plugging into (A.1), we obtain (25).

29

APPENDIX B

PROOF OF THEOREM 2

The rate coverage probability of a typical MU associated with the kth tier SBS is evaluated

following the similar steps as of Theorem 1 with the PDF of the distance between a typical DL

MU and its serving SBS is given by [46]

f|Xo,k|(x)|DL =2πλbkrexp −π

K

X

j=2

λbj(Pjrαk/Pk)2/αj−πλbMPMΨ

Pk2/αM

r2αk/αM,(B.1)

where Ψis given in (18).

APPENDIX C

PROOF OF THEOREM 3

The UL rate coverage probability of a typical MU associated with the kth tier SBS is evaluated

following the similar steps as of Theorem 1 with the PDF of the distance between a typical UL

MU and its serving SBS is given by [46]

f|Xo,k|(x)|U L = 2π

K

X

j=2

λbjxexp n−π

K

X

j=2

λbjx2o.(C.1)

APPENDIX D

PROOF OF THEOREM 4

Based on (42), the DL rate coverage probability of the macrocell with massive multiuser

MIMO at the BSs, can be given as

CM(RDL)mM =

∞

Z

0

FIuM

0mMPMβ

γDL

MmM|Xo,uM

0|αM−N0f|Xo,M|(x) dx, (D.1)

where we resort to apply the Gil-Pelaez inversion theorem [54] and the CDF of the interference

FIuM

0mM

(.)can be derived as

FIuM

0mM

(x) =1

2−1

π

∞

Z

0

Im"LIMmM,uM

0

(−jw)LIS,uM

0

(−jw)LIus

ul,uM

0

(−jw)

exp jwPMβ

γDL

MmMxαM−N0#dw

w,(D.2)

where Im(.)represents the imaginary part of the argument. In (D.2), the Laplace transform of

IMmM,uk

0can be derived as under

LIMmM,uM

0

(−jw) (a)

= exp −2πλbM

∞

Z

x1−exp{−(−jw)PMβrαM}rdr,(D.3)

30

where (a) follows from the probability generating functional of PPP. Solving the integral in

(D.3) we derive LIMmM,uk

0

(−jw). In (D.2), the Laplace transforms of IS,uM

0and Ius

ul,uM

0can be

evaluated as (A.6) and (A.7), respectively. Finally substituting LIMmM,uM

0(−jw),LIS,uM

0

(−jw),

and LIus

ul,uM

0

(−jw) into (D.2), and plugging (D.2) into (D.1), we obtain Theorem 4.

REFERENCES

[1] J. Andrews, S. Buzzi, W. Choi, S. Hanly, A. Lozano, A. Soong, and J. Zhang, “What will 5G be?” IEEE J. Sel. Areas

Commun., vol. 32, no. 6, pp. 1065–1082, Jun. 2014.

[2] S. Parkvall, A. Furuskar, and E. Dahlman, “Evolution of LTE towards IMT-advanced,” IEEE Commun. Mag., vol. 49,

no. 2, pp. 84–91, Feb. 2011.

[3] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral efﬁciency of very large multiuser MIMO systems,”

IEEE Trans. Commun., vol. 61, no. 4, pp. 1436–1449, Apr. 2013.

[4] A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges

and opportunities,” IEEE J. Sel. Areas Commun., vol. 32, no. 9, pp. 1637–1652, Sept. 2014.

[5] T. Snow, C. Fulton, and W. J. Chappell, “Transmit-receive duplexing using digital beamforming system to cancel self-

interference,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 12, pp. 3494–3503, Dec. 2011.

[6] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback self-interference in full-duplex MIMO relays,” IEEE

Trans. Signal Process., vol. 59, no. 12, pp. 5983–5993, Dec. 2011.

[7] T. Riihonen, S. Werner, and R. Wichman, “Hybrid full-duplex/half-duplex relaying with transmit power adaptation,” IEEE

Trans. Wireless Commun., vol. 10, no. 9, pp. 3074–3085, Sep. 2011.

