Joint Load Balancing and Interference CoordinationCan Double Heterogeneous Network Capacity

Sivarama VenkatesanAccess Technologies Research

Bell Labs, Alcatel-Lucent

Holmdel, NJ 07733, U.S.A.

Email: Venkat.Venkatesan@alcatel-lucent.com

Abstract—In a heterogeneous network (HetNet) obtained byoverlaying low-power picocells on a conventional macrocellnetwork, two important problems must be addressed in orderto obtain significant downlink capacity gains. These are inter-cell load balancing, i.e., determining which cell must serveeach user so as to utilize all cells effectively, and inter-cellinterference coordination, i.e., determining when and for how longmust each macrocell be silenced so as to mitigate interferencecaused to nearby picocell users. We formulate a centralizedoptimization framework for these coupled problems, and proposea simple solution to them based on convex relaxation and twoapplications of the Frank-Wolfe algorithm. Simulation resultsshow that the proposed solution has near-optimal performancein a canonical “hotspot” network model, and can double boththe 50th percentile and 5th percentile user rates, when comparedto a baseline HetNet with no load balancing and no interferencecoordination. Further, both interference coordination and loadbalancing are necessary; each by itself yields much lower gains.

I. INTRODUCTION

Cellular network operators are expected to rely heavily on

increased cell density in achieving the required future growth

in system capacity. In this context, the overlay of picocells on

an existing macrocell network has attracted particular atten-

tion. Compared to macrocells, picocells have lower transmitter

power, and lower-gain antennas mounted at lower heights

relative to the surrounding clutter. The large difference in

coverage capabilities between the two tiers of cells results in a

so-called heterogeneous network (HetNet), and poses several

technical challenges in realizing the full capacity potential of

the picocell deployment [1], [2], [3], [4].

Our focus here is on the downlink of such a HetNet, with the

picocells assumed to operate co-channel with the macrocells.

One issue that arises in this context is that of inter-cell loadbalancing, i.e., determining which cell (macro or pico) should

serve each of a given set of users in the network. Simply

associating each user with the cell it receives at maximum

signal-to-noise ratio (SNR) results in only a tiny fraction of

users being offloaded to the picocells, limiting the gain in

system capacity from their deployment. Note that it could well

be advantageous for a user to be served by a cell that is not

its strongest but is lightly loaded, since the user would then

face less contention for channel resources.

So we are led to consider user-to-cell association rules

that artificially expand the areas served by the picocells, by

deviating from the maximum-SNR approach. However, users

near the edge of a picocell’s expanded service area are ex-

posed to strong interference from nearby macrocells. Without

further measures to mitigate this interference, it is impossible

to expand a picocell’s service area significantly since, e.g.,

reliable control channel performance will require the signal-to-

interference-plus-noise ratio (SINR) of the picocell to exceed

a threshold at any user it serves.

One way to mitigate severe macro-to-pico interference is

through inter-cell interference coordination, i.e., silencing

each macrocell for certain time periods, so as to create low-

interference conditions for nearby picocell users. The question

that arises here is when and for how long must each macrocell

be silenced, so that the right balance is struck between mit-

igating interference to picocell users, and retaining adequate

resources at the macrocells for their own users to be served.

Our contribution in this paper is a centralized optimization

framework for jointly solving the load balancing and interfer-

ence coordination problems identified above. Our formulation

lends itself naturally to a conceptually simple algorithmic

solution, based on convex relaxation and two applications of

the Frank-Wolfe algorithm [5]. We show by system simula-

tions that, in a canonical “hotspot” model for a HetNet, our

algorithm achieves near-optimal performance, and yields large

user rate gains (exceeding 100% at both the 50th percentile

and the 5th percentile, at a high enough picocell density)

over a baseline HetNet with no load balancing (i.e., each

user is served by a maximum-SNR cell) and no interference

coordination (i.e., all macrocells are active all the time).

Further, both load balancing and interference coordination are

necessary; each by itself yields much lower gains.

In HetNets based on the 3GPP LTE standard (Release 10

onwards), load balancing is achieved through an associa-tion bias in favor of picocells, and interference coordination

through almost-blank subframes from the macrocells [2]. We

expect that the approach proposed here could be adapted to

determine dynamically the association bias values and almost-

blank subframe patterns for such a network.

Our work is closely related to that in [6], which was the first

to address systematically through optimization the coupled

problems of load balancing and interference coordination in a

HetNet. However, there are significant differences between our

problem formulation and that in [6] (mainly in the framework

for interference coordination), with pros and cons for both.

