-
Contents lists available at ScienceDirect
Computer Communications
journal homepage: www.elsevier.com/locate/comcom
Analytical evaluation of heterogeneous cellular networks under
flexible userassociation and frequency reuse☆
Mehdi Fereydoonia, Masoud Sabaei⁎,a, Mehdi Dehghana, Gita
Babazadeh Eslamloub,Markus Ruppb
a Amirkabir University of Technology, Iranb Technische
Universität Wien, Austria
A R T I C L E I N F O
Keywords:Heterogeneous cellular networksPoisson point
processesFrequency reuseUser association
A B S T R A C T
Offloading mobile users from highly loaded macro base stations
(BSs) to lightly-loaded small cell BSs is criticalfor utilizing the
full potential of heterogeneous cellular networks (HCNs). However,
to alleviate the signal-to-interference-plus-noise ratio (SINR)
degradation of so called biased users, offloading needs to be
activated inconjunction with an efficient interference management
mechanism. Fractional frequency reuse (FFR) is an at-tractive
interference management technique due to its bandwidth efficiency
and its suitability to orthogonalfrequency division multiple access
based cellular networks. This paper introduces a general
mathematical modelto study the potential benefit of load balancing
in conjunction with two main types of FFR interference
co-ordination: Strict-FFR and soft frequency reuse (SFR)- in the
downlink transmissions of HCNs. For some specialbut realistic cases
we were able to reduce the rather complex general mathematical
expressions to much simplerclosed-forms that reveal the basic
properties of BS density on the overall coverage probability. We
show thatalthough Strict-FFR outperforms the SFR mechanism in terms
of SINR and rate coverage probability, it fails toprovide the same
spectral efficiency. Finally, we present a novel resource
allocation mechanism based on the BSsbias values and FFR thresholds
that achieves an even higher minimum user throughput and rate
coverageprobability.
1. Introduction
Cellular network operators are forced to increase their
networkcapacity in order to cope with the rapidly rising demand on
data rate. Inthe recent years the mobile data usage has grown up to
200% perannum [1]. Introducing new tiers that comprise of base
stations (BSs)with smaller transmission ranges (called small cell
BSs), is a potentialand cost effective approach to increase the
capacity of cellular networks[2].
The design of cellular networks with optimal parameter
settingsrequires to have efficient methods and models to analyze
the perfor-mance of heterogeneous cellular networks (HCNs). Because
of the un-controlled nature of small cell BS distributions in HCNs,
conventionalmodels such as Wyner [3] and hexagonal models are
considered to beobsolete for modeling. A recent approach to
describe the random natureof BS locations is to apply point process
theories, leveraging techniquesfrom stochastic geometry [4]. Its
accuracy in abstracting realistic BSdeployments has been validated
in numerous contributions [5–7]. In
[5], the authors showed that even in a single tier network
Poisson pointprocesses (PPPs) provide an accuracy at least as high
as grid models.However, the grid model does not exhibit the same
level of analyticaltractability as PPPs. Owing to their advantages
in tractability and ac-curacy, PPPs have been extensively employed
to model and analyzeHCNs [8] in recent years.
The authors in [5] derived important performance metrics such
ascoverage probability and average rate for a given system model in
asingle-tier cellular network. Their analysis is generalized in [9]
for a K-tier cellular network. They proved that in open access and
interference-limited networks with identical target
signal-to-interference-plus-noiseratio (SINR) for all tiers the
overall coverage probability is independentof the number of tiers
and the density of BSs. The model is furtherdeveloped in [10] to
incorporate a user-based load definition into theanalysis. In [11],
the authors presented a framework to evaluate thecoverage
probability of indoor users in urban two-tier cellular net-works.
In this model, environments are partitioned by walls with acertain
penetration loss to distinguish between the outdoor BSs in line
https://doi.org/10.1016/j.comcom.2017.11.014Received 26 April
2017; Received in revised form 17 August 2017; Accepted 24 November
2017
☆ This work has been supported by the INWITE project.⁎
Corresponding author.E-mail addresses: [email protected] (M.
Fereydooni), [email protected] (M. Sabaei), [email protected] (M.
Dehghan), [email protected] (G. Babazadeh
Eslamlou),
[email protected] (M. Rupp).
Computer Communications 116 (2018) 147–158
Available online 06 December 20170140-3664/ © 2017 Elsevier B.V.
All rights reserved.
T
http://www.sciencedirect.com/science/journal/01403664https://www.elsevier.com/locate/comcomhttps://doi.org/10.1016/j.comcom.2017.11.014https://doi.org/10.1016/j.comcom.2017.11.014mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://doi.org/10.1016/j.comcom.2017.11.014http://crossmark.crossref.org/dialog/?doi=10.1016/j.comcom.2017.11.014&domain=pdf
-
of sight (LOS) and non-line of sight (NLOS). They showed that an
in-creasing number of small cell BSs can reduce the impact of the
buildingsafeguard against the aggregate interference.
Most of the users in HCNs are assigned to macro BSs due to
thelower transmission power of small cell BSs [12]. Hence, the
networkencounters an imbalanced load distribution among its tiers.
A commonapproach to tackle this issue is biasing the users, as
proposed by 3rdGeneration Partnership Project (3GPP) in Release 10
[13]. Biasingfollows the idea of pushing the user assignment
towards low-power BSsby artificially increasing their coverage
region.
In [14,15] the authors considered the problem of user
associationand power allocation in the downlink of HCNs with the
goal of max-imizing the network energy efficiency. The authors in
[16] evaluatedthe performance of cellular networks under an
interference coordina-tion mechanism when the users are associated
to the closest BS. In [17],the authors developed a model to compute
the coverage probability andthe average throughput in a K-tier
cellular network using a flexible userassociation. They showed that
biasing the users without an efficientinterference coordination
mechanism always reduces the overall cov-erage probability of HCNs.
It is a direct result of forcing the users with alower SINR to a
BS. In [18] the authors proposed a model to evaluatethe performance
of a two-tier cellular network under a simple resourcepartitioning
mechanism. They assumed that a ξ fraction of resources isallocated
to the macro cell users and the unbiased small cell users.
Theremaining − ξ(1 ) fraction of the resources, in which the macro
cellshuts down the transmission, is assigned to the biased small
cell users.Since the interference level in the fraction of
resources for the biasedusers is lower than the shared part, those
users experience a better SINRdistribution.
The interference coordination mechanism under consideration
in[18] is not bandwidth efficient due to reserving a fixed fraction
of re-sources solely for the biased users. Achieving a high
spectral efficiency,the use of the total bandwidth in all cells, is
one of the key objectives oflong-term evolution (LTE) systems [19].
Fractional frequency reuse(FFR) is a popular interference
coordination strategy due to its goodspectral efficiency and its
suitability to orthogonal frequency divisionmultiple access (OFDMA)
based cellular networks [20–22]. It is in-cluded in the 3GPP-LTE
standard since Release 8 [23]. The FFR me-chanism partitions the
cell into two regions (interior and edge regions)and applies
different frequency reuse factors to each region [24,25]. Inthis
paper we consider two main types of FFR mechanism: Strict-FFRand
soft frequency reuse (SFR).
Strict-FFR mechanism: In a Strict-FFR system the entire
frequencyband W is partitioned into a common part =W βWInFFR and a
reuse part
= −W β W(1 )eFFR where 0≤ β≤ 1. The common part of resources
isshared by the cell interior users of each tier. The reuse part is
dividedamong the network tiers and W ke,FFR is utilized by the kth
tier, where
= ∑ =W WkK
keFFR
1 e,FFR. Hence, the cell edge users of a different tier use
a
disjoint set of resources. Furthermore, the reuse part of the
kth tier isfurther partitioned into Δk sub-bands and each BS
randomly choosesone sub-band to transmit to the cell edge
users.
SFR mechanism: In an SFR system the entire frequency band in
thekth tier is divided into Δk sub-bands. One of the Δk sub-bands
is ran-domly allocated to the cell edge users of each tier, and the
cell interiorusers share the rest of the sub-bands with the edge
users of other cells.Since the BS employs the sub-bands used for
the edge users of othercells to connect to its own cell interior
users, the downlink data of cellinterior users is typically
transmitted with a lower transmit power todecrease the interference
level of the other cells. Each tier has twopossible power levels,
i.e., a high power level Pk and a low power levelmkPk where 0
-
K-tier cellular networks. Fig. 1 shows the network architecture
in asample two-tier cellular network. Let us denote k (where ∈k ,K
and= … K{1, 2, 3, , })K the index of tier k, and assume that the
BSs in tier k
are spatially distributed as a PPP, �∈ϕ ,k 2 of density λk. The
locationsof the users in the network are modeled by another
independenthomogeneous PPP, �∈ϕ ,u 2 with a non-zero density λu.
