TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 12, Decembe
r
2014, pp. 82
4
6
~ 825
1
DOI: 10.115
9
1
/telkomni
ka.
v
12i12.66
94
8246
Re
cei
v
ed
No
vem
ber 2
1
, 2013; Re
vi
sed
Febr
uary 18,
2014; Accept
ed March 6, 2
014
Optimal Threshold of LTE-Femtocell Network Based
Bayes-
Nash Equilibrium Theory
Hao C
h
en*
1
, Ying Liu
2
, Jianfu Te
ng
3
1
School of Co
mputer an
d Informatio
n
Engi
n
eeri
ng,
T
i
anjin
Che
ngj
ian U
n
iv
ersit
y
, T
i
anji
n
, P.R.Chin
a
2
school of Com
puter Scie
nces
,
T
i
anjin U
n
iver
sit
y
of Scie
nce
and T
e
chno
log
y
, T
i
anjin 3
0
0
2
22, Chi
na
3
School of Elec
tronic an
d Infor
m
ation En
gi
ne
erin
g, T
i
anjin U
n
iversit
y
, No.9
2 W
e
ijin R
o
a
d
, Nank
ai District,
T
i
anjin, P.R.Ch
ina.
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
haoch
e
n
111
44@
163.com
1
,
jfteng@tj
u.ed
u
.
cn
3
A
b
st
r
a
ct
T
o
incre
a
se
L
T
E (lon
g ti
me
evol
ution)
n
e
tw
orks s
pectru
m
utili
z
a
t
i
o
n
a
nd in
terfere
n
ce
itig
ation,
a
LTE system
overlai
d
w
i
th fem
t
ocells
is
studied. This paper w
ill focu
s
a self-opti
m
i
z
e
d
pow
er c
ontr
o
l
sche
m
e f
o
r LT
E-femtoc
ell
net
w
o
rks, in w
h
ic
h the
trans
mi
tt
ed pow
er
of a
f
e
mtoc
el
l
b
a
se station is
a
d
jus
t
ed
base
d
on th
e
opti
m
a
l
SINR
thresh
old. It
is
know
n th
at
ga
me
the
o
ry is
a
usefu
l
to
ol for
an
aly
z
i
n
g
out
a
g
e
prob
abi
lities
an
d opti
m
a
l
pow
e
r
in w
i
reless ne
tw
orks.
In
this pap
er, Bayes-
Nash e
qui
libr
i
u
m
the
o
ry is us
ed
to deriv
e
a o
p
ti
ma
l SINR
(sig
n
a
l-int
e
rferenc
e-
nois
e
-ratio)
thr
e
sho
l
d fro
m
ea
ch fe
mtoce
ll. T
he p
o
w
e
r co
ntrol
sche
m
e ca
n be app
lie
d to realistic LT
E-fe
mtocell n
e
tw
orks to enabl
e robu
st commun
icati
on ag
ainst cro
ss-
tier interfere
n
c
e
thereby o
b
tai
n
in
g a substa
ntial li
nk qu
ality.
Ke
y
w
ords
:
L
T
E-femtocel
l, g
a
me theory, Ba
yes-Nas
h
equ
il
ibriu
m
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
LTE is being
stand
ardi
ze
d by 3GPP to
provi
de multi
-
megabit ba
nd
width, more e
fficient
use
of the radio net
work, l
a
tency
reduct
ion, improved
mobility, and potentially
lower
cost per
bi
t
[1]. Femtocel
ls a
r
e l
o
w-p
o
w
er a
c
cess
points that
op
e
r
a
t
e in
lic
e
n
s
ed
s
p
ec
tr
um a
n
d
pr
o
v
id
e
mobile cove
rage and ca
p
a
city
over
in
ternet-g
ra
de
backh
aul. In
ord
e
r to
im
prove th
e L
T
E
netwo
rk thro
u
ghput
s an
d spectrum effici
ency, LT
E
-
fe
mtocell
t
w
o-ti
red netwo
rks [2-5]
have be
en
studie
d
.