2013 IEEE 24th International Symposium on Personal, Indoor and Mobile Radio Communications: Mobile and Wireless Networks

978-1-4577-1348-4/13/$31.00 ©2013 IEEE 1957

Details are omitted here due to lack of space. The solution

proposed in [6] is also different; while also employing convex

relaxation, it is based on a primal-dual subgradient algorithm.

For other related work, see [7], [8], [9], [10] (the last of which

contains a distributed algorithm to solve just the load balancing

component of our problem).

As for the rest of the paper, Section II contains a precise

formulation of the problem of interest, Section III the details

of the proposed solution, Section IV some performance eval-

uation results, and Section V some concluding remarks.

II. PROBLEM FORMULATION

We are given a heterogeneous cellular network comprised

of a set M of macrocells, a set P of picocells, and a set

U of users to be served on the downlink by these cells.

We denote by C the set of all cells in the network, i.e.,

C =M∪P . Communication from the cells to the users occurs

over a channel divided into B frequency subbands, indexed

1, 2, . . . , B, for the purposes of user scheduling. Our interest

is in solving jointly the problems of inter-cell load balancingand inter-cell interference coordination in this network. We

propose a centralized optimization framework for doing so,

which we describe and make precise below.

A. Inputs

We take as given the following inputs to the optimization

problem of interest:

a) Cu ⊆ C is the set of candidate serving cells for user

u ∈ U . Each user must be served by a single cell, and each

cell can serve at most one user at a time in each subband. Thegoal of inter-cell load balancing is to determine for each userwhich of its candidate serving cells must actually serve it.

b) Γ ⊆ 2C is a collection of cell groups, each cell group

being a subset of cells allowed to be simultaneously active

and transmitting. In other words, at any time, the cells that are

active must constitute one of these cell groups. The goal ofinter-cell interference coordination is to determine the fractionof time for which each of these cell groups must be active.

c) R(G)u,c (b) is the rate in bit/s that user u ∈ U would achieve

in subband b ∈ {1, 2, . . . , B} if it were the only user served by

cell c ∈ C, and only the cells in cell group G ∈ Γ were active

(set R(G)u,c (b) = 0 if c /∈ G ∩ Cu). These rates serve as channel

quality metrics that guide the load balancing and interferencecoordination decisions.

B. Optimization variables

We now specify the optimization variables and the con-

straints on them:

1) Let cu denote the serving cell chosen for user u ∈ U .

We must have cu ∈ Cu.

2) Let y(G) ≥ 0 denote the fraction of time for which cell

group G ∈ Γ is active. Since exactly one of the cell groups is

active at a time, we must have∑G∈Γ y

(G) = 1.

3) Let t(G)u,c (b) ≥ 0 denote the fraction of time for which

cell group G ∈ Γ is active and cell c ∈ C serves user u ∈ Uin subband b ∈ {1, 2, . . . , B}. Since each cell can serve no

more than one user at a time in each subband, we must have∑u∈U t

(G)u,c (b) ≤ y(G).

4) Let ru ≥ 0 denote the overall rate in bit/s that user

u ∈ U achieves. This rate can be expressed as the sum of parts

corresponding to different active cell groups and subbands:

ru =∑G∈Γ

∑Bb=1 t

(G)u,cu(b)R

(G)u,cu(b).

To avoid trivialities, we will require for each u ∈ U that

R(G)u,c (b) > 0 for some c ∈ Cu, G ∈ Γ, and b ∈ {1, 2, . . . , B};

otherwise user u would be incapable of achieving a nonzero

rate, and could be regarded as being in outage.

C. Optimization problem of interest

The problem of interest, which we label Problem LB-IC, is to maximize a system objective function of the form∑

u∈U Fu(ru), where Fu is a utility function for user u. For

concreteness, we will assume that Fu(r) = log(r) for all users,

which corresponds to requiring proportional fairness in the

overall user rates. Extensions to more general utility functions

are straightforward. A formal problem statement now follows.

Problem LB-IC:

max{cu},{y(G)},{t

(G)u,c(b)

},{ru}

∑u∈U

log(ru) subject to

cu ∈ Cu ∀ u ∈ U ; (1)

y(G) ≥ 0 ∀ G ∈ Γ,∑G∈Γ

y(G) = 1; (2)

t(G)u,c (b) ≥ 0 ∀ u ∈ U∑u∈U

t(G)u,c (b) ≤ y(G)

⎫⎪⎬⎪⎭∀ G ∈ Γ, c ∈ C, b ∈ {1, 2, . . . , B};

(3)

ru =∑G∈Γ

B∑b=1

t(G)u,cu(b)R(G)u,cu(b) ∀ u ∈ U . (4)

We regard the cell group time fractions{y(G)

}and the

user time fractions{t(G)u,c (b)

}as continuously variable. Ac-

commodating any discreteness constraints on them will require

suitably rounding off the solution obtained here.

