TELKOM
NIKA
, Vol.13, No
.1, March 2
0
1
5
, pp. 299~3
0
4
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i1.117
299
Re
cei
v
ed
Jul
y
19, 201
4; Revi
sed
Jan
u
a
r
y 6, 20
15; Accepted
Jan
u
a
ry 2
4
, 2015
Internet Pricing on Bandwidth Function Diminished
with Increasing Bandwidth Utility Function
Indra
w
ati
*
, Irmeil
y
a
na, Fit
r
i Ma
y
a
Puspita, Ok
y
Sa
nja
y
a
Jurusan Matematika, Faku
lta
s
Matematika d
an Ilmu Pen
get
ahu
an Al
am, Universit
a
s Sri
w
i
j
a
y
a
Jln. Ra
ya Pra
b
u
muli
h KM 32 Inder
ala
y
a O
g
a
n
Ilir Sumatera
Selata
n Indo
ne
sia
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: indra
w
a
t
i1
00
6@gma
il.com
A
b
st
r
a
ct
In this paper
w
e
analy
z
e
th
e intern
et prici
ng sche
m
es b
a
sed o
n
ban
d
w
idth function
di
min
i
sh
ed
w
i
th increas
in
g
ban
dw
idth
util
ity functio
n
w
i
th 3
pr
ici
ng str
a
tegi
es for h
o
m
o
g
e
neo
us a
n
d
het
erog
en
eo
us
consu
m
er. Th
e n
e
w
prop
os
ed
prici
ng sc
h
e
mes w
i
th th
is
utility fu
nctio
n
w
ill g
i
ve th
e i
n
formatio
n
to t
he
intern
et servic
e provi
ders (ISP) in maxi
mi
z
i
n
g
pr
of
its an
d provi
de b
e
tter service q
u
a
lity for users. T
h
e
Mode
ls on ev
e
r
y type of consumer ar
e a
ppl
i
ed to the data t
r
affic in Pale
mban
g loc
a
l serv
er. LINGO 11.0 is
used to co
mpu
t
e the non
li
nea
r progra
mming
probl
e
m
to
ge
t the opti
m
al s
o
luti
on. T
he re
sults show
ed t
h
a
t
for each
case
base
d
o
n
3-
pricin
g sch
e
m
e, ISPs get
b
e
tter profit by
choos
in
g al
l
three sch
e
m
e
s
i
n
consu
m
ers typ
e
of
ho
mo
ge
no
us cas
e
w
h
il
e f
o
r h
e
terog
e
n
e
ous c
a
ses
on w
i
ll
ing
ness
to pa
y and
b
a
sed
o
n
dema
nd of the
consum
ers
, ISPs can sel
e
ct flat fee sche
m
e
to gain
hi
g
her
profit rather tha
n
those tw
o oth
e
r
schem
es.
Ke
y
w
ords
: uti
lity functio
n
s,
the functi
on
o
f
di
min
i
sh
ed b
andw
idth
w
i
th
incre
a
si
ng b
andw
idth, prici
n
g
sche
m
es, cons
umer ho
moge
n
eous, het
erog
e
neo
us cons
u
m
ers
1. Introduc
tion
Internet ha
s
an impo
rtant
role in the
eco
nomy an
d edu
cation
arou
nd the
world. The
Internet is a
multimedia
libra
ry, beca
u
se it ha
s a lot of complete information. Com
p
lete
informatio
n a
nd q
u
ickly m
a
ke
con
s
ume
r
s intereste
d
i
n
be
comi
ng
a
co
nsume
r
int
e
rnet
se
rvice
s
.
Con
s
um
ers
who
ma
ke a l
o
t of Internet
Service P
r
ovi
ders (ISPs)
compete to
provide service
s
of
the hig
h
e
s
t q
uality (Q
uality of Se
rvice
)
a
nd
the
optimal prices for
con
s
um
ers [1]-[3]. In addition
in maintai
n
in
g the
quality
of se
rvice a
n
d
optimal
p
r
ices fo
r
co
nsu
m
ers, Interne
t
Service
Pro
v
ider
(ISP)
sho
u
ld also con
s
id
er
profits.
The
re
sea
r
ch
on diffe
renti
a
ted net
wo
rk in g
ene
ral
netwo
rk archi
t
ecture
with
quality of
servi
c
e a
r
e d
ue to [4]-[7] whi
c
h then a
r
e improve
d
b
y
[8] in multi
QoS netwo
rks and [9] in multi
servi
c
e networks.
