TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4468 ~ 4
4
7
4
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.548
4
4468
Re
cei
v
ed
De
cem
ber 2
4
, 2013; Re
vi
sed
Febr
uary 18,
2014; Accept
ed March 5, 2
014
The Design of Fine-grained Network QoS Controller and
Performance Research with Network Calculus
Hu Jia*, Zho
u Jinhe
Dep
a
rtment of
Information a
n
d Commu
nicati
on Eng
i
n
eeri
n
g
,
Beiji
ng Informa
tion Scie
nce a
nd T
e
chnol
og
y Universit
y
,
No.35, Beis
ih
u
an Z
hon
gl
u Ro
ad, Cha
o
y
a
ng
District, Beijin
g
,
telp:188
104
5
249
9
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: huji
a28
11
@g
mail.com
A
b
st
r
a
ct
In this
pap
er a
netw
o
rk QoS
control
l
er
is d
e
s
ign
ed. W
e
us
e the
contro
lle
r to qu
antific
ati
on
all
y
control QoS u
n
der the ra
nd
o
m
n
e
tw
ork traffic. T
he c
ontrol
l
e
r is bu
ilt by
mi
n-pl
us al
gebr
a
base
d
on
netw
o
rk
calcul
us. T
he
control
l
er i
n
cl
u
des tw
o parts,
the l
o
ssl
ess f
r
actal re
gul
ato
r
and t
he fi
nit
e
vari
abl
e stor
age
shaper (FVSS). We can get the arr
i
val c
u
rv
e with t
he loss
less fractal r
e
gulato
r. Then the FVSS control
netw
o
rk traffic to real
i
z
e
the fin
e
-ga
i
ne
d QoS control
l
er. At la
st,
w
e
analy
z
e
the perfor
m
a
n
c
e
of the netw
o
r
k
QoS control
l
er.
Rese
arch res
u
lts show
that t
he re
la
tio
n
sh
ip
betw
een
pack
e
t losses, p
a
ck
et del
ays a
nd t
h
e
buffer stora
ge.
W
e
clear
ly o
b
s
e
rve the
cha
n
g
e
of th
e
pack
e
t loss a
nd t
he
p
a
cket d
e
lay w
i
t
h
the stor
ag
e o
f
buffer. T
he results can be ap
plie
d to eval
ua
te the c
ongesti
on an
d flow
control strategi
es
, as
w
e
ll as thes
e
are refere
nces
to desig
n netw
o
rk
control d
e
vi
ce para
m
eters.
Ke
y
w
ords
: fin
e
-grai
n
e
d
, cont
roller, QoS, net
w
o
rk calculus,
shap
er
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
The Inte
rnet
is a
com
p
lex an
d h
uge
syste
m
. The
compl
e
xity is reflected
i
n
uncontroll
abili
ty topology, traffic bur
stine
ss, net
work p
r
otocol dive
rsity and compl
e
xity of network
behavio
r an
d
other a
s
pe
cts [1]. The
views h
a
ve
attracted
e
x
tensive atte
ntion in net
work
resea
r
ch field that structu
r
e determi
ning
func
tion, traffic burstine
s
s impacting p
e
r
forma
n
ce. We
sho
u
ld
clea
r
the relatio
n
sh
ip between traffic ch
ara
c
te
ristic pa
ramet
e
rs and
th
e perfo
rman
ce of
netwo
rk. T
h
e
n
we
control
and supe
rvise the incomin
g
traffic at the point of inte
rnet u
s
er
access
and net
wo
rk
conve
r
ge
nce
by controlli
n
g
the netwo
rk traffic. The
works are e
ffective ways to
increa
se the
fairne
ss of n
e
twork resou
r
ce al
l
o
cation
, avoid cong
estion a
nd i
m
prove n
e
twork
perfo
rman
ce.