Without loss ofgenerality, we perform all of the analysis on a
typical mobile user lo-cated at the origin, which is possible via a
striking property of PPPsexplained by the Slivnyak theorem [28].
Each BS in tier k is offered amaximum transmit power Pk, and τk
denotes the tier’s SINR threshold.More precisely, a mobile user can
reliably communicate with a BS in thekth tier, only if the downlink
SINR of the BS at the mobile user is greaterthan τk. The noise
power is represented by σ2 and the distance basedpath loss function
is = −l y y( ) ,αk where αk is the path loss exponent forthe BSs of
tier k. The downlink SINR of a user when it connects to a BSx∈ ϕk
is
=+
−
k PP h y
I σSINR( , ) ,k
k kx kxα
2
k
(1)
while the interference at such user is computed by
∑ ∑== ∈ ∖
−I P h y ,j
K
z ϕ xj jz jz
α
1 j
j
(2)
where hjz denotes the random fading, following an exponential
dis-tribution with unit mean (hjz∼ exp (1)). Furthermore, yjz is
the dis-tance between the user under consideration and BS z in tier
j. Table 1summarizes the paper notations.
We consider a Strict-FFR and an SFR interference
managementmechanism, and similar to [20] we divide the users into
cell edge andcell interior users by means of a certain frequency
reuse threshold τk, FR.It should be noted that, a user in tier k is
considered as cell interior userif it’s received SINR exceeds a
threshold τk, FR, otherwise it is consideredas cell edge user.
2.1. Tier association probability and distance distribution
Each user chooses a BS as its serving BS, if it provides the
maximumlong-term biased received power [17]. The user association
policy isgiven by:
∈ =∈
−u i P B Rif arg max ,ii
i i iαiU
K (3)
where iU denotes the user’s set of BSs of the ith tier and Bi is
a positivebias value which is identical for all the BSs of tier i.
Employing a biasvalue Bi>1 by the BSs of the ith tier, extends
its coverage area.
Strict-FFR mechanism: The tier association probability Ai,
thenumber of user associated to tier i N(i), and the probability
distributionfunction (PDF) of distance between the user and it’s
associated BS fi(x)under the Strict-FFR system are obtained via
[17, Lemma 1], [17,Lemma 2] and [17, Lemma 3], respectively.
∫ ∑ ⎜ ⎟= ⎛⎝⎜ −
⎛⎝
⎞⎠
⎞
⎠⎟
∞
=
A λ π B PB P
r πrλ drexp 2 ,ik
K
kk k
i i
α
i01
2/ k αkαi
2
(4)
∫ ∑ ⎜ ⎟= ⎛⎝⎜ −
⎛⎝
⎞⎠
⎞
⎠⎟
∞
=
N i λ π B PB P
r πrλ dr( ) exp 2 ,k
K
kk k
i i
α
u01
2/ k αkαi
2
(5)
∑ ⎜ ⎟= ⎛⎝⎜−
⎛⎝
⎞⎠
⎞
⎠⎟
=
f x πλ xA
π λ B PB P
x( ) 2 exp .ii
i k
K
kk k
i i
α
1
2/ k αkαi
2
(6)
SFR mechanism: In an SFR system the BS uses two different
powerlevels to transmit to cell edge and cell interior users. Since
the userassociation mechanism is based on the maximum long-term
biased re-ceived power, the two power levels of the SFR mechanism
have a directinfluence on the user association. Consider the case
in which based onuser association policy (3), the typical user
chooses the BS at tier i asserving BS, i.e., >− ∈ ≠ −P B R P B
Rmaxi i i α k k i k k k α,i kK . Since each tier hastwo transmit
power levels, the user could be in two different states. Inone
case, the user is located somewhere that even if the BSs of tier i
usea low transmit power level miPi, while all the BSs of other
tiers use ahigh transmit power level Pk, ∈k ,K k≠ i, again tier i
provides themaximum long-term biased received power for the user,
i.e.,
>−∈ ≠
−m P B R P B Rmax .i i i i αk k i
k k kα
,i k
K
Let us call this type of users Category I users.In the second
case, when the BSs of tier i apply a low power level
miPi the typical user of this tier receives its maximum
long-term biased
Macro BS Small cell BS
Fig. 1. Topology of a two-tier network.
Table 1Notation of frequently used parameters.
Symbol Meaning Symbol Meaning
Pk Maximum transmit power of the kth tier mk Power control
factor of the kth tierBk Bias value of the kth tier τk SINR
threshold of the kth tierτk, FR Frequency reuse threshold of the
kth tier γk Rate threshold of the kth tierλk Density of the kth
tier BSs distribution λu Density of the user distributionϕk Point
process of the kth tier BSs ϕu Point process of usersΔk Reuse
factor of the kth tier Rk Distance between the user and its nearest
BSyk, z Distance between the user and the BS z σ2 Noise powerK Set
of the network tiers U Set of the usershk, z Fading between the
user and the BS z αk Path loss exponent of the kth tierβ Strict-FFR
cell interior user resource fraction ρtotal Total number of
sub-bandsW Channel bandwidth K Number of tiers
WInFFR Bandwidth of Strict-FFR cell interior user W ke,FFR
Bandwidth of Strict-FFR kth tier cell edge user
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
149
-
received power from BSs of tier k where k≠ i,
> ∈ = −
> ∈∈
F x R x uR x u
u( ) 1 [ ] 1
[ , ][ ]
.i i ii i
i,2 ,2
,2
,2U
U
U (13)
We calculate the numerator of (13) as
� ∫ ∑
∑
⎜ ⎟
⎜ ⎟
> ∈ = ⎧⎨⎩
⎛
⎝⎜−
⎛⎝
⎞⎠
⎞
⎠⎟
− ⎛
⎝⎜−
⎛⎝
⎞⎠
⎞
⎠⎟
−⎫⎬⎭
∞
=
= ≠
R x u πrλ π λ B PB P
r
π λ B Pm B P
r
πλ r dr
[ , ] 2 exp
exp exp
( ) ,
i i x ik
K
kk k
i i
α
k k i
K
kk k
i i i
α
i
,21
2/
1,
2/
2
k αkαi
k αkαi
2
2
U
(14)
plugging (14) and (8) into (13) and derivation of the results
withrespect to x, results in (12). □
3. Coverage probability
The coverage probability is the probability that the SINR of a
userlocated at the origin is greater than a predefined target SINR
value. Inthe following we derive the coverage probability for two
different in-terference management mechanisms.
3.1. SFR interference coordination
This section provides the coverage probability of the cell edge
usersof a K-tier cellular network under biasing and an SFR
interferencemanagement as well as the coverage probability of the
cell interiorusers. Regarding our system model, cell edge users are
divided into twocategories and we shall separately compute the
coverage probability ofeach category. Since the two categories are
disjoint and each user isassociated with at most one of them, the
coverage probability of a celledge user is computable by the law of
total probability.
Theorem 3.1. The coverage probability of tier i cell edge user
in a K-tiercellular network with the SFR interference coordination
mechanism andbiased user association is
= +i τ A S i τ A S i τ( , ) ( , ) ( , ),i i i i ieSFR 1, 1,eSFR
2, 2,eSFRS (15)
where
∫
∫
∏
∏ ∏
∏
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜
⎟
= ⎡
⎣⎢ −
⎛⎝
⎛⎝
⎞⎠⎞⎠
⎛⎝
⎛⎝
⎞⎠
⎞⎠
⎛⎝
+ ⎞⎠
⎤
⎦⎥
= ⎛
⎝⎜
⎞
⎠⎟
∞
=
′
= ≠ = ≠
∞
=
′
S i τ δ τ Q τ ω Q ττm
Qτm
ω Q τ ω δ τ
τm
f r dr
S i τ δ τ Q τ ω f r dr
( , ) ( ) (Ψ ( ), ) Ψ ( ), Ψ
Ψ , (Ψ ( ), )
( ) ,
( , ) ( ) (Ψ ( ), ) ( ) ,
i i ik
K
k i i k i i i i i ii
i
k k i
K
k ii
ik i
k k i
K
k i i k i i i
i
ii
i i ik
K
k i i k i i
1,eSFR
01
, , , ,,FR
1,,
,FR,
1,, ,
,FR1,
2,eSFR
01
, , 2,
and
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
150
-
∫
∫⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
= = − +
= ⎛⎝− ⎞
⎠
= ⎛⎝−
+⎞⎠
= ⎛⎝− ⎛
⎝−
+ +⎞⎠
⎞⎠
= ⎛⎝
⎞⎠
= ⎛⎝
⎞⎠
∞
−
′ ∞
− −
′
x rP
η P x η m
δ x r σP
x Q a ω
πλ ya y
dy
Q a b πλa y b y
ydy
ω r B Pm B P
ω r B PB P
Ψ ( ) , (Δ 1) 1Δ
,
( ) exp , ( , )
exp 21 ( )
,
( , ) exp 2 1 11
11
,
, .
k iα
ik k k
k k
k
iα
ik
k ω kα
k k i r kα
kα
k iα
k k
i i i
α
k i
αk k
i i
α
,
2
1
,
1/
,
1/
i
i
k
i i
i k i k
Proof. See Appendix A. □
The coverage probability of tier i cell interior users S i τ( ,
)iInSFR iscomputed by following the same procedure. A cell interior
user is a userwhose received SINR in the shared sub-band is higher
than the fre-quency reuse threshold τi, FR. As mentioned in Section
2, only thoseusers that lie in Category I can be classified as cell
interior users. Usingthis fact we compute the coverage probability
of cell interior users.