Con
v
entional po
wer control work
ties
in
cellul
a
r
networks an
d p
r
i
o
r
wo
rk on
utility
optimizatio
n
based o
n
g
a
m
e theo
ry. Result
s in
Fo
schini
et al.[6], Zand
er [7],
Gran
dhi
et al
. [8
]
and Bamb
os
et al [9].
2. Contribu
ti
on
Prior
wo
rk a
bout femto
c
e
ll po
wer cont
rol
h
a
s p
r
op
ose
d
to
use
the utility-ba
sed n
on-
coo
perative femtocell SI
NR ad
aptation
[10]. In t
hat literature, SINR threshold
o
f
each fe
mto
c
ell
is pre-esta
blishe
d. And th
en Nash equil
i
brium
can
be
cal
c
ulated. B
u
t in reality, much i
n
form
a
t
ion
is in
compl
e
te
informatio
n
and g
a
me
s a
r
e a
s
ymmetry
.
The key co
ntribution
s
in
our p
ape
r i
s
that
use in
com
p
le
te informatio
n game
s
the
o
ry
－
Baye
s-Nash eq
uilib
riu
m
theory to study two-ti
re
d
femtocell
p
o
w
er co
ntrol. Bayes-Na
sh equilib
ri
um t
heory i
s
em
ployed to fin
d
optimal SINR
threshold. O
w
ing to the B
a
yes-Na
sh e
quilibri
um the
o
ry, the adap
tation minimu
m SINR targ
ets
can b
e
found.
An optimal chann
el-d
epe
n
dant SINR
thresh
old is o
b
tained at ea
ch
femtocell.
3. Sy
stem Model
The syste
m
con
s
i
s
ts of a single central macro
c
ell
0
B
serving a re
gi
on
C
, providing
a
cellul
a
r coverage ra
diu
s
c
R
.
i
B
,
N
i
1
.The LTE macrocell is un
derlai
d
with
N
co-chan
nel
femtocell
s
APs. The sy
ste
m
module i
s
sho
w
n in Fig
u
re 1.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Threshol
d of LTE-Fem
t
ocell Networ
k
Based Bayes-Nash E
quilibrium
…
(Hao
Chen)
8247
Figure 1. System Module
Femtocell users a
r
e lo
cat
ed on the ci
rcumfe
ren
c
e
of a disc of radiu
s
f
R
cente
r
ed at
their femto
c
el
l AP. Ortho
g
o
nal u
p
lin
k
sig
naling
is a
s
su
med in
ea
ch
slot
(1
sched
uled
active
u
s
er
per
cell
d
u
rin
g
ea
ch
signal
ing slot), wh
e
r
e a slot
m
a
y refe
r to
a
time o
r
frequ
en
cy re
sou
r
ce
(the
ensuing
an
al
ysis l
eadi
ng
up to
The
o
re
m 1
apply
eq
ually well ove
r
the
do
wnli
n
k
).
Duri
ng
a gi
ven
s
l
ot, let
}
.....
2
,
1
{
N
i
deno
te the sch
ed
uled u
s
er
co
nne
cted to its BS
i
B
. D
e
s
i
gna
te
u
s
er
s
i
'
transmitting p
o
we
r to be
i
p
Watts
. Let
2
be the varian
ce o
f
AWGN (Ad
d
itive White Gau
ssi
an
Noi
s
e) at
i
B
.
Definition 1.
The re
ceive
d
SINR
i
of user
i
at
i
B
is given as:
,
2
,
ii
i
ii
ji
j
ji
pg
pg
(
1
)
Whe
r
e
i
re
p
r
es
e
n
t
s
th
e SINR
thr
e
sh
o
l
d fo
r
us
er
i
at
i
B
.The term
j
i
g
,
denote
s
the cha
nnel
gain
betwe
en u
s
e
r
j
and BS
i
B
, but it really is interferen
ce term for user
i
at
i
B
.The term
i
i
g
,
ca
n
also a
c
cou
n
t for post
-
p
r
ocessing SINR gain
s
.
