A general model of fractional frequency reuse: Modelling and performance analysis
VNU Journal of Science: Comp. Science & Com. Eng, Vol. 36, No. 1 (2020) 38-45
Original Article
A General Model of Fractional Frequency Reuse:
Modelling and Performance Analysis
Lam Sinh Cong1,*, Nguyen Quoc Tuan1, Kumbesan Sandrasegaran2
1Faculty of Electronics and Telecommunications, VNU University of Engineering and Technology,
Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
Received 01 November 2018
Revised 27 December 2018; Accepted 23 April 2019
Abstract: Fractional Frequency Reuse (FFR) is a promising to improve the spectrum e ciency in
the LongTerm Evolution (LTE) cellular network. In the literature, various research works have
been conducted to evaluate the performance of FFR. However, the presented analytical approach
only dealt with the special cases in which the users are divided into 2 groups and only two power
levels are utilised. In this paper, we consider a general case of FFR in which the users are
classified into
groups and each group is assigned a serving power level. The mathematical model
of the general FFR is presented and analysed through a stochastic geometry approach. The derived
analytical results in terms of average coverage probability can covered all the related well-known
results in the literature.
Keywords: Fractional Frequency Reuse, LongTerm Evolution, coverage probability, stochastic geometry.
1. Introduction *
which represents 70% of the global population.
This will make mobile data traffic experience
eight-fold over the next five years. Therefore,
the requirement of spectral efficiency
improvement is a big challenge for the network
designers and operators.
One of the most popular to improve spectral
efficiency relates to frequency resource
allocation in which all Base Stations (BSs) are
allowed to operate on all Resource Blocks
(BSs). It is reminded that in Long Term
Evolution (LTE) network, each RB is defined
as having a time duration of 0.5ms and a
bandwidth of 180kHz made up of 12 sub-
In recent years, there has been a rapid rise
in the number of mobile users and mobile data
traffic. According to Cisco report [1], the
number of mobile users has a 5-fold growth
over the past 15 years. In 2015 more than a half
of a million devices have joined the cellular
networks. It is predicted that the number of
mobile users will reach 5.5 billion by 2020
_______
* Corresponding author.
E-mail address: [email protected]n
38
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
39
carriers with a sub-carrier spacing of 15kHZ.
Due to sharing RBs between BSs, InterCell
Interference (ICI) which is caused by using the
same RB at adjacent cells at the same time
becomes a main negative factor to limit
the network performance. Therefore, Fractional
Frequency Reuse (FFR) algorithms have
been introduced to control the reuse of
frequency [2].
Hence in this paper, we consider the FFR
algorithm in which the users and RBs are
N
groups ( 2). Thus,
classified into
power levels are deployed, in which each user
group is served by a group of RB with a specific
power level. Figure 1 is an example of the
= 3
proposed model with frequency reuse
.
The basic idea of FFR algorithm is to divide
the active users as well as the allocated RBs
into some groups so each group of users is
served by a specific group of RBs.
As recommendations of 3GPP [3,4,5], the BS
can utilise a lower power level to serve the user
with better wireless channel, a higher power level
to sever other users. By this way, the main
benefits are expected to achieve as follows:
• Reduce the power consumption of the
BSs. Some users with good communication
links such as low propagation path loss,
low fading can obtain their desired performance
with low power levels. Thus, the BSs do not
need to use high power levels to serve
those users.
• Improve system performance. It is
obvious that when a BS cuts its transmit power
off, its interfering power at the adjacent cell
will be reduced. Thus, the system performance
can be improved.
In the literature, there are a lot of research
works on modelling and performance analysis
of FFR in LTE networks by utilizing the
simulators such as [6,7,8,9,10] or stochastic
geometry models such as [11,12,13]. However,
these works only considered two groups of
users and thus only two power levels were
utilised. In a real network, the users as well as
RBs can be partitioned into more than two
groups. For example, a macro cell with radius
= 3
Figure 1. A proposed FFR algorithm with
.
The operational discipline of the system
model can be described as follows:
= 3
cells use the same frequency
km
• Every
reuse pattern.
from 1 - 20
can cover a huge area of up to
km2
400
. Thus, the users associated with that
• The users are classified into 3 groups by
two SINR thresholds. There power levels are
macro cell experiences a wide range of SINR
and consequently they should be classified in
more than two groups to achieve better network
performance as well as save the power
consumption of BSs.
P
.
3
denoted by
P
,
P
and
1
2
• The resource and power allocations are
presented in Table 1.
