TU e Introduction into Network Calculus

TU e Introduction into Network Calculus
Introduction into
Network Calculus
Stephan Recker
[email protected]
TU/e
eindhoven university of technology
1
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
2
The Standard Linear Theory
+
x(t)
+
y(t)
-
β(t)
-
LTI filter in conventional algebra (R, +, ×)
Input signal = electrical voltage x(t)
System = circuit (filter) with impulse response β(t)
Output = convolution of x(t) and β(t) :
y(t) = ∫ β(t − s)x(s)ds
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eindhoven university of technology
3
Network Calculus uses Min-Plus Linear Theory
x(t)
y(t)
bit rate c
A linear system in min-plus algebra (R, min, +)
Input = arrived traffic in [0,t]: x(t)
System = server with rate c : β(t) = ct
Output = convolution of x(t) and β(t):
y(t) = inf {β(t − s) + x(s)}
s∈
β(t), x(t) = 0 ∀ t < 0
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eindhoven university of technology
4
Two Key Concepts: Arrival and Service Curve
IntServ and DiffServ use the concepts of
arrival curve and service curves
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eindhoven university of technology
5
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
6
Cumulative Flows
Cumulative flow R(t) ∈ F , t real or integer
F = {x(t) is non decreasing and x(t) = 0 ∀ t < 0}
Examples:
bits
R1(t)
12
567
bits
time t
Fluid model (continuous)
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bits
R2(t)
1
55.5
time t
Packet model
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7
R3(t)
12
56
time t
Discrete-time model
Arrival Curves
Arrival curve Â(t) :
For any times 0 ≤ s ≤ t, the cumulative flow R(t) satisfies
ˆ − s)
R (t) − R (s) ≤ A(t
Example 1: affine arrival curve γr,b
Â(t) = γ r ,b (t) = r ⋅ t + b ∀ t > 0
data
data
slope r
-s)
A( t
+
)
F(s
)
(t-0
A
)+
F(0
b
b
time t
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eindhoven university of technology
0
8
s
F(t)
t
time t
Arrival Curves
Example 2: stair arrival curve kv T,τ
t + τ
Â(t) = kv T ,τ (t) = k 
 T 
with T = period, τ = tolerance, k = constant packet size
Characterizes flows that are periodic stream of packets of same size k
(cells), that suffers a variable delay <= τ
bits
All packets of size k.
If R conforms to kv T,τ
then R conforms to γ r,b
with r = k/T and
b = k(τ+T)/T
t + τ
kv T,τ (t) = k 
 T 
4k
3k
2k
k
T-τ
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eindhoven university of technology
9
2T-τ
3T-τ
time
Leaky Bucket
All packets of flow R are declared conformant by a leaky
buket controller of rate r and size b
R conforms to Â(t) = γ r ,b (t) = r ⋅ t + b ∀ t > 0
R(t)
γr,b
slope r
R(t)
R(t)
b
b
x(t)
b
x(t)
t
r
TU/e
eindhoven university of technology
10
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
11
Arrival Curves and Min-Plus Convolution
Arrival Curve property
means for all 0 ≤ s ≤ t:
ˆ − s)
x(t) − x(s) ≤ A(t
ˆ − s) ∀ 0 ≤ s ≤ t
⇔ x(t) ≤ x(s) + A(t
ˆ − u) 
⇔ x(t) ≤ inf  x(u) + A(t

