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IEEE ACCESS - SUBMITTED FOR PUBLICATION 1

Full-Duplex Small Cells for Next Generation

Heterogeneous Cellular Networks: A Case Study of

Outage and Rate Coverage Analysis

M. Omar Al-Kadri, Student Member, IEEE, Yansha Deng, Member, IEEE, Adnan Aijaz, Member, IEEE,

and Arumugam Nallanathan, Fellow, IEEE

Abstract—Full-duplex (FD) technology is currently under con-

sideration for adoption in a range of legacy communications

standards due to its attractive features. On the other hand,

cellular networks are becoming increasingly heterogeneous as

operators deploy a mix of macrocells and small cells. With

growing tendency towards network densiﬁcation, small cells are

expected to play a key role in realizing the envisioned capacity

objectives of emerging 5G cellular networks. From a practical

perspective, small cells provide an ideal platform for deploying

FD technology in cellular networks due to its lower transmit

power, and lower cost for implementation compared with the

macrocell counterpart. Motivated by these developments, in this

paper, we analyze a two-tier heterogeneous cellular networks

(HCNs) wherein the ﬁrst tier comprises half-duplex (HD) macro

base stations (BSs) and the second tier consists of FD small cells.

Through a stochastic geometry approach, we characterize and

derive the closed-form expressions for the outage probability and

the rate coverage. Our analysis explicitly accounts for the spatial

density, the SI cancellation capabilities, and the interference

coordination based on enhanced inter-cell interference coordina-

tion (eICIC) techniques. Performance evaluation investigates the

impact of different parameters on the outage probability and the

rate coverage in various scenarios.

Index Terms—full-duplex, enhanced inter-cell interference co-

ordination, heterogeneous cellular networks, outage probability,

rate coverage.

I. INTRODUCTION

IN half-duplex (HD) wireless communications systems, bi-

directional communications between a pair of nodes is

achieved with either frequency division duplexing (FDD) or

time division duplexing (TDD). The former technique employs

different frequency bands for the uplink (UL) and downlink

(DL), whereas, in the latter technique, a single channel is

shared in the time domain for both UL and DL. Such

techniques however are not suitable to fulﬁl the envisioned

requirements of next generation wireless systems [1]. His-

torically, simultaneous transmission and reception in wireless

communications was deemed infeasible in practice due to the

so called self-interference (SI), which is the interference gener-

ated by the transmitter of a radio on its own receiver. Recent

developments in SI cancellation techniques [2]–[6] have led

M. O. Al-Kadri, Y. Deng, and A. Nallanathan are with the Centre

for Telecommunications Research, King’s College London, London, UK

(email:{yansha.deng, arumugam.nallanathan, mhd_omar.alkadri}@kcl.ac.uk).

A. Aijaz was with the Centre for Telecommunications Research, King’s

College London, London, UK. He is now with the Telecommunications

Research Laboratory, Toshiba Research Europe Ltd., Bristol, UK (email:

adnan.a.aijaz@ieee.org).

to the practical realization of FD radios. The feasibility of

single-array FD transceivers has been presented in [7], [8].

FD technology has a number of attractive features e.g., it

can potentially double (theoretically) the ergodic capacity [9],

[10], reduce the feedback delay [11], decrease the end-to-end

delay [12], improve the network secrecy [13] and increase the

efﬁciency of network protocols (e.g., medium access control

[14]).

On the other hand, small cells are gaining increasing pop-

ularity in the next generation cellular systems. Small cells

provide an easy and cost-efﬁcient deployment solution for

capacity and coverage improvements over the conventional

macro-centric networks [15], [16]. The low-powered nature of

small cells make them the ideal candidate for FD deployment

considering that the self-interference (SI) is more manageable

compared to the conventional high-power macro counterparts.

This inspires and motivates the investigation for the feasibility

and performance gains of FD small cells underlay heteroge-

neous cellular networks (HCNs).

An important issue for HCNs is the inter-cell interference,

which arises due to the dense unplanned deployments of small

cells, loud neighbors, and the closed subscriber group access.

To mitigate this interference, 3GPP has recently standardized

the enhanced Inter-Cell Interference Coordination (eICIC)

technique in Release 10 [17]. eICIC provides interference

cancellation techniques in time, frequency, and power control

domains. When the subframes of macro cells and small cells

are aligned, their control and data channel overlap with each

other. Therefore, eICIC mitigates the interference on the

control channel of small cells through Almost Blank Sub-

frames (ABS) at small cells. During ABS, the macro BSs only

transmits the reference signals, which allows small cell BSs

to schedule the associated users without interference from the

macrocells.

A. Related Work

Recent studies on modeling and analysis of HCNs heavily

rely on stochastic geometry framework [18]–[21]. Using these

tools, comprehensive modeling and analysis of legacy HCNs

has been carried out in [22]–[25]. In [26], the authors have pre-

sented the outage probability, the average ergodic rate, and the

minimum average user throughput for a downlink HD multi-

tier HCNs. They have concluded that neither the number of

BSs nor the tiers affect outage probability or average ergodic

IEEE ACCESS - SUBMITTED FOR PUBLICATION 2

rate in an interference-limited full-loaded HCNs with unbiased

cell association. These conclusions, however, may not hold

in environments which are prone to higher interference, like

HCNs comprising FD nodes.

FD-enabled wireless networks have been attracting growing

interest, recently [27]. In [28], the authors have derived the

expression for the throughput of hybrid duplex heterogeneous

networks composed of multi-tier networks, with access points

(APs) operating either in bi-directional FD mode or downlink

HD mode in each tier. The authors have concluded that having

tiers with hybrid duplex BSs degrades the performance, while

higher throughput was achieved when each tier operates in the

same duplex, either HD or FD rather than a mixture of both.

This motivates further research on two-tier HCNs with FD

small cells and HD macrocells, instead of considering hybrid

scenarios. In [29], the authors have derived the downlink rate

coverage probability of a user in a single-tier FD small cell

network with massive MIMO wireless backhauls. In [30],

the authors have introduced an FD-assisted cross-tier inter-

cell interference (ICI) mitigation scheme called fICIC, which

operates on small cells compared to the standardized eICIC

that operates on the macrocells. Such a change may lead

to modiﬁcations on the current backhaul affecting feasibility

of application. This motivates further investigation on the

application of eICIC on FD-enabled HCNs to avoid legacy

network modiﬁcations. In [31], the authors consider a hybrid

scenario where all BSs operate in FD mode. They derived

a closed-form expression for the critical value of the self-

interference attenuation power, which is required for the FD

users to outperform HD users. In [32], the authors have

considered a single tier mixed small cell network, where BSs

operate in either HD or FD, with all users operating in HD.

The effect of FD cells on the performance of the mixed system

was presented, however, inter-cell interference coordination

was not considered and only single tier was investigated.

B. Contributions and Outline

Different from the aforementioned studies, our objective in

this paper is to model and analyze an interference-coordinated

two-tier HCN with FD small cells. The key contributions of

this paper can be summarized as follows.

•We formulate a tractable model for the interference

coordinated two-tier HCN with FD small cells, wherein

tier 1 comprises legacy HD macrocells and tier 2 consists

of FD small cells. By explicitly accounting for spatial

distribution of base stations, self-interference, transmit

power, cell association, uplink power control and ABS

factor, we provide signal-to-interference-plus-noise ratio

(SINR) expressions for users in the corresponding two

tiers. Speciﬁcally, the underlying model captures the DL

scenario for tier 1 and both UL and DL scenarios for tier

2, since tier 1 and tier 2 operate in HD and FD mode,

respectively.