In parti
cula
r,
the re
cent
r
e
s
e
ar
ch focus
on
w
i
r
e
less
mesh QoS
network
architectu
re
a
r
e d
ue to
[10
],[11] that mainly
discu
s
s th
e adva
n
ced t
e
ch
nolo
g
y in
comm
uni
cati
on
netwo
rk.
There are
some assumpti
ons for
utility func
tion to
be applied in the model
but the
resea
r
chers
usu
a
lly use the band
width
function with
fixed loss a
nd delay and
follow the ru
les
that margin
al
utility as band
width fun
c
tion
dimini
shing
with incre
a
si
n
g
band
width [
4
-8]. The oth
e
r
reason deali
ng
with the choices
of utility
functi
on i
s
that t
he utility function
should be
differentiabl
e
and
ea
sily t
o
be
an
alyze
d
the
hom
og
eneity an
d h
e
terog
eneity
that impa
cts
the
choi
ce
of pri
c
ing
structu
r
e for the
co
mpanie
s
. Ke
l
l
y [12]-[14] a
l
so
cont
e
n
d
s
that the util
ity
function al
so
can be a
ssumed to be
increa
sing fu
nction, st
rictl
y
concave a
nd co
ntinuo
u
s
ly
differentiabl
e.
In [15], the f
i
nding
of int
e
rnet
ch
argin
g
is
ba
sed
o
n
an
alytical
step
s a
nd o
n
Cobb
-
Dou
g
la
ss util
ity function. Other u
s
eful
utility
functions
are
pro
v
ided, but o
n
ly a few were
discussed. S
o
metimes, it is
more li
kely hav
e a
go
od a
d
vantag
e if de
aling
with findi
ng t
h
e
solutio
n
num
erically rathe
r
than analytically,
if involving many vari
able
s
and p
a
rameters.
So, we provi
de to sea
r
ch
the optimal solutio
n
s n
u
m
eri
c
ally for
three inte
rnet
prici
ng
scheme
s
wh
ich
are flat
fee, u
s
ag
e
-
ba
sed,
and
two
-
pa
rt ta
riff for
hom
ogen
eou
s
a
nd
hetero
gen
eo
us
con
s
um
e
r
s b
a
sed o
n
functio
n
of band
widt
h dimini
she
d
with incre
a
s
ing
band
width u
s
i
ng LING
O 11
.0 [16].
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 1, March 2
015 : 299 – 3
0
4
300
2. Rese
arch
Metho
d
In this pape
r, the internet prici
ng sch
e
m
es will be
compl
e
ted by
the program
LINGO
11.0 to o
b
tai
n
the o
p
timal
sol
u
tion. Th
e solution
obt
ained
will
hel
p dete
r
mine
the optim
al p
r
ice
on the flat fee, usage
-ba
s
e
d
, and two-pa
rt
tariff for internet p
r
ici
ng
scheme
s
.
3. Model
The general form of utility f
unction based on the F
u
nction of
Diminished Bandwi
dth with
Incre
a
si
ng Ba
ndwi
d
th
:
In
j
class is divi
ded into cl
asses du
ring p
e
a
k
hou
r (
X
) a
n
d
off-pea
k ho
ur (
Y
) to obtai
n:
In
(1)
In
(2)
w
h
er
e
(3)
Then, it will be
,
In
In
(4)
,
I
n
I
n
(5)
Then, Eq(5
) b
e
com
e
s
,
I
n
I
n
(6)
This chan
ge wa
s made to simplify the calcul
ation wh
en the minim
u
m con
s
um
ption level
dan
resp
ectively, as wel
l
as the level of
consum
ption durin
g pe
ak hou
rs and
off-peak
hours
dan
can pro
d
u
c
e a
minimum val
ue of 0 than creating n
egati
v
e value.
For the case of homoge
ne
ous
con
s
um
e
r
s,
Max
I
n
I
n
(7)
Subject to
(8)
(9)
I
n
I
n
0
(10
)
0
o
r
1
(11
)
For the case
of heterog
en
eou
s upp
er a
nd lo
wer cl
ass co
nsume
r
s,
supp
ose tha
t
there
are m
con
s
u
m
ers up
per
class (
i
= 1)
a
nd n lo
we
r cl
ass con
s
ume
r
s
(
i
= 2
)
. It is a
s
sumed t
hat
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Internet Pri
c
ing on Bandwi
dth Function Dim
i
nish
ed with Increasing Bandw
idth .... (Indrawati)
301
each of these heterog
ene
ous con
s
ume
r
s have a limit on the same
the
level o
f
consum
ptio
n
durin
g pea
k h
ours and
the level of con
s
u
m
ption du
ring
off-pea
k hou
rs
dan
.