0≤ u ≤ t
ˆ − u)  with x(t), A
ˆ ∈F
⇔ x(t) ≤ inf  x(u) + A(t

u∈
ˆ (t)
⇔ x(t) ≤ x ⊗ A
(
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)
eindhoven university of technology
Min-Plus Convolution
12
Properties of Min-Plus Convolution
(f ⊗ g) ∈ F
⊗ is associative
⊗ is commutative
Neutral element: δ0 : f ⊗ δ0 = f δ0(t) = 0 for t = 0 and δ0(t) = ∞ for t > 0
⊗ is distributive with respect to min (∧
∧)
Functions passing through the origin (f(0) = g(0) = 0): f ⊗ g ≤ f ∧ g
Concave functions passing through the origin: f ⊗ g = f ∧ g
Convex piecewise linear functions: f ⊗ g is the convex piecewise linear
function obtained by putting end-to-end all linear pieces of f and g,
sorted by increasing slopes
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eindhoven university of technology
13
Min-Plus Convolution of Convex Functions
convex piecewise linear wide-sense increasing, passing
by origin: put segments end to end with increasing
slope
r2
r2
r1
⊗
s1
t1
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=
s2
eindhoven university of technology
r1
s1
u1 u1+t1
u1
14
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
15
Sub-Additive Functions
f is sub-additive f (t) + f(s) ≥ f(t+s)
f is concave with f(0) = 0 ⇒ f is sub-additive
f,g are sub-additive and pass through the origin: f(0) =
g(0) = 0 ⇒ f ⊗ g is sub-additive
bits
bits
b
γr,b(t)
vT,τ (t)
4k
3k
2k
k
slope r
time
Τ−τ 2T-τ 3T-τ 4T-τ
vT,ττ is not concave, but is
sub-additive
γr,b is concave
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eindhoven university of technology
time
16
Min-Plus Deconvolution
Definition:
h (t ) ≤ ( f g )(t ) = sup u [f (t + u ) − g (u ) ]
( f g )(t )
f(t)
g(t)
t
-g(-t)
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eindhoven university of technology
17
Properties of Min-Plus Deconvolution
(f 
g) ∉ F in general
(f 
f ) is sub-additive with (f 
f )(0) = 0
(f 
f)∈ F
(f 
g) h = f (g ⊗ h) 
Duality with ⊗ : f 
g ≤ h ⇔f ≤ g⊗h
TU/e
eindhoven university of technology
18
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
19
Service Curves
data
f(t) = R·[t-T]
Service curve f(t) :
Minimum service
offered.
x(t)
y (t ) ≥ y ( s ) + f (t − s )
in case of backlog
y(t)
time t
T
For all t there exists at least one s ≤ t, such that:
y (t ) ≥ x ( s ) + f (t − s )
(i.e. s is the start of the last backlogged period)
y (t ) ≥ inft ≥s ≥0 [ x(s ) + f (t − s )]
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eindhoven university of technology
20
Service Curve Example
Shaper
R(t)
x(t)
σ
If σ is sub-additive and σ (0) = 0 ⇒ x(t) = (R ⊗ σ )(t)
The service curve of a shaper is thus f(t)=σ(t).
Standard Internet Router Model: Rate-Latency
bits
r
f (t) = R (t − T)
T
time
TU/e
eindhoven university of technology
21
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
22
Summary of Network Calculus
g (t )
f (t )
h(t )
Lower Service Bound
h(t ) ≥ (f ⊗ g )(t ) = infu [f (t − u ) + g (u )]
Upper Service Bound
h(t ) ≤ (f g )(t ) = supu [f (t + u ) − g (u )]
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eindhoven university of technology
23
Tight Bounds on Delay and Backlog
If flow has arrival curve A and node offers service
curve f then
backlog ≤ sup (A(s) -f(s))
delay ≤ h(A, f)
A
h(A,f)
backlog
f
TU/e
eindhoven university of technology
24
Overview
1.
2.
3.
4.
5.
6.
7.
TU/e
What is Network Calculus?
Arrival Curves
Min-Plus Convolution
Min-Plus Deconvolution
Service Curves
Single Server Analysis
Multiple Server Analysis – Composition Theorem
eindhoven university of technology
25
The Composition Theorem
Theorem: the concatenation of two network
elements each offering service curve fi offers the
service curve f1 ⊗ f2
f1
f2
f1 ⊗ f2
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eindhoven university of technology
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Example: Tandem of Routers
R1
⊗
T1
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R1
=
R2
T2
T2
27
T1+T2
Pay Bursts Only Once
A
D1
D2
f1
f2
D11 +D22 ≤ (2b + RT11)/ R + T11 + T22
D
A
f1⊗ f2
D
D ≤≤ bb /R
/R ++ T
T11 ++ T
T22
end
end to
to end
end delay
delay bound
bound is
is less
less
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eindhoven university of technology
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Summary: Linear Systems and Network Calculus
Linear System Theory
Network Calculus
g (t )
g (t )
f (t )
f (t )
h(t )
f (t )
h(t )
f (t )
g (t )
g (t )
h(t )
h(t )
Min-plus convolution
Convolution
+∞
h(t ) = (f ∗ g )(t ) =
∫ f (t − u)g(u)du
h(t) ≥ (f ⊗ g)(t) = inf u [f (t − u) + g(u)]
−∞
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eindhoven university of technology
29
Summary of Network Calculus
g (t )
f (t )
h(t )
Lower Service Bound
h(t ) ≥ (f ⊗ g )(t ) = infu [f (t − u ) + g (u )]