•Based on the system model for two-tier HCN with

FD small cells, we derive closed-form expressions for

outage probability of different tiers. The ﬁnal expressions

explicitly account for interference coordination.

Fig. 1: Example cells of the system model, where macro BS

operates in HD mode, and small cells operate in FD mode.

•We adopt the notion of rate coverage from [28], and

derive closed-form expressions for the corresponding two

tiers.

•We conduct a comprehensive performance evaluation

through numerical as well as simulation studies. We

investigate the impact of various design parameters on

network performance in various scenarios.

The rest of the paper is organized as follows. Section II

provides the system model. In Section III, we analyze the

outage probability of two-tier HCNs with FD small cells. This

is followed by rate coverage analysis in section IV. Numerical

and simulation results are given in Section V. Finally, the paper

is concluded in Section VI.

II. SY ST EM MO DE L

We consider a two-tier HCNs, where tier 1 comprises macro

BSs operating in HD mode, and tier 2 consists of small cells

operating in FD mode, as illustrated in Fig. 1. Both tiers are

spatially distributed in R2following homogeneous Poisson

point processes (HPPPs) ΦS1and ΦS2, with intensities λS1and

λS2, respectively. All users operate in HD mode. The UL small

cell users are spatially located in R2following the HPPP ΦU2,

with intensity λu2. Assuming that the intensity of DL users is

high enough, and each user has data ready for transmission,

such that saturated trafﬁc conditions hold. We also assume that

each small cell BS serves single active uplink user and single

downlink user per channel, and each macrocell BS serves

single active downlink user per channel. This assumption

is justiﬁed due to the conclusions in [28], that the highest

network performance is achieved when each tier in the network

operates in the same duplex, rather than having hybrid tiers.

We assume that the UL users share the macro DL frequency

to minimize the interference on the DL users, considering that

the density of small cells is usually signiﬁcantly higher than

the density of macrocells. The full frequency reuse scenario

is assumed, such that all the cells use the same frequency

band. We assume that the channel coefﬁcients are invariant

in each block and vary between different blocks. Moreover,

we assume that the channel hi,j between any pair of nodes

iand jis impaired by Rayleigh fading, and the path loss is

assumed to be inversely proportional to distance with the path

loss exponent α.

IEEE ACCESS - SUBMITTED FOR PUBLICATION 3

TABLE I: Frequently Used Notations

Notation Deﬁnition

ΦSxHPPP of base stations in tier x

ΦUxHPPP of users in tier x

λSxSpatial density of base stations in tier x

λuxSpatial density of users in tier x

SxBase station of tier x∈(1,2)

S∗

xAssociated BS of tier x∈(1,2)

uxUsers of tier x∈(1,2)

u0

xThe user at origin of tier x∈(1,2)

RSI Residual self-interference

PyxTransmission power of y∈(S, u)of tier x∈(1,2)

ha,b Small scale fading channel coefﬁcient between aand b

Ra,b Distance between aand b

αa,b Pathloss exponent between aand b

IZ

yInterferences caused by y∈(S, u)in Z∈(U L, DL)

N0Additive Gaussian noise

ASxUsers association probability with BS of tier x∈(1,2)

We assume that the FD small cells are equipped with

a single antenna and achieve FD capability through the

techniques mentioned in [33], [34]. A node in FD mode

receives interference from its transmitted signal, and performs

SI interference cancellation. Since the amount of SI depends

on the transmission power at the receiver PS2, we deﬁne the

residual self-interference (RSI) power after performing the SI

cancellation as [28], [35], [36],

RSI =PS2HS I ,(1)

where HSI =|hS I |2is the RSI channel gain of the small cell

BS, and indicates the SI cancellation capability of that BS,

where hSI is the SI channel of the BS. Note that RSI = 0

denotes perfect cancellation capability.

The residual self-interfering channel gain HSI in (1) needs

to be characterized based on the applied SI cancellation

algorithm. Here, we consider the digital-domain cancellation,

where hSI can be presented as hSI =hSIc−ˆ

hSIcwhere

hSIcand ˆ

hSIcare the self-interfering channel and its estimate

as the self-interference is subtracted using the estimate [28],

[35]–[37], which allows HSI to be modeled as a constant

value, such that HSI =σ2

efor the estimation error variance

σ2

e[28], [35], [37]. Other SI cancellation algorithms, such as

analogue domain algorithms [38]–[40] or propagation domain

algorithms [36], [41], [42] will make the modeling of HSI

challenging. Therefore, in our analysis, we consider HSI to be

a constant value. Please note that the analysis can still be easily

extended to the case of random HSI within our framework.

For instance, once the probability density function (PDF) of

HSI is available for a certain SI cancellation algorithm, by

averaging the analytic results presented in this paper, over the

distribution of HSI , the results for the random HSI can be

derived.

We consider the maximum received power cell association

rule in the downlink transmission of HCNs, adopting the

ﬂexible cell association without biasing [26]. In our case, the

association probability Afor the macrocells and the small cells

can be expressed by

AS1= 1 − AS2= 1 − 1 + λS1

λS2PS1

PS22/α2!−1

.(2)

and

AS2=PPr

S2> P r

S1= 1 + λS1

λS2PS1

PS22/α1!−1

,(3)

respectively, In (2) and (3), Pr

S1and Pr

S2are the received

power at the associating user from the macrocell and small

cell BSs, respectively. Moreover, α1and α2are the path loss

exponents of macrocells and small cells, respectively.

In this paper, we assume that the HCNs employs eICIC

technique for interference mitigation due to it’s wide usage

and popularity, with ABS transmission factor of ρdeﬁned as

the ratio of ABS transmitted over the total transmitted frames.

A. Downlink SINR of Macrocell User

For a typical macrocell downlink user located at the origin

u0

1, associated with its serving macrocell BS S∗

1, the SINR is

expressed as

SI N RDL

u1=PS1|hS∗

1,u0

1|2RS∗

1,u0

1

−α1

IUL

u2+IS2+IDL

S1+N0

,(4)

where

IUL

u2=X

u2∈ΦU2

Pu2|hu2,u0

1|2Ru2,u0

1

−α2,

IS2=X

S2∈ΦS2

PS2|hS2,u0

1|2RS2,u0

1

−α2,

IDL

S1=X

S1∈ΦS1\S∗

1

PS1|hS1,u0

1|2RS1,u0

1

−α1.

given IUL

u2is the interference from small cell uplink users,

IS2is the interference from small cell BSs and IDL

S1is the

interference from other macrocell BSs.

In (4), Pu2is the transmit power of UL user associated

with small cell, hS∗

1,u0

1,hu2,u0

1,hS2,u0

1, and hS1,u0

1denote the

small scale fading channel coefﬁcient for the channels of the

typical downlink user and its serving macrocell BS, small cell

users, small cell BSs and other non-associated macrocell BSs,

respectively. Moreover, RS∗

1,u0

1,Ru2,u0

1,RS2,u0

1, and RS1,u0

1

denote the distances between the typical downlink macrocell

user and its associated macrocell BS, small cell users, small

cell BSs, and other interfering macrocell BSs, respectively.