Theorem 3.2. The coverage probability of tier i cell interior
user in a K-tiercellular network with an SFR interference
coordination mechanism andbiased user association is
∫ ∏ ∏⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜
⎟
= ⎛⎝
⎛⎝
⎞⎠
⎞⎠
⎛⎝
⎛⎝⎞⎠
⎞⎠⎛⎝
+ ⎞⎠
∞
= =S i τ Q
τm
ω Q τm
ω δ τm
τm
f r dr
( , ) Ψ , Ψ ,
( ) .
ik
K
k ii
ik i
k
K
k ii
ik i i
i
i
i
ii
InSFR
01
,,FR
,1
, ,
,FR1,
(16)
Proof. The proof is carried out along the lines of Theorem 3.1.
□
Using Theorems 3.1 and 3.2, the overall coverage probability
underSFR is
∑= +=
S S i τ A S i τ( , ) ( , ).i
K
i i iSFR
1eSFR
1, InSFR
(17)
3.2. Strict-FFR interference coordination
In this section we provide the coverage probability of HCNs
under aflexible user association and a Strict-FFR interference
coordinationmechanism.
Theorem 3.3. The coverage probability of ith tier cell edge user
in a K-tiercellular network with biased user association and
Strict-FFR is
∫ ∫
∫
∏
⎜ ⎟=⎡
⎣⎢
⎛
⎝⎜−
⎛⎝− ⎞
⎠
⎞
⎠⎟
− +
⎛
⎝⎜⎜−
⎡
⎣⎢⎢−
+
⎤
⎦⎥⎥
⎞
⎠⎟⎟
⎤
⎦
⎥⎥
′
′
∞ ∞
= ≠
′
∞
−
′
′
S i τ δ τ πλ Q y ydy
δ τ τ Q τ ω
πλ Q yτ y
ydy f r dr
( , ) ( )exp 2 1 ( )
( ) (Ψ ( ), )
exp 2 1 ( )1 Ψ ( )
( ) ,
i i i i r
i i ik k i
K
k i i k i
i r i i iα i
eFFR
0
,FR1,
, ,FR ,
, ,FR i
where
⎜ ⎟= =⎛
⎝⎜ −
⎛
⎝−
+⎞
⎠
⎞
⎠⎟
′−
′x P r
Px Q y
τ yΨ ( ) , ( ) 1 1
Δ1 1
1 Ψ ( ).k i k
α
i i i i iα,
,
i
i
Proof. See Appendix B. □
Theorem 3.4. The coverage probability of ith tier cell interior
user in a K-tier cellular network with biased user association and
Strict-FFR is
∫ ∏ ∏= ⎡⎣⎢ +
⎤
⎦⎥
∞
=
′
=
′
S i τ
δ τ τ Q τ ω Q τ ω f
r dr
( , )
( ) (Ψ ( ), ) (Ψ ( ), )
( ) .
i
i i ik
K
k i i k ik
K
k i i k i i
InFFR
0 ,FR1
, ,1
, ,FR ,
(18)
Proof. The proof is carried out along the lines of Theorem 3.3.
□
Using Theorems 3.3 and 3.4, the overall coverage probability
underthe Strict-FFR mechanism is
∑= +=
S A S i τ S i τ( ( , ) ( , )).i
K
i i iFFR
1eFFR
InFFR
(19)
4. Spectral efficiency
4.1. Average ergodic rate
Finding the average ergodic rate (average cell throughput) is
im-portant for the planning and design of the cellular network. The
averageergodic rate of network under an SFR interference management
SFRR isgiven as
∑= +=
i i( ( ) ( )).i
KSFR
1eSFR
InSFRR R R
(20)
The average ergodic rate of tier i cell edge users i( )eSFRR and
tier i cellinterior users i( )InSFRR are computed by
= +i A i A i( ) ( ) ( ),i ieSFR 1, 1,eSFR 2, 2,eSFRR R R
(21)
=i A i( ) ( ),iInSFR 1, InSFRR R (22)
where i( ),1,eSFRR ,2,eSFRR and i( )InSFRR are the rate of cell
edge users inCategory I, cell edge users in Category II, and the
rate of cell interiorusers of tier i under an SFR interference
coordination mechanism, re-spectively.
Theorem 4.1. The average ergodic rate of network under an
SFRinterference coordination mechanism is equal to
∫
∫
∫
∑
∑
∑
= −
+ −
+ −
=
∞
=
∞
=
∞
A S i ν dν
A S i ν dν
A S i ν dν
( , exp( ) 1)
( , exp( ) 1)
( , exp( ) 1) .
i
K
i
i
K
i
i
K
i
SFR
11, 0 1,e
SFR
12, 0 2,e
SFR
11, 0 SFR,In
R
(23)
Proof. The average ergodic rate of tier i cell edge user is
given as
�
�
= + ′ < ∈
+ + ′ ∈
i A i P i m P τ u
A i P u
( ) [log(1 SINR ( , )), SINR( , ) ]
[log(1 SINR ( , )) ],
i i i i i i
A
i i i
eSFR
1, ,FR 1,
2, 2,
R U
U
(24)
since the SINR value is positive, the first part of (24) is
computed as
�∫= ′ > − < ∈∞A i P ν i m P τ u dν[SINR ( , ) exp( ) 1,
SINR( , ) ] ,i i i i i0 ,FR 1,U
∫= −∞ S i ν dν( , exp( ) 1) .a0 1,e
SFR
The final term (a) is obtained by using the results of Theorem
3.1. Byfollowing the same approach for the other parts, we reach
(23). □
Theorem 4.2. The average ergodic rate of network under a
Strict-FFRinterference coordination mechanism is equal to
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
151
-
∫
∫
∑
∑
= −
+ −
=
∞
=
∞
A S i ν dν
A S i ν dν
( , exp( ) 1)
( , exp( ) 1) .
i
K
i
i
K
i
FFR
10 e
FFR
10 In
FFR
R
(25)
Proof. The proof is carried out along the lines of Theorem 4.1.
□
4.2. Average user throughput
Another important metric considered in the design of the
cellularnetworks is the average user throughput. Assuming the
resources arefairly shared between the users associated to a BS,
the average cell edgeuser throughput of tier i under SFR is given
as
=ii
N i( )
( )Δ ( )
.i
u,eSFR e
SFR
eSFRR
R
(26)
Also, the average user throughput of cell interior users of tier
i iscomputed as
=−
ii
N i( )
(Δ 1) ( )Δ ( )
,ii
u,InSFR In
SFR
InSFRRR
(27)
where N i( )eSFR and N i( )InSFR are the average number of cell
edge and cellinterior users associated to a BS of tier i under an
SFR mechanism, re-spectively.