In early times, the resea
r
che
r
s b
ega
n to st
udy the sha
per. Whil
e the origin
al
work by
Cru
z
in de
fi
n
e
s a lea
k
y bu
cket sha
per
b
a
se
d on min
-
pl
us al
geb
ra. The min-plu
s
algeb
ra defin
es
a
serie
s
of a
rrival cu
rve
a
nd servi
c
e curve
to
de
scribe
co
nci
s
el
y the bou
nd
arie
s of
network
perfo
rman
ce
[2]. The traffic shapi
ng
strategy on a
ccount of lea
k
y
bucket
sh
ap
er i
s
ad
opted
by
the se
rvice m
odel of IETF
netwo
rk and
most in
du
stri
es, it monito
rs the
rate of f
l
ow by
cha
n
g
i
ng
the pa
ram
e
te
rs
of shap
er,
So that the
bl
ocked
data i
s
dro
ppe
d o
r
cach
ed by
sh
a
per
and
then
is
transfe
rred a
gain at the a
ppro
p
ri
ate time. While th
e lea
k
y bu
cket simplifie
s the pa
ram
e
ters o
f
traffic co
ntrol,
it isn’t enou
gh to reg
u
lat
e
the pra
c
tical traffic. Re
cently, the research of sha
per
with network calculu
s
ob
tains g
r
eat p
r
og
re
ss.
Fo
r instan
ce, Z
hang xinmin
g system
atically
studie
s
the
g
eneral mo
del
of t
he gree
dy sha
per,
shape
r with
n
o
buffer
and
sha
per in fi
xed
-
length
storag
e [3]. Zhan
g
lianming
ha
s the res
earch of lo
ssl
ess sh
ape
r, he
prop
oses a
n
e
w
traffic sh
apin
g
model an
d arrive the
correspon
ding
p
e
rform
a
n
c
e e
v
aluation, whi
l
e the cap
a
cit
y
of
sha
per i
s
g
e
nerally limite
d
in fact, th
en he
studi
e
s
lo
ssy
sha
p
e
r an
d e
s
ta
blish
a ge
ne
ral
mathemati
c
al
model of lossy sha
per b
a
s
ed o
n
the de
termine
d
net
work calculu
s
.
The pa
pe
r studies fin
e
-g
rained
netwo
rk QoS
cont
roller a
nd
co
ntrolle
r pe
rfo
r
man
c
e
cha
r
a
c
teri
stics. We
de
scrib
e
t
hat co
ntroll
ing the net
wo
rk
QoS by ch
angin
g
the st
orag
e len
g
th
of
buffer. We a
nalyse the pe
rforma
nce pa
ramete
rs of
p
a
cket delay a
nd packet lo
ss with min-pl
us
algeb
ra. The
n
we di
scuss
how the p
a
cket delay
and
packet lo
ss
chang
e with variabl
e sto
r
ag
e.
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TELKOM
NIKA
ISSN:
2302-4
046
The De
sig
n
o
f
Fine-g
r
aine
d Network Qo
S Controll
er a
nd Perform
a
n
c
e Research
with… (Hu Ji
a)
4469
2. Net
w
o
r
k
Calculus
Network cal
c
ulus i
s
a coll
ection of re
sult
s ba
sed o
n
Min-Plu
s
al
gebra, whi
c
h
can be
applie
d to determini
stic q
u
euing
system
s found in
co
mmuni
cation
netwo
rks. It is a set of re
cent
developm
ent
s
whi
c
h
provi
de a
de
ep i
n
sight i
n
to
flo
w
p
r
obl
em
s
encounte
r
ed
in net
workin
g
[4].
Network calculus i
s
ba
se
d
on the
idea t
hat given a re
gulated flo
w
of traffic into the network, o
n
e
can
qua
ntify the characte
ri
stics of
the fl
ow a
s
it trave
l
s from
nod
e
to node th
rou
gh the n
e
two
r
k.
This m
ean
s t
hat traffic flo
w
s
are
co
nst
r
ained
by
sh
a
pers an
d del
a
y
ed by the no
des'
sche
dule
r
s.
In netwo
rk calcul
us
a n
o
de be
haviou
r
is
cha
r
a
c
teri
zed
by a fun
c
tion
calle
d t
he servi
c
e
curve
whi
c
h de
note
s
ho
w long a
packet mu
st be se
rvice
d
a
fter an arrival to a node [5]. The input traf
fi
c
is ch
ara
c
te
rized by a wide
-sen
se in
crea
sing fun
c
ti
on
of time and it is so
-called t
he arrival cu
rve
.
This fun
c
tion
quanti
fi
es
co
nstrai
nts o
n
the num
ber
of bits of pa
cket
fl
ow in the time interval at
servi
c
e n
ode.
