1
A Game Theoretic Distributed Algorithm for
FeICIC Optimization in LTE-A HetNets
Ye Liu, Member, IEEE, Chung Shue Chen, Senior Member, IEEE, Chi Wan Sung, Senior Member, IEEE, and
Chandramani Singh, Member, IEEE
Abstract—In order to obtain good network performance in
Long Term Evolution-Advanced (LTE-A) heterogeneous net-
works (HetNets), enhanced inter-cell interference coordination
(eICIC) and further enhanced inter-cell interference coordination
(FeICIC) have been proposed by LTE standardization bodies
to address the entangled inter-cell interference and the user
association problems. We propose distributed algorithms based
on the exact potential game framework for both eICIC and
FeICIC optimizations. We demonstrate via simulations a 64%
gain on energy efficiency (EE) achieved by eICIC and another
17% gain on EE achieved by FeICIC. We also show that FeICIC
can bring other significant gains in terms of cell-edge throughput,
spectral efficiency (SE) and fairness among user throughputs.
Moreover, we propose a downlink scheduler based on a cake-
cutting algorithm that can further improve the performance of
the optimization algorithms compared to conventional schedulers.
Index Terms—LTE/LTE-A, heterogeneous networks, resource
allocation, distributed optimization, potential game.
I. INTRODUCTION
According to an estimate of the growth of mobile data
volume [1], more capacity must be added to the current cellular
networks. Cell densification, due to its ability of reusing spec-
trum geographically and its property of preserving signal-to-
interference-plus-noise ratio (SINR) [2], serves as a promising
candidate solution to meet the demand of mobile users [3].
Contrary to the traditional cell densification where more high-
power base stations (BSs) are added, it is more practical to
add low-power BSs due to the high cost of installing macro
BSs and the shortage of available sites suitable for macro BSs
[4], which gives rise of the development of heterogeneous
networks (HetNets).
The emergence of HetNets gives rise to two challenging net-
work management problems. First, because pico BSs transmit
Y. Liu is with the Wolfson School of Mechanical, Manufacturing and Elec-
trical Engineering, Loughborough University, Leicestershire, United Kingdom,
LE11 3TU (email: y.liu6@lboro.ac.uk).
C. S. Chen is with the Mathematics of Dynamic & Complex Net-
works department in Nokia Bell Labs, 91620 Nozay, France (e-mail:
chung shue.chen@nokia.com).
C. W. Sung is with the Department of Electronic Engineering, College of
Science and Engineering, City University of Hong Kong, Kowloon, Hong
Kong (e-mail: albert.sung@cityu.edu.hk).
C. Singh is with the Department of Electronic Systems Engineering, Indian
Institute of Science (e-mail:chandra@iisc.ac.in).
Part of the work was done when Y. Liu was with Nokia Bell Labs, Centre
de Villarceaux, 91620 Nozay, France.
This work has been partially supported by ANR project IDEFIX under grant
number ANR-13-INFR-0006, a grant from the Research Grants Council of
the Hong Kong Special Administrative Region, China, under Project CityU
121713, and by the Engineering and Physical Science Research Council of
the UK, EPSRC, under the grant EP/M015475.
at low power levels compared to macro BSs, mobile users
who are physically located near pico BSs may be attracted
to macro BSs, which can create underutilized pico BSs and
overcrowded macro BSs. Therefore, in order to fully utilize
the available resources in BSs with different transmission
power, careful treatment is needed when performing user
association. Second, the surrounding macro BSs of a pico BS
can generate large interference to a user associated to the pico
BS, and such inter-cell interference must be well-managed
in order to prevent pico BSs’ users from suffering very low
downlink throughputs. To solve these issues, enhanced inter-
cell interference coordination (eICIC) has been proposed in
Release-10 of the 3GPP LTE standards, where
1) Cell selection bias (CSB) is used to offset the received
signal power from BSs to a user so that a user is not
necessarily associated with the BS that provides the
strongest received power, and
2) Almost blank subframe (ABS) can be configured in
macro BSs so that the macro BSs cease data transmis-
sions in certain time slots, which reduces interference to
pico BSs.