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
40
= 3
In stead of assuming that there are only two
Table 1. Power allocation in the case of
groups of users, we classified users into
groups by 1 SINR thresholds. The user
j
Cell 1 Cell 2 Cell 3
P
RB group 1
RB group 2
RB group 3
P
P
1
3
2
is assigned to group if its downlink SINR on
j
P
P
P
1
the control channel satisfy the following
condition
3
2
P
P
P
2
3
1
Tj1 < SINR < Tj (3)
in which Tj is the SINR threshold
, ,
T0 = 0
j
2. System model
T = , and Tj1 < Tj for
.
0 < j
We consider a single tier cellular network in
which the locations of BSs follows a spatial
2.2. Communication phase
Poison Point Process (PPP) with mean . The
We denote the transmit power used to serve
users in group is j . Since the high power
levels are used to serve users with the lower
SINR on the control channel, Pj1 < Pj for
. We denote the ratio between the
user prefers a connection with the nearest BS.
According to 3GPP recommendations at
[4, 5], the operation of FFR includes two
phases, called establishment phase and
communication phase. The detail of these
phases are described as follows:
P
j
1< j N
power levels and the lowest power level
P is
1
2.1. Establishment phase
j = Pj /P1 . It is noted that the transmit power
The users measure and report the received
SINRs on the downlink control channels [4, 5]
for user classification purpose. Every BS is
continuously transmitting downlink control
information, and subsequently each control
channel experiences the ICI from all adjacent
BSs. Furthermore, since all BSs are assumed to
transmit on the control channels at the same
power, the ICI of the measured SINR during
this phase is given by.
Pj and j
number.
,
are a constant
(0 < j N)
Due to sharing the RBs between cells, each
user experiences ICI from all neighbouring
cells. The total ICI power at the typical user is
given by:
I =
P
g r (4)
k
j j
k=1
j
k
in which k is the set of interfering BSs
I = Pg(o)r (1)
0
j
j
transmitting at Pj power level. The density of
j
where g(jo) and rj are the channel power
BSs in k is
.
gain and distance between BS
respectively.
and the user,
j
Equation 4 can be considered as the general
case of the well-known FFR algorithm
modelling in the literature. For examples:
• When =1, Equation 4 degrades into
The reported SINR on the control channel is
given by
Pg(o)r
I = Pg r (5)
j
j
SINR =
(2)
2 Pg(o)r
j
j
j
In Equation 5,
BSs. This equation has been found in the
literature such as [15, 16].
consists of all adjacent
j
in which
g
and
is the channel power gain
r
and the distance from the user to its serving BS.
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
41
• When only two power levels are deployed
(only one SINR threshold is required): for
example group 1 is served by transmit power
Therefore, the probability in which the
typical user is under the network coverage at a
given time slot is given by
P
and 1 remaining groups are served by
1
P =
P(T < SINR < T )
n1 n
transmit power P2 , Equation 4 degrades into
c
(9)
n=1
1
I = Pg r
P g r (6)
ˆ
P(SINR > T)
1
j
j
2 j j
j
k=1 j
k
1
It is reminded that the coverage probability
Due to the thinning properties of PPP [16],
each BS in 1 is distributed independently to
in Equation 9 is a function of random variables
such as channel power gain
g
, j , distance
g
any BS in k
(
). Therefore, Equation 6 is
j 1
from the user to other BSs. Thus, to obtain the
average coverage probability of the typical user,
the expected value of
rewritten as
I = Pg r
P g r (7)
0
1
j
j
2
j j
P
should be computed.
c
j
j
1
Therefore, the average coverage probability of
the user in the network is defined as following
in which the density of BSs in 1 and 2 are
/
and
respectively.
(1)/
equation:
Equation 7 is exactly the ICI of Soft FR.
The reported SINR on the data channel
ˆ
P(T) = E (P(T < SINR < T )
n1
n
during the communication phase is given by
(10)
n=1
Pgr
ˆ
P(SINRn > T))
SINR =
(8)
P
g r
Using the definition of SINR in Equation 2,
k
j j
k=1
j
k
P(Tn1 < SINR < Tn )
in which
g
and
is the channel power
r
gain and the distance from the user to its
serving BS.
g(o)r
= P T
<
< T
n
n1
g(o)r
j
j
j
r
rj
r
rj
3 Performance evaluation
= P T
g(o)
< g(o) < T g(o)
n1
n
j
j
j
j
In this section, we derive the average
coverage probability of the typical user, which
can be classified into one of
At a given time slot, the user at a distance
from its serving BS is assigned to group if
groups.