III. PROPOSED SOLUTION

A. Multi-cell convex relaxation

What makes Problem LB-IC difficult to solve optimally is

the combinatorial nature of the constraint (1) that each user

must be served by a single cell. An important observation

to make here (as in [6], [10]) is that relaxing this constraint

and allowing each user to be served simultaneously by one ormore of its candidate serving cells leads to a tractable convex

program (in fact, one with linear constraints). We refer to

this multi-cell relaxation as Problem Rel-LB-IC and state it

formally below:

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Problem Rel-LB-IC:

max{y(G)},{t

(G)u,c(b)

},{ru}

∑u∈U

log(ru) subject to (2), (3), and

ru =∑c∈Cu

∑G∈Γ

B∑b=1

t(G)u,c (b)R(G)u,c (b) ∀ u ∈ U . (5)

Note that each user’s rate in the multi-cell relaxation has

potential contributions from each of its candidate serving cells.

B. Outline of proposed solution

Our approach for identifying a good (though not necessarily

optimal) solution to Problem LB-IC with acceptable computa-

tional complexity can be summarized succinctly as follows:

1) First solve the multi-cell relaxation Problem Rel-LB-IC optimally. Denote the resulting optimal solution by({

y(G)},{t(G)u,c (b)

}, {ru}

).

2) For each user u ∈ U , find the candidate serving cell cu ∈Cu that contributes the most to its rate in the above solu-

tion, i.e., cu = argmaxc∈Cu∑G∈Γ

∑Bb=1 t

(G)u,c (b)R

(G)u,c (b).

3) Solve Problem Rel-LB-IC optimally once again, but now

with the set of candidate serving cells for each user u ∈ Uset to Cu = {cu}. Denote the resulting optimal solution

by({

y(G)},{t(G)u,c (b)

}, {ru}

).

4) Take({cu} ,

{y(G)

},{t(G)u,c (b)

}, {ru}

)to be the final

(suboptimal) solution to Problem LB-IC.

So we solve two instances of the multi-cell relaxation

Problem Rel-LB-IC. We fix each user’s serving cell based on

the solution to the first instance, and then obtain the cell group

time fractions with the second instance (which has a trivial

load balancing component, i.e., single candidate serving cell

for each user). Note that the first instance yields an upper

bound on the optimal value of Problem LB-IC, which is useful

in assessing the optimality gap of the final solution obtained.

C. Solving multi-cell relaxation by Frank-Wolfe

It remains only to describe an efficient algorithm for solving

the multi-cell relaxation Problem Rel-LB-IC. Since this is a

convex program, any number of generic convex optimization

techniques could be employed to solve it [5]. It turns out,

however, that the specific structure of Problem Rel-LB-ICmakes the Frank-Wolfe (or conditional gradient) algorithm

[5] particularly well-suited to it. For completeness, we now

describe this algorithm in detail.

We wish to obtain feasible cell group time fractions{y(G)

},

user time fractions{t(G)u,c (b)

}, and user rates {ru} that are

guaranteed to be within a tolerable gap from the optimal

solution to Problem Rel-LB-IC. It is more natural to express

this gap in terms of the geometric mean of the user rates, i.e.,

exp{|U|−1∑u∈U log(ru)

}, rather than the objective function

value∑

u∈U log(ru) itself. Accordingly, we will require that

the geometric mean of the user rates at the determined solution({y(G)

},{t(G)u,c (b)

}, {ru}

)be at least a fraction 1− ε of that

at the optimal solution (e.g., ε = 10−2). It will be convenient

here to denote by Uc the subset of users for which cell c is a

candidate serving cell, i.e., Uc = {u ∈ U : c ∈ Cu}.1) Initialization: We initialize

({y(G)

},{t(G)u,c (b)

}, {ru}

)

to an arbitrary candidate solution satisfying ru > 0 for all

u ∈ U . For example, let y(G) = |Γ|−1 for each G ∈ Γ

(all cell groups are given equal time); t(G)u,c (b) = |Uc|−1 for

each G ∈ Γ, c ∈ C, u ∈ Uc, and b ∈ {1, 2, . . . , B}(whichever cell group is active, in each subband, each cell

gives equal time to all the users it can serve); and ru =∑c∈Cu |Uc|

−1∑G∈Γ

∑Bb=1R

(G)u,c (b) for each u ∈ U .