The use of ABSs can help reduce the interference from
macro BSs to pico BSs. However, the restriction that macro
BSs must mute their data transmissions entirely in ABSs
may result in the inefficient use of the increasingly scarce
resources. In Release-11, further enhanced inter-cell interfer-
ence coordination (FeICIC) has been proposed, where instead
of offering ABSs, macro BSs allocate reduced power almost
blank subframes (RP-ABSs) to serve their users at reduced
power levels.
Clearly, the configurations of CSB values and ABS patterns
in eICIC optimization are coupled, because the amount of
ABSs depends on the load on pico BSs which depends on the
CSB values. To achieve the maximum possible performance
gain using eICIC, joint optimization in ABS patterns and CSB
values is required. Similarly, we must jointly consider RP-ABS
patterns and CSB values when doing FeICIC optimization.
While eICIC optimization algorithms have been studied in
[5]–[15], little attention is paid on the algorithm that performs
FeICIC optimization.
In this paper, we propose an exact potential game framework
that is suitable for performing both eICIC and FeICIC opti-
mizations. Specifically, we make the following contributions:
1) A distributed optimization framework: Based on the
exact potential game framework, we propose a scalable
distributed algorithm that can either jointly optimize
2
ABS and CSB patterns or jointly optimize RP-ABS and
CSB patterns. The game theoretic framework can adapt
itself to various system optimization targets, such as
proportional fairness (PF) and sum rate maximization.
2) Performance evaluation: We evaluate the performance
gain due to FeICIC and eICIC optimizations. Simula-
tion results show that, compared to the case when no
optimization is performed, FeICIC can nearly double
the energy efficiency (EE) while eICIC provides about a
64% improvement on EE. Also, FeICIC provides higher
fairness in the throughputs of the users and better cell-
edge throughputs compared to eICIC.
3) A better downlink scheduler: We propose a downlink
scheduler based on a cake-cutting algorithm. Simulation
results show that the proposed scheduler can further
improve the EE and spectral efficiency (SE) by 10%
compared to conventional schedulers, can provide better
fairness in SE, and is about 20 times faster than con-
ventional convex algorithms in terms of simulation run
time.
A. Related work
A number of eICIC optimization algorithms have been pro-
posed in the literature. Tall et al.’s algorithm in [5] decouples
the ABS optimization and CSB optimization, where the ABS
patterns are simplified as fractional numbers. A centralized
algorithm is proposed by Deb et al. in [6], where ABS and
CSB patterns are jointly optimized and the surrounding macro
BSs of a pico BS must offer ABSs on the same subframes. In
[7], a distributed algorithm is proposed by Pang et al. where
the number of ABSs is determined without considering CSB.
Thakur et al. considered the problem of CSB optimization and
power control in [8]. Bedekar and Agrawal, in [9], simplify
the joint ABS and CSB optimization problem so that the
optimization of ABS ratios and user attachment are solved
separately. Simsek et al. propose a learning algorithm that
optimizes CSB patterns in frequency domain in [10] and
further extend the idea to optimizing CSB patterns in both
time and frequency domain in [11]. Liu et al., in [12],
propose to optimize the probability that a macro BS offers
almost blank resource blocks on both time and frequency
dimensions. Potential game based solutions for distributed
eICIC optimization are considered in [13]–[15].
The benefit of FeICIC against eICIC has been analyzed in
[16] using stochastic geometric approach, where the expres-
sions for SE and cell-edge throughputs have been derived as
a function of the power reduction factor on the RP-ABSs.
However, the power reduction factor on all RP-ABSs are
assumed to be the same in [16]. An optimization algorithm
that can dynamically adjust the transmission power on each
RP-ABS has not been considered to our best knowledge.