B. Downlink SINR of Small Cell User

For a typical small cell downlink user located at the origin

u0

2, associated with its serving small cell BS S∗

2, the SINR

expression is given by

SI N RDL

u2=PS2|hS∗

2,u0

2|2RS∗

2,u0

2

−α2

IUL

u2+IS2+IDL

S1+N0

,(5)

during non ABS transmission, while SINR expression during

ABS transmission is given by

SI N RDL_AB S

u2=PS2|hS∗

2,u0

2|2RS∗

2,u0

2

−α2

IUL

u2+IS2+N0

,(6)

IEEE ACCESS - SUBMITTED FOR PUBLICATION 4

where

IUL

u2=X

u2∈ΦU2

Pu2|hu2,u0

2|2Ru2,u0

2

−α2,

IS2=X

S2∈ΦS2\S∗

2

PS2|hS2,u0

2|2RS2,u0

2

−α2,

IDL

S1=X

S1∈ΦS1

PS1|hS1,u0

2|2RS1,u0

2

−α1.

given IUL

u2is the interference from small cell uplink users,

IS2is the interference from small cell BSs and IDL

S1is the

interference from other macrocell BSs.

In (5) and (6), hS∗

2,u0

2,hu2,u0

2,hS2,u0

2, and hS1,u0

2denote

the small scale fading channel coefﬁcient for the channels

of the downlink typical small cell user and its serving small

cell BS, small cell users, small cell BSs and macrocell BSs,

respectively. Further, RS∗

2,u0

2,Ru2,u0

1,RS2,u0

1, and RS1,u0

1

denote the distances between the typical small cell downlink

user and its associated small cell BS, small cell users, other

interfering small cell BSs, and macrocell BSs, respectively.

C. Uplink SINR of Small Cell BS

We assume that UL users utilize distance-proportional frac-

tional power control of the form Rα

x[43], where ∈[0,1] is

the power control factor. Therefore, as users moves closer to

the associated BS, the transmit power required to maintain the

same received signal power decreases, which is a key issue for

battery-limited users.

For a typical small cell BS in the uplink located at the origin

S0

2, the SINR can be expressed as

SI N RU L

S2=Pu2|hu∗

2,S0

2|2Ru∗

2,S0

2

α2(−1)

RSI +IU L

u2+IS2+IDL

S1+N0

,(7)

during non ABS transmission, while SINR expression during

ABS transmission is given by

SI N RU L_ABS

S2=Pu2|hu∗

2,S0

2|2Ru∗

2,S0

2

α2(−1)

RSI +IU L

u2+IS2+N0

,(8)

where

IUL

u2=X

u2∈ΦU2

Pu2|hu2,S0

2|2Ru2,S0

2

α2,

IS2=X

S2∈ΦS2\S∗

2

PS2|hS2,S0

2|2RS2,S0

2

α2,

IDL

S1=X

S1∈ΦS1

PS1|hS1,S0

2|2RS1,S0

2

α1.

given IUL

u2denotes the interference from other small cell

uplink users, IS2is the interference from other small cell BSs

and IDL

S1is the interference from macrocell BSs.

When = 1, the numerator of (7) becomes Pu2|hu∗

2,S0

2|2,

with the pathloss completely inverted by the power control,

and when = 0, no channel inversion is performed and all

the nodes transmit using the same power.

In (7), hu∗

2,S0

2,hu2,S0

2,hS2,S0

2, and hS1,S0

2denote the small

scale fading channel coefﬁcient for the channels of small

cell uplink BS and its associated small cell uplink user,

other interfering small cell uplink users, other small cell BSs

and macrocell BSs, respectively. Moreover, Ru∗

2,S0

2,Ru2,S0

2,

RS2,S0

2, and RS1,S0

2denote the distances between the typical

small cell uplink BS and its associated small cell uplink user,

other interfering small cell uplink users, other small cell BSs

and macrocell BSs, respectively.

III. OUTAG E PROBABILITY ANALYS IS

In this section, we analyze the outage probability of two-tier

HCNs with FD small cells, which is a metric that represents

the average fraction of the cell area that is in outage at any

time. We deﬁne the outage probability Oas the probability

that the instantaneous SINR of a randomly located user is less

than a target SINR τ. Since the typical user is associated with

at most one tier, from the law of total probability, the outage

probability is given as

O=

K

X

k=1

OkAk,(9)

where Akis the per-tier association probability given in (3)

and (2), and Okis the outage probability of a typical user

associated with kth tier, and Kdenotes the number of tiers.

For a target SINR τkand a typical user SINRk(x)at a distance

xfrom its associated BS, the outage probability is given by

Ok=E[P[SI N Rk(x)< τk]] .(10)

Considering the chosen network model of HD macrocells

and FD small cells, the expression of the outage probability

becomes

O=ODL

1A1+ (ODL

2+OUL

2)A2,(13)

where ODL

1,ODL

2and OUL

2denote the outage probability of

macrocell downlink user, small cell downlink user, and small

cell uplink BS, respectively, and are derived in the following

section.

A. Outage Probability of Macrocell Downlink User

The probability density function (PDF) of the distance be-

tween the typical macrocell user and the associated macrocell

BS RS∗

1,u0

1[26], is given by

fRS∗

1,u0

1

(r) = 2πλS1

AS1

rexp (−π

2

X

j=1

λj(PSj

PS1

)

2

αj/α1),

(14)

where AS1is given in (2).

Theorem 1. The outage probability ODL

1in HCNs com-

prised of HD macrocell and FD small cell, is deﬁned as

the probability that the instantaneous SINR of a randomly

located macrocell downlink user is lower than a target τ1,

and expressed as

IEEE ACCESS - SUBMITTED FOR PUBLICATION 5

ODL

2=1 −(2π(1 −ρ)λS2

AS2Z∞

0

rexp n−rα2PS2

−1N0τ2−π(η1r2)+(η2r2

α2/α1)+(η3r2

α1/α2)odr

+2πρλS2

AS2Z∞

0

rexp n−rα2PS2

−1N0τ2−π(η1r2)+(η2r2

α2/α1)odr),(11)

OUL

2=1 −(2π(1 −ρ)λS2Z∞

0

rexp n−rα2(−1)Pu2

−1PS2σ2

eN0τ3−π(Γ1r2) + (Γ2r2) + (Γ3r2

α1/α2)odr

+ 2πρλS2Z∞

0

rexp n−rα2(−1)Pu2

−1PS2σ2

eN0τ3−π(Γ1r2) + (Γ2r2)odr),(12)

ODL

1= 1 −(2πλS1

AS1Z∞

0

rexp n−rα1PS1

−1N0τ1

−π(Ψ1r2

α2/α1) + (Ψ2r2

α2/α1) + (Ψ3r2)odr),(15)

where

Ψ1=λu2Pu2

PS12/α22τ1

α2−22F1[1,1−2

α2

; 2 −2

α2

;−τ1],

Ψ2=λS2PS2

PS12/α22τ1

α2−22F1[1,1−2

α2

; 2 −2

α2

;−τ1],

Ψ3=λS1PS1

PS∗

12/α12τ1

α1−22F1[1,1−2

α1

; 2 −2

α1

;−τ1],

with 2F1[·]denote the Gauss hypergeometric function, and the

pathloss exponents αj>2.

Proof. See Appendix A.

B. Outage Probability of Small Cell Downlink User

The PDF of the distance between the typical small cell

downlink user and its associated BS RS∗

2,u0

2[26], is given

by

fRS∗

2,u0

2

(r) = 2πλS2

AS2

rexp n−π

2

X

j=1

λj(PSj/PS2)

2

αj/α2o.