No
w we i
n
trodu
ce
some i
m
porta
nt
tool
s an
d con
c
lu
sion
s of n
e
twork
cal
c
ul
us
as
follows
[6, 7]
Definition 1
WIF: wide
-se
n
se in
crea
sin
g
function.
If
f(x
)
is
a function, for any
s
t
,if
()
(
)
f
sf
t
, f(x)is
a wide-sens
e
inc
r
eas
i
ng func
tion.
Definition 2
WIFS: wide
-sense increa
si
ng functio
n
set.
if
{(
)
|
(
)
0
,
0
;
(
0
)
0
;
(
)
(
)
,
,,
[
0
,
]
}
F
f
t
ft
t
f
f
s
ft
st
s
t
(1)
F is a wide
-sense increa
si
ng functio
n
set.
Definition
3. Min-pl
us con
v
olution.
Let f and g be
two WIFS. The min-plu
s
c
onvolution of
f and g is the function:
0
inf
t
-
s
+g(
s
)
}
{
st
fg
t
f
(2)
Definition 4.
Min-pl
us d
e
convolution.
Let f and g be
two WIFS. The min-plu
s
conv
olution of
f and g is the function:
0
{
f
g
t
=
s
up
t
+
s
-
g(s)}
(3)
Theo
rem 1.
Gene
ral p
r
op
erties of
.
Let f, g and h be two WIFS.
Rule 1 (Closure of
)
()
f
gF
.
Rule 2 (A
ss
oc
iativity of
)
()
(
)
f
gh
f
g
h
.
Rule 3 (Commutativity of
)
f
gg
f
.
Rule 4 (Di
s
tri
butivity of
with res
p
ec
t to
∧
)
()
(
)
(
)
f
gh
f
h
g
h
Definition 5.
Arrival cu
rve.
Give a
WIF
α
defined
for a
sha
per,
a flo
w
f i
s
con
s
tra
i
ned
by
α
if
an
d only if fo
r
al
l
st
,
R
tR
s
t
s
.
We say
that R
ha
s
α
a
s
a
n
arrival curv
e, or al
so tha
t
R is
α
-smo
oth. The a
rriv
a
l cu
rve
actually defin
es an u
ppe
r b
ound o
n
the a
rrival rate of a
flow to a part
i
cula
r nod
e.
Definition
6. Service curve
.
If a system S
has
an in
put
flow R(t) a
n
d
output flo
w
R*(t), th
en S
offers to th
e
flow
a
serv
i
c
e c
u
rv
e
β
, if and only if
β
is wide sense increa
si
ng,
(0
)
0
for all
0
t
,
*
RR
.
A service cu
rve is a lowe
r boun
d on the
depa
rture rat
e
from a network n
ode.
Definition 7.
Subadditivity.
Let f be two WIFS. If
()
f
ts
f
t
f
s
, f is
sub
additive functio
n
.
Definition 8.
Sub-a
dditive clo
s
ure.
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4468 – 4
474
4470
Let f be a fun
c
tion o
r
a se
quen
ce of F.
Den
o
te
n
f
the functio
n
obtai
ned by re
pea
ting
(n-1)
convol
u
t
ions of f wit
h
itself. By convention,
0
0
f
,so that
1
f
f
,
2
f
ff
.
Then the
sub
-
additive cl
osure of f, deno
ted by
*
f
, is defined by
*
inf
{
}
n
f
f
.
Theo
rem 2.
Let f(t)
be
two
WIFS. If it’s sub
-
add
itive closure
*
f
sat
i
sf
ies
*
f
f
an
d
*
f
is
sub
additive functio
n
.
Definition 9. L
i
near ali
quot
s operato
r
.
Let f and
σ
be two WIFS.
The linea
r ap
iquoto
s
ope
ra
tor is:
0
in
f
{
-
(
)}
st
hf
t
t
s
f
s
(4)
Theo
rem 3.
Sub-a
dditive clo
s
ure of linear aliq
uots o
perato
r
H
Q
,
H
H
LQ
12
11
2
,,
,
,
i
nf
{
i
nf
{
,
(
,
)
(
,
)
}}
n
n
nN
uu
u
Ht
s
H
t
u
H
u
u
H
u
s
(5)
Corollary 1.