In this work, we address the FeICIC optimization problem
based on exact potential game models. We adapt the game
theoretic frameworks in [14], [15] such that power control
on each time-frequency slot, i.e., physical resource block
(PRB), are included during the optimization process. Also, we
rigorously discuss the necessary assumptions which are needed
for the validity of the exact potential game formulations
and evaluate the effect of such assumptions. Moreover, we
evaluate the performance of a downlink scheduler based on
a cake-cutting algorithm and compare it against conventional
schedulers.
B. Organization and notation
The rest of the paper is organized as follows. Section
II gives the system model of the LTE-A HetNets. Section
III formulates the eICIC and FeICIC optimization problems.
Section IV develops the exact potential game framework that
is suitable for eICIC and FeICIC optimizations. Section V
describes the strategy sets and the better response dynamics
of the games for eICIC and FeICIC optimization. Section
VI introduces the cake-cutting downlink scheduler and other
benchmark schedulers. Section VII presents the numerical
studies. Finally, Section VIII concludes the paper.
Unless otherwise specified, we use small letters such as a to
denote scalars, bold small letters such as a to denote vectors,
calligraphy letters such as A to denote sets. Also, |A| returns
the number of elements in set A and ∅ denotes the empty set.
A \B gives the elements in set A that are not in set B.
II. SYSTEM MODEL
Consider a randomly generated HetNet as shown in Fig. 1
which consists of macro BSs and pico BSs, where the squares
represent macro BSs and the triangles represent pico BSs.
Denote M and P as the set of all macro BSs and the set
of all pico BSs, respectively. Also, denote M
c
and P
c
as
the macro BSs in the center cluster of the HetNet and pico
BSs in the center cluster of the HetNet, respectively, where
the center cluster is surrounded by bolded borders in Fig. 1.
Six clusters which are identical to the center clusters are
placed around the center cluster. We make such distinction
between the center cluster and other clusters because we only
care about the optimization of the BSs in the center cluster,
and the surrounding clusters are generated only to realize the
interference as encountered in practice. We assume that there
is only one macro BS located at the center of each hexagon,
and each hexagon has the same number of pico BSs, e.g., one
pico BS per hexagon in Fig. 1.
Let N(i, n) be BS i’s neighboring BSs that are located in
the n-th layer of hexagons w.r.t.
1
the hexagon in which BS
i is located, where i ∈ M ∪ P. The 0-th layer of hexagons
w.r.t. the hexagon ξ is ξ itself, and the n-th layer of hexagons
w.r.t. ξ are the hexagons
1) that are adjacent to the (n −1)-th layer of hexagons of
ξ, and
2) that are further away from ξ than the hexagons of the
(n −1)-th layer.
For example, in Fig. 1, N(1, 0) gives {101},
N(101, 0) gives {1}, both N(1, 1) and N(101, 1) give
{2, 3, 4, 5, 6, 7, 102, 103, 104, 105, 106, 107}, and both
N(1, 2) and N(101, 2) give the set of BSs in the center
cluster except the BSs in {1, 101} ∪ N(1, 1). The definition
1
w.r.t. stands for with respect to.
3
−3000 −2000 −1000 0 1000 2000 3000
−3000
−2000
−1000
0
1000
2000
3000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
x−axis coordinate in meters
y−axis coordinate in meters
109
110
113
114
116
117
118
102
107
101
106
103
104
115
112
111
119
108
105
Fig. 1: An example of a hexagonal HetNet layout. The squares
represent the macro BSs and the triangles represent the pico
BSs. Users are not displayed for the sake of clarity.
of N(i, n) can also be easily extended to the case where i
represents a set of BSs located in the same hexagon.
Let U be the set of all users in the system. Denote m
u
as
the macro BS that is located in the same hexagon as user u .
We assume that only the BSs in the same hexagon or in the
adjacent hexagons can serve a user. In other words, the set of
candidate BSs that can serve user u is given as:
O
u
, {m
u
} ∪N(m
u
, 0) ∪N(m
u
, 1).