(16)

Theorem 2. The outage probability ODL

2in HCNs com-

prised of HD macrocell and FD small cell, is deﬁned as

the probability that the instantaneous SINR of a randomly

located small cell downlink user is lower than a target τ2,

during transmission of both ABS and non-ABS subframes, and

expressed as (11) at the top of this page, where

η1=λu2Pu2

PS22/α22τ2

α2−22F1[1,1−2

α2

; 2 −2

α2

;−τ2],

η2=λS2PS2

PS∗

22/α22τ2

α2−22F1[1,1−2

α2

; 2 −2

α2

;−τ2],

η3=λS1PS1

PS22/α12τ2

α1−22F1[1,1−2

α1

; 2 −2

α1

;−τ2],

for the pathloss exponents αj>2.

Proof. See Appendix B

C. Outage Probability of Small Cell Uplink BS

Since macrocells can only service one DL active user at a

time, the UL users can only be associated to the FD small

cells. Therefore, we assume that UL users are associated with

the small cells based on the nearest BS association rule, where

the PDF of the distance between the UL users and the small

cells RS2,uUL

2[26], is given as

fRS2,uUL

2

(r) = e−λ2πr22πλ2r. (17)

Theorem 3. The outage probability OUL

2in HCNs comprised

of HD macrocell and FD small cell, is deﬁned as the proba-

bility that the instantaneous SINR of a randomly located UL

small cell BS is lower than a target τ3during both ABS and

non-ABS subframes is given by (12) at the top of this page,

where

Γ1=λu2

2τ3

α2−22F1[1,1−2

α2; 2 −2

α2−τ3]

Γ2=λS2PS2

Pu22/α22τ3

α2−22F1[1,1−2

α2; 2 −2

α2−τ3]

Γ3=λS1PS1

Pu22/α12τ3

α1−22F1[1,1−2

α1; 2 −2

α1;−τ3],

for αj>2.

Proof. See Appendix C.

IEEE ACCESS - SUBMITTED FOR PUBLICATION 6

2π(1 −ρ)λS2

AS2Z∞

0

P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I> $!rexp

−π

2

X

j=1

λj(Pj

PS2

)

2

αj/α2r

2

αj/α2

dr

+2πρλS2

AS2Z∞

0

P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I0> $!rexp

−π

2

X

j=1

λj(Pj

PS2

)

2

αj/α2r

2

αj/α2

dr. (18)

IV. RATE COVERAGE ANALYS IS

In this section, we analyze the rate coverage of two-tier

HCNs with FD small cells. The rate coverage is deﬁned in [44]

that the probability that a randomly chosen user can achieve

a target rate $, which is given by

Θ,P(R > $).(19)

Since the DL users can associate with either macro cells or

small cells in open-access mode, the overall rate coverage for

the chosen user in two-tier HCNs is given by

Θo=AS1P(RS1> $|AS1) + AS2P(RS2> $|AS2),(20)

where AS1and AS2denote the probability that a user

is associated with the macrocell or the small cell, and

P(RS1> $|AS1)and P(RS2> $|AS2)denote the rate cov-

erage conditioned on the association with the former and the

latter, respectively.

The rate achieved by a user associated with the tagged BS

in the xth-tier is given by

Rx=W

Nx

log2(1 + SI N Rx),(21)

where Wis the bandwidth of the frequency band, Nxis a

random variable which denotes the average number of users

associated with the tagged base station in the xth-tier, and

SI N Rxis the received signal-to-interference-plus-noise-ratio

from the serving base station for a user.

A. Rate Coverage for Macrocell Users in the Downlink

In Rayleigh fading environments, the rate coverage for a

macrocell DL user is given by

P(RS1> $|AS1)

=ENS1PSI N RDL

u1>2

$Nf

W−1|AS1

=X

n≥0

P PS1hS∗

1,u0

1RS∗

1,u0

1

−α1

I> $|AS1!

×P(NS2=n+ 1) ,(22)

where I=IUL

u2+IS2+IDL

S1+N0is the cumulative interference

from small cell UL users along with macrocell and small cell

BSs, and the additive Gaussian noise. $= 2

$NS1

W−1.

According to [44], the distribution of the load associated

with the xth-tier is given by

P(Nx=n+ 1) =

3.53.5

n!

Γ (n+ 4.5)

Γ (3.5) λuAx

λxn3.5 + λuAx

λx−n−4.5

,(23)

with the mean load E[Nx] = 1 + 1.28λuAx

λx, where Γ (x) =

R∞

0tx−1e−tdt is the Gamma function, and Axdenotes the

association probability of the xth-tier. Hence,

P PS1hS∗

1,u0

1RS∗

1,u0

1

−α1

I> $|AS1!

=Z∞

0

P PS1hS∗

1,u0

1RS∗

1,u0

1

−α1

I> $!fRS∗

1,u0

1

(r)

=2πλS1

AS1Z∞

0

P PS1hS∗

1,u0

1RS∗

1,u0

1

−α1

I> $!r

×exp

−π

2

X

j=1

λj(Pj

PS1

)

2

αj/α1r

2

αj/α1

dr. (24)

Using the derivation of outage probability for macrocell DL

users and (23), the ﬁnal expression for DL rate coverage of

macrocell users can be obtained through (22).

B. Rate Coverage for Small Cell Users in the Downlink

Following the same derivation approach, the rate coverage

for DL small cell users is given by

P(RS2> $|AS2)

=ENS2ρPSI N RDL_AB S

u2>2

$Nf

W−1|AS2

+ (1 −ρ)PSI N RDL

u2>2

$Nf

W−1|AS2

="(1 −ρ)X

n≥0

P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I> $|AS2!

+ρX

n≥0

P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I0> $|AS2!#

×P(NS2=n+ 1) ,(26)

where I=IUL

u2+IS2+IDL

S1+N0denote the cumulative

interference from small cell UL users along with macrocell

and small cell BSs, receptively. I0=IUL

u2+IS2+N0is the

cumulative interference during ABS transmission, and $=

2

$NS1

W−1.

IEEE ACCESS - SUBMITTED FOR PUBLICATION 7

2π(1 −ρ)λS2Z∞

0

P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I+RSI > $!rexp

−π

2

X

j=1

λj(Pj

PS2

)

2

αj/α2r

2

αj/α2

dr

+ 2πρλS2Z∞

0

P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I0+RSI > $!rexp

−π

2

X

j=1

λj(Pj

PS2

)

2

αj/α2r

2

αj/α2

dr. (25)

Using the load distribution given in (23), (27) and (28), we

obtain (18) at the top of this page

(1 −ρ)P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I> $|AS2!= (1 −ρ)

×Z∞

0

P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I> $!fRS∗

2,u0

2

(r).

(27)

ρP PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I0> $|AS2!=

ρZ∞

0

P PS2hS∗

2,u0

2RS∗

2,u0

2

−α2

I0> $!fRS∗

2,u0

2

(r).

(28)

Finally, Using the derivation of outage probability for small

cell DL users and (23), the ﬁnal expression for rate coverage

of small cell DL users can be obtained through (26).

C. Rate Coverage for Small cell BS in the Uplink

Similarly, the rate coverage for UL small cell BS is given

by

P(RS2> $|AS2)

=ENS2ρPSI N RU L_ABS

u2>2

$Nf

W−1|AS2

+ (1 −ρ)PSI N RU L

u2>2

$Nf

W−1|AS2.