2) Linear program: We now find the feasible solution({y(G)

},{t(G)u,c (b)

}, {ru}

)that maximizes the first-order ap-

proximation∑

u∈U[log(ru) + r−1u (ru − ru)

]of the objective

function at the current candidate solution. Clearly, this is a

linear program. The key to the attractiveness of the Frank-

Wolfe algorithm in our setting is that this linear program can

be solved readily by inspection. The solution corresponds to

a single cell group G ∈ Γ being active all the time, and each

cell c ∈ C serving a single user uc(b) ∈ Uc all the time in

each subband b ∈ {1, 2, . . . , B}. Here,

G = argmaxG∈Γ

∑c∈G

B∑b=1

maxu∈Uc

[R(G)

u,c (b)/ru

], (6)

uc(b) = argmaxu∈Uc

R(G)u,c (b)/ru, (7)

with all ties resolved arbitrarily. Thus, y(G) equals 1 if G =

G, and 0 otherwise. Further, t(G)u,c (b) equals 1 if G = G and

u = uc(b), and 0 otherwise. The resulting rate in bit/s for

user u ∈ U is ru =∑

(c,b):uc(b)=u R(G)u,c (b).

3) Termination check: The geometric mean of the user

rates at the optimal solution to Problem Rel-LB-IC must lie

between M = exp{|U|−1∑u∈U log(ru)

}(the geometric

mean of the user rates at the current candidate solution)

and M = exp{|U|−1∑u∈U

[log(ru) + r−1u (ru − ru)

]}(this

follows from the concavity of the objective function). It can

be shown further [5] that the ratio M/M of these bounds

must approach 1 as the algorithm progresses. Accordingly,

we terminate the algorithm if M > (1 − ε)M ; we are then

guaranteed that the geometric mean of the user rates at the

final candidate solution is at least a fraction 1 − ε of that at

the optimal solution. Otherwise, we proceed to the next step.

4) Line search: Next, we identify a suitable point on the

line joining the current candidate solution and the linear-

program solution, and update the current candidate solution

to this point: y(G) ← (1 − δ∗)y(G) + δ∗y(G), t(G)u,c (b) ←

(1 − δ∗)t(G)u,c (b) + δ∗t(G)u,c (b), ru ← (1 − δ∗)ru + δ∗ru.

Here, δ∗ ∈ (0, 1) can be chosen by the Armijo rule [5]:

pick constants μ, σ ∈ (0, 1), e.g., μ = 0.5 and σ =0.01, and set δ∗ = μk∗

, where k∗ is the smallest inte-

ger k ≥ 0 such that∑

u∈U log[(1− μk

)ru + μkru

] ≥∑u∈U

[log(ru) + σμk (ru − ru) /ru

]. Having performed this

update, we return to the linear program step.

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TABLE IMACRO AND PICO PARAMETERS

Macrocells PicocellsTx power 40 W 1 WAntenna gain 14 dBi 5 dBiAntenna pattern 3D [11], 15◦ downtilt 2D [11], omniAntenna height 32 m N/ATx antennas/cell 1 1Path loss at 1 km 128.1 dB 140.7 dBPath loss exponent 3.76 3.67Shadowing std. dev. 8 dB 10 dBMin. user distance 35 m 10 m

TABLE IIUSER PARAMETERS

Penetration loss 20 dBShadowing correlation 50% across sitesAntenna pattern Omni, 0 dBi gainAntenna height 1.5 mRx antennas/user 2Noise power density -174 dBm/HzRx noise figure 9 dB

IV. PERFORMANCE EVALUATION

In this section, we present simulation results to illustrate

the HetNet performance gains obtainable with the joint load

balancing and interference coordination algorithm of this pa-

per. The parameters of the simulated system are based on a

“hotspot” network model in [11].

A. Network model

The underlying macrocellular network has 19 sites on a

regular hexagonal lattice, with an inter-site distance of 500 m.

Wraparound is implemented to avoid edge effects. Each site

hosts 3 macrocells covering different 120◦ sectors (with their

main lobes pointing at neighboring sites). In the hexagonal

Voronoi region of each site, Np = 3, 6, 12, 15, or 30 picocells

are placed at random locations, subject to a minimum site-

to-pico distance of 75 m and pico-to-pico distance of 40 m.