Con
s
um
er O
p
timization Problem
s:
Max
,
,
I
n
I
n
0
(12
)
Subject to
(13
)
(14
)
I
n
I
n
0
(15
)
0
o
r
1
(16
)
As for the ca
se of heterog
eneo
us hig
h
le
vel of usag
e and low u
s
age level co
nsum
ers,
sup
p
o
s
e a
ssumed t
w
o types
of co
nsu
m
ers,
co
nsu
m
er
con
s
um
ption level i
s
high (
i
= 1) with
a
maximum co
nsum
ption rate of
and
and
con
s
um
er u
s
age rate is lo
w (i = 2
)
with
a maximum
con
s
um
ption rate
of
dan
. There
are
m
con
s
um
ers of
type 1 and t
y
pe 2
n
con
s
umers
with
dan
.
4. Result a
n
d Analy
s
is
Pricin
g sche
mes inte
rnet
probl
ems
solved usi
ng
the sam
e
m
odel by [17]
with the
para
m
eter val
ues
use
d
are
in Table 1
-
3 b
e
low.
Table 1. Para
meter Value
s
Use
d
in Ca
se 1-3
Parameter
Case 1
Case 2
Case 3
ɑ
4 4
4
b
3 3
3
x
2656.17
2656.17
2656.17
y
5748.88
5748.88
5748.88
20.89
20.89
20.89
49.43
49.43
49.43
P
x
0
0.2 x 10
-
1
0.2 x 10
-
1
P
y
0 0
0
P
58.5 0
1.8
Z
1 1
1
1 1
1
Table 2. Para
meter Value
s
Use
d
in Ca
se 4-6
Parameter
Case 4
Case 5
Case 6
2656.17
283.8350
282.5491
5748.88
212.6416
211.6775
2314.40
69.21977
69.14995
2406.87
45.82485
45.77830
20.89
20.89
20.89
49.43
49.43
49.43
P
x
0 0.1
0.1
P
y
0 0.1
0.1
P
49.1 0
0.1
Z
1
1 1
1
Z
2
1 1
1
U
01
1 1
1
U
02
1 1
1
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 1, March 2
015 : 299 – 3
0
4
302
Table 3. Para
meter Value
s
Use
d
in Ca
se 7-9
Parameter
Case 7
Case 8
Case 9
X
1
2656.17
222.93
221.61
Y
1
5748.8
148.30
147.42
X
2
2314.4
69.937
69.847
Y
2
2406.8
46.303
46.243
X
m
20.89
20.89
20.89
Y
m
49.43
49.43
49.43
P
x
0 0
0.1
P
y
0 0.1
0.1
P
49.16
0
0.1
Z
1
1 1
1
Z
2
1 1
1
U
01
1 1
1
U
02
1 1
1
The value
s
of the param
eters a
r
e
sub
s
ti
tuted into the model, then
we have:
Case 1: For f
l
at fee prici
n
g
sch
eme
s
, se
t
0
,
0
,
a
n
d
0
meani
ng th
at the prices
use
d
by the service provide
r
ha
s no e
ffect on
the time of use.
Ca
se 2: For usage
-ba
s
e
d
pricin
g sch
e
me by setting
0
,
0
,
a
n
d
0
, meanin
g
that
servi
c
e p
r
ovi
ders delive
r
differentiated
price
s
,
the p
r
ice of
con
s
u
m
ption du
rin
g
pea
k hou
rs and
at off-pea
k ho
urs.
Cas
e
3: For pric
ing sc
heme with a
two-part tariff
, s
e
t
0
,
0
,
a
n
d
0
means that
servi
c
e
provi
ders d
e
liver
differentiated
pri
c
e, i.
e. th
e pri
c
e
of co
nsum
ption
d
u
ring
pea
k
h
ours
and off-pe
ak
hours.
Ca
se 4: Fo
r
prici
ng
sche
me
by setting
a flat fee, then
0
,
0
,
a
n
d
0
, meanin
g
that
the prices used by
the
service pr
ovider has
no
effect on the time
of use, then consumers
will
cho
o
se the maximum co
nsumption rate
,
,
,a
n
d
.