="(1 −ρ)X

n≥0

P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I+RSI > $|AS2!

+ρX

n≥0

P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I0+RSI > $|AS2!#

×P(NS2=n+ 1) ,

(29)

where I=IUL

u2+IS2+IDL

S1+N0is the cumulative interference

from UL small cell users along with macrocell and small cell

BSs, and the Gaussian additive noise. I0=IUL

u2+IS2+N0

is the cumulative interference during ABS transmission, and

$= 2

$NS1

W−1.

Using the load distribution given in (23) and both (30) and

(31) we obtain (25) at the top of this page

(1 −ρ)P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I+RSI > $|AS2!=

(1 −ρ)Z∞

0

P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I+RSI > $!fRS2,uUL

2

(r).

(30)

TABLE II: Parametric Values (unless otherwise speciﬁed)

Parameter Value

λx∀x(π×5002)−1

PS1[dBm] 43 dBm (20 W)

PS,2[dBm] 23 dBm (200 mW)

Puy∀y[dBm] 23 dBm (200 mW)

W[Hz] 107

αk∀k4

τn∀n[dB] 0dB

RSI PS210LdB /10

LdB [dB] −38 dB

ρ0.3

0.2

ρP PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I+RSI > $|AS2!=

ρZ∞

0

P PS2hu0

2,S∗

2Ru0

2,S∗

2

−α2(−1)

I+RSI > $!fRS2,uUL

2

(r).

(31)

Finally, Using the derivation of outage probability for UL

small cell BS and (23), the ﬁnal expression for rate coverage

of UL small cell BS can be obtained through (29).

V. NUMERICAL AND SIMULATION RESULTS

In this section, we evaluate the performance of two-tier

HCNs with FD small cells. Speciﬁcally, we investigate how

different parameters affect network performance in terms of

the outage probability and the rate coverage. The simulation

methodology comprises independent realization of PPP dis-

tributions for the BSs of two tiers, followed by realization of

user distribution and the association process. After that, outage

probability and rate coverage are calculated based on the

cumulative interference. The parameters used for the analysis

and simulation are stated in Table II. Monte Carlo simulations

have been conducted to obtain the results, averaged over 10000

iterations, which are then compared with numerical evaluation

of the derived expressions.

Fig. 2 plots the outage probability of a typical DL user

associated with macrocell BS, small cell BS, and random type

of BS in the DL, as a function of small cell BSs density λ2.

We observe that the outage probability of macrocell DL user

increases with increasing the small cell BS density. This results

from the increase in aggregate interference from the small

cell BSs, as shown in (4). Additionally, the outage probability

of macrocell DL user decreases with increasing the transmit

power at the macrocell BS, which is due to the increase in

SINR at the typical downlink user associated with macrocell

IEEE ACCESS - SUBMITTED FOR PUBLICATION 8

12345678910

Small Cell BS Density λ2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Downlink Outage Probability

Small cell DL PS1 =40 dBm

Small cell DL PS1 =43 dBm

Macrocell DL PS1 =40 dBm

Macrocell DL PS1 =43 dBm

Random DL PS1 =40 dBm

Random DL PS1 =43 dBm

Simulation

× ( � 500 )-1

2

Fig. 2: Outage probability of macrocell and small cell down-

link as a function of small cell density λ2.

12345678910

Small Cell BS Density λ2×-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Outage Probability

Macrocell DL

Small cell DL

Small cell UL, =0.2

Small cell UL, =0.7

Random User, =0.2

Random User, =0.7

Simulation

( � 500 )

2

Fig. 3: Outage probability as a function of small cell density

λ2.

BS, as shown in (4). Interestingly, for the typical downlink user

associated with small cell BS, the outage probability decreases

with increasing the small cell BS density. This is because

densiﬁcation of tier 2 reduces the inter-link distances between

the typical downlink user and the associated small cell BS,

as shown in (5). In addition, the outage probability of typical

small cell DL user increase with increasing the transmit power

at the macrocell BS, which is due to the increase in aggregate

interference caused by macrocell BSs, as shown in (5). Finally,

outage probability of a random DL user, which is deﬁned as

ODL

1A1+ODL

2A2, increases with both the increase of small

cell density, and the decrease of transmit power of macrocell

BS. This is because ODL

1in the expression is lower than ODL

2,

therefore the expression reﬂects such tendency. Note that the

simulation results closely follow the analytical results, and

therefore, validate the analytical modeling.

12345678910

Small Cell BS Density λ2 ×( � 500 )

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Rate Coverage Probability

UL PS2 =23 dBm, =0.7

UL PS2 =23 dBm, =0.2

UL PS2 =26 dBm, =0.7

UL PS2 =26 dBm, =0.2

DL PS2 =23 dBm

DL PS2 =26 dBm

Simulation

2-1

Fig. 4: Rate coverage as a function of small cell density λ2.

Fig. 3 plots the outage probability of macrocell DL user,

small cell DL user, small cell UL BS, and a randomly located

user versus the density of small cell BSs. In this ﬁgure,

we focus on the impact of small cell BSs density on the

outage probability of a randomly located user. Interestingly,

we observe that the outage probability of a randomly located

user is not signiﬁcantly affected by the increase in the small

cell BS density. It suffers from slight increase that results from

aggregate interference from the small cell BSs, as shown in

(13). We also evaluate the impact of uplink power control

factor, on outage. As shown by the results, a higher value of

results in a higher outage probability, for small cell user in the

uplink, due to reduced uplink transmit power as a consequence

of more aggressive power control. The simulation results also

closely follow the analytical results.

Fig. 4 plots rate coverage for a random DL user and UL

small cell BS versus the small cell BSs density. We note that

the rate coverage of a random DL user decrease as the density

of small cell BSs increases. This is because of increase in

aggregate interference caused by small cell BSs, as seen in

(4). Similarly, the rate coverage of a random DL user decreases

with increasing the transmission power of small cell BSs. On

the contrary, rate coverage of an UL small cell BS increase

with increase of small cell BSs density. This is due to the

fact that densiﬁcation reduces the inter-link distance between

a user and it’s associated BS, which can be veriﬁed by (7).

Similarly, the rate coverage of an UL small cell BS increases

with increasing the transmission power of small cell BSs due

to higher SINR of small cell UL BS as can be veriﬁed by (7).

We also evaluate the impact of uplink power control factor,

on rate coverage. As shown by the results, a higher value of

results in a lower rate coverage probability, for small cell

user in the uplink, due to reduced uplink transmit power as a

consequence of more aggressive power control.

Fig. 5 plots the rate coverage of UL small cell BSs as a

function of the ABS factor ρ. We note that the rate coverage of

UL small cell BSs increases as ρincreases. This is because of

the aggregate interference caused by macrocell BSs, which can

IEEE ACCESS - SUBMITTED FOR PUBLICATION 9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ABS factor ρ

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

UL Rate Coverage Probability

λ2 = λ1

λ2 = 2λ1

λ2 = 5λ1

λ2 = 10λ1

Simulation

Fig. 5: UL rate coverage probability as a function of ABS

factor ρ.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ABS factor ρ

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

DL Rate Coverage Probability

λ2 = λ1

λ2 = 2λ1

λ2 = 5λ1

λ2 = 10λ1

Simulation

Fig. 6: DL rate coverage probability as a function of ABS

factor ρ.

be seen in (7) and (8). Similarly, in Fig. 6, the rate coverage

of random DL users increases with increasing ρfor the same

reason, which can be seen in (5) and (6). Fig. 7 shows that the

outage probability of a random user decreases as ρincreases.