Define vector γ
O
u
as the CSB values of all BSs in O
u
and
let γ
O
u
(i) gives the CSB value of BS i, where i ∈ O
u
. The
set C contains all possible values that γ
O
u
(i) can take. Let
P
Rx
i,u
be the reference signal received power (RSRP) of user u
from BS i when the BS is transmitting at its full power. The
exact value of P
Rx
i,u
depends on the distance between BS i and
user u and the loss due to shadow fading. The effect of fast
fading is assumed to be averaged out for P
Rx
i,u
. The following
equation gives the BS that serves user u:
g(u, γ
O
u
) , arg max
i∈O
u
(P
Rx
i,u
+ γ
O
u
(i)). (1)
Let U
B
be the set of users who are associated with BSs in the
set B, i.e.,
U
B
, {u|g(u, γ
O
u
) ∈ B}.
Clearly, U
B
is a function of the CSB values of the BSs in B
and their nearby BSs. Let γ denote the vector which specifies
all BS’s CSB values.
Suppose each BS has N
T
subframes in the time domain
and N
F
resource blocks (RBs) in the frequency domain. All
subframes have the same duration and all RBs are identical in
terms of bandwidth. A PRB is formed by a pair of subframe
and RB, and we denote N
B
:= N
T
· N
F
as the total number
of PRBs available at each BS. It is assumed that all subframes
and RBs of all BSs are synchronized.
Let the length N
T
vector α
m
specify the ABS pattern
of macro BS m, where all the entries in α
m
are binary.
Let A contain all possible ABS patterns that a macro BS
can adopt, where each element in A consists of a binary
vector of length N
T
. Also, let
ˆ
A be a subset of {1, 2, ..., N
T
}
which contains the indices of subframes which can be an
ABS as indicated in any element in A. For example, suppose
A = {(0, 1, 1, 1), (0, 1, 0, 1), (0, 0, 0, 1)}, then
ˆ
A = {1, 2, 3}
because subframes 1, 2, and 3 are possible ABSs.
Let τ(b) be the subframe index of PRB b. Moreover, let
ˆ
α
m
be a vector of length N
T
× N
F
whose elements specify the
power allocation of macro BS m on each PRB, where
ˆ
α
m
(b)
is a real number between 0 and 1 for τ (b) ∈
ˆ
A and
ˆ
α
m
(b) is
fixed to be one for τ (b) ∈ {1, 2, ..., N
T
}\
ˆ
A. The vector
ˆ
α
m
then defines the RP-ABS pattern of macro BS m. Note that
although it is not necessary to assume that a macro BS offers
RP-ABS only in the subframes specified by
ˆ
A, the definition
of
ˆ
α
m
aims at offering a fair comparison between FeICIC
optimization and eICIC optimization.
In this paper, we assume that only the macro BSs would
offer ABSs/RP-ABSs while the pico BSs always transmit on
all subframes. Such an assumption is reasonable because
1) The macro BSs have much more transmission power
than the pico BSs. Consequently, the macro BSs are the
main source of interference in the network.
2) The complexity of the resulting eICIC/FeICIC optimiza-
tion is reduced compared to the case where all stations
offer ABSs/RP-ABSs.
Also, we assume that only the pico BSs may set their CSB
values to some positive numbers while the macro BSs fix their
CSB values to zeros. This is because, in general, it is the
coverage range of a pico BS which needs to be extended in
order to better utilize the resources from the pico BS.