Ca
se 5: For
usa
ge-ba
sed
prici
ng sch
e
m
e by setting
0
,
0
,
a
n
d
0
, then a maximum
con
s
um
ption
rate
,
,
,a
n
d
. Then con
s
ume
r
s
will cho
o
se th
e
maximum co
nsum
ption rate
,
,
,a
n
d
.
Ca
se 6: Fo
r two-p
a
rt tari
ff pricing
scheme, set
0
,
0
,
a
n
d
0
, with a maxi
mum
con
s
um
ption
rate
,
,
,a
n
d
. Then con
s
ume
r
s
will cho
o
se th
e
maximum co
nsum
ption rate
,
,
,a
n
d
.
Ca
se 7: For flat fee pricing schem
es
set
0
,
0
,
a
n
d
0
, by choosin
g the level of
con
s
um
ption
,
,a
t
a
u
,
.
Ca
se 8: F
o
r
u
s
ag
e-b
a
sed p
r
icin
g
schem
e, set
0
,
0
,
a
n
d
0
, by ch
oosi
ng the
le
vel of
con
s
um
ption
,
,a
t
a
u
,
.
Cas
e
9: For
Pric
ing s
c
heme
with a two-part tariff, s
e
t
0
,
0
,
a
n
d
0
, by choo
si
ng the
level of con
s
u
m
ption
,
,a
t
a
u
,
.
Table 4 b
e
lo
w explain
s
th
e data u
s
ag
e
fo
r pea
k an
d off-pea
k h
ours served
by local
serv
e
r
.
Table 4. Data
Usa
ge for Pe
ak Hours an
d
Off-Peak Ho
urs
M
a
il
(
by
te
)
M
a
il
(
kb
ps
)
2719914.01
2656.17
2369946.51
2314.40
21388.28
20.89
5886849.92
5748.88
2464637.66
2406.87
50619.47
49.43
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Internet Pri
c
ing on Bandwi
dth Function Dim
i
nish
ed with Increasing Bandw
idth .... (Indrawati)
303
w
h
er
e
1.
o
r
is the maximum possibl
e level of con
s
u
m
ption du
ring
peak h
ours b
o
th in units of
kilo
bytes per
se
cond.
2.
is the maximum possibl
e level of con
s
u
m
ption du
ring
off-peak h
o
u
r
s in unit
s
of kilo byte
s
per second.
3.
is the most l
o
w level of co
nsum
ption du
ring pe
ak h
o
u
r
s in unit
s
of kilo bytes p
e
r second.
4.
o
r
is the maxim
u
m po
ssi
ble l
e
vel of con
s
u
m
pt
ion both
durin
g pe
ak
hours in
units of kil
o
bytes per
se
cond.
5.
maximum possible level
of consump
t
ion durin
g p
eak h
ours in
units of kilo
bytes pe
r
se
con
d
.
6.
is the mo
st
low level of
con
s
um
ptio
n duri
ng off-pea
k ho
urs i
n
units
of kil
o
bytes p
e
r
se
con
d
.
Table
5. describe the optimal soluti
on of
using utility function of the function
of
band
width di
minish
ed with
incre
a
si
ng b
and
width.
Table 5. Solu
tion for Utility Functio
n
s of t
he Fun
c
tion Bandwi
d
th Di
minish
ed with
Increa
sin
g
Bandwi
d
th
Case
1 2 3
4 5 6
7 8 9
Profit 58.511
58.511
58.511
245.815
171.952
171.754
204.46
134.621
134.428
We
can
see
from Ta
ble 5
that in ho
m
ogen
ou
s case, we
obtain
the same m
a
ximum
profit for all case of flat fee, usag
e based and
two p
a
rt tariff sch
e
m
es. In other case, when
we
deal
with het
erog
ene
ou
s h
i
gh en
d an
d l
o
w e
nd u
s
e
r
con
s
um
ers, the maximu
m
profit is
achie
v
ed
whe
n
we app
ly the usa
ge
based. The l
a
st case whe
n
dealin
g wit
h
high a
nd lo
w dem
and
users,
again, the u
s
age ba
se
d yield the maximum profit.
If we
com
pare the
re
sult i
n
[18],[19], we hav
e
sli
ghtl
y
differen
c
e.