[8] G. Liu, F. R. Yu, H. Ji, V. C. M. Leung, and X. Li, “In-band full-duplex relaying: A survey, research issues and challenges,”

IEEE Communications Surveys Tutorials, vol. 17, no. 2, pp. 500–524, Secondquarter 2015.

[9] H. Ju, E. Oh, and D. Hong, “Improving efﬁciency of resource usage in two-hop full duplex relay systems based on resource

sharing and interference cancellation,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 3933–3938, Aug. 2009.

[10] D. Nguyen, L. N. Tran, P. Pirinen, and M. Latva-aho, “On the spectral efﬁciency of full-duplex small cell wireless systems,”

IEEE Trans. Wireless Commun., vol. 13, no. 9, pp. 4896–4910, Sep. 2014.

[11] H. S. Dhillon, M. Kountouris, and J. G. Andrews, “Downlink MIMO hetnets: Modeling, ordering results and performance

analysis,” IEEE Trans. Wireless Commun., vol. 12, no. 10, pp. 5208–5222, Oct. 2013.

[12] A. K. Gupta, H. S. Dhillon, S. Vishwanath, and J. G. Andrews, “Downlink multi-antenna heterogeneous cellular network

with load balancing,” IEEE Trans. Commun., vol. 62, no. 11, pp. 4052–4067, Nov. 2014.

[13] J. Hoydis, K. Hosseini, S. T. Brink, and M. Debbah, “Making smart use of excess antennas: Massive MIMO, small cells,

and TDD,” Bell Labs Technical Journal, vol. 18, no. 2, pp. 5–21, Sept. 2013.

[14] C. Li, J. Zhang, J. G. Andrews, and K. B. Letaief, “Success probability and area spectral efﬁciency in multiuser MIMO

hetnets,” IEEE Trans. Commun., vol. 64, no. 4, pp. 1544–1556, Apr. 2016.

[15] J. Lee and T. Q. S. Quek, “Hybrid full-/half-duplex system analysis in heterogeneous wireless networks,” IEEE Trans.

Wireless Commun., vol. 14, no. 5, pp. 2883–2895, May. 2015.

[16] H. Alves, C. H. M. de Lima, P. H. J. Nardelli, R. D. Souza, and M. Latva-aho, “On the average spectral efﬁciency of

interference-limited full-duplex networks,” in Proc. CROWNCOM, Jun. 2014, pp. 550–554.

[17] B. Li, D. Zhu, and P. Liang, “Small cell in-band wireless backhaul in massive mimo systems: A cooperation of next-

generation techniques,” IEEE Trans. Wireless Commun., vol. 14, no. 12, pp. 7057–7069, Dec. 2015.

31

[18] Y. Li, P. Fan, L. Anatolii, and L. Liu, “On the spectral and energy efﬁciency of full-duplex small cell wireless systems

with massive MIMO,” IEEE Trans. Veh. Technol., vol. PP, no. 99, pp. 1–1, 2016.

[19] H. Tabassum, A. H. Sakr, and E. Hossain, “Analysis of massive MIMO-enabled downlink wireless backhauling for full-

duplex small cells,” IEEE Trans. Commun., vol. 64, no. 6, pp. 2354–2369, Jun. 2016.

[20] T. K. Vu, M. Bennis, S. Samarakoon, M. Debbah, and M. Latva-aho, “Joint in-band backhauling and interference mitigation

in 5G heterogeneous networks,” in European Wireless 2016; 22th European Wireless Conference, May 2016, pp. 1–6.

[21] A. AlAmmouri, H. ElSawy, O. Amin, and M. S. Alouini, “In-band α-duplex scheme for cellular networks: A stochastic

geometry approach,” IEEE Trans. Wireless Commun., vol. 15, no. 10, pp. 6797–6812, Oct. 2016.

