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Computer Communications

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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

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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


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)


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


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)


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


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


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).


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 .


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 .


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


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.


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

analytical evaluation of heterogeneous cellular networks ... · bias values and ffr thresholds that...

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