Given the above definitions, the signal-to-noise-plus-
interference ratio (SINR) of user u on PRB b when associated
with macro BS m can be calculated as:
SINR
m
u,b
=
h
m
u,b
P
Rx
m,u
·α
m
(τ(b))
P
IF
I
m
,u,b
+N
0
W
, BS m offers ABS, (2a)
h
m
u,b
P
Rx
m,u
·
ˆ
α
m
(b)
P
IF
I
m
,u,b
+N
0
W
, BS m offers RP-ABS,(2b)
where h
m
u,b
gives the fast fading gain on PRB b from macro
BS m to user u, τ(b) returns the subframe index of PRB b, I
m
denotes the set of BSs whose transmission will interfere the
users located in the same hexagon as macro BS m, P
IF
I
m
,u,b
is
the sum of interference at user u received from BSs in I
m
at
PRB b, N
0
denotes the additive white Gaussian noise (AWGN)
spectral density and W is the bandwidth of a PRB. Similarly,
the SINR of user u on PRB b when associated with pico BS
p is given by:
SINR
p
u,b
=
h
p
u,b
P
Rx
p,u
P
IF
I
p
,u,b
+ N
0
W
, (3)
where a pico BS does not offer ABS/RP-ABS as discussed
before. Let r
u,b
be the achieved rate of user u at PRB b,
where b ∈ [1, N
B
]. It is assumed that the serving BS knows
4
the achieved rate of user u at PRB b, and the achieved rate is
calculated by Shannon’s capacity formula, i.e.,
r
u,b
=
W · log
2
(1 + SINR
m
u,b
), g(u, γ
O
u
) = m ∈ M,
W · log
2
(1 + SINR
p
u,b
), g(u, γ
O
u
) = p ∈ P.
Table I summarizes the notation used in this paper.
III. PROBLEM FORMULATION
Let x
u,b
be a binary variable indicating whether PRB b
is allocated to user u by its serving BS, where x
u,b
= 1
means that PRB b is allocated to user u and x
u,b
= 0 means
otherwise. To discriminate the importance of different users,
positive weighting factors are applied, where we denote w
u
as
the weighting factor for user u.
We formulate the eICIC optimization problem as follows
MAXPFUTILITY-I
maximize
i∈M∪P
u∈U
i
w
u
·ln
N
B
b=1
(x
u,b
·r
u,b
), (4a)
subject to
u∈U
m
x
u,b
=α
m
(τ(b)),
∀m∈M
c
, b∈[1, N
B
], α
m
∈A, (4b)
u∈U
p
x
u,b
=1, ∀p∈P
c
, b∈[1, N
B
], (4c)
x
u,b
∈{0, 1}, ∀u∈U, b∈[1, N
B
], (4d)
γ(i)∈C, ∀i∈P
c
, (4e)
where (4b) specifies that a macro BS can adopt one of the ABS
patterns in A and only non-ABS PRBs can be assigned to the
users such that at most one user can occupy a PRB, (4c) states
that all PRBs from pico BSs can be allocated to the users and
at most one user can occupy a PRB, and (4e) means that a
pico BS can adopt one of the CSB values specified in C.
For the FeICIC optimization in which macro BSs may offer
RP-ABSs, we aim at solving the following problem
MAXPFUTILITY-II
maximize
i∈M∪P
u∈U
i
w
u
·ln
N
B
b=1
(x
u,b
·r
u,b
), (5a)
subject to
ˆ
α
m
(b)∈[0, 1], τ (b)∈
ˆ
A, ∀m∈M
c
, (5b)
ˆ
α
m
(b)=1, τ (b)∈{1, 2, ..., N
T
}\
ˆ
A, ∀m∈M
c
, (5c)
u∈U
i
x
u,b
=1, ∀i∈M
c
∪P
c
, b∈[1, N
B
], (5d)
(4d) and (4e),
where (5b) means that power allocation is optimized on PRBs
whose subframe indices are in
ˆ
A and (5c) means that no power
optimization is performed on PRBs whose subframe indices
are not in
ˆ
A. Because there is no restriction on a macro BS
that it must completely mute its transmission on a subframe
in FeICIC optimization, every PRB from a macro BS can be
allocated to at most one user as specified in (5d).
The objective functions of both (4) and (5) are defined as
the sum of logarithm of users’ throughputs. Such an objective
achieves the proportional fairness among the users’ achiev-
able rates, which strikes a good trade-off between aggregate
network throughput and user fairness [17]. Also, different
realizations of γ will affect the elements in {U
i
|i∈P
c
}, which
is how CSB optimization comes into the problems (4) and (5).
TABLE I: Summary of notation.