If Q
1
is a linear aliqu
o
ts op
erato
r
of F to F, and
F
11
Qh
h
Q
h
(6)
3. Finite and Variable Sto
r
age Shap
er
Traffic gen
erated
by so
urce
s
can
not
be exp
e
cte
d
to natu
r
ally
sati
sfy so
m
e
a
prio
ri
arrival c
u
rve c
o
ns
traint
.If
we want
to ens
u
re
in a network
s
o
me Qo
S guarantees
. We mus
t,
fi
rst
of all, con
s
tra
i
n its input
fl
o
w
s. Thi
s
con
d
ition can be
ac
hi
eved by
sha
p
ing the i
nput traf
fi
c wi
t
h
sha
per. A sh
aper i
s
used to force
a flow to satisfy so
me arrival
cu
rve con
s
traint.
Definition 10.
Finite and variable sto
r
ag
e sha
per
FVSS has a finite storage, we can change
the length of the
st
orage. Because the
stora
ge i
s
lim
ited, so the
shape
r can’t g
uara
n
tee
p
a
cket lo
ss to b
e
zero. Whil
e the outp
u
t flow of
the sha
per i
s
maximum accepta
b
le valu
e.
In order to
g
e
t a
gene
ral
con
c
lu
sio
n
, this
se
ction
d
oes not
assu
me that th
e t
y
pe of
netwo
rk t
r
affic flows. We consi
der th
e varietie
s
of pa
cket loss a
nd
packet d
e
lay
whe
n
the
sha
p
e
r
cha
nge
s the
length of sto
r
age. If the length is se
t too small, mo
st of packets
will be dropp
ed.
While th
e le
ngth is set too la
rge, the
pa
cket
del
a
y
is in
creasi
ng, both th
e
setting
s of f
i
nite
stora
ge shap
er affect t
he n
e
twork pe
rformance.
We a
s
sume t
hat L(t) is the
total pac
ket loss at the time t, and the L(0)=0.
Lemma 1.
If R(t) is th
e i
nput flow at t
he time t, an
d
α
(t
) is
the arrival
c
u
rve
of FVSS, s
o
the total
packet lo
ss a
s
the follow:
()
()
()
,
0
Lt
R
t
t
t
(7)
Theo
rem 4.
We suppose that the s
e
rvic
e curve of FVSS is
β
, th
e length of buffer is B, the arrival
curv
e i
s
α
, the c
u
mulative pack
et loss
is
:
21
2
2
1
2
ΓΓ
1
{s
u
p
{
[
(
)
(
)
]
}
}
sup
n
ii
i
i
i
Lt
tt
t
t
n
B
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The De
sig
n
o
f
Fine-g
r
aine
d Network Qo
S Controll
er a
nd Perform
a
n
c
e Research
with… (Hu Ji
a)
4471
s
.t.
12
2
...
}
|
{0
n
tt
t
t
n
N
(8)
Proof: By definition 9,
∀
s,
0
≤
s
≤
t, the input flow of FVS
S
at the time
t is
R(t).
0
()
i
n
f
{
()
(
)
(
)
}
(
(
)
)
st
R
tt
s
R
s
h
R
t
And
()
(
)
(
(
)
)
{
(
)
}
Rt
t
h
x
t
t
B
So
((
))
(
(
)
)
Rh
B
h
h
B
. By theore
m
3
an
d
co
rolla
ry 1,
we
kno
w
:
()
(
)
(
)
()
(
(
)
)
()
Lt
t
R
t
t
h
B
t
()
()
i
n
f
{
{
(
)
}
}
(
)
n
th
B
t
21
2
2
1
2
1
su
p
{
(
)
in
f
{
(
)
(
)
}
(
)
(
)
}
}
n
ii
i
i
i
tt
t
t
t
t
B
21
2
2
1
2
0
sup
{
sup
{
(
)
(
)
(
)]}
}
n
ii
i
i
i
tt
t
t
n
B
Theo
rem 5.
We
assum
e
t
hat the m
a
ximum p
e
rmi
s
sible of
total
p
a
cket lo
ss is
P, the se
rvice curve
is
β
, the arrival curve is
α
. Then we get the storage length of FVSS.
22
1
2
2
1
1
1
su
p
{
[
]
}
n
Li
i
i
i
i
Bt
t
t
P
t
P
t
n
s
.t.
12
2
...