[22] A. H. Sakr and E. Hossain, “On cell association in multi-tier full-duplex cellular networks,” CoRR, vol. abs/1607.01119,

2016. [Online]. Available: http://arxiv.org/abs/1607.01119

[23] Y. Deng, L. Wang, M. Elkashlan, M. D. Renzo, and J. Yuan, “Modeling and analysis of wireless power transfer in

heterogeneous cellular networks,” IEEE Transactions on Communications, vol. 64, no. 12, pp. 5290–5303, Dec 2016.

[24] S. Akbar, Y. Deng, A. Nallanathan, M. Elkashlan, and A. H. Aghvami, “Simultaneous wireless information and power

transfer in K-tier heterogeneous cellular networks,” IEEE Transactions on Wireless Communications, vol. 15, no. 8, pp.

5804–5818, Aug 2016.

[25] A. M. Hunter, J. G. Andrews, and S. Weber, “Transmission capacity of ad hoc networks with spatial diversity,” IEEE

Trans. Wireless Commun., vol. 7, no. 12, pp. 5058–5071, Dec. 2008.

[26] N. Jindal, J. G. Andrews, and S. Weber, “Multi-antenna communication in ad hoc networks: Achieving MIMO gains with

SIMO transmission,” IEEE Trans. Commun., vol. 59, no. 2, pp. 529–540, Feb. 2011.

[27] M. Kountouris and J. G. Andrews, “Downlink SDMA with limited feedback in interference-limited wireless networks,”

IEEE Trans. Wireless Commun., vol. 11, no. 8, pp. 2730–2741, Aug. 2012.

[28] S. Goyal, C. Galiotto, N. Marchetti, and S. Panwar, “Throughput and coverage for a mixed full and half duplex small cell

network,” in Proc. IEEE ICC, May 2016, pp. 1–7.

[29] D. Bharadia and S. Katti, “Full duplex MIMO radios,” Proc. 11th USENIX Symp. NSDI, pp. 359–372, 2014.

[30] T. D. Novlan, H. S. Dhillon, and J. G. Andrews, “Analytical modeling of uplink cellular networks,” IEEE Trans. Wireless

Commun., vol. 12, no. 6, pp. 2669–2679, Jun. 2013.

[31] H. ElSawy and E. Hossain, “On stochastic geometry modeling of cellular uplink transmission with truncated channel

inversion power control,” IEEE Trans. Wireless Commun., vol. abs/1401.6145, 2014.

[32] C. Cox and E. Ackerman, “Demonstration of a single-aperture, full-duplex communication system,” in 2013 IEEE Radio

and Wireless Symposium, Jan. 2013, pp. 148–150.

[33] M. E. Knox, “Single antenna full duplex communications using a common carrier,” in Proc. IEEE WAMICON, Apr. 2012,

pp. 1–6.

[34] L. Wang, K. K. Wong, M. Elkashlan, A. Nallanathan, and S. Lambotharan, “Secrecy and energy efﬁciency in massive

MIMO aided heterogeneous C-RAN: A new look at interference,” IEEE J. Sel. Top. Signal Process., vol. 10, no. 8, pp.

1375–1389, Dec. 2016.

[35] A. Ghosh, N. Mangalvedhe, R. Ratasuk, B. Mondal, M. Cudak, E. Visotsky, T. A. Thomas, J. G. Andrews, P. Xia, H. S.

Jo, H. S. Dhillon, and T. D. Novlan, “Heterogeneous cellular networks: From theory to practice,” IEEE Commun. Mag.,

vol. 50, no. 6, pp. 54–64, Jun. 2012.

[36] A. He, L. Wang, M. Elkashlan, Y. Chen, and K. K. Wong, “Spectrum and energy efﬁciency in massive mimo enabled

hetnets: A stochastic geometry approach,” IEEE Commun. Lett., vol. 19, no. 12, pp. 2294–2297, Dec 2015.

[37] A. Shojaeifard, K. K. Wong, M. D. Renzo, G. Zheng, K. A. Hamdi, and J. Tang, “Massive MIMO-enabled full-duplex

cellular networks,” CoRR, vol. abs/1611.03854, 2016. [Online]. Available: http://arxiv.org/abs/1611.03854

32

[38] S. M. Yu and S. L. Kim, “Downlink capacity and base station density in cellular networks,” in 11th International Symposium

and Workshops on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), May 2013, pp. 119–124.