This is due to the fact that interference originated by macrocell

BSs decreases with increasing ρ, as seen in (4) and (6).

Fig. 8 plots the relation between small cell UL rate cov-

erage probability and the residual SI cancellation RSI =

PS2.10LdB/10 , where LdB is the ratio of RSI after interfer-

ence cancellation is applied to the transmission power at the

receiver. We observe that outage probability of a randomly

located user is initially high, especially when SI cancellation

capability is low (LdB <−15 ), then it decreases with increas-

ing LdB, until it nearly stabilise beyond (LdB >−37 ). This

is because the high SI cancellation capabilities improve the

performance of FD links as can be seen in (7). Additionally, we

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ABS factor ρ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Outage Probability

λ2 = λ1

λ2 = 2λ1

λ2 = 5λ1

λ2 = 10λ1

Simulation

Fig. 7: Outage probability in relation to ABS transmission

factor ρ.

-50-40-30-20-100

LdB

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Outage Probability

λ2 = λ1

λ2 = 2λ1

λ2 = 5 λ 1

λ2 = 10λ1

Fig. 8: Outage probability as a function of the SI cancellation

capability LdB.

observe that the outage probability in high small cell densities

is more sensitive to LdB variations. This is due to increased

FD links in higher small cell densities since only the small

cell BSs operate in FD mode.

In Fig. 9, we plot the relation between small cell UL rate

coverage probability and SI cancellation capability LdB. Since

only the small cell BSs operate in FD mode, SI only applies

to those BSs. We note that the rate coverage increases with

the increase of LdB. This is because higher SI cancellation

improves the performance of FD links, as can be seen in (7)

and (8). Moreover, increasing the density of small cell BSs

increases the rate coverage. This is due to the fact that more

FD links exist in higher small cell densities.

VI. CONCLUSIONS

Unprecedented technological developments like network

densiﬁcation and FD communications will be crucial in shap-

IEEE ACCESS - SUBMITTED FOR PUBLICATION 10

-50-40-30-20-100

LdB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

UL Rate Coverage Probability

λ2 = λ1

λ2 = 2λ1

λ2 = 5λ1

λ2 = 10λ1

Fig. 9: UL rate coverage probability as a function of the self-

interference cancellation capability LdB.

ing 5G radio access networks for achieving the envisioned

capacity objectives. Realizing the FD capability at small cells

is particularly attractive due to simplicity, superior SI cancel-

lation (compared to macrocells), and widespread deployment.

In this paper, we have investigated the performance of two-

tier interference-coordinated HCNs with FD small cells. We

have derived closed-form expressions for outage probability

and rate coverage in two-tier HCNs with FD small cells

explicitly accounting for interference coordination between

macro and small cells. Performance evaluation investigates the

impact of different network parameters on both outage and rate

coverage probabilities. The results demonstrate that the outage

probability and the rate coverage improves with higher ABS

factor and better underlying SI cancellation capabilities of FD

small cells.

APPENDIX

A. Proof of Theorem 1.

From (10), the outage probability ODL

1is given as

ODL

1=EPSI N RDL

u1< τ1

=1 −Z∞

0

PSI N RDL

u1> τ1fRS∗

1,u0

1

(r)dr

=1 −2πλS1

AS1Z∞

0

PSI N RDL

u1> τ1r

×exp n−π

2

X

j=1

λj(PSj

PS1

)

ζ1

rζ1odr, (32)

where ζ1=2

αj/α1.

By setting Q1=IUL

u2+IS2+IDL

S1+N0, We rewrite

SI N RDL

u1as hS∗

1,u1

PS1

−1rα1Q1. Therefore,

PSI N RDL

u1> τ1=PhS1,u1> PS1

−1rα1τ1Q1

= exp −rα1PS1

−1N0τ1LIUL

u2(rα1PS1

−1τ1)

× LIS2(rα1PS1

−1τ1)LIDL

u1(rα1PS1

−1τ1).(33)

Starting with the Laplace transform of the interference

originated from small cell UL users on the macrocell DL user,

presented in (33), we have

LIUL

u2(rα1PS1

−1τ1) = EIUL

u2exp −rα1PS1

−1τ1IUL

u2

=EΦU2hexp n−rα1Pu2

PS1

τ1X

u2∈ΦU2

hu2,u0

1Ru2,u0

1

−α2oi

(a)

=

exp n−2πλu2Z∞

0

1− LhS1,u1rα1Pu2

PS1

τ1x−α2xdxo

= exp n−2πλS1Z∞

01−1

1 + rα1PS1

PS∗

1

τ1x−α1xdxo

= exp n−2πλu2Z∞

0

x

1 + rα1Pu2

PS1τ1−1xα2

dxo.(34)

where (a) is provided in [45]. Note that the integration limits

in (34) are from 0 to ∞since the small cell UL users can

be at any distance from the DL macrocell users. Now, with a

change of variables

v1= (rα1Pu2

PS1τ1)−2/α2x2,We express

LIUL

u2(rα1PS1

−1τ1) = exp n−πλu2Pu2

PS12/α2

×Z1(τ1, α2)r2

α2/α1o,(35)

where

Z1(τ1, α2) = τ2/α2

1Z∞

x1

1

1 + vα2/2

1

dv1

=2τ1

α2−22F1[1,1−2/α2; 2 −2/α2;−τ1].(36)

In (36), 2F1[·]denote the Gauss hypergeometric function

and x1= (1/τ1)2/α2. The expression holds for α2>2.

Similarly, we can derive the Laplace transform for the

interference from small cells BSs expressed in (33), as

LIS2(rα1PS1

−1τ1) = exp n−πλS2(PS2/PS1)2/α2

×2τ1

α2−22F1[1,1−2/α2; 2 −2/α2;−τ1]o,(37)

for α2>2. We ﬁnally derive the Laplace transform for the

interference originated from macrocell BSs expressed in (33)

using similar approach, as

LIDL

S1(rα1PS1

−1τ1) = exp n−πλS1PS1/PS∗

12/α1

×2τ1

α1−22F1[1,1−2/α1,−τ1]o,(38)

for α1>2. Now, plugging (35), (37) and (38) into

P[SI N RDL

u1> τ1]we obtain

IEEE ACCESS - SUBMITTED FOR PUBLICATION 11

P[SI N RDL

u1> τ1] = exp n−rα1PS1

−1N0τ1

−πΨ1r2

α2/α1+Ψ2r2

α2/α1+Ψ3r2o.(39)

given

Ψ1=λu2Pu2

PS12/α22τ1

α2−22F1[1,1−2/α2; 2 −2/α2;−τ1],

Ψ2=λS2PS2

PS12/α22τ1

α2−22F1[1,1−2/α2; 2 −2/α2;−τ1],

Ψ3=λS1PS1

PS∗

12/α12τ1

α1−22F1[1,1−2/α1; 2 −2/α2;−τ1],

where αj>2. Therefore, the ﬁnal expression for a randomly

located macrocell DL user is given by (15).