Lemma 4.1. The average number of cell edge users of a BS in tier
i under anSFR mechanism is given by
∫ ∏ ⎜ ⎟⎜ ⎟ ⎜ ⎟
= ⎡
⎣⎢ +
− ⎛
⎝⎜⎛⎝
⎞⎠
⎛⎝
⎛⎝
⎞⎠
⎞⎠
⎞
⎠⎟
⎤
⎦⎥
∞
=
N i λλ
A A
A δτm
Qτm
ω f r dr
( )
Ψ , ( ) .
u
ii i
i ii
i k
K
k ii
ik i i
eSFR
2, 1,
1, 0,FR
1,
,FR, 1,
(28)
Proof. Under an SFR mechanism we have two types of cell edge
users.The SINR of Category I cell edge users falls below the
frequency reusethreshold. Consequently the number of Category I
cell edge users is
∫ ∏ ⎜ ⎟⎜ ⎟ ⎜ ⎟
= ⎡
⎣⎢
− ⎛
⎝⎜⎛⎝
⎞⎠
⎛⎝
⎛⎝
⎞⎠
⎞⎠
⎞
⎠⎟
⎤
⎦⎥
∞
=
N iλ A
λ
δτm
Qτm
ω f r dr
( ) 1
Ψ , ( ) .
u i
i
ii
i k
K
k ii
ik i i
1,eSFR 1,
0,FR
1,
,FR, 1,
(29)
All users that lie in Category II are considered as cell edge
users, i.e.,=N i( ) λ Aλ2,e
SFR u ii
2, . The total number of tier i cell edge users is
= +N i N i N i( ) ( ) ( )eSFR 1,eSFR 2,eSFR . □
Similarly, we can compute the mean number of interior users
as-sociated to a BS in tier i.
Lemma 4.2. The average number of users associated as cell
interior users ofa BS in tier i under an SFR mechanism is
∫ ∏ ⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎛⎝⎜⎛⎝
⎞⎠
⎛⎝
⎛⎝
⎞⎠
⎞⎠
⎞
⎠⎟
∞
=
N i λλ
A δτm
Qτm
ω f r dr( ) Ψ , ( ) .ui
i ii
i k
K
k ii
ik i iIn
SFR1, 0
,FR
1,
,FR, 1,
(30)
Plugging (28), and (21) into (26) provides the average cell edge
userthroughput, and the average cell interior user throughput is
derived byputting (30), and (22) into (27). Finally, using (31), we
compute theminimum rate achievable by each user
=∈
i imin( ( ), ( )).i
minSFR
u,eSFR
u,InSFRR R R
K (31)
The average cell edge user throughput under a Strict-FFR
interferencecoordination is
=iW i
WN i( )
( )Δ ( )
,ii
u,eFFR e,
FFReFFR
eFFRR
R
(32)
where
∫ ∏= ⎡⎣⎢ −
⎤
⎦⎥
∞
=
′N i λ Aλ
δ τ Q τ ω f r dr( ) 1 ( ) (Ψ ( ), ) ( ) .u ii
i ik
K
k i i k i ieFFR
0 ,FR1
, ,FR ,(33)
The average cell interior user throughput under a Strict-FFR
inter-ference coordination is given as
=iW i
WN i( )
( )( )
,u,InFFRInFFR
InFFR
InFFRRR
(34)
where
∫ ∏= ∞=
′N i λ Aλ
δ τ Q τ ω f r dr( ) ( ) (Ψ ( ), ) ( ) .u ii
i ik
K
k i i k i iInFFR
0 ,FR1
, ,FR ,(35)
4.3. Rate coverage probability
We now compute the rate coverage probability of a user located
atthe origin with the same approach as before for computing the
coverageprobability. The following theorem provides the per-tier
rate coverageprobability under an SFR interference
coordination.
Theorem 4.3. The rate coverage probability of tier i under the
SFRinterference coordination for a rate threshold γi is given
as
= + +i γ A i γ A i γ A i γ( , ) ( , ) ( , ) ( , ),i i i i i i
iSFR 1, 1,eSFR
2, 2,eSFR
1, InSFRP P P P (36)
where
⎜ ⎟=⎛
⎝⎜
⎛
⎝
⎞
⎠− ⎞
⎠⎟i γ S i
γ N iW
( , ) , expΔ ( )
1 ,ii i
1,eSFR
1,eSFR 1,e
SFR
P
⎜ ⎟=⎛
⎝⎜
⎛
⎝
⎞
⎠− ⎞
⎠⎟i γ S i
γ N iW
( , ) , expΔ ( )
1 ,ii i
2,eSFR
2,eSFR 2,e
SFR
P
⎜ ⎟= ⎛
⎝⎜
⎛⎝ −
⎞⎠− ⎞
⎠⎟i γ S i
γ N iW
( , ) , expΔ ( )
(Δ 1)1 .i
i i
iInSFR
InSFR In
SFRP
Proof. To compute the rate coverage probability of tier i, let
us firstconcentrate on the rate coverage probability of Category I
cell edgeusers of tier i, i.e., i γ( , )i1,e
SFRP
�
� ⎜ ⎟
⎜ ⎟
= ⎡
⎣⎢ + ′ ≥
< ∈ ⎤
⎦⎥
= ⎡
⎣⎢ ′ ≥
⎛
⎝
⎞
⎠−
< ∈ ⎤
⎦⎥
= ⎛
⎝⎜
⎛
⎝
⎞
⎠− ⎞
⎠⎟
i γ WN i
i P γ i m P
τ u
i Pγ N i
Wi m P
τ u
S iγ N i
W
( , )Δ ( )
log(1 SINR ( , )) , SINR( , )
,
SINR ( , ) expΔ ( )
1, SINR( , )
,
, expΔ ( )
1 ,
ii
i i i i
i i
ii i
i i
i i
i i
1,eSFR
1,eSFR
,FR 1,
1,eSFR
,FR 1,
1,eSFR 1,e
SFR
P
U
U
the rate coverage probability of Category II cell edge users i
γ( , )i2,eSFRP
and the cell interior users, i γ( , ),iInSFRP are obtained by
following the
same approach as computing i γ( , )i1,eSFRP . □
Theorem 4.4. The rate coverage probability of tier i under a
Strict-FFRinterference coordination for a rate threshold γi is
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
152
-
= +i γ A i γ A i γ( , ) ( , ) ( , ),i i i i iFFR eFFR
InFFRP P P (37)
where
⎜ ⎟=⎛
⎝⎜
⎛
⎝
⎞
⎠− ⎞
⎠⎟i γ S i
γ N iW
( , ) , expΔ ( )
1 ,ii i
ieFFR
eFFR e
FFR
e,FFRP
⎜ ⎟= ⎛
⎝⎜
⎛⎝
⎞⎠− ⎞
⎠⎟i γ S i
γ N iW
( , ) , exp( )
1 .ii
InFFR
InFFR In
FFR
InFFRP
5. Discussion of the results
5.1. Special cases of interest
As the general coverage probability has not been derived in a
closedform in the previous section, this section provides more
insight by somespecial cases where =α 4 for all the tiers and =σ 02
. Because of thehigh BS density in HCNs, in general the noise power
is negligiblecompared to the interference power (wireless networks
are interferencelimited systems). Besides, the choice of the path
loss exponent =α 4 iscommonly accepted in practice as long as users
are not too close to theBS. Due to the space limitation we just
provide explicit results forspecial cases of cell edge users under
an SFR mechanism; correspondingresults for the cell interior users
of an SFR mechanism and a Strict-FFRmechanism are easily computable
by employing the same approach.
Corollary 5.1. Consider =σ 02 and =α 4. The coverage probability
of ithtier cell edge users in Category I under an SFR mechanism in
Theorem 3.1 is
=+
−+ + +( )( )
S i τ λA C i τ C i
λ
A C i C i τ C i C i
( , )( ( , ) ( ))
( ) ( , ) , ( ),
ii
i i
i
i iτ
m
1,eSFR
1, 2 4
1, 1 3 3 4i
i
,FR
(38)
where
=⎛⎝
⎞⎠−
−
−−
− ( )( )
C iτ τ η
λ( )
arctan arctan( )
( ),
iη τ
mτ
m i i
im
η τ τ τm
τ
1
3/23/2
1 1
i i
i
i
i
i
i i i ii
i
,FR ,FR
,FR ,FR
∑
∑
∑ ∑
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
= ⎛⎝
⎞⎠
⎧⎨⎩⎛⎝
⎞⎠⎫⎬⎭
= ⎛⎝
⎞⎠
⎧⎨⎩⎛⎝
⎞⎠⎫⎬⎭
= ⎛
⎝⎜ +
⎛⎝
⎞⎠
⎞
⎠⎟ =
⎛⎝
⎞⎠
=
= ≠
= ≠ =
C i x λη P x
Pm B η x
B
C i x λη P x
Pm B η x
B
C i λ λ B Pm B P
C i λ B PB P
( , ) arctan ,
( , ) arctan ,
( ) , ( ) .
k
K
kk k
i
i i k
k
k k i
K
kk k
i
i i k
k
ik k i
K
kk k
i i i k
K
kk k
i i
21
1/2 1/2
31,
1/2 1/2
41,
1/2
51
1/2
Corollary 5.2. Consider =σ 02 and =α 4. The coverage probability
of ithtier cell edge users in Category II under a SFR interference
coordinationmechanism in Theorem 3.1 is
=+
−+( ) ( )
S i τm C i C i m C i C i
( , ), ( ) , ( )
.i
λA
iτm
λA
iτm
2,eSFR
2 5 2 4
ii
ii
ii
ii
2, 2,
(39)
This result is opposing [5,9,17], where it is argued that in
inter-ference limited networks when all tiers have the same path
loss ex-ponent, the coverage probability is independent of the BS
density andthe number of tiers. The reason for this difference is
that the other workdoes not employ any interference coordination
mechanism among theusers. As evident from the coverage probability
expressions (38) and(39), by employing an FFR interference
coordination, even in an
unbiased user association mechanism the overall coverage
probabilitydepends indeed on the BS distribution density λi.