Definition 2. The term
)
(
i
i
p
I
represe
n
ts the interferen
ce value of use
r
j
(
i
j
) at
i
B
.
In orde
r to accord with the
terms of ga
m
e
theory,
i
den
otes elem
ent sets oth
e
r tha
n
i
.
2
,
()
ii
j
i
j
ji
Ip
p
g
(2)
Usi
ng Equati
on (1
) and Eq
uati
on (2
), the received SINR
i
ca
n be re
written in Equ
a
tion
(3).
,
()
ii
i
i
ii
p
g
I
p
(
3
)
4. The Opti
mal SINR Threshold
w
i
th Incomplete Information
Auction
gam
e is on
e type
of Bays-NE
theory. It
will
be a
pplie
d to
find the
opti
m
al SINR
threshold
sol
u
tion, that is
Bayes-NE
sol
u
tion. T
he i
n
complete info
rmation fa
ctors mai
n
ly incl
u
de:
wheth
e
r
one
user t
r
an
sm
itting sig
nals or
not
a
nd
t
r
an
smitting si
gnal power. The
two
fa
ctors
compl
e
tely are ra
ndom. In
orde
r to
conv
eniently an
al
yze, we
assu
me that
a fe
w conditio
n
s
that
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8246 – 82
51
8248
are al
so ve
ry close to th
e actu
al co
n
d
it
ions. Ea
ch
femtocell B
S
transmitted
powe
r
i
P
is a
rand
om vari
a
b
le. The te
rm
i
p
denote
s
a n
u
m
eri
c
al value
with a rand
o
m
variable
m
appin
g
s to a
given sample
spa
c
e.
*
i
is a optimal sol
u
tion of
i
. We n
e
e
d
to find an
o
p
timal SINR t
h
re
shol
d
*
i
.
Whe
n
Bayes-NE can b
e
arrived, strate
g
y
function
i
S
is given as:
*
()
ii
i
Sp
(
4
)
In the sectio
n, our task is to ca
lcul
ate
the strate
gy functio
n
set
}
...
,
{
,
2
1
N
set
S
S
S
S
,
whe
n
all elem
ents of L
T
E
-
femtoce
ll n
e
tworks have a
rri
ved Bayes-NE.
Assuming 1:
within any
F
A
P
coverage area
, the probability that
every user whether
transmitting sign
als o
r
no
t is indepe
nd
ent identical
l
y
distributed
(i.i.d). Ho
wev
e
r
,
for dif
f
ere
n
t
femtocell
systems (F
AP
a
nd femtocell
use
r
s),
the d
i
stributio
n fun
c
tion may b
e
dif
f
erent.
Th
e
distrib
u
tion fu
nction i
s
user transmitting
sign
al po
wer
function.
()
(
'
)
1
()(
)
i
i
D
i
str
i
bu
tio
n
F
u
n
c
itio
n
F
p
d
o
n
t
t
ra
ns
m
ittin
g
s
ign
a
l
s
D
i
s
t
rib
u
t
i
on
F
u
n
c
tio
n
F
p
t
ra
ns
m
ittin
g
s
ign
a
l
s
：
：
Definition 3:
The term
)
(
r
P
denotes the p
r
o
bability. The term
)
(
F
denotes
distrib
u
tion
function. The
relation of two function
s is
given as:
()
(
)
rX
PX
x
F
x
(
5
)
Assu
mption
2
:
In order to
maximize
the
interfe
r
en
ce
mitigation, in
every unit tim
e
, only
one femtocell
BS of max p
o
we
r is in tra
n
smitting stat
e.
Assu
mption
3
:
For
any u
s
e
r
, only
whe
n
i
t
s tran
smittin
g
po
we
r i
s
m
a
ximum of all
of FAP
receiving si
gn
als, this u
s
er
can ta
ke opti
m
iz
ation effe
ct on FAP receiver SINR th
reshold.