(a)
=
exp
Tn1g(jo)rj r
j
r
j
exp T g(o)r r
its downlink SINR satisfies Equation 3. The
(11)
n
j
j
corresponding
probability
is
j
P(T < SINR < T )
.
n1
n
in which
due to g(o) has a exponential
(a)
The user in group
is under the network
j
distribution.
coverage if its SINR during the communication
SINR
Similarity, using the definition of
in
SINR
phase, denoted by
, is greater than the
Equation 8, we have
ˆ
coverage threshold
. Thus, the coverage
T
ˆ
P(SINR > T)
ˆ
P(SINR > T)
probability is
.
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
42
P gr
Evaluating the fist element with notice that
n
ˆ
= P(
> T)
into
k is a subset of , we divide
P
g r
k
j
j
independent subsets k with the densities of
k=1
j
k
/
. Thus, the first element in
BSs are
P
= P g > T
g r r
Equation 14 can be rewritten as follows:
k
ˆ
P
j j
k=1
j
k
n
(b)
1
1
P
exp T gjrj r (12)
k
ˆ
E =
E
=
1
r
k
r
rj
P
P rj
n=1
k
k=1 j
P
ˆ
k=1 j
n
1T
1Tn1
k
n
where
due to is a exponential random
g
(b)
Employing the properties of the Probability
variable.
Generating Function [19], we obtain
Substituting Equations 11 and 12 into
Equation 10, the average coverage probability
2
1
1
ˆ
1
r dr
P(T ) is given by
j
j
P
r
r
r
r
r
k
ˆ
1T
1T
n1
P
n
j
j
E =
E
e
P
1
exp T gjrj r
k
ˆ
n=1
k=1
k1
P
n
jk
exp T g(o)r r
E
(13)
n1
j
j
n1
Using a change of variable y = (rj/r)2
,
E1
j
exp T g(o)r r
can be rewritten as follows
n
j
j
j
2 1
2r
1
1
dy
1
P
/2
/2 1T
y
y
k
ˆ
Since all channel power gains are
independent exponential random variables
whose the Moment Generating Function (MGF)
1
1T
n1
P
n
E =
E
e
1
n=1
k=1
is M X = E[esx ] =
, taking the expected
1 s
value of Equation with respect to g(jo) and g j
Taking the expected value with respect to
,
r
,
E1 is given by
ˆ
P(T ) is obtained by
2
r
ˆ
(T
,T ,P )
n
n1
k
k=1
2
re r2e
dr
1
1
0
E
n=1
P r
r
rj
n=1
k=1 j
j
k
ˆ
in which
k
1T
1Tn1
P rj
n
1
1
ˆ
n (Tn1,T, P ) =
1
dy
1
1
k
1Tn1 y/2
1
P
E
(14)
1T
y/2
k
P r
P rj
r
rj
ˆ
n=1
k=1 j
P
1T
j 1Tn
k
ˆ
k
n
n
Similarly, the second element of Equation 14 is
given by
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
43
2
m 2
Equation 15 with T0 = 0,T ,
Tm =
1
r
ˆ
(T ,T ,P )
n n k
re 2 e
r
k=1
and P = P
, we obtain
m,n > 2
E2 = 2
dr
m
n
0
n=1
1
ˆ
P(T) =
Substituting E1 and E2 into Equation 14 and
employing a change of variable in which
y = r2 , the average coverage probability
1
1
ˆ
ˆ
1
1(0,T, P )
1(0,T, P )
1
1
1
ˆ
P(T ) is given by:
1
1
ˆ
ˆ
1
2 (T ,T, P)
2 (T ,T, P )
1
1
1
1
2
ˆ
P(T) =
1
n=1 1
ˆ
n (Tn1,T, P )
k
k=1
1
(17)
1
(15)
1
1
ˆ
ˆ
1
1
1(T ,T, P) 1(T ,T, P )
n=1 1
ˆ
n (Tn ,T, P )
1
1
1
2
k
k=1
Equation 15 provides the mathematical
expression of the average coverage probability
of the typical user in LTE network using FFR
The corresponding result for Soft Frequency
Reuse algorithm has been found in [17].
with reuse factor
in which users are
classified into user groups. This result can be
4 Simulation and discussion
considered as the general form of the published
results in the literature. Take two special cases,
= 3
=1 and
, for example
Special case 1: =1
In this case, T0 = 0 and T = , then
1
1
ˆ
n (0,T, P ) =
1
dy
k
1
P
/2
y
k
ˆ
1T
P
n
ˆ
and
by
.
n(,T,P ) = 0
k
The average coverage probability is given
1
ˆ
P(T) =
(16)
ˆ
n=1 1
n (Tn1,T, P )
k
The expression in Equation 16 is the well-
known result on the average coverage
probability of the typical user in LTE network
Figure 2. Comparison between simulation and
analytical results.