Corollary 5.3. Consider =σ 0,2 =α 4, and =m ϵi (ϵ is an
infinitely smallvalue). The coverage probability of users under an
SFR interferencecoordination mechanism is simplified to
⎜ ⎟
=∑
∑ ⎛⎝
+ ⎞⎠
=
= { }( ) ( )( )
( )S i τ
λ
λ( , )
arctan 1.i
kK
kB PB P
kK
kB PB P
B τB
B τB
SFR 1
1/2
1
1/2
Δ
1/2
Δ
1/2
k ki i
k ki i
i ik k
i ik k (40)
We observe that by using larger values for reuse factor Δk of
all tiers,the coverage probability increases as well. When the
frequency reusefactor Δk of all tiers goes up, the coverage
probability of tier i becomes
=S i τ( , ) 1iSFR . By a sufficient increase of Δk, almost all
users are coveredby the BSs. Besides, in case of general Δk values,
the overall coverageprobability only loosely depends on the BS
density.
5.2. Resource allocation
The optimal resource allocation among the cell edge and cell
in-terior users is one of the main concerns in FFR literature. Most
of theformer works determined optimal values of FFR system
parameters byutilizing advanced techniques such as convex
optimization [29–31],game-theoretic approaches [32], or exhaustive
search [33].
In this paper, we determine the number of sub-bands of cell
edgeand cell interior users depending on the chosen bias values and
FFRthresholds of different tiers. Considering ρtotal as the total
number ofavailable sub-bands, ρe is the number of cell edge user
sub-bands and ρInthe number of cell interior user sub-bands, the
optimal resource allo-cation of Strict-FFR is
=∑
∑=
=
βN k
N k
( )
( ),k
K
kK1 In
FFR
1 (41)
= ⌈ ⌉ρ βρ ,InFFR
total (42)
=⎢
⎣⎢⎢
−
∑
⎥
⎦⎥⎥=
ρ iρ ρ N i
N k( )
( ) ( )
Δ ( ),
i kKe
FFR total InFFR
eFFR
1 eFFR
(43)
while the optimal resource allocation of SFR is
= ⌈ − ⌉ρ ρ ρ i( ) ,InSFR
total eSFR
(44)
= ⎢⎣⎢
⎥⎦⎥
ρ iρ
( )Δ
.i
eSFR total
(45)
Intuitively, applying this resource allocation mechanism will
enforce amore efficient resource allocation among the users based
on the numberof cell edge and cell interior users and the SINR
distribution. Hence, theuser will experience a higher throughput
and rate coverage probability.
6. Numerical results
For the verification of our results we carried out Monte Carlo
si-mulations under 3GPP compliant scenarios. In particular, we
consider atwo-tier cellular network (macro cell BSs and small cell
BSs). In oursimulation of two-tier cellular networks, we assume =W
10 MHz andfor both tiers we set the SINR threshold to − 3 dB, the
frequency reusethreshold to 3 dB, and the rate threshold to 1
Mbit/s. The macro cell BSsare distributed with density = −λ m
π11
5002
2 over a field of10 000× 10 000 m2 with a user density of =λ
λ100u 1. We also set thebias value of macro cell BSs to =B 01 dB.
The macro BSs and small cellBSs transmit powers are set to 46 dBm
and 26 dBm, respectively. Wecompare the results of SFR and
Strict-FFR interference coordination
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
153
-
mechanisms against the joint resource partitioning and user
association(RPUA) mechanism of [18]. In the RPUA simulation we set
the resourcepartitioning fraction to =ξ 0.47 which is shown that is
optimal in [18].This means 47% of resources are allocated to the
macro cell users alongwith unbiased small cell users and the rest
of resources, i.e., 53%, goesto the biased small cell users.
6.1. Verification of analysis
In this part, we provide numerical results in order to validate
therepresented analytical expressions. As shown in Fig. 2, the
proposedanalytical evaluation matches very well the simulation
results. Fig. 2(a)compares the coverage probability of the
analytical expressions fromSection 3 to the simulation results. The
overall coverage probability ofusers under an SFR mechanism
decreases with larger bias values.However, the overall coverage
probability of a network under a Strict-FFR mechanism remains
almost constant at one for different bias va-lues. Since in a
Strict-FFR mechanism we have better resource parti-tioning among
the users, we observe higher SINR values compared toan SFR
mechanism. Also, we observe that the overall coverage prob-ability
of the network is higher when the user experiences a larger
pathloss from the small cell BSs, specially for larger bias values.
When weemploy a lower path loss exponent for the small cell BSs,
most of theusers will receive the highest long-term biased received
power from thesmall cell BSs.
Fig. 2(b) compares the average minimum user rate under
varyingsmall cell BS densities. Intuitively, a denser deployment of
small cellBSs increases the average minimum user rate under SFR and
Strict-FFRmechanisms. A denser deployment of low-power BSs
decreases thenumber of users associated with each BS and
consequently increases theavailable resources for each user. Also
clearly under the SFR me-chanism, the users experience a higher
rate compared to Strict-FFR dueto the better spectral efficiency of
SFR. The final observation is thatwhen small cell BSs have a larger
path loss exponent, a denser de-ployment of small cell BSs is more
effective on the minimum user re-ceived rate. A growing density of
small cell BS increases the impact ofsmall cell BS interference.
Hence, scenarios with larger small cell pathloss exponent
experience a higher average minimum rate.
6.2. Impact of small cell density and bias value on the rate
coverageprobability
This part presents numerical results for comparing the rate
coverageprobability of users under various system settings.
Fig. 3(a) depicts the impact of different bias values on the
overallrate coverage probability of a user, computed in (36) and
(37). In thissimulation the small cell density is set to =λ λ102
1.
Since in RPUA mechanism a fraction of the resources is dedicated
tousers with low SINR, this mechanism exhibits the highest increase
byusing larger bias values.
Fig. 3(b) represents the rate coverage probability of a flexible
userassociation under various small cell BS densities. In this
simulation weuse a biased user association by employing =B 82 dB
for all the BSs ofthe second tier. Clearly employing a higher small
cell BS density, in-creases the rate coverage probability of
cellular networks by bringingthe users closer to the BSs. Besides,
we observe that in dense deploy-ment of small cell BSs the
Strict-FFR mechanism outperforms the SFRmechanism in terms of rate
coverage probability. Increasing the smallcell BS density leads to
an increase in the interference. Due to betterresource partitioning
among the users in Strict-FFR and RPUA me-chanisms, they experience
a higher increase in the rate coverageprobability by a denser
deployment of small cell BSs.
6.3. Impact of small cell density and bias value on the average
ergodic rate
Fig. 4 compares the average ergodic rate of users under
differentinterference management techniques. Fig. 4(a) depicts the
impact of thebias value on the average ergodic rate of the users.
An SFR mechanismwith power factors m∈ {0.1, 0.4, 0.9} outperforms
the Strict-FFR andRPUA mechanisms in terms of average ergodic rate.
Although the RPUAand Strict-FFR mechanisms outperform the SFR
mechanism with re-spect to rate coverage probability, the SFR
technique shows a betterspectral efficiency due to its more
efficient frequency reuse. However,SFR with a power factor ϵ
(infinitely small value) does not exhibit thesame spectral
efficiency by placing almost all users into the cell edgeuser
set.
In Fig. 4(b) by fixing the small cell bias value to =B 82 dB,
wechange the small cell density and compare the average ergodic
rate ofthe users in different interference management techniques.
Simulationresults show that the average ergodic rate with the SFR
and Strict-FFRmechanisms remain almost constant with respect to
different small celldensities. Also the RPUA mechanism observes a
decrease in the averageergodic rate when employing a higher
density. Since by increasing thesmall cell density most of the
users are assigned to the small cell BSwithout biasing, RPUA wastes
resources by allocating a fraction of themto the biased users.
Fig. 2. Lines denote results from simulations, and markers refer
to results from analysis ( =B 01 dB, = −λ m ,π11
50022 = =m m 0.1,1 2 = −σ 102 dBm).