The optimization
mo
del of FAP
re
ceive
r
th
re
shol
d i
s
based
au
ctio
n gam
e m
o
d
e
l. If all
the u
s
ers tra
n
smit
sign
al,
and th
e u
s
e
r
tran
sm
itting
power
maxim
u
m, this eve
n
t probability
as
follows
:
j
ji
YM
a
x
P
(
6
)
1
()
[
1
(
)
]
N
ri
i
Pp
Y
F
p
(
7
)
Equation (7
) indicate
s that the
th
i
tran
smitting po
wer is big
ger than other use
r
s
transmitting p
o
we
r.
Definition 4.
Strategy func
tion of femtocell use
r
s.
i
0
i
T
h
e
u
s
e
r
c
a
n
n
o
t
ta
k
e
o
p
t
im
iz
a
tio
n
e
ff
e
c
t
o
n
F
A
P
r
e
c
e
iv
e
r
th
r
e
s
h
o
l
d
St
r
a
t
e
gy
F
unct
i
o
n
The
u
s
e
r
c
a
n
t
a
ke
opt
i
m
i
z
at
i
o
n
e
f
f
ect
on
F
A
P
r
ecei
ver
t
hr
es
hol
d
Combi
n
ing
st
rategy function with
Equati
on (7), the
utility function of
th
i
use
r
optimi
z
ing
FAP receive
r
threshold
can
be expre
s
se
d on Equatio
n (8).
Definition 4.
The term
i
U
rep
r
esents Bay
s
-NE utility function, and it is given as:
1
[1
(
)
]
(
)
1
N
ii
i
i
UF
p
i
N
(
8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Threshol
d of LTE-Fem
t
ocell Networ
k
Based Bayes-Nash E
quilibrium
…
(Hao
Chen)
8249
Equation
(8
)
physi
cal m
e
a
n
ing i
s
: T
he
u
t
ility f
unction i
s
exp
r
e
s
sed t
hat differe
nce
of the
real FAP receiver SINR a
nd the FAP receive
r
SI
NR threshold. T
he big
ger th
e
differen
c
e i
s
, the
better the F
AP receive
r
comm
uni
cati
on qu
ality is. Becau
s
e th
e more difference can p
r
ovide
greate
r
re
dun
dan
cy for outage commu
ni
cation.
For all
N
i
1
use
r
s within any FAP area, the pr
o
c
e
ss that
they transmit
signal
s is
incom
p
lete i
n
formation
ga
me. Equation
(9) an
d Equ
a
tion (10) ca
n optimi
z
e th
e FAP re
ceiv
er
threshold.
1
i
M
ax
U
i
N
(
9
)
*
ar
g
ii
p
Ma
x
U
(
1
0
)
Whe
n
the po
wer
co
ntrol g
a
me re
ache
s Bays-NE,
*
i
p
is as a sym
bol
for optimal p
o
we
r
solutio
n
. For
all users with
in any FAP area, t
hei
r transmitting
si
gnal p
r
ob
abili
ty distribution
is
i.i.d. So, when all
femtocell sy
stem
i
s
in equilibrium, optimal
power
*
i
p
is t
h
e same value.
However, for different FAP, the optimal so
lution
may be different. Because the probability
distribution
may be different. For example, for di
ff
erent femtocell users, the probability that
whether users transmitting signal
s or not
is
Bernoulli distribution
or Poisson
distribution. So, for
different FAP
,
the optimal
thre
shol
d is also
di
fferent. Our objec
t
ive is
to
deduc
e the optimal
SINR thr
e
s
h
o
l
d
*
i
function
b
y
using
*
i
p
. Usi
ng Equatio
n (3), Equation
(8)
and Eq
ua
tion (10
)
,
Equation (11) and Equatio
n (12
)
are d
e
rived as follo
ws:
,
*1
ar
g
[
1
(
)
]
(
)
[
0
,
]
()
ii
i
N
ii
i
i
M
a
x
ii
gp
pM
a
x
F
p
p
p
Ip
(
1
1
)
,
*1
arg
[
1
(
)]
(
)
[
0
,
]
()
ii
N
ii
i
i
i
i
M
a
x
i
ii
g
pM
a
x
F
p
G
p
p
p
a
n
d
G
I
p
(12
)
Eric M
a
skin
and
Joh
n
Ril
e
y have p
r
ov
ed Equ
a
tion
(12
)
that Bay
e
s-Na
sh
equi
librium
exists only fo
r optimal
solu
tion [11-1
2
]. Takin
g
the first-o
r
de
r de
rivative of
i
U
with respec
t to
i
p
and a
pplying
Equation
(8
), the optimal
so
lution
can
b
e
de
rived. Mo
reove
r
, wh
en
*
i
i
p
p
, Eq.(4)
will be
set up
. At the same time, it means th
at differentiating
wit
h
re
spe
c
t to
i
p
of
*
i
can b
e
achi
eved.