If using
the
modified
Cob
b
-Dou
gla
s
s utility fun
c
tion, the
maxi
mum p
r
ofit a
c
hieved
wh
en
we
apply th
e
flat fee a
nd t
w
o
part tariff
scheme
s
for
h
o
moge
nou
s
ca
se. Fo
r he
teroge
neo
us
ca
se, maxim
u
m profit occu
rs
whe
n
we ap
p
l
y the flat fee and two pa
rt
tariff sc
heme
s
. In ou
r utility function, th
e thre
e sch
e
m
es
yield the
sam
e
p
r
ofit in
ho
mogen
eou
s case,
whil
e in
hetero
gen
eo
us
ca
se
we
o
b
tain hi
ghe
r
p
r
ofit
if
we apply usa
ge ba
sed
.
The advant
age
s
of usi
n
g the
utility functio
n
we
choo
se th
at the
provide
r
ha
s other
choi
ce
s in a
pplying
prici
ng
sche
mes that attract the
cu
sto
m
er to joi
n
the
scheme
s
.
5. Conclusio
n
Acco
rdi
ng to
above result we
can
con
c
l
ude that
if ISP intends to
obtain maxim
u
m profit,
ISP can cho
o
s
e all three schem
es if de
aling with h
o
m
ogen
ou
s ca
se. For
heterogen
eou
s ca
se
based on
willi
ngne
ss to pa
y and ba
sed
on dem
and o
f
the con
s
um
ers, ISP can
adopt flat fee
to
gain m
a
ximum profit. For further
re
search, we can consider
other utility functions
that fit with ISP
choi
ce
s to maximum their
benefit.
Ackn
o
w
l
e
dg
ement
The re
se
arch
leading to th
is study
wa
s financ
i
a
lly su
pporte
d by Directo
r
ate of
High
er
Educatio
n Ind
one
sia (DIKTI) for su
ppo
rt throu
gh “Hiba
h
Funda
ment
al Tahun II”, 2014.
Referen
ces
[1]
He H, K
Xu, Y
Liu. Inter
net re
source
prici
n
g
mode
ls, mech
a
n
isms, an
d met
hods
. N
e
tw
orki
ng Sci
enc
e.
201
2; 1(1-4): 4
4
-68.
[2]
Malin
o
w
ski
K, E Nie
w
i
a
doms
k
a S, P Jakól
a
. Price meth
od
and
net
w
o
rk c
o
ngesti
on c
ontr
o
l
. Jour
nal
of
T
e
leco
mmunic
a
tions a
nd Info
rmati
on T
e
ch
n
o
lo
gy.
201
0; 2.
[3]
Wu Y, et al.
Qo
S-Reve
nue
T
r
a
deoff w
i
th T
i
me
-Constrai
ne
d I
SP Pricin
g
. 2
0
10. [cited
3 Au
gust 20
10];
Avail
abl
e from: http://scenic.prin
ceto
n.ed
u/pa
per/IW
QoS_Dr
aft.pdf.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 1, March 2
015 : 299 – 3
0
4
304
[4]
Yang W
,
et al.
An Auction P
r
icing Strate
gy
for Differentia
ted Service N
e
tw
ork
. Proceedin
g
s of the
IEEE Global Telec
o
mmun
i
cat
i
ons C
onfere
n
c
e
, IEEE. 2003.
[5]
Yang W
.
Prici
ng Net
w
o
r
k R
e
sourc
e
s in D
i
fferentiated S
e
rvice Net
w
o
r
ks
.
School
of el
ectrical a
n
d
Comp
uter Engi
neer
ing.
Ph
d T
hesis
. Georg
i
a
Institute of
T
e
chno
log
y
. 2
004:
1-111.
[6]
Yang W, H Ow
en, DM Blough.
A C
o
mpar
i
s
on of A
u
ctio
n
and
F
l
at Pric
i
ng for
Different
iated
Servic
e
Netw
orks
. Proceedings
of the IEEE
International Conference
on Communic
a
tions. 2004.
[7]
Yang W
,
H
L
O
w
e
n
, DM Bl
oug
h.
Deter
m
i
n
in
g Differ
enti
a
ted S
e
rvices
Netw
ork Prici
ng T
h
ro
ug
h
Auctions
. N
e
t
w
orkin
g
-ICN 2
0
0
5
, 4th Internati
ona
l Co
nfer
e
n
c
e
on N
e
t
w
orki
n
g
April
20
05 Pr
ocee
din
g
s,
Part I. Reunio
n
Island, F
r
ance
.
Springer-V
erl
ag Berl
in He
id
elb
e
rg. 20
05.