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
154
-
6.4. Impact of small cell density and bias value on the minimum
averageachievable rate
In Fig. 5(a) we evaluate the impact of biasing on the
minimumaverage user rate as derived in Section 4.B. By employing a
higher biasvalue, more users are pushed from highly loaded macro
BSs to lightlyloaded small cell BSs. Therefore, the minimum rate of
users increaseswith increasing bias value until it reaches a
certain optimal point wherethe situation changes and a further
increase in the bias value decreasesthe minimum achievable rate of
the users by overloading the small cellBSs. Because of a more
efficient bandwidth reuse in SFR, its users ex-perience a higher
minimum average rate compared to other mechan-isms.
In Fig. 5(b) we change the small cell density for the network
anddisplay the average minimum user rate when applying a small cell
biasvalue of =B 82 dB. Intuitively, by increasing the small cell
density, theaverage minimum user rate increases as well. This is
mainly due todecreasing the number of users associated with each
BS. Regarding toSection 4.B, by decreasing the user number
associated with each BS, theBS has more resources available for
each user and we observe a higheraverage minimum rate. Again the
SFR mechanism with m∈ {0.1, 0.4,
0.9} exhibits the best average minimum user rate because of the
betterfrequency reuse between the BSs. However, by setting =m ϵ,
almost allthe users lie in the cell edge user set and the network
does not benefitfrom a frequency reuse between different BSs.
7. Conclusion
This paper proposes a general framework for the
performanceanalysis of HCNs with flexible user association and FFR
mechanisms.We evaluate the per-tier and overall coverage
probability, as well as therate coverage probability of such
networks, and the average user andcell throughput for the cell edge
and the cell interior users. Supportedby extensive numerical
results, it is shown that SFR outperforms othermechanisms in terms
of spectral efficiency and average minimum rateof users. However,
because of dedicating a fraction of resources to userswith low
SINR, RPUA and Strict-FFR provide better rate coverageprobability
at large bias values. The presented model can be utilized byfinding
proper optimal values for the design parameters. A naturalextension
of this work is evaluating the performance of cellular net-works by
relaxing some assumptions of this paper such as cooperating aload
definition into analysis or considering multi antenna BSs.
Fig. 3. (a) Impact of bias value on the rate coverage
probability. (b) Impact of small cell density on the rate coverage
probability with = =α α 41 2 .
Fig. 4. (a) Impact of small cell bias value on the average
ergodic rate. (b) Impact of small cell density on the average
ergodic rate with = =α α 41 2 .
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
155
-
Appendix A. Proof of Theorem 3.1
Using the law of total probability we have
� �
� �
� �
∑
∑
= ∈ > ∈
= ∈ > < ∈
+ ∈ > ∈
=
′
=
′
′
S u i P τ u
u i P τ i m P τ u
u i P τ u
( ) [SINR ( , ) ],
( ) [SINR ( , ) , SINR( , ) ]
( ) [SINR ( , ) ],
i
K
i i i i
a
i
K
i i i i i i i
i i i i
eSFR
1e, e,
11, ,FR 1,
2, 2,
U U
U U
U U
where (a) is obtained using the Bayes’ theorem and considering
the independency of probability of locating user in Category I and�
∑ +
< ∈ ⎤
⎦⎥
= ⎡
⎣⎢ >
∑ +<
∑ +∈ ⎤
⎦⎥
=⎡
⎣⎢⎢
⎛
⎝⎜−
∑ + ⎞
⎠⎟⎛
⎝⎜ −
⎛
⎝⎜−
∑ + ⎞
⎠⎟⎞
⎠⎟⎤
⎦⎥⎥
= ⎡⎣⎢
⎛⎝− ⎞
⎠⎤⎦⎥
⎡
⎣⎢
⎛
⎝⎜−
⎞
⎠⎟⎤
⎦⎥
−⎡
⎣⎢⎢
⎛
⎝⎜−
⎛
⎝⎜ +
∑ ⎞
⎠⎟⎞
⎠⎟⎤
⎦⎥⎥
⎡
⎣⎢
⎛⎝− ⎛
⎝+ ⎞
⎠⎞⎠⎤
⎦⎥
−
=
−
=
= =
= =
=
=
=
S i τP h r
η P I στ
m P h rη P I σ
τ u
hτ η P I σ
Pr h
τ η P I σm P
r d u
rτ η P I σ
Pr
τ η P I σm P
r τP
σ r τP
η P I
rP
τ η P Iτ η P I
mr σ
Pτ
τm
( , ) , ,
( ),
( ),
exp( )
1 exp( )
,
exp exp
exp exp ,
ii i x
α
kK
k k ki
i i i xα
kK
k k ki i
i xi k
Kk k k
i
αi x
i kK
k k k
i i
αi
aI h
α i kK
k k k
i
α i kK
k k k
i i
hα
i
ih
αi
i k
K
k k k
hα
ii
k
K
k k ki k
Kk k k
ih
α
ii
i
i
1,eSFR ,
12
,
12 ,FR 1,
,1
2
,,FR 1
2
1,
,1
2,FR 1
2
Φ,2
Φ,1
Φ,1
,FR 1Φ,
2 ,FR
i i
i i
i i
i i
i i
U
U
(46)
̂ ̂ ̂ ̂ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟= ⋯ − ⎛⎝
⋯ ⎛⎝
⎞⎠⋯ ⎛
⎝⎞⎠⎞⎠⎛⎝+ ⎞
⎠⋯ ⋯ ⋯δ τ τ τ τ τ
τm
τm
δ ττm
( ) (Ψ ( ), ,Ψ ( )) Ψ ( ), ,Ψ ( ), Ψ , ,Ψ ,b
i i I I i i K i i
A
I I I I i i K i i ii
iK i
i
i
B
i ii
i, , 1, , , , , , , 1, , 1,
,FR,
,FR ,FRK K K1 1 1L L
(47)
where (a) follows from the exponential distribution of fading
with unit mean, and (b) is obtained by considering =x xΨ ( )k ir η
P
P,αi k k
iand
= −( )δ x x( ) expi r σPαi i 2 . The Laplace transform of the
second part of (47) is computed as
Fig. 5. (a) The average minimum achievable user rate as a
function of bias value. (b) The average minimum achievable user
rate as a function of small cell BS density with = =α α 41 2 .
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
156
-
�
�
� �
̂ ̂
̂
̂⎜ ⎟
∑ ∑
∑ ∑
⎡⎣⎢∏ ⎛
⎝∑ ⎞
⎠⎤⎦⎥
∏ ∑
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
=⎡
⎣⎢⎢
⎛
⎝⎜− −
⎛⎝
⎞⎠− − ⎛
⎝⎞⎠
⎞
⎠⎟⎤
⎦⎥⎥
=⎡
⎣⎢⎢
⎛
⎝⎜− −
⎛⎝
⎞⎠
⎞
⎠⎟⎤
⎦⎥⎥
−⎡
⎣⎢⎢
⎛
⎝⎜−
⎛⎝
⎞⎠
⎞
⎠⎟⎤
⎦⎥⎥
= ≠ = ≠
∈ ∖
−
∈ ∖
−
= ≠ ∈
−
= ≠ ∈
−
B τ Iτm
I τ Iτm
I
τ h yτm
h y
τ h yτm
h y
exp Ψ ( ) Ψ Ψ ( ) Ψ ,
exp Ψ ( ) Ψ
exp Ψ ( ) exp Ψ ,
h i i i i i ii
ii
k k i
K
k i i kk k i
K
k ii
ik
h i i iz x
iz izα
i ii
i z ϕ xiz iz
α
hk k i
K
k i iz
kz kzα
hk k i
K
k ii
i z ϕkz kz
α
Φ, , ,,FR
1,,
1,,
,FR
Φ, ,Φ
,,FR
Φ,1,
,Φ
Φ,1,
,,FR
i
i
i
i
k
k
k
k
using the Laplace transform of exponential random variables with
unit mean, we have,
�
� �
∏
∏ ∏ ∏ ∏
=⎡
⎣
⎢⎢ + +
⎤
⎦
⎥⎥
⎡
⎣
⎢⎢ +
⎤
⎦
⎥⎥
⎡
⎣⎢ +
⎤
⎦⎥
∈ ∖ −−
= ≠ ∈ − = ≠ ∈−
( )
( )
y τ y
y τ y
1
1 Ψ
11 Ψ ( )
1
1 Ψ
11 Ψ ( )
.
z x i iτ
m izα i i i iz
α
k k i
K
z k iτ
m kzα k k i
K
z k i i kzα
ΦΦ , ,
1,Φ
Φ , 1,Φ
Φ ,
ii
ii i
ki
ik k
k
,FR
,FR
Assuming ∫= − ∞+ −( )Q a ω πλ dy( , ) exp 2 ,k k ω ya y1 ( )k
αk1 and applying the probability generating functional (PGFL) of a
PPP [28], we reach
∫ ∏ ∏⎜ ⎟⎜ ⎟=⎛
⎝
⎜⎜−
⎡
⎣
⎢⎢−
+ +
⎤
⎦
⎥⎥
⎞
⎠
⎟⎟
⎛⎝
⎛⎝
⎞⎠
⎞⎠
∞
− − = ≠ = ≠( )πλ
y τ yydy Q
τm
ω Q τ ωexp 2 1 1
1 Ψ
11 Ψ ( )
Ψ , (Ψ ( ), ),i ri i
τm
α i i iα
k k i
K
k ii
ik i
k k i
K
k i i k i, , 1,
,,FR
,1,
, ,i
ii i,FR
(48)
if ∈u i1,U then we have = ( )ωk i r B Pm B P α, 1/αi k ki i i k.