*2
*
*
*
*
1
'
*
(
1
)[
1
(
)]
(
1
)
(
)[
(
)
]
[
1
(
)]
[
(
)]
0
NN
i
ii
i
i
i
i
i
i
i
u
NF
p
f
p
G
p
S
p
F
p
G
S
p
p
(13
)
Whe
r
e
)
(
*
i
p
f
de
not
es pdf of
i
p
and
)
(
*
'
i
p
S
re
pre
s
e
n
ts the first-o
r
d
e
r
derivative
of
*
i
w.r.t
i
p
.
Since
0
)]
(
1
[
1
*
N
i
p
F
，
this term can be removed from
bot
h sid
e
s of
Equation (13), yields:
'*
*
*
*
*
()
()
()
[()
1
]
ii
i
i
i
i
Sp
M
p
S
p
G
M
p
p
(
1
4
)
and
)
(
)]
(
1
)[
1
(
)
(
*
1
*
*
i
i
i
p
f
p
F
N
p
M
Lemma 1:
(The ge
neral
solution
of first-o
r
de
r lin
ear n
on-hom
ogen
eou
s dif
f
erential
equatio
n).
If the normal
form of
differential eq
uatio
n:
)
(
)
(
'
x
q
y
x
p
y
,then the g
eneral
solutio
n
is
as
follows
:
()
(
)
((
)
)
p
x
dx
p
x
dx
ye
q
x
e
d
x
C
(
1
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8246 – 82
51
8250
Whe
r
e
C
is a consta
nt coeffi
cient.
The lem
m
a 1
provid
es
a
most imp
o
rta
n
t me
thod to
solve Eq
uati
on (14). So
Equation
(16
)
sh
ows a
s
follows, that is the solutio
n
of Equation
(14).
*
**
*
1
*
1
1
0
()
{
[
(
1
()
]
[
1
(
)
]
[
1
(
)
]
}
i
p
NN
ii
i
i
i
Sp
G
F
p
F
p
F
t
d
t
C
(16
)
Acco
rdi
ng to physi
cal mea
n
ing
s
, it is assume
d as
0
*
i
p
,
0
)
(
*
i
p
S
.
1
C
can be
derived. Th
erefore, the strategy set fun
c
tion
*
i
is
rewritten as
follows:
*
**
*
1
*
1
1
0
()
{
[
(
1
()
]
[
1
(
)
]
[
1
(
)
]
1
}
i
p
NN
ii
i
i
i
Sp
G
F
p
F
p
F
t
d
t
(17
)
As a re
sult, whe
n
Equatio
n (4) h
a
s b
e
e
n
exp
licitly determine
d, the
strategy function set
set
S
will consist
s
of different v
a
lues
of Equation (17) according to
parameters of each femtocell
.
It will ensure each femtocell further to
optimize
their power
cont
rol, sp
ectrum utilization
and
mitigate interf
eren
ce.
5. Simulatio
n
Resul
t
and
the An
aly
s
is
In this section, we will
sim
u
late the result
of utilizing optimal thre
shold to power control
in matlab 7.0
platform
s. We li
st the p
a
ram
e
ters ta
ble, give the
results of B
a
ys-Na
s
h b
a
s
ed
experim
ents.
