(Min,+) algebra. ¢. Overview. ¢. Network calculus. ¢. Hop-by-hop feedback control analysis. ¢. Fluid flow modelling.
Compartmental and (min,+) modelling of network elements in communication systems
V. Guffens , G. Bastin , H. Mounier UCL/CESAME (Belgium) - Universite´ Paris Sud (France)
Compartmental and (min,+) modelling of network elements in communication systems – p.1/17
(Min,+) algebra
Overview Network calculus Hop-by-hop feedback control analysis
Fluid flow modelling
Outline
Equivalent fluid flow model Alternative model Hop-by-hop feedback control analysis
Conclusion
Compartmental and (min,+) modelling of network elements in communication systems – p.2/17
Conventional addition and multiplication are replaced by :
(Min,+) theory overview
is a commutative dioid Distributivity of w.r.t
(min)
Compartmental and (min,+) modelling of network elements in communication systems – p.3/17
Conventional addition and multiplication are replaced by :
(Min,+) theory overview
is a commutative dioid Neutral for and
(min)
Compartmental and (min,+) modelling of network elements in communication systems – p.3/17
Conventional addition and multiplication are replaced by :
(Min,+) theory overview
(min)
is a commutative dioid etc.
Compartmental and (min,+) modelling of network elements in communication systems – p.3/17
Pioneer work Cruz91,A calculs for network delay. Exhaustive Book Leboudec, Thiran, Network calculus
# number of incoming packets up to time
V(t)
U(t)
x(t)
Uses cumulative rate
V(t)
# number of outgoing packets up to time
U(t)
wide-sense increasing functions
x(t)
[p]
FIFO buffer
Network calculus
time [s]
Compartmental and (min,+) modelling of network elements in communication systems – p.4/17
is said to be constrained by
At most (bits or packets) can be released on ANY time interval
A flow
Flow constraints
[p]
Constant bit rate server
[p]
R(t) = µ.t
Token leaky bucket R(t) = σ+ µ.t µ.
µ.
σ time [s]
time [s]
Compartmental and (min,+) modelling of network elements in communication systems – p.5/17
(Min,+) system theory
V(t)
U(t)
x(t) R(t)
FIFO buffer [p]
# !" $
V(t)
%"
U(t)
R(t)
In the case of a “greedy system”
by causality:
Convolution !
time [s]
Compartmental and (min,+) modelling of network elements in communication systems – p.6/17
Hop-by-hop feedback control
S
y0
v2
v1
v0 x1
y2
y1 x2
v(n+1) x(n+1)
A token/credit is fed back for every transmitted packet Conservation of packet/token around feedback loop
Compartmental and (min,+) modelling of network elements in communication systems – p.7/17
Hop-by-hop feedback control analysis W0
W1
Vn−1
Wn−1 Vn
R µn+1
Vn+1
σn
0
.
-
', &
*
CBR server,
bucket size,
R µn
σ2
/*
* ) +
R µ2
σ1
*
*
.
('&
σ0
V1
*
R µ1
*
V0
S
Compartmental and (min,+) modelling of network elements in communication systems – p.8/17
Hop-by-hop feedback control analysis as a
1
!
The relationship expressing the output function of the input is : V0
R µ1
W0
V1
R µ2
W1
Vn−1
R µn
Wn−1 Vn
R µn+1
Vn+1
1
S
!
!
σ0
σ1
σ2
σn
!
!
-
represents a constant bit rate serpackets per second.
) )
&
where vice rate of
!
6 * 7
89& ,
0
2 & -
&
!
*
3
3
.
.
3 4
!
!
.
.
!
!
5
with
Compartmental and (min,+) modelling of network elements in communication systems – p.9/17
Hop-by-hop feedback control analysis W0
V1
R µ2
W1
Vn−1
R µn
Wn−1 Vn
R µn+1
Vn+1
σ2
σn
>; >A >?
>?
:A
@
A
@ >?
@
:
;
@ >A
@
:=
? B
σ1
>?
σ0
!
!