Figure 2 presents the comparison between
the simulation and analytical results with
with frequency reuse factor =1
.
3.1 Special case 2: Only two power levels
are deployed
different values of path loss coefficient
and
ˆ
coverage threshold
The following parameters are selected for
= 3
T
.
This model is usually called Soft Frequency
Reuse [18] in which the users and RBs are
simulation: the frequency reuse factor
,
the Rayleigh fading with a unit power, the
divided into
equal groups. Using the result in
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
44
= 3
users. Meanwhile the BSs in the case of
utilize P/3 and P/9. Thus, it can be said
that the BSs in the case of = 2 consume
= 3
= 0.025
(
density of BSs
2 ) and the
BS/km
SNR = 10
P
,
signal-to-noise ratio
dB.
1
1
1
As shown in Figure 2, the Monte Carlo
simulation results perfectly match with the
analytical results that can confirm the accuracy
of the analytical approach.
As indicated in Figure 2, the average
coverage probability of the typical user
. This conclusion also has
been found in the literature and can be
explained as follows:
• Since the user is assumed to associate with
the nearest BS. The distance from the user to
the interfering BSs must be greater than that
from the user the serving BS.
• The path loss is proportional to the path
loss coefficient and the distance. Hence, when
the path loss exponent increases, the interfering
signals experience higher path loss than the
serving signal. In other words, SINR and
consequently average coverage probability
more energy than that in the case
It is observed that the average coverage
probability reduces when increases. This
phenomenon is reasonable since the user
achieves the higher performance with high
serving power. However, in order to compare
the performance of frequency reuse algorithms,
various parameters and scenarios should be
considered [7].
.
increases with
0.8
0.75
0.7
0.65
=2
=3
=4
0.6
increase with the path loss exponent
Figure 3 compares the average coverage
probability of the typical user with different
.
0.55
values of
and SINR threshold. The selection
0.5
of parameters are as the following table:
0.45
T3
-10
-5
0
5
10
15
T
T2
1
SINR Threshold
-10 (dB)
= 2
= 3
= 4
Figure 3. Comparison average coverage probability
with different values of
-10 (dB) 0 (dB)
-10 (dB) 0 (dB)
.
10 (dB)
Serving Power
of each group
P
P/3
1
1
5 Conclusion
Table 2. Analytical parameters of Figure 3.
In this paper, the general model of FFR in the
LTE network was modelled and analysed under
Rayleigh fading environment in which the BSs
are distributed according to a spatial Poisson
process. Instead of assuming that there are only
two power levels are used to serve the associated
It is assumed that all users in Group 1 have
the same serving power and the users with high
SINRs will be served with lower transmit
powers. Thus, the serving power of the adjacent
group of users with high SINRs is reduced by 3
times. From Table 2, it is observed that the total
energy that is used by the BSs to serve the
user, this paper considered
power levels in
which each power level is utilised to serve a
specific user group. The analytical results which
are verified by Monte Carlo simulation can be
considered as the general expressions of the
typical user performance since they contain all the
associated users reduces with . For example,
the BSs in the case of = 2 will transmit at
two levels
P
and P/3 to serve the associated
1 1
L.S. Cong et al. / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 36, No. 1 (2020) 38-45
45
Cellular Networks, IEEE Commun, Surveys &
Tutorials 15(4) (2013) 1642-1670.
related results in the literature. For practical
perspective, based on the relationships between
frequency reuse factor
, SINR threshold Tj ,
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blocks allocation strategies on downlink
interference and sir distributions in lte networks:
density of BSs and the network performance that
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user performance.
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[11] H. ElSawy, E. Hossain, M. Haenggi, Stochastic
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Acknowledgments
This work has been supported/partly
supported by VNU University of Engineering
and Technology under project number
CN18.01.
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(2013)
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[12] W. Bao, B. Liang, Stochastic Analysis of Uplink
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