Following from the second Laplace transform of (47), the first
Laplace transform is given as∫∏ ⎜ ⎟= ⎛
⎝−
+⎞⎠=
∞
−A πλy
τ ydyexp 2
1 (Ψ ( )).
k
K
k ω k i iα
1 ,1k i k, (49)
The coverage probability of edge users in Category I is obtained
by combining (48) and (49) into (47). By following the same
procedure, for the celledge users in Category II we have,
�̂
∫∏
=⎡
⎣⎢⎢
⎛
⎝⎜−
∑ + ⎞
⎠⎟⎤
⎦⎥⎥
=+
=
=
∞
−′
S i τr τ η P I σ
P
δ τ πλ yτ y
dy
( , ) exp( )
,
( ) 21 (Ψ ( ))
,
i h
αi k
Kk k k
i
i ik
K
k ω k i iα
2,eSFR
Φ,1
2
1 ,1
i
k i k,
∫ ∏= ⎛⎝⎜
⎞
⎠⎟
∞
=
′S i τ δ τ Q τ ω f r dr( , ) ( ) (Ψ ( ), ) ( ) .i i ik
K
k i i k i i2,eSFR
01
, , 2,(50)
If ∈u i2,U we have =′ ( )ωk i r B PB P α, 1/αi k ki i k.
Appendix B. Proof of Theorem 3.3
Similar to the proof of Theorem 3.1 and applying the law of
total probability, the coverage probability of cell edge user under
the Strict-FFRsystem is defined as,
� � �
� �
̂
̂
̂ ̂
∑
⎜ ⎟
⎜ ⎟
= > < ∈ = ⎡⎣⎢
⎛⎝− ⎞
⎠⎤⎦⎥
−
−⎡
⎣⎢⎢
⎛
⎝⎜−
⎛
⎝⎜ +
⎞
⎠⎟⎞
⎠⎟⎤
⎦⎥⎥
⎡⎣⎢
⎛⎝− + ⎞
⎠⎤⎦⎥
= − ⋯ +
′
=
⋯
S i τ i P τ i P τ u r τP
σ r τ I
rP
τ P I τ P I r σP
τ τ
δ τ τ τ τ τ δ τ τ
( , ) [SINR ( , ) , SINR( , ) ], exp [exp( ) ]
exp exp ( ) ,
( ) (Ψ ( )) (Ψ ( ), Ψ ( ), ,Ψ ( )) ( ) ,
i i i i i i hα
i
ih
αi i
hα
ii i i i
k
K
k k hα
ii i
i i I i i i I I I i i i i i K i i i i i
A
eFFR
,FR Φ,2
Φ,
Φ, ,FR1
Φ,2
,FR
, , , , , 1, ,FR , ,FR ,FR
ii
i i
i i K1
U
L L
(51)
where =x xΨ ( )k iP r
P,k αi
i . The second Laplace transform part of (51) is computed as
� �̂ ⎜ ⎟ ⎜ ⎟⎡⎣⎢⎛⎝
∑ ⎞⎠⎛⎝
∑ ⎞⎠⎤⎦⎥∏ ⎡
⎣⎢⎛⎝
∑ ⎞⎠⎤⎦⎥
= − = − −∈ ∖
−
∈
−
= ≠ ∈
−1A τ ρ ρ h y τ h y τ h yexp Ψ ( ) ( ) exp Ψ ( ) exp Ψ ( ) ,h i
i iz x
x z iz izα
i i iz ϕ
iz izα
k k i
K
h k i iz ϕ
kz kzα
Φ, ,Φ
, ,FR1,
Φ, , ,FRi
i
i
i
k
k
� �∏ ∏ ∏= ⎡⎣⎢⎢ +
⎛
⎝⎜ −
⎛
⎝⎜ − +
⎞
⎠⎟⎞
⎠⎟⎤
⎦⎥⎥
⎡
⎣⎢ +
⎤
⎦⎥
∈ ∖− −
= ≠ ∈−τ y τ y τ y
11 Ψ ( )
1 1Δ
1 11 Ψ ( )
11 Ψ ( )
.z x i i i iz
α i i i izα
k k i
K
z k i i kzαΦ
Φ , ,FR , 1,Φ
Φ , ,FRii i
kk
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
157
-
where =1 ρ ρ( )x z takes on the value 1 when BS x and BS z use
the same sub-band. Using the PGFL of PPPs and following the
procedure ofTheorem 3.1, we find the coverage probability of cell
edge users under the Strict-FFR mechanism.
References
[1] H. Zhang, X. Chu, W. Guo, S. Wang, Coexistence of Wi-Fi and
heterogeneous smallcell networks sharing unlicensed spectrum, IEEE
Commun. Mag. 53 (3) (2015)158–164.
[2] H. Zhang, C. Jiang, R.Q. Hu, Y. Qian, Self-organization in
disaster-resilient het-erogeneous small cell networks, IEEE Netw.
30 (2) (2016) 116–121.
[3] A. Wyner, Shannon-theoretic approach to a Gaussian cellular
multiple-accesschannel, IEEE Trans. Inf. Theory 40 (6) (1994)
1713–1727, http://dx.doi.org/10.1109/18.340450.
[4] M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, M.
Franceschetti, Stochastic geo-metry and random graphs for the
analysis and design of wireless networks, IEEE J.Sel. Areas Commun.
27 (7) (2009) 1029–1046,
http://dx.doi.org/10.1109/JSAC.2009.090902.
[5] J.G. Andrews, F. Baccelli, R.K. Ganti, A tractable approach
to coverage and rate incellular networks, IEEE Trans. Commun. (11)
(2011) 3122–3134,
http://dx.doi.org/10.1109/TCOMM.2011.100411.100541.
[6] A. Guo, M. Haenggi, Spatial stochastic models and metrics
for the structure of basestations in cellular networks, IEEE Trans.
Wirel. Commun. 12 (11) (2013)5800–5812,
http://dx.doi.org/10.1109/TWC.2013.100113.130220.
[7] B. Blaszczyszyn, M.K. Karray, H.P. Keeler, Using Poisson
processes to model latticecellular networks, Proceedings of IEEE
INFOCOM Conference, (2013), pp.
773–781,http://dx.doi.org/10.1109/INFCOM.2013.6566864.
[8] H. ElSawy, E. Hossain, M. Haenggi, Stochastic geometry for
modeling, analysis, anddesign of multi-tier and cognitive cellular
wireless networks: a survey, IEEECommun. Surv. Tutor. 15 (3) (2013)
996–1019, http://dx.doi.org/10.1109/SURV.2013.052213.00000.
[9] H.S. Dhillon, R.K. Ganti, F. Baccelli, J.G. Andrews,
Modeling and analysis of K-tierdownlink heterogeneous cellular
networks, IEEE J. Sel. Areas Commun. 30 (3)(2012) 550–560,
http://dx.doi.org/10.1109/JSAC.2012.120405.
[10] M. Fereydooni, M. Sabaei, M. Dehghan, G.B. Eslamlou, M.
Rupp, Coverage dis-tribution of heterogeneous cellular networks
under unsaturated load, InternationalWorkshop on Link-and System
Level Simulations (IWSLS), (2016), pp. 1–5.
[11] M. Taranetz, R.W. Heath, M. Rupp, Analysis of urban
two-tier heterogeneous mo-bile networks with small cell
partitioning, IEEE Trans. Wirel. Commun. 15 (10)(2016)
7044–7057.
[12] J. Andrews, S. Singh, Q. Ye, X. Lin, H. Dhillon, An
overview of load balancing inhetnets: old myths and open problems,
IEEE Wirel. Commun. 21 (2) (2014)
18–25,http://dx.doi.org/10.1109/MWC.2014.6812287.