R µ1
1
V0
S
t Compartmental and (min,+) modelling of network elements in communication systems – p.10/17
H
G
%
E
%
E
L
)
P Q
for
D
L
Single equilibrium at
L
D
)
if
O
D
L
M
)
if
N
D
L
M
D
L K
is equivalent to
&
I
,
D
Fluid-flow model: The convolution operation
%
EJ
H
%
EF
G
Equivalent fluid-flow model
Compartmental and (min,+) modelling of network elements in communication systems – p.11/17
Alternative fluid-flow model Experiment with exponentially distributed inter-packet arrival time x [p]
R
0
)
0.6
ST
S
0.7
0.5 0.4 0.3 0.2 Discrete event simulator
0.1 0 0
10
20
30
40
50
60
70 time [s]
Compartmental and (min,+) modelling of network elements in communication systems – p.12/17
Alternative fluid-flow model Experiment with exponentially distributed inter-packet arrival time x [p]
R
0
)
0.6
ST
S
0.7
0.5 0.4
Alternative model :
0.3
L
D
L K
Discrete event simulator
0.1
)
0.2
10
20
30
40
50
60
70 time [s]
L
0
0
Compartmental and (min,+) modelling of network elements in communication systems – p.12/17
Alternative fluid-flow model Experiment with exponentially distributed inter-packet arrival time x [p]
R
0
)
0.6
ST
S
0.7
0.5 0.4
0.2
D
L K
Discrete event simulator
0.1
)
Fluid Flow Model
L
Alternative model :
0.3
20
30
40
50
60
70 time [s]
10
W
L
or
X & Y
U
V
/
R
Equilibrium :
V
&U
W
0
L
0
Compartmental and (min,+) modelling of network elements in communication systems – p.12/17
t v
{
}
_
]
h
u
}
_
a
~
u
]
d
j ] `
j
m l n
c
c
g
f
g
f
q
s
s
t
h u v
qs r p o
b
_
k ` Q j ]
]
d
b
]
i d `
h
d c e
_ ] `
_
g
f
x2
{
_
c
_ ] `
_
k j Q
a
x1
}
{ d |
b
i y
a
v2
q
o
_ ] `
x
_ ^ ` ] \
Z
v1
s
u
_
j ] `
y
b
i y
h
w
j ^ ` ] \
[
y1
s
qs
m l n
}
b
j
u
z
a
i d `
v0
_
j ] `
~
c
y0
}
b
y
h
a
d
S
j
z
j ] `
h
u
j
j
Hop-by-hop feedback control analysis y2 v(n+1)
x(n+1)
Compartmental and (min,+) modelling of network elements in communication systems – p.13/17
Compartmental modelling
I
L
I
0
I
0
I
L
L
K
The system can be rewritten :
g q
s
s
qs
i d q
a
d
the
g
f ]
input vector and
f
5 g {
]
the state vector,
q
s
s
_ ] qs `
a
with
]
f
5
/
0
I
I
...
...
the compartmental matrix Compartmental and (min,+) modelling of network elements in communication systems – p.14/17
Results Integral of the rate
16
140
14
120
12
100
10 80
8
60
6
40
x1 buffer occupancy
4
20
2
0 0.4
0
0.8
1.2
1.6
2.0
2.4
2.8
3.2
0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Results from (Min,+) analysis are recovered good adequacy with experimental results Compartmental and (min,+) modelling of network elements in communication systems – p.15/17
Results I/S caracteristic 16 14 12 10 x4
8 6
x1,x2,x3
4 2 0 10 20 30 40 50 60 70 80 90 100
0
Fluid-flow models offer additional interesting results Compartmental and (min,+) modelling of network elements in communication systems – p.16/17
Fluid-flow models over complementary view on network calculus Alternative fluid flow models allow discovering new aspects
New results might be obtained stability of hop-by-hop feedback control (ECC’03) Optimal control studies (submitted to CDC’04)
Conclusion
Compartmental and (min,+) modelling of network elements in communication systems – p.17/17
Fluid-flow models over complementary view on network calculus Alternative fluid flow models allow discovering new aspects
New results might be obtained stability of hop-by-hop feedback control (ECC’03) Optimal control studies (submitted to CDC’04)
Conclusion
The end Compartmental and (min,+) modelling of network elements in communication systems – p.17/17