}
|
{0
n
tt
t
t
n
N
(9)
Proof: By definition 10, we
kno
w
the follo
w:
()
()
(
(
)
)
L
P
tt
t
B
,then
()
(
)
(
(
)
)
L
P
tt
t
B
,s
o
()
()
()
i
n
f
{
(
(
)
)
}
n
L
Pt
t
t
B
,
()
i
n
f
{
()
()
}
L
nB
t
t
P
t
22
1
2
2
1
1
(
)
i
n
f
{
(
)
(
(
)
(
))}
n
Li
i
i
i
i
nB
t
t
t
P
t
P
t
From the fo
regoin
g
, the uppe
r bo
und
of t
he lengt
h storage
sa
tisfies the fo
llowing
formula.
21
2
2
1
1
1
s
u
p
{
(
)
[
(
)
(
)
(
)]}
n
Li
s
i
i
i
i
Bt
t
t
P
t
P
t
n
Theo
rem 6.
We s
u
ppos
e
that
the s
t
orage
length
of
FVSS is
B, t
he arrival curve of the flow i is
i
,
and the servi
c
e curve of F
VSS is
i
.The total numbe
r
of sch
eduli
n
g
queue
s is m.
Then we g
e
t
the maximum
packet delay
s and the ave
r
age p
a
cket d
e
lays
avg
d
.
max
,1
0
1
sup
{
sup
{
(
)
(
)
0
}
}
n
ii
iN
i
m
k
dt
n
B
(10)
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Vol. 12, No. 6, June 20
14: 4468 – 4
474
4472
0
11
1/
{
s
u
p
{
(
)
(
)
0
}
}
mn
ii
av
g
ik
dm
t
n
B
(11)
12
1
0
kk
n
n
tt
t
t
t
t
t
Proof: At the
time t, the
queu
e length
of the flow i is
()
i
Ct
,and
00
i
C
. By
the
corollary 1,
((
)
(
(
)
)
ii
i
i
i
CB
B
, we kno
w
th
at the packet
delay sati
sfies the
follow:
0
0
in
f
{
(
)
(
)
}
s
u
p
{
(
)
(
)
0
}
ii
i
i
i
t
dC
t
t
C
t
t
0
s
u
p
{
(
(
)
(
(
)
))
(
(
))
0
}
ii
i
L
t
tt
B
t
()
()
0
sup
{
(
)
(
(
)
)
(
(
)
0
}
ii
n
n
tt
B
t
()
(
)
0
s
u
p{
(
)
(
(
)
)
(
(
)
)
0}
ii
n
n
tt
B
t
0
1
s
u
p{
(
)
(
)
0}
n
ii
i
tn
B
So that:
0
00
11
l
i
m{s
u
p{
(
)
(
)
0}
}
s
u
p
{
(
)
(
)
0
}
nn
ii
i
i
i
t
kk
dt
n
B
t
n
B
4. Performan
ce Analy
s
is
on the Net
w
ork QoS Con
t
roller
In this sectio
n we
analy
z
e the pe
rformanc
e of th
e the net
work QoS
cont
roller
with
rand
om n
e
twork traffic. T
he controll
er inclu
d
es
two part
s
, one part is
the loss
les
s
frac
tal
regul
ator, the another is the FVSS. Th
e
lossless fract
a
l is
used
to provide spec
i
f
ic arrival
curve
.
The FVSS control
s
the network QoS
by the spec
if
ic arrival
curve and servi
c
e curve. Many
resea
r
chers
b
u
ild mo
del
s t
o
stu
d
y the
a
c
tual
network
traffic
,
and
obtain that
the
network traffic
is
self-simila
r [8, 9]. In
the literature, Zhan
g
li
anming ha
s a research a
bout upp
er b
ound mo
del
s of
perfo
rman
ce
in self-simil
ar network
based on
fractal shap
er.
After passi
ng traffic sh
aper,
envelop
e
curve of the
tra
ffic is a li
nea
r fun
c
ti
on.
T
he
sha
per in
trodu
ce
s m
o
re cha
r
a
c
teri
stic
para
m
eter to
descri
be the
self-simila
r traffic accu
rate
ly. He prop
osed a lo
ssl
ess
fractal regul
ator.
The en
d-to
-e
nd delay an
d
the length st
orag
e in buf
f
e
r do
n’t incre
a
se
with usi
n
g lossle
ss fra
c
ta
l
regul
ator in n
e
twork.