[13] Kyocera, Potential Performance of Range Expansion in
Macro-Pico Deployment(r1-104355), Technical Report, (2010).
[14] G. Ye, H. Zhang, H. Liu, J. Cheng, V.C. Leung, Energy
efficient joint user associationand power allocation in a two-tier
heterogeneous network, Proceedings of IEEEGLOBECOM Conference,
IEEE, 2016, pp. 1–5.
[15] H. Zhang, S. Huang, C. Jiang, K. Long, V.C. Leung, H.V.
Poor, Energy efficient userassociation and power allocation in
millimeter wave based ultra dense networkswith energy harvesting
base stations, IEEE J. Sel. Areas Commun. (2017).
[16] A. Ijaz, S.A. Hassan, S.A.R. Zaidi, D.N.K. Jayakody, S.M.H.
Zaidi, Coverage and rateanalysis for downlink hetnets using
modified reverse frequency allocation scheme,IEEE Access 5 (2017)
2489–2502.
[17] H.-s. Jo, Y.J. Sang, P. Xia, J.G. Andrews, Heterogeneous
cellular networks with
flexible cell association: a comprehensive downlink SINR
analysis, IEEE Trans.Wirel. Commun. 11 (10) (2012) 3484–3495,
http://dx.doi.org/10.1109/TWC.2012.081612.111361.
[18] S. Singh, J.G. Andrews, Joint resource partitioning and
offloading in heterogeneouscellular networks, IEEE Trans. Wirel.
Commun. 13 (2) (2014) 888–901,
http://dx.doi.org/10.1109/TWC.2013.120713.130548.
[19] H. Zhang, C. Jiang, J. Cheng, V.C. Leung, Cooperative
interference mitigation andhandover management for heterogeneous
cloud small cell networks, IEEE Wirel.Commun. 22 (3) (2015)
92–99.
[20] T.D. Novlan, R.K. Ganti, A. Ghosh, J.G. Andrews, Analytical
evaluation of fractionalfrequency reuse for OFDMA cellular
networks, IEEE Trans. Wirel. Commun. 10 (12)(2011) 4294–4305,
http://dx.doi.org/10.1109/TWC.2011.100611.110181.
[21] M. Qian, W. Hardjawana, Y. Li, B. Vucetic, J. Shi, X. Yang,
Inter-cell interferencecoordination through adaptive soft frequency
reuse in LTE networks, Proceedings ofIEEE WCNC Conference, (2012),
pp. 1618–1623, http://dx.doi.org/10.1109/WCNC.2012.6214041.
[22] H. Zhuang, T. Ohtsuki, Analytical evaluation of fractional
frequency reuse forMIMO heterogeneous cellular networks,
Proceedings of IEEE GLOBECOMConference, (2015), pp. 4275–4280,
http://dx.doi.org/10.1109/GLOCOM.2014.7037479.
[23] N. Himayat, S. Talwar, A. Rao, R. Soni, Interference
management for 4G cellularstandards [WiMax/LTE update], IEEE
Commun. Mag. 48 (8) (2010) 86–92.
[24] K. Doppler, C. Wijting, K. Valkealahti, Interference aware
scheduling for soft fre-quency reuse, Proceedings of IEEE 69th VTC
Conference, 1(April) (2009), pp.
1–5,http://dx.doi.org/10.1109/VETECS.2009.5073608.
[25] M. Fereydooni, G.B. Eslamlou, M. Rupp, Performance
evaluation and resource al-location in hetnets under joint
offloading and frequency reuse, Proceedings of IEEESPAWC Confernce,
(2017), pp. 370–374.
[26] W. Li, W. Zheng, H. Zhang, T. Su, X. Wen, Energy-efficient
resource allocation withinterference mitigation for two-tier OFDMA
femtocell networks, Proceedings ofIEEE PIMRC Conference, IEEE,
2012, pp. 507–511.
[27] T.D. Novlan, R.K. Ganti, A. Ghosh, J.G. Andrews, Analytical
evaluation of fractionalfrequency reuse for heterogeneous cellular
networks, IEEE Trans. Commun. 60(2012) 2029–2039,
http://dx.doi.org/10.1109/TCOMM.2012.061112.110477.
[28] F. Baccelli, B. Baszczyszyn, Stochastic Geometry and
Wireless Networks, I NOW:Foundations and Trends in Networking,
2009.
[29] H. Zhang, C. Jiang, X. Mao, H.-H. Chen,
Interference-limited resource optimizationin cognitive femtocells
with fairness and imperfect spectrum sensing, IEEE Trans.Veh.
Techol. 65 (3) (2016) 1761–1771.
[30] H. Zhang, Y. Nie, J. Cheng, V.C. Leung, A. Nallanathan,
Sensing time optimizationand power control for energy efficient
cognitive small cell with imperfect hybridspectrum sensing, IEEE
Trans. Wirel. Commun. 16 (2) (2017) 730–743.
[31] H. Zhang, C. Jiang, N.C. Beaulieu, X. Chu, X. Wen, M. Tao,
Resource allocation inspectrum-sharing OFDMAfemtocells with
heterogeneous services, IEEE Trans.Commun. 62 (7) (2014)
2366–2377.
[32] H. Zhang, C. Jiang, N.C. Beaulieu, X. Chu, X. Wang, T.Q.
Quek, Resource allocationfor cognitive small cell networks: a
cooperative bargaining game theoretic ap-proach, IEEE Trans. Wirel.
Commun. 14 (6) (2015) 3481–3493.
[33] M. Qian, W. Hardjawana, Y. Li, B. Vucetic, X. Yang, J. Shi,
Adaptive soft frequencyreuse scheme for wireless cellular networks,
IEEE Trans. Veh. Techol. 64 (1) (2015)118–131.
M. Fereydooni et al. Computer Communications 116 (2018)
147–158
158
http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0001http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0001http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0001http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0002http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0002http://dx.doi.org/10.1109/18.340450http://dx.doi.org/10.1109/18.340450http://dx.doi.org/10.1109/JSAC.2009.090902http://dx.doi.org/10.1109/JSAC.2009.090902http://dx.doi.org/10.1109/TCOMM.2011.100411.100541http://dx.doi.org/10.1109/TCOMM.2011.100411.100541http://dx.doi.org/10.1109/TWC.2013.100113.130220http://dx.doi.org/10.1109/INFCOM.2013.6566864http://dx.doi.org/10.1109/SURV.2013.052213.00000http://dx.doi.org/10.1109/SURV.2013.052213.00000http://dx.doi.org/10.1109/JSAC.2012.120405http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0010http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0010http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0010http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0011http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0011http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0011http://dx.doi.org/10.1109/MWC.2014.6812287http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0013http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0013http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0014http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0014http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0014http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0015http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0015http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0015http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0016http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0016http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0016http://dx.doi.org/10.1109/TWC.2012.081612.111361http://dx.doi.org/10.1109/TWC.2012.081612.111361http://dx.doi.org/10.1109/TWC.2013.120713.130548http://dx.doi.org/10.1109/TWC.2013.120713.130548http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0019http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0019http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0019http://dx.doi.org/10.1109/TWC.2011.100611.110181http://dx.doi.org/10.1109/WCNC.2012.6214041http://dx.doi.org/10.1109/WCNC.2012.6214041http://dx.doi.org/10.1109/GLOCOM.2014.7037479http://dx.doi.org/10.1109/GLOCOM.2014.7037479http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0023http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0023http://dx.doi.org/10.1109/VETECS.2009.5073608http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0025http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0025http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0025http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0026http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0026http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0026http://dx.doi.org/10.1109/TCOMM.2012.061112.110477http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0028http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0028http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0029http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0029http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0029http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0030http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0030http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0030http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0031http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0031http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0031http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0032http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0032http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0032http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0033http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0033http://refhub.elsevier.com/S0140-3664(17)30478-4/sbref0033
Analytical evaluation of heterogeneous cellular networks under
flexible user association and frequency reuseIntroductionSystem
modelTier association probability and distance distribution
Coverage probabilitySFR interference coordinationStrict-FFR
interference coordination
Spectral efficiencyAverage ergodic rateAverage user
throughputRate coverage probability
Discussion of the resultsSpecial cases of interestResource
allocation
Numerical resultsVerification of analysisImpact of small cell
density and bias value on the rate coverage probabilityImpact of
small cell density and bias value on the average ergodic rateImpact
of small cell density and bias value on the minimum average
achievable rate
ConclusionProof of Theorem 3.1Proof of
Theorem 3.3References