Relevant parameters pertaining to the macrocells and picocells

are summarized in Table I.

In each of the 19 hexagons, 90 users are dropped. Of these,

60/Np are dropped at random locations within 40 m of each

of the Np picocells in that hexagon (these are the “hotspot

users”), and the remaining 30 are dropped at random locations

within the entire hexagon. All user locations are, in addition,

subject to the minimum distances specified in Table I. Relevant

user parameters are summarized in Table II.

The channel is assumed time-invariant and spatially uncor-

related but frequency-selective, with the Extended Typical Ur-

ban (ETU) power-delay profile. Orthogonal frequency-division

multiplexing (OFDM) is employed, with 600 tones spaced 15

kHz apart. The OFDM symbol rate is 14 ksym/s. Each cell

transmits with a fixed power whenever active, divided equally

among all the tones. The number of frequency subbands Bfor user scheduling is 10 (so each subband has 60 tones).

B. Test cases

For each value of Np, the number of picos per hexagon, we

compare the following ways of operating the HetNet:

1) Baseline: No load balancing, no interference coordination

2) LB only: Load balancing, no interference coordination

3) IC only: Interference coordination, no load balancing

4) LB and IC: Load balancing and interference coordination

Here, “no load balancing” means that each user must

be served by a maximum-SNR cell, while “no interference

coordination” means that all cells must be active all the time

(i.e., Γ = {C}). In contrast, “load balancing” allows each user

to be served by any cell whose SNR is within 15 dB of that

of the maximum-SNR cell, while “interference coordination”

allows all the macrocells to be silenced synchronously for

certain time periods (i.e., Γ = {C,P}).As for the channel quality metrics, we set R

(G)u,c (b) = 0 if the

average SINR of cell c at user u when cell group G is active

is below −6 dB; this is intended to model the requirement

of control channel decodability. Otherwise, we set R(G)u,c (b) =

W ·∑f min{6, log2

[1 + γ

(G)u,c (f)

]}(the sum is over all tones

f in subband b), where W = 14 × 103 is the number of

OFDM symbols per second, and γ(G)u,c (f) is the SINR in tone

f of cell c at user u when cell group G is active, assuming a

linear MMSE receiver with perfect channel state information.

The upper limit of 6 bit/sym is to preclude per-tone spectral

efficiency values that would be difficult to achieve with typical

mobile transceiver hardware.

Finally, any user u such that∑B

b=1R(G)u,c (b) = 0 for all

c ∈ Cu and G ∈ Γ is declared to be in outage and excluded

from the set U of users to which the algorithm is applied (the

rate of any such user is simply set to 0). With the assumed

minimum average SINR requirement of −6 dB, the fraction

of such users was under 1% in all simulated cases.

C. Simulation results

Figure 1 shows the geometric mean of the user rates

(averaged over many drops of picos and users) in all four cases,

versus the number of picocells per macrocell (i.e., Np/3).

We have also overlaid as dashed lines the corresponding

upper bounds from the multi-cell convex relaxation, but they

are virtually indistinguishable from the solutions obtained by

the algorithm, indicating the latter’s near-optimality in this

evaluation scenario (this near-optimality was observed in a

number of other evaluation scenarios also). It appears that,

for the vast majority of users, a clear-cut choice of a single

serving cell emerges from the solution to the relaxation; there

is little to be gained by allowing multiple serving cells.

Figure 2 and Figure 3 show, respectively, the 50th (median)

and 5th (edge) percentiles of the user rate distribution across

many drops in all four cases, versus the number of picocells

per macrocell. It can be seen that the joint load balancing and

interference coordination algorithm (“LB and IC”) yields very

significant gains, increasing with the picocell density, over the

baseline HetNet; with 10 picocells per macrocell, the median

and edge user rates are both doubled.

1960

0 2 4 6 8 102

3

4

5

6

7

8

Picocells per macrocell

Geo

. mea

n of

use

r ra

tes

(Mbi

t/s)

LB and ICLB onlyIC onlyBaseline

Fig. 1. Geometric mean of user rates vs. pico density

0 2 4 6 8 101

2

3

4

5

6

7

8

9

Picocells per macrocell

50th

−pc

t. us

er r

ate

(Mbi

t/s)

LB and ICLB onlyIC onlyBaseline

Fig. 2. Median user rate vs. pico density

Further, the combination of load balancing and interference

coordination is crucial. Load balancing alone (“LB only”)

gives a much lower gain over the baseline HetNet, especially

at higher picocell densities. This is contrary to the conclusion

in [4]; the discrepancy is due to the −6 dB minimum average

SINR requirement we have imposed, which limits the ability

to offload users to picocells in the absence of interference

coordination. Finally, interference coordination alone (“IC

only”) results in no discernible gain, since it makes little

sense to take resources away from the macrocells without first

expanding each picocell’s service area.