Upper Service Bound h(t ) ≤ (f g )(t ) = supu [f (t + u ) − g(u )]
f (t )
g (t )
h(t ) = ( f ⊗ g )(t ) = inf u [ f (t − u ) + g (u )]
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eindhoven university of technology
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Conclusion
Network Calculus is a set of tools and theories for the
deterministic analysis of communication networks
A new system theory, which applies min-plus algebra to
communication networks
Applicability needs to be evaluated carefully as still worst
case performance is determined
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eindhoven university of technology
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Further Reading
ica1www.epfl.ch/PS_files/NetCal.htm
Cheng-Shang Chang,
Performance Guarantees in
Communication Networks,
ISBN:1852332263
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eindhoven university of technology
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Legendre Transform
Eigenfunctions:
+∞
Convolution
e
j2π st
∗ g (t ) =
j2π s ( t −u )
j2π st
e
g
(
u
)
du
e
=
⋅ G( s )
∫
−∞
Min-Plus
De-convolution
(b + s ⋅ t ) g (t ) = supu [ b + s ⋅ (t + u ) − g (u )] = (b + s ⋅ t ) + G(s )
Convex conjugate
G(s ) = supu [s ⋅ u − g (u )]
Concave conjugate
H (s ) = infu [s ⋅ u − h(u )]
Fenchel Conjugates
Fenchel Conjugate of a continuously differentiable function
is called Legendre Transform
Min-plus de-convolution
h(t )
=
f ( t ) g (t )
Min-plus convolution
h(t )
f concave,
g convex
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f (t ) ⊗ g (t )
H ( s ) = F ( s ) + G( s )
H ( s ) = F ( s ) − G( s )
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=
33
f convex,
g convex
Conjugate Arrival Curves
data
data
r
rate s
-b
r
pe
s lo
b
time t
backlog bound at a constant rate server with rate s
data
r
R
rate s
data
-b
b
sl
R ope
r
pe
o
l
s
time t
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eindhoven university of technology
backlog bound at a constant rate server with rate s
34
Conjugate Service Curves
data
data backlog bound for a constant rate flow with rate s
sl
e
op
R
RT
time t
T
R
data
data backlog bound for a constant rate flow with rate s
e
op
l
s
R
RT
(R-r)τ
er
slop
T
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rate s
τ
eindhoven university of technology
time t
r
35
R
rate s
Server Output
data
r
rate s
data
-b
RT
R
data
r
R
rate s
rate s
data
r
pe
o
l
s
-b
b+RT
-RT
time t
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eindhoven university of technology
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Concatenation
data
data
R2T2
R1T1
rate s
R1
rate s
R2
data
data
R2T2
m
pe
s lo
R1T1
R1
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R2
eindhoven university of technology
rate s
T1+T2
37
]
,R 2
1
R
in[
time t
Fenchel Duality Theorem
minimize f1 (x) − f 2 (x)
s.t.
x∈
minimize
s.t.
f1,-f2 convex
f1 (y) − f 2 (z)
y−z =0
L(y, z,s) = f1 (y) − f 2 (z) + s(y − z)
q(s) = inf {f1 (y) − f 2 (z) + s(y − z)}
y,z∈
F2 Concave Fenchel Conjugate
F1 Convex Fenchel Conjugate
= inf {sz − f 2 (z)} − sup {sy − f1 (y)}
z∈
y∈
Duality Theory:
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Maximize q(s) = F2 (s) − F1 (s)
s.t.
s∈
38
Conjugate Performance Bounds
Conjugate Delay Bound:
maximize
d
s.t.
α(t) = β(t + d)
L(t,s) = d + s(α(t) − β(t + d))
Biconjugates of closed functions
A(s) = inf [st − α (t) ]
t∈
α (t) = inf [ ts − A(s) ]
s∈
∂α(t) ∂β(t + d)
=
→ tangents in one s
∂t
∂t
∂L
→ α(t) = β(t + d) → Identical Ordinates
∂s
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eindhoven university of technology
39
Essence of Conjugate Network Calculus
R.T. Rockafellar, Convex Analysis,
Princeton University Press, 1972:
M. Fidler, S. Recker, A Dual Approach to
Network Calculus Applying the Legendre
Transform, QoSIP 2005, Catania, Italy:
f concave, g convex
f convex, g convex
F (s )
G (s )
G (s )
F (s )
H (s ) ≤ F (s ) − G(s )
H(s) = F(s) + G(s)
= inft [st − f (t )] − supt [st − g (t )]
= sup t [st − f (t)] + sup t [st − g(t)]
M. Fidler, S. Recker, Network
Calculus and Conjugate Duality
in Network Performance
Analysis, ITC19, Beijing, China:
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eindhoven university of technology
H (s )
Backlog Bound:
Q = − sup [ A(s) − B(s)]
s∈
Delay Bound:
40
Difference of slopes of tangents
to A(s*) and B(s*),
which have the same ordinate
Token Bucket – Latency-Rate Example
With p ≥ r ≥ ρ
B(s)
σ−M
p−ρ
σ σ−M ρ
=d+ +
ρ p−ρ r
~ TA(s) =
A(s)
σ−M
>d:
p−ρ
σ−M
Q = σ + ρd + 
(ρ − r )

 p−ρ 
M
σ−M p−r
D=
+d+
r
p−ρ r
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Q = σ + ρd −
D =d+
=d+
41
~ TB(s)
σ−M
(r − ρ )
p−ρ
σ σ−M ρ 
+
 − 1
ρ p−ρ  r

σ ( p − r ) − M(ρ − r)
r(p − ρ)
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