B. Proof of Theorem 2.

From (10), the outage probability ODL

2, considering the

adapted eICIC mechanism is given by

ODL

2=(1 −ρ)EhPSI N RDL

u2< τ2i

+ρEhPSI N RDL_AB S

u2< τ2i,(40)

where ρis the ABS transmission ratio.

Starting by the ﬁrst term of (40), we have

(1 −ρ)EPSI N RDL

u2> τ2

= 1 −(1 −ρ)Z∞

0

PSI N RDL

u2> τ2fRS∗

2,u0

2

(r)dr

= 1 −2π(1 −ρ)λS2

AS2Z∞

0

PSI N RDL

u2> τ2r

×exp n−π

2

X

j=1

λj(Pj

PS2

)

ζ2

rζ2odr,

(41)

where ζ2=2

αj/α2.

By setting Q2=IUL

u2+IS2+IDL

S1+N0, we get

PSI N RDL

u2> τ2=PhS2,u2> PS2

−1rα2τ2Q2

=Z∞

0

exp{−rα2PS2

−1τ2q}fQ2(q)dq

=EQ2exp{−rα2PS2

−1τ2q}

= exp −rα2PS2

−1N0τ2LIUL

u2(rα2PS2

−1τ2)

× LIS2(r2PS2

−1τ2)LIDL

S1(rα2PS2

−1τ2).(42)

Following the approach presented in Appendix A, we can

obtain the Laplace transforms in (42), starting with the Laplace

transform of the interference originated from UL to DL small

cell users as follows

LIUL

u2(rα2PS2

−1τ2) = exp n−πr2λu2Pu2

PS22/α2

Y1(τ2, α2)o

(43)

given

Y1(τ2, α2) = τ2/α2

2Z∞

τ2−(2/α2)

1

1 + yα2/2

1

dy1

=2τ2

α2−22F1[1,1−2

α2

; 2 −2

α2

;−τ2],(44)

where α2>2, and y1= (rα2PS2

PS∗

2

τ2)−2/α2r2.

Similarly, the second Laplace transform in (42) of the

interference originated from small cell BS on DL small cell

user is given as

LIS2(rα2PS2

−1τ2) = exp n−πr2λS2PS2

PS∗

22/α2

Y1(τ2, α2)o

(45)

The Laplace transform of the interference originated from

macrocell BS on DL small cell user is given as

LIDL

S1(rα2PS2

−1τ2) = exp n−πλS1PS1

PS22/α1

×Y2(τ2, α1)r2

α1/α2o,(46)

given

Y2(τ2, α1) = τ2/α1

2Z∞

τ2−(2/α1)

1

1 + yα1/2

2

dy2

=2τ2

α1−22F1[1,1−2

α1

; 2 −2

α1

;−τ2],(47)

where α1>2, and y2= (rα2PS1

PS2τ2)−2/α1r2.

Plugging (43), (45) and (46) into PSI N RDL

u2> τ2we

obtain

PhSI N RDL

u2> τ2i= exp n−rα2PS2

−1N0τ2−π(η1r2)

+ (η2r2

α2/α1)+(η3r2

α1/α2)o,(48)

where

η1=λu2Pu2

PS22/α2

Y1(τ2, α2)

η2=λS2PS2

PS∗

22/α2

Y1(τ2, α2)

η3=λS1PS1

PS22/α1

Y2(τ2, α1).

Similarly, the analysis of the second term in (40) of the

outage probability during ABS subframes transmission is

given as follows.

First, we consider the SINR expressed in (6) for the ABS

subframes transmission. By setting Q∗

2=IUL

u2+IS2+N0, we

have

PSI N RDL_AB S

u2> τ2=PhS2,u2> PS2

−1rα2τ2Q∗

2

=Z∞

0

exp{−rα2PS2

−1τ2q}fQ∗

2(q)dq

=EQ∗

2exp{−rα2PS2

−1τ2q}

= exp −rα2PS2

−1N0τ2LIUL

u2(rα2PS2

−1τ2)

× LIS2(rα2PS2

−1τ2).(49)

IEEE ACCESS - SUBMITTED FOR PUBLICATION 12

Since we have previously derived LIUL

u2(rα2PS2

−1τ2)

and LIS2(rα2PS2

−1τ2), we can obtain the probability

PSI N RDL_AB S

u2> τ2as

PSI N RDL_AB S

u2> τ2= exp n−rα2PS2

−1N0τ2

−π(η1r2)+(η2r2

α2/α1)o.(50)

Therefore, the ﬁnal expression for the outage probability for

a randomly located DL small cell user, considering eICIC is

given in (11).

C. Proof of Theorem 3.

From (10), The outage probability OUL

2can be obtained by

OUL

2=(1 −ρ)EhPSI N RU L

S2< τ3i

+ρEhPSI N RU L_ABS

S2< τ3i(51)

Starting by the ﬁrst term of (51), we have

(1 −ρ)EPSI N RU L

S2> τ2

= 1 −(1 −ρ)Z∞

0

PSI N RU L

S2> τ2fRS2,uUL

2

(r)dr

= 1 −2π(1 −ρ)λS2Z∞

0

PSI N RU L

S2> τ2r

×exp −πλ2r2dr,(52)

Following the same steps used in previous derivations,

taking into account the power control factor , we obtain the

ﬁnal expression of the UL small cell outage probability as

given in (12).

REFERENCES

[1] A. Gohil, H. Modi, and S. K. Patel, “5G technology of mobile commu-

nication: A survey,” IEEE ISS, pp. 288–292, Mar. 2013.

[2] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characteri-

zation of full-duplex wireless systems,” IEEE Trans. Wireless Commun.,

vol. 11, no. 12, pp. 4296–4307, May 2012.

[3] D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” SIGCOMM

Comput. Commun. Rev., vol. 43, no. 4, pp. 375–386, Aug. 2013.

[4] M. Duarte, A. Sabharwal, V. Aggarwal, R. Jana, K. Ramakrishnan,

C. W. Rice, and N. Shankaranarayanan, “Design and characterization

of a full-duplex multiantenna system for wiﬁ networks,” IEEE Trans.

Veh. Technol, vol. 63, no. 3, pp. 1160–1177, Nov. 2014.

[5] H. Ju, X. Shang, H. V. Poor, and D. Hong, “Rate improvement of

beamforming systems via bi-directional use of spatial resources,” in

IEEE GLOBECOM, Dec. 2011, pp. 1–5.

[6] B. P. Day, A. R. Margetts, D. W. Bliss, and P. Schniter, “Full-duplex

MIMO relaying: Achievable rates under limited dynamic range,” IEEE

J. Sel. Areas Commun., vol. 30, no. 8, pp. 1541–1553, Sep. 2012.

[7] C. Cox and E. Ackerman, “Demonstration of a single-aperture, full-

duplex communication system,” IEEE RWS, pp. 148–150, Jan. 2013.

[8] M. Knox, “Single antenna full duplex communications using a common

carrier,” IEEE WAMICON, pp. 1–6, Apr. 2012.

[9] H. Ju, X. Shang, H. V. Poor, and D. Hong, “Bi-directional use of

spatial resources and effects of spatial correlation,” IEEE Trans. Wireless

Commun., vol. 10, no. 10, pp. 3368–3379, Nov. 2011.

[10] M. Al-Kadri, A. Aijaz, and A. Nallanathan, “Ergodic capacity of

interference coordinated hetnet with full-duplex small cells,” EW, pp.

1–6, May 2015.