Notation Description
α
m
ABS pattern of macro BS m
ˆ
α
m
RP-ABS pattern of macro BS m
γ Vector specifying CSB values of all BSs
γ
O
u
Vector specifying CSB values of BSs in O
u
g(u, γ
O
u
) The BS that user u is associated with
m
i
The macro BS located in the same hexagon as an object
with index i, where the object can be a user or a pico BS
r
u,b
Achieved rate of user u at PRB b
τ (b) The subframe index of PRB b
w
u
Weighting factor on the achieved rate of UE u
x
u,b
Indicator of whether user u occupies PRB b of the
serving cell
N
0
Noise power spectral density
N
B
Number of PRBs
N
F
Number of RBs (in frequency domain)
N
T
Number of subframes (in time domain)
V
i
The payoff function of player i
W Bandwidth per RB
A Set of vectors from which macro BSs can
choose their ABS patterns
ˆ
A Set containing indices of subframes which can be ABSs
C Set of CSB values from which a pico BS can choose from
I
m
Set of BSs whose transmissions interfere the users
located in the same hexagon as BS m
L The set of players in the potential game model
M Set of all macro BSs
M
c
Set of macro BSs in the center cluster
N (i, n) The set of BS i’s neighboring BSs located in the
n-th layer of hexagons w.r.t. the hexagon that contains i
O
u
Candidate BSs who can serve user u
P Set of all pico BSs
P
c
Set of pico BSs in the center cluster
S
i
The strategy set of player i
U Set of all users in the system
U
i
Set of users associated with BSs in set i or with BS i
We now show the NP-hardness of (4) and (5).
Theorem 1. Both (4) and (5) are NP-hard.
Proof: Consider the case where no ABS/RP-ABS is
applied in any macro BS and all pico BSs fix their CSB values
to zeros, and assume there is only one element in M
c
∪ P
c
.
We then obtain a special case for both (4) and (5) where the
only problem left is to decide how to allocate the PRBs of a
single BS. We denote this special case as:
PRB-ALLOCATION
maximize
u∈U
i
w
u
· ln
N
B
b=1
(x
u,b
· r
u,b
), (6a)
subject to
u∈U
i
x
u,b
= 1, b ∈ [1, N
B
], (6b)
x
u,b
∈ {0, 1}, ∀u ∈ U
i
, b ∈ [1, N
B
]. (6c)
It is shown in [18] that (6) is NP-hard. Therefore, both (4) and
(5) are NP-hard because a special case of the two problems is
NP-hard.
In the next section, we propose a potential game based
framework which can be applied to both (4) and (5) to solve
the problems distributedly and heuristically.
IV. EXACT POTENTIAL GAME FORMULATION
In this section, we frame the eICIC and FeICIC optimization
problems as exact potential games. Our approach is motivated
5
by the successful application of potential games to another
scenario in [19] for BS power control and user association.
A. Preliminary
A finite game consists of a finite set of players, a finite
set of strategies of each player, and the payoff functions of
the players, where the payoff of a player is a function of the
strategies played by all the players. A strategy profile gives the
strategies adopted by all the players, and a Nash equilibrium
is a strategy profile s
∗
such that no player can improve its
payoff by playing a different strategy than the one specified
in s
∗
while other players keep their strategies same.
A game is called an exact potential game if there exists an
exact potential function such that change in the value of the
exact potential function due to a change of a player’s strategy
is the same as the change of the player’s payoff. In a finite
exact potential game, a Nash equilibrium can be achieved if
players take turns randomly and play their best responses or
better responses [20], where, given that all other players fix
their strategies,
1) A
best response
is a player strategy that maximizes the
player’s payoff function.
2) A better response is a player strategy that improves the
payoff function of the player.
As demonstrated later, that being able to formulate the eICIC
and FeICIC optimization problems as exact potential games
will allow us to solve them distributively using simple algo-
rithms based on best/better response dynamics.