[39] K. Hosseini, W. Yu, and R. S. Adve, “Large-scale MIMO versus network MIMO for multicell interference mitigation,”

IEEE J. Sel. Areas Commun., vol. 8, no. 5, pp. 930–941, Oct. 2014.

[40] E. Bj ¨

ornson, J. Hoydis, M. Kountouris, and M. Debbah, “Massive MIMO systems with non-ideal hardware: Energy

efﬁciency, estimation, and capacity limits,” IEEE Transactions on Information Theory, vol. 60, no. 11, pp. 7112–7139,

Nov 2014.

[41] T. L. Marzetta, “How much training is required for multiuser MIMO?” in Proc. 40th Asilomar Conference on Signals,

Systems and Computers, Paciﬁc Grove, CA, Oct. 2006, pp. 359–363.

[42] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback self-interference in full-duplex MIMO relays,” IEEE

Trans. Signal Process., vol. 59, no. 12, pp. 5983–5993, Dec. 2011.

[43] T. Riihonen, S. Werner, and R. Wichman, “Hybrid full-duplex/half-duplex relaying with transmit power adaptation,” IEEE

Trans. Wireless Commun., vol. 10, no. 9, pp. 3074–3085, Sept. 2011.

[44] D. Kim, H. Ju, S. Park, and D. Hong, “Effects of channel estimation error on full-duplex two-way networks,” IEEE Trans.

Veh. Technol., vol. 62, no. 9, pp. 4666–4672, Nov. 2013.

[45] A. C. Cirik, Y. Rong, and Y. Hua, “Achievable rates of full-duplex MIMO radios in fast fading channels with imperfect

channel estimation,” IEEE Trans. Signal Process., vol. 62, no. 15, pp. 3874–3886, Aug. 2014.

[46] H.-S. Jo, Y. J. Sang, P. Xia, and J. G. Andrews, “Heterogeneous cellular networks with ﬂexible cell association: A

comprehensive downlink SINR analysis,” IEEE Trans. Wireless Commun., vol. 11, no. 10, pp. 3484–3495, Oct. 2012.

[47] K. Hosseini, W. Yu, and R. S. Adve, “Large-scale MIMO versus network MIMO for multicell interference mitigation,”

IEEE J. Sel. Areas Commun., vol. 8, no. 5, pp. 930–941, Oct. 2014.

[48] A. AlAmmouri, H. ElSawy, O. Amin, and M. S. Alouini, “In-band full-duplex communications for cellular networks with

partial uplink/downlink overlap,” in Proc. IEEE GLOBECOM, Dec. 2015, pp. 1–7.

[49] W. C. Cheung, T. Q. S. Quek, and M. Kountouris, “Throughput optimization, spectrum allocation, and access control in

two-tier femtocell networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 3, pp. 561–574, Apr. 2012.

[50] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, “An Aloha protocol for multihop mobile wireless networks,” IEEE Trans.

Inf. Theory, vol. 52, no. 2, pp. 421–436, Feb. 2006.

[51] D. Bethanabhotla, O. Y. Bursalioglu, H. C. Papadopoulos, and G. Caire, “User association and load balancing for cellular

massive MIMO,” in Information Theory and Applications Workshop (ITA), 2014, Feb. 2014, pp. 1–10.

[52] M. Haenggi, Stochastic geometry for wireless networks. Cambridge University Press, 2012.

[53] S. Singh, X. Zhang, and J. G. Andrews, “Joint rate and SINR coverage analysis for decoupled uplink-downlink biased cell

associations in hetnets,” IEEE Trans. Wireless Commun., vol. 14, no. 10, pp. 5360–5373, Oct 2015.

[54] J. G. Wendel, “The non-absolute convergence of Gil-Pelaez’ inversion integral,” Ann. Math. Statist., vol. 32, no. 1, pp.

338–339, Mar. 1961.