[11] D. Kim, S. Park, H. Ju, and D. Hong, “Transmission Capacity of Full-

Duplex-Based Two-Way Ad Hoc Networks With ARQ Protocol,” IEEE

Trans. Veh. Technol, vol. 63, no. 7, pp. 3167–3183, Sep. 2014.

[12] H. Ju, E. Oh, and D. Hong, “Catching resource-devouring worms in

next-generation wireless relay systems: two-way relay and full-duplex

relay,” IEEE Commun. Mag., vol. 47, no. 9, pp. 58–65, Oct. 2009.

[13] G. Zheng, I. Krikidis, J. Li, A. P. Petropulu, and B. Ottersten, “Improving

physical layer secrecy using full-duplex jamming receivers,” IEEE Trans.

Signal Process., vol. 61, no. 20, pp. 4962–4974, Jun. 2013.

[14] M. Al-Kadri, A. Aijaz, and A. Nallanathan, “An Energy-Efﬁcient

Full-Duplex MAC Protocol for Distributed Wireless Networks,” IEEE

Wireless Commun. Lett., vol. 5, no. 1, pp. 44–47, Feb. 2016.

[15] J. G. Andrews, “Seven ways that hetnets are a cellular paradigm shift,”

IEEE Commun. Mag, vol. 51, no. 3, pp. 136–144, Mar. 2013.

[16] V. Chandrasekhar, J. Andrews, and A. Gatherer, “Femtocell networks:

a survey,” IEEE Commun. Mag, vol. 46, no. 9, pp. 59–67, Sep. 2008.

[17] R1-104968, “Summary of the Description of Candidate eICIC Solu-

tions,” 3GPP std., Madrid, Spain, Aug. 2010.

[18] M. Win, P. Pinto, and L. Shepp, “A mathematical theory of network

interference and its applications,” Proceedings of the IEEE, vol. 97,

no. 2, pp. 205–230, Feb. 2009.

[19] H. ElSawy, E. Hossain, and M. Haenggi, “Stochastic geometry for

modeling, analysis, and design of multi-tier and cognitive cellular

wireless networks: A survey,” Commun. Surveys Tuts., vol. 15, no. 3,

pp. 996–1019, Jul. 2013.

[20] R. Heath, M. Kountouris, and T. Bai, “Modeling heterogeneous network

interference using poisson point processes,” IEEE Trans. Signal Process.,

vol. 61, no. 16, pp. 4114–4126, Aug. 2013.

[21] U. Schilcher, S. Toumpis, M. Haenggi, A. Crismani, G. Brandner, and

C. Bettstetter, “Interference functionals in poisson networks,” IEEE

Trans. Inf. Theory, vol. 62, no. 1, pp. 370–383, Jan. 2016.

[22] H. Dhillon, R. Ganti, F. Baccelli, and J. Andrews, “Modeling and

analysis of K-tier downlink heterogeneous cellular networks,” IEEE J.

Sel. Areas Commun., vol. 30, no. 3, pp. 550–560, Apr. 2012.

[23] T. Novlan, H. Dhillon, and J. Andrews, “Analytical modeling of uplink

cellular networks,” IEEE Trans. Wireless Commun., vol. 12, no. 6, pp.

2669–2679, Jun. 2013.

[24] H. Dhillon, R. Ganti, and J. Andrews, “Load-aware modeling and

analysis of heterogeneous cellular networks,” IEEE Trans. Wireless

Commun., vol. 12, no. 4, pp. 1666–1677, Apr. 2013.

[25] Y. S. Soh, T. Quek, M. Kountouris, and H. Shin, “Energy efﬁcient

heterogeneous cellular networks,” IEEE J. Sel. Areas Commun., vol. 31,

no. 5, pp. 840–850, May 2013.

[26] H.-S. Jo, Y. J. Sang, P. Xia, and J. Andrews, “Heterogeneous Cellular

Networks with Flexible Cell Association: A Comprehensive Downlink

SINR Analysis,” IEEE Trans. Wireless Commun., vol. 11, no. 10, Oct.

2012.

[27] D. Kim, H. Lee, and D. Hong, “A survey of in-band full-duplex

transmission: From the perspective of phy and mac layers,” Commun.

Surveys Tuts, vol. 17, no. 4, pp. 2017–2046, Fourthquarter 2015.

[28] J. Lee and T. Quek, “Hybrid full-/half-duplex system analysis in het-

erogeneous wireless networks,” IEEE Trans. Wireless Commun, vol. 14,

no. 5, pp. 2883–2895, May 2015.

[29] H. Tabassum, A. H. Sakr, and E. Hossain, “Massive MIMO-Enabled

Wireless Backhauls for Full-Duplex Small Cells,” IEEE GLOBECOM,

pp. 1–6, Dec. 2015.

[30] S. Han, C. Yang, and P. Chen, “Full duplex-assisted intercell interference

cancellation in heterogeneous networks,” IEEE Trans. Commun., vol. 63,

no. 12, pp. 5218–5234, Dec. 2015.

[31] A. AlAmmouri, H. ElSawy, and M. S. Alouini, “Flexible design for

α-duplex communications in multi-tier cellular networks,” IEEE Trans.

Commun., vol. 64, no. 8, pp. 3548–3562, Aug 2016.

[32] S. Goyal, C. Galiotto, N. Marchetti, and S. Panwar, “Throughput and

coverage for a mixed full and half duplex small cell network,” in 2016

IEEE ICC, May 2016, pp. 1–7.

[33] C. Cox and E. Ackerman, “Demonstration of a single-aperture, full-

duplex communication system,” IEEE RWS, pp. 148–150, Jan. 2013.

[34] M. Knox, “Single antenna full duplex communications using a common

carrier,” IEEE WAMICON, pp. 1–6, Apr. 2012.

[35] D. Ng, E. Lo, and R. Schober, “Dynamic Resource Allocation in MIMO-

OFDMA Systems with Full-Duplex and Hybrid Relaying,” IEEE Trans.

Commun., vol. 60, no. 5, pp. 1291–1304, May 2012.

[36] 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.

[37] 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.

IEEE ACCESS - SUBMITTED FOR PUBLICATION 13

[38] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characteri-

zation of full-duplex wireless systems,” IEEE Trans. Wireless Commun,

vol. 11, no. 12, pp. 4296–4307, Dec. 2012.

[39] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “Achieving

single channel, full duplex wireless communication,” ACM MobiCom,

pp. 1–12, Sep. 2010.

[40] M. Jain, J. I. Choi, T. Kim, D. Bharadia, S. Seth, K. Srinivasan, P. Levis,

S. Katti, and P. Sinha, “Practical, real-time, full duplex wireless,” ACM

MobiCom, pp. 301–312, Nov. 2011.

[41] 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.

[42] P. Lioliou, M. Viberg, M. Coldrey, and F. Athley, “Self-interference

suppression in full-duplex MIMO relays,” in IEEE ASILOMAR, Nov.

2010, pp. 658–662.

[43] T. D. Novlan, H. S. Dhillon, and J. G. Andrews, “Analytical modeling

of uplink cellular networks,” IEEE Transactions on Wireless Communi-

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

[44] S. Singh, H. Dhillon, and J. Andrews, “Ofﬂoading in heterogeneous

networks: Modeling, analysis, and design insights,” IEEE Trans. Wireless

Commun, vol. 12, no. 5, pp. 2484–2497, May 2013.

[45] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to

coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59,

no. 11, pp. 3122–3134, Nov. 2011.