In view of th
ese
advanta
g
e
s
of lossle
ss fra
c
tal
reg
u
l
ator, we introdu
ce the
re
gulator to
our system. Our network QoS
controll
er
i
s
sho
w
n
i
n
Figu
re
1. Whe
n
the
net
work traffic e
n
ters
the co
ntrolle
r, the traffic is
sha
ped by lo
ssl
ess fr
a
c
tal
regul
ator. S
o
that
the arri
val curve
of the
FVSS is
frac
t
a
l c
u
rve
α
. Th
e servi
c
e curve of the FVSS is set
β
.
Figure 1. Net
w
ork Q
o
S Co
ntrolle
r
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TELKOM
NIKA
ISSN:
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046
The De
sig
n
o
f
Fine-g
r
aine
d Network Qo
S Controll
er a
nd Perform
a
n
c
e Research
with… (Hu Ji
a)
4473
The arrival cu
rve
α
is:
()
,
0
tr
t
b
t
(12)
*1
(1
)
2
1
H
H
rH
r
H
（）
(13)
*
(1
)
2
(
)
1
H
H
H
b
H
(14)
In the formula
,
r is the long-term avera
g
e
of input traffic,
σ
is stand
a
r
d deviation a
nd H is
self-similar param
e
ter,
γ
i
s
a positive co
nstant. We
se
t that
γ
is 6.
The s
e
rvice curve of the FVSS is
β
.
,0
tR
t
t
(15)
R is
the servic
e rate
of FVSS.
We
s
e
t
the
parameter of t
he FVSS
based on
the arrival c
u
rve
higher than the
s
e
rvic
e
curve. T
he p
a
ram
e
ters is t
hat r=700
kbit/
s
,
σ
=1
00
kbit, R=400M
bit/s. At the time t,
3
1.
2
1
0
ts
,
we
ob
serve
the vari
ation
of pa
cket lo
ss a
nd
p
a
cket
delay
with t
he vari
able
l
ength
sh
ape
r in
buffer. The variation
s
a
r
e
sho
w
n in Fig
u
re 2 an
d Fig
u
re 3.
Figure 2. The
Variation of Packet Loss
Fi
gure 3. The
Variation of Packet Delay
From the
abo
ve two figure
s
, we d
r
a
w
some con
c
lu
si
ons.
With the
appropri
a
te l
ength in
buffer, the FVSS c
an effec
t
ively improve network
performanc
e. For ins
t
anc
e
,
the pack
e
t loss
and p
a
cket d
e
lay are
able
to be tolerabl
e. Whe
n
the l
ength
stora
g
e
increa
sing, t
he pa
cket del
ay
is incre
a
si
ng
and the p
a
cket loss i
s
red
u
cin
g
. While
with the oversize st
o
r
ag
e, the packet lo
ss
will not red
u
ce, the design
of hard
w
a
r
e is difficu
lt. Wh
en the length
stora
ge redu
cing, the packet
delay i
s
redu
cing
an
d the
pa
cket lo
ss is i
n
cr
ea
sin
g
.
Whil
e with
too small st
orag
e,
mo
st of
packet
s
will b
e
drop
ped, so
that network
perfo
rman
ce
deterio
rate
sh
arply.
5. Conclusio
n
s
The pap
er st
udie
s
the fine-g
r
ain
ed ne
twor
k QoS controlle
r. We
propo
se a
gene
ral
mathemati
c
al
descriptio
n
of the design
base
d
on n
e
twork calcul
us. Throug
h the re
sea
r
ch, we
get the
relati
onship b
e
twe
en the
pa
cket loss a
nd
pa
cket d
e
lay wit
h
the le
ngth
of storage. T
hen
we di
scu
ss
h
o
w the
pa
cke
t
delay and
p
a
cket lo
ss
ch
ange
with va
riable
sto
r
ag
e. The works in
the pa
per ha
ve pra
c
tical
significa
nce to
evaluate th
e
control of t
r
af
fic an
d the
de
sign
of net
wo
rk
0.
5
0.
5
5
0.
6
0.
65
0.
7
0.
75
0.
8
0.
8
5
0.
9
0.