[8] Irmeily
a
na,
Indra
w
ati,
F
M
Pu
spita, L
Herd
a
y
a
na
T
he N
e
w
Impr
oved
Mo
dels
of Sin
g
l
e
Link Int
e
rne
t
Pricing Sc
he
me in Multip
le QoS Netw
ork
.
In
ternatio
nal C
o
nferenc
e Rece
nt treads in En
gin
eeri
ng
&
T
e
chnolog
y (IC
R
ET
’2014), Batam (Indon
esi
a
). 2014.
[9]
Irmeil
yan
a
, Ind
r
a
w
ati, FM Pu
spita, RT
Amelia.
Gener
ali
z
e
d
Mod
e
l
and
Optima
l So
luti
on of Inter
n
e
t
Pricing Sc
he
me i
n
Sin
g
le
Link un
der
Multiservic
e
N
e
tw
orks
. 1st Internati
o
n
a
l C
onfere
n
ce o
n
Comp
uter Sci
ence
an
d En
gin
eeri
ng. Pa
l
e
mba
ng,
So
uth Sumater
a
, Indo
nesi
a
. Jur
u
san S
i
stem
Komputer U
n
iv
ersitas Sri
w
i
j
a
y
a. 2014.
[10]
Che
n
L, G Qing, N Z
hen
yu, J Kai
y
uan.
A
T
h
reshold
Based Ha
nd
over T
r
iggerin
g Scheme i
n
Hetero
gen
eo
u
s
Wireless
Ne
t
w
orks
. T
E
LK
OMNIKA T
e
le
communic
a
tio
n
Co
mp
utin
g El
ectronics
an
d
Contro
l.
201
4; 12(1): 16
3-1
7
2
.
[11]
Satria MH, JB
Yunus, E
Su
pr
i
y
anto. Em
erg
enc
y
Pr
enat
al
T
e
lemonitori
ng
S
y
stem
in
W
i
r
e
less M
e
sh
Net
w
ork
.
TEL
K
OMNIKA Teleco
mmu
n
icati
o
n Co
mp
utin
g Electron
ics an
d Contro
l.
20
1
4
; 12(1): 12
3-
134.
[12]
Kell
y F
P
. Effec
t
ive b
and
w
i
dt
h
s
at multi-c
l
as
s qu
eues
. Qu
e
uei
ng Syste
m
s
:
T
heory a
n
d
Appl
icatio
ns.
199
1; 9(1-2): 5
-
16.
[13]
Kell
y F
.
Charg
i
ng an
d rate co
ntrol for elastic
traffic
. European T
r
ansacti
on
s on T
e
leco
mmu
n
ic
ations
.
199
7; 8: 33-37.
[14] Kell
y
F
.
Notes
on Effective Bandw
idths.
S
t
ochastic Net
w
orks:
T
heor
y
and Ap
pl
icatio
ns of Ro
ya
l
Statistical Soci
et
y
L
e
cture Not
e
s Series. 19
9
6
; 4.
[15] Indra
w
at
i,
Irmeil
ya
na, FM Puspita, MP Le
st
ari. Cobb-
Do
ugl
ass Utilit
y
Function i
n
Optimizin
g
th
e
Internet Prici
n
g Schem
e Mode
l
.
T
E
LKOMNIKA T
e
lec
o
mmunic
a
tio
n
Co
mputi
ng E
l
ectron
ics an
d
Contro
l.
201
4; 12(1): 22
7-2
4
0
.
[16] LINGO.
LINGO 11.0
. LINDO Systems, Inc: Chica
go. 20
11.
[17]
Puspita FM, K Seman, BM
T
a
ib, Z Shafii.
An
improv
ed
optimiz
atio
n
mode
l of
inter
net ch
argi
n
g
scheme i
n
mu
lti service
net
w
o
rks
.
T
E
LKOMNIKA T
e
leco
mmu
n
icati
on
Co
mp
uting E
l
e
c
tronics a
n
d
Contro
l.
201
2; 10(3): 59
2-5
9
8
.
[18]
W
u
SY, PY Ch
en, G. Ananda
l
i
ng
am.
Optim
a
l Pricing Schem
e fo
r Infor
m
a
t
ion Servic
es
. Univers
i
t
y
of
Penns
yl
van
i
a Phil
ade
lp
hia.
2
002.
[19]
W
u
SY, RD Banker. Best Pr
icing Strat
e
g
y
for Information
Services
.
Jou
r
nal of the As
sociati
on fo
r
Information System
s
. 20
10; 1
1
(6): 339-
36
6.
Evaluation Warning : The document was created with Spire.PDF for Python.