In order to realize the process by which a macro BS adapts
its ABS/RP-ABS pattern when a pico BS in the same hexagon
optimizes its CSB value, it is convenient to define a player
as a union of a macro BS and the pico BSs within the same
hexagon. Let L be the set of players, where each element in L
consists of a set that contains the macro BS and the pico BSs in
a hexagon in the center cluster. We can then denote the game as
Γ , ⟨L, {S
i
: i ∈ L}, {V
i
, : i ∈ L}⟩, where S
i
is the strategy
set of player i and V
i
is the payoff function of player i. Note
that the game structure Γ can be applied to both eICIC and
FeICIC optimization problems because the two problems have
the same players and the same objective functions. The only
difference between the eICIC optimization and the FeICIC
optimization is the power allocation constraint on the PRBs,
and this difference can be captured by the definitions of the
respective strategy sets. The details of the strategy sets and
payoff functions will be discussed later.
When a player changes its strategy during the game for
eICIC optimization, users that are associated with the BSs
represented by the player and are associated with other nearby
BSs would be affected. A similar situation applies to the
game for FeICIC optimization. Consequently, to achieve a
good system performance for both (4) and (5), the payoff
function of a player should take users who are located in
nearby hexagons into account, even if these users are not being
served by the player. On the other hand, the transmission of
a BS can, in theory, interfere users located very far away.
To ensure accuracy, the payoff function of a player should
then consider all users in the system. However, such a payoff
function will introduce high complexity to the optimization
process and at the same time deviate from the intention of
designing a distributed algorithm. Some approximation on
the interference is necessary for a low complexity distributed
algorithm. It is therefore important to first identify the impact
of changing ABS/RP-ABS and CSB patterns before defining
a payoff function that leads to an exact potential game and
facilitates low-complexity distributed designs.
In the following, we first discuss which neighboring BSs
of player i can be affected by changes in player i’s CSB
values
2
. We then define the payoff function of players and
identify an exact potential function based on some interference
approximation. Details of the strategy sets, the algorithms that
converge to a Nash equilibrium, and the downlink schedulers
will be given in later sections.
B. Neighboring sets of a player
As mentioned in previous discussion, for scalability, we
make an approximation that the interference range of a BS
is limited only to some of its neighboring hexagons, because
the interference power from a BS to a user is negligible if the
user is located far away from the BS. We use N
IF
i
to specify
the set of BSs whose hexagons are interfered by player i. More
precisely, it means that a user is interfered by the transmission
of the BSs represented by player i if and only if he is located
in the hexagon of a BS that belongs to N
IF
i
.
Let N
Att
i
contains the BSs whose user attachment patterns
depend on the CSB values of the pico BSs represented by
player i. Clearly, the actual serving BS of a user depends on
the CSB values of the pico BSs represented by player i, if a
BS represented by player i is a candidate serving BS of that
user. Moreover, because user u can be attached to any BS
in O
u
, the actual serving BS of user u depends on the CSB
values of all BSs in O
u
. Therefore,
N
Att
i
=
{∀u|{i}⊂O
u
}
O
u
. (7)
The next proposition shows which elements constitute N
Att
i
.
Proposition 1. N
Att
i
= i ∪N(i, 1) ∪ N(i, 2).
Proof: See Appendix A.
Define the utility of player i as
U
i
(s) ,
u∈U
i
w
u
· ln
N
B
b=1
(x
u,b
· r
u,b
), (8)
where s is the strategy vector that specifies the strategies
played by all players. Let N
i
contain player i and player i’s
neighboring BSs whose downlink users’ SINRs and/or whose
user attachment patterns can be affected by changing the
ABS/RP-ABS patterns and CSB values of player i. The next
proposition shows the elements in N
i
when N
IF
i
= i∪N(i, 1).
Proposition 2. Suppose i ∈ L and N
IF
i
= i∪N(i, 1). Keeping
s
−i
unchanged, changes in s
i
may affect U
j
only if j ∈ N
Att
i
.
In other words, N
i
= N
Att
i
.
Proof: See Appendix B.
2
More accurately, by changes of the CSB values of the pico BSs represented
by player i.
![TABLE II: Parameters for generating HetNet topologies [27].](/figures/table-ii-parameters-for-generating-hetnet-topologies-27-13kaks7t.webp)