V. CONCLUSIONS

In this paper, we have proposed an optimization framework

and a simple algorithm for the coupled problems of load

balancing (through appropriate user-to-cell association) and

interference coordination (through time-domain resource par-

titioning among cells) on the downlink of a heterogeneous cel-

lular network. The proposed algorithm achieves near-optimal

performance in several canonical evaluation scenarios. The

large gains it exhibits in both median and edge user rates

0 2 4 6 8 100.5

1

1.5

2

2.5

3

Picocells per macrocell

5th−

pct.

user

rat

e (M

bit/s

)

LB and ICLB onlyIC onlyBaseline

Fig. 3. Edge user rate vs. pico density

demonstrate the importance of well-designed load balancing

and interference coordination algorithms in achieving the full

potential of heterogeneous networks. A distributed version of

the proposed algorithm would be of interest. Some other areas

for future work include investigating performance with a more

refined collection of cell groups that allows macrocells to be

silenced selectively (i.e., not all at once), and dynamically

determining such a collection of cell groups.

REFERENCES

[1] A. Damnjanovic, J. Montojo, Y. Wei, T. Ji, T. Luo, M. Vajapeyam,T. Yoo, O. Song, and D. Malladi, “A Survey on 3GPP HeterogeneousNetworks,” Wireless Communications, IEEE, vol. 18, no. 3, pp. 10–21,Jun. 2011.

[2] L. Lindbom, R. Love, S. Krishnamurthy, C. Yao, N. Miki,and V. Chandrasekhar, “Enhanced Inter-Cell Interference Coordi-nation for Heterogeneous Networks in LTE-Advanced: A Survey,”http://arxiv.org/abs/1112.1344v2, Dec. 2011.

[3] D. Lopez-Perez, I. Guvenc, G. De La Roche, M. Kountouris, T. Quek,and J. Zhang, “Enhanced Inter-Cell Interference Coordination Chal-lenges in Heterogeneous Networks,” http://arxiv.org/abs/1112.1597v1,Dec. 2011.

[4] J. G. Andrews, “Seven ways that HetNets are a cellular paradigm shift,”Communications Magazine, IEEE, vol. 51, no. 3, pp. 136–144, 2013.

[5] D. Bertsekas, Nonlinear Programming. Athena-Scientific, 1999.[6] S. Deb, P. Monogioudis, J. Miernik, and J. Seymour, “Algorithms for

Enhanced Inter-Cell Interference Coordination (eICIC) in LTE HetNets,”http://arxiv.org/abs/1302.3784, Feb. 2013.

[7] R. Madan, J. Borran, A. Sampath, N. Bhushan, A. Khandekar, and T. Ji,“Cell Association and Interference Coordination in Heterogeneous LTE-A Cellular Networks,” Selected Areas in Communications, IEEE Journalon, vol. 28, no. 9, pp. 1479 –1489, Dec. 2010.

[8] M. Shirakabe, A. Morimoto, and N. Miki, “Performance Evaluationof Inter-Cell Interference Coordination and Cell Range Expansion inHeterogeneous Networks for LTE-Advanced Downlink,” in WirelessCommunication Systems (ISWCS), 2011 8th International Symposiumon, 2011, pp. 844–848.

[9] C. S. Chen and F. Baccelli, “Gibbsian Method for the Self-Optimizationof Cellular Networks,” http://arxiv.org/abs/1207.3704, Aug. 2012.

[10] Q. Ye, B. Rong, Y. Chen, C. Caramanis, and J. Andrews, “Towardsan Optimal User Association in Heterogeneous Cellular Networks,”in Global Telecommunications Conference, 2012. IEEE GLOBECOM2012. IEEE, 2012.

[11] 3GPP, “Evolved Universal Terrestrial Radio Access (E-UTRA): FurtherAdvancements for E-UTRA Physical Layer Aspects,” 3rd GenerationPartnership Project (3GPP), TR 36.814, 2010. [Online]. Available:http://www.3gpp.org/ftp/Specs/html-info/36814.htm

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