Firstly, som
e
para
m
eters a
r
e gived. Secondly, simu
lat
i
on re
sult is ill
ustrate
d
.
Table 1.
System paramete
r
s
Variable Signal
Parameter
(Unit)
Value
R
Femtocell Radius (m)
30
f
FAP carrier f
r
equ
enc
y
(
GHz)
2
P
ma
x
Femtocell user tr
ansmission pow
e
r(W)
10
M
pseudo rando
m
cy
cle numbe
r B
y
Monte Carlo me
t
hod
100
K
po
w
e
r iter
ative number
100
T
FAP number in t
w
o
-
tier net
woks
20
N
femtocell user number in a FAP a
r
ea
10
F(
x)
Whether femtoce
ll user transmissi
on signal or not
uniform distribution
n
Number
of ever
y femtoce
ll user tra
n
smission signal
10
q
Failure possibility
of transmission signal
0.3
k
Failure Numbe
r
of transimission
signal of ever
y
u
s
er in one FAP a
r
ea
4
Γ
ma
x
Max SINR
thresh
old
Γ
*
+2dB
Γ
mi
n
Min SINR thresh
old
Γ
*
-2
dB
i
cost factor
0.1
2
additive gaussia
n
white noise po
w
e
r (W
)
1×10
-
4
Assu
ming th
at each FAP use
r
send
s a
sign
al
is u
n
iform di
strib
u
tion, and e
a
ch
use
r
i
s
indep
ende
nt and ide
n
ticall
y distribution.
Followi
ng th
e
experim
ents
－
Monte Carl
o
sim
u
lation, we
us
e the B
a
yes-uility function to
solve th
e o
p
timal SINR th
resh
old.
F
r
om
Figure 2, it
sh
ows that
the
*
exits only val
ue, when
the
para
m
eter i
s
given, su
ch a
s
i
G
,
)
(
F
and
N
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Threshol
d of LTE-Fem
t
ocell Networ
k
Based Bayes-Nash E
quilibrium
…
(Hao
Chen)
8251
02
46
8
1
0
0
2
4
6
8
10
Assume
the F
(
.)
is uni
for
m
dis
t
r
i
bu
tion
betw
een
p
i
P
Max
]
Ba
y
e
s
-
ut
ili
ty
fu
nc
t
i
on
(
d
B
)
*
(dB)
Figure 2. The
optimal SINR thre
shol
d
]
,
0
[
max
P
p
i
6. Conclusio
n
In this pa
pe
r, the po
we
r control a
nd int
e
rfer
en
ce miti
gation i
s
sue
s
of femtocell
s
in two
-
tier LTE macro-femto net
works a
r
e discussed. A
novel powe
r
cont
rol sche
me is propo
se
d ba
sed
on Baye
s-Na
sh e
quilib
riu
m
and ite
r
ati
v
e algorith
m
. Whe
n
Baye
s-Na
sh
equili
brium
are
u
s
ed in
femtocell
s
, the optimal SINR targ
et and
optimal tr
a
n
smit powe
r
ca
n be d
e
rived,
whi
c
h i
s
critica
l
to mitigate interferen
ce betwe
en nei
ghbo
ring fe
mtocell
s
and
improve iterative algorit
hm
efficien
cy. Th
e sim
u
lation
results
sh
ow tha
t, by u
s
i
ng the
prese
n
ted
scheme
,
optimal SI
NR
target and
op
timal transmit powe
r
of femtocell
s
ca
n
be obtain
ed
and sufficient
SINR to mitigate
interferen
ce can be provid
ed. In concl
u
sion, t
he su
g
geste
d sche
me can ma
ke femtocell p
o
we
r
control more efficient than
that of the power
cont
rol wi
thout usin
g Bayes-Na
sh e
quilibri
um.
Ackn
o
w
l
e
dg
ements
This work wa
s su
ppo
rted
by the Natio
nal Natu
ral Scien
c
e Fo
und
ation of Chin
a Grant
No.11
301
382
.
Referen
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he F
e
mto F
o
rum. Interference Man
a
g
e
m
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p
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