9
5
1
0
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
s
e
lf
-
s
im
ila
r
p
a
r
a
t
e
m
e
r
(
H
)
p
a
cke
t
l
o
ss
(
L
/M
b
i
t
)
bu
f
f
e
r
l
eng
t
h
s
m
a
l
l
e
r
bu
f
f
e
r
l
eng
t
h
m
i
ddl
e
b
u
f
f
e
r
l
e
n
g
th
l
e
n
g
th
l
a
r
g
e
r
0.
5
0.
5
5
0.
6
0.
65
0.
7
0.
75
0.
8
0.
8
5
0.
9
0.
9
5
1
20
0
40
0
60
0
80
0
100
0
120
0
140
0
s
e
lf
-
s
im
ila
r
p
a
r
a
t
e
m
e
r
(
H
)
pa
c
k
e
t
del
a
y
(
d
/
M
bi
t
)
bu
f
f
e
r
l
eng
t
h
s
m
a
l
l
e
r
bu
f
f
e
r
l
eng
t
h
m
i
ddl
e
b
u
f
f
e
r
l
e
n
g
th
l
e
n
g
th
l
a
r
g
e
r
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Vol. 12, No. 6, June 20
14: 4468 – 4
474
4474
device
pa
ram
e
ters.
The
rel
a
ted
re
sults can b
e
u
s
e
d
fo
r q
uantitative
analysi
s
of th
e pe
rforman
c
e
para
m
eters o
f
network dev
ice
s
.
Ackn
o
w
l
e
dg
ements
This work was suppo
rte
d
by Beijing Natural Sci
ence Foun
d
a
tion (41
310
03) an
d
Nation
al Natu
ral Scie
nce Found
ation of Chin
a (61
271
198) .
Referen
ces
[1]
Z
hang l
i
anm
in
g. Stud
y
o
n
up
per bo
un
d mod
e
ls of per
forma
nce in se
lf-simi
l
ar net
w
o
rk cal
c
ulus. 20
06.
[2]
Che
n
zhi
gan
g. Performance
ana
l
y
sis of ge
n
e
raliz
ed
proce
ssor shari
ng b
a
sed o
n
fractal
leak
y b
u
ck
e
t
regulators.
Jou
r
nal o
n
Co
mmunic
a
tions
. 2
0
0
6
; 27(6): 29-3
5
.
[3]
Z
hang
xi
nmin
g
,
Chen g
uol
ian
g
. On the computatio
n
of en
d
-
to-end d
e
l
a
y
b
oun
d in g
uara
n
t
eed servic
e
b
y
net
w
o
rk
ca
l
c
ulus.
Jour
nal of
Softw
are.
2001; 12(6): 8
89-
893.
[4]
Boun
dec, JY L
E
,
T
h
iran P. Net
w
o
r
k Ca
lcul
u
s
. Berlin: Sprin
ger Verl
ag. 20
04.
[5]
W
ang zij
un,
Xu
w
e
is
he
ng. R
e
searc
h
on t
h
e determ
i
ni
stic
del
a
y
calc
ulus
theor
y of co
ntrol n
e
t
w
o
r
ks
.
Acta Electronic
a
Sinic
a
.
200
6; 34(2): 380-
38
4.
[6]
Cruz RL. A ca
lculus for
net
w
o
rk del
a
y
,
part
I: net
w
o
rk
ele
m
ents in is
olat
ion.
IEEE Transactions
on
Information Theory.
199
1; 37(
1): 114-1
31.
[7]
Cruz RL. A calculus for net
w
ork
delay
,
part II: net
w
o
rk analy
sis.
IEEE T
r
ansactions
on Information
T
heory
. 199
1; 37(1): 13
2-1
4
1
.
[8]
Wu YM, NING ZR. Revi
e
w
o
f
self-simil
arit
y mode
ls of
net
w
o
rks traffic.
J
ourn
a
l
of Ch
in
a Institute
of
Co
mmun
icati
o
n
. 2004; 2
5
(3): 97-1
04.
[9]
Gao Li
ng, L
i
Z
eng z
h
i, W
a
n
g
Z
heng. Mi
n-pl
u
s
alg
ebra r
epr
e
s
entatio
n for p
e
rf
ormanc
e par
ameters
w
i
t
h
finite flo
w
shaper in buffer.
Jo
urna
l of Xi an J
i
ao T
o
n
g
Univ
e
r
sity
. 2005; 39(
10): 106
8-1
071
.
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