An efficient
ans simple
class
M. Boyer
An efficient and simple class of functions to
model arrival curve of packetised flows
Network
calculus
Shaping,
packetization
and
computation
time
Marc Boyer, Jörn Migge, Nicolas Navet
Swaping
between
function
classes
Experiment
Conclusion
M. Boyer (ONERA,France)
Journée WEED
23 Novembre 2011
An efficient ans simple class
WEED - Nov. 2011
1 / 22
Outline
An efficient
ans simple
class
M. Boyer
1 Network calculus
Network
calculus
2 Shaping, packetization and computation time
Shaping,
packetization
and
computation
time
3 Swaping between function classes
Swaping
between
function
classes
Experiment
4 Experiment
5 Conclusion
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
2 / 22
Outline
An efficient
ans simple
class
M. Boyer
1 Network calculus
Network
calculus
2 Shaping, packetization and computation time
Shaping,
packetization
and
computation
time
3 Swaping between function classes
Swaping
between
function
classes
Experiment
4 Experiment
5 Conclusion
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
3 / 22
Network calculus overview
An efficient
ans simple
class
M. Boyer
Network
calculus
Two basic objects:
Flow:
modelling: R ∈ F = {R+ → R+ , non-decreasing}
semantics: R(t), cumulative amount of data up to t
Server:
S
modelling: S ∈ F × F: R −
→ R 0 =⇒ R 0 ≤ R
semantics: relation between some input and some output,
no loss, output comes after input (R 0 (t) ≤ R(t))
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Experiment
Conclusion
delay:
R
d(t)
b(t)
R’
v(R,R’)
max h(R, R 0 )
S
R−
→R 0
0
h(R, R ) : horizontal deviation
d(R, S)
=
h(R,R’)
t
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
4 / 22
Contract modelling
An efficient
ans simple
class
Flow contract: arrival curve α
R ≺ α ⇐⇒ ∀t, ∆ ∈ R+ R(t + ∆) − R(t) ≤ α(∆)
M. Boyer
⇐⇒ R ≤ R ∗ α
Network
calculus
Server contract: service curve
simple service of curve β
S
R−
→ R 0 ⇐⇒ R 0 ≥ R ∗ β
strict service of curve β
for all backlogged period [t, t + ∆[
(i.e.∀x ∈ [t, t + ∆[: R 0 (x) < R(x)):
R 0 (t + ∆) − R 0 (t) ≥ β(∆)
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Experiment
S
Results: R −
→ R 0 , R ≺ α, S has service curve β:
Conclusion
R0 ≺ α β
d(R, S) ≤ h(α, β)
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
5 / 22
Outline
An efficient
ans simple
class
M. Boyer
1 Network calculus
Network
calculus
2 Shaping, packetization and computation time
Shaping,
packetization
and
computation
time
3 Swaping between function classes
Swaping
between
function
classes
Experiment
4 Experiment
5 Conclusion
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
6 / 22
Shaping on links
An efficient
ans simple
class
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
A link is shared by a set of flows: what is the throughput of
this set ?
Principle: whatever the applicative throughput is, is it
limited by the links capacity
Also known has:
Serialisation: the frames of the different flows can not be
sent at the same time
Grouping: computes per-group throughput, not per-flow
Interest: considering long term rate ρ and instantaneous
burst b
applicative flows: small ρ, big b
link: big ρ, null b
Experiment
Conclusion
Impact: up to 40% in industrial system
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
7 / 22
Shaping and network calculus
An efficient
ans simple
class
Kb
Shaping
M. Boyer
Group sum
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
ms
Let S be a server, with shaping curve σ, then, the output is
constrained by σ.
If the output is constrained by α0 , it is by α0 ∧ σ.
Experiment
Conclusion
S
R−
→ R 0 =⇒ R 0 ≺ σ
R ≺ α, S D β =⇒ R 0 ≺ σ ∧ (α β)
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
8 / 22
Modelling a packetized flow
An efficient
ans simple
class
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Common example: sporadic flow
inter emission “period”: T
frame size (fixed on min): b
Two modelling:
fluid (“token bucket”): affine function, continuous
packetized: stairs functions, discontinuous
Kb
Swaping
between
function
classes
Fluid
Experiment
Frame
Conclusion
Packet
Size
ms
T
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
9 / 22
Fluid modelling: the virtual burst problem
An efficient
ans simple
class
M. Boyer
Network
calculus
Jitter “shifts” the arrival curve:
if jitter < period: instantaneous burst unchanged
in fluid modelling: creation of virtual burst =⇒ increase
bounds
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Experiment
Kb
Fluid
Virtual
Burst
Frame
Packet
Size
Conclusion
ms
T − Jitter
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
10 / 22
Putting all together
An efficient
ans simple
class
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Experiment
Conclusion
fluid + shaping: concave piecewise linear function (CPL)
Efficient min, max, sum
Implementation in floating points
stair-case modelling: general class (UPP)
Complex min, max, sum
Implementation in exact rationals (Q)
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
11 / 22
Outline
An efficient
ans simple
class
M. Boyer
1 Network calculus
Network
calculus
2 Shaping, packetization and computation time
Shaping,
packetization
and
computation
time
3 Swaping between function classes
Swaping
between
function
classes
Experiment
4 Experiment
5 Conclusion
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
12 / 22
Getting the better of each class
An efficient
ans simple
class
M. Boyer
Classes strengths/weaknesses:
jitter effect: stair-case class
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
summing (“grouping”): CPL class
shaping: CPL class
Idea:
keeping stair-case for individual flow constraint
converting into CPL when summing and shaping
Experiment
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
13 / 22
From stair-case to CPL
An efficient
ans simple
class
M. Boyer
γ
b
T −τ
,b
γ b ,b(1+τ /T )
νT ,τ
Network
calculus
b
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
T
T −τ
t
Figure: CPL overapproximation of a stair-case function
Experiment
Conclusion
M. Boyer (ONERA,France)
cpl(bνT ,τ ) = γ
b
nT −τ
,nb
An efficient ans simple class
∧ γ b ,b(1+τ /T )
(1)
T
WEED - Nov. 2011
14 / 22
Algorithm adaptation
An efficient
ans simple
class
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Experiment
Conclusion
Adaptation: replace
M. Boyer (ONERA,France)
P
Fik ∈F
An efficient ans simple class
αik by
P
Fik ∈F
cpl(αik )
WEED - Nov. 2011
15 / 22
Outline
An efficient
ans simple
class
M. Boyer
1 Network calculus
Network
calculus
2 Shaping, packetization and computation time
Shaping,
packetization
and
computation
time
3 Swaping between function classes
Swaping
between
function
classes
Experiment
4 Experiment
5 Conclusion
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
16 / 22
Testbed configuration
An efficient
ans simple
class
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
industrial (Thales) configuration
104 nodes
8 switches
974 multicast flows
6501 end-to-end bounds
Experiment
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
17 / 22
Comparing methods
An efficient
ans simple
class
12000
M. Boyer
10000
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
8000
CPL
CPL/nu
UPP
6000
4000
2000
Experiment
Conclusion
0
326
652
M. Boyer (ONERA,France)
978
1304
1630
1956
2282
2608
2934
3260
3586
An efficient ans simple class
3912
4238
4564
4890
5216
5542
5868
6194
WEED - Nov. 2011
18 / 22
Zoom on worst delays
An efficient
ans simple
class
11000
M. Boyer
10000
Network
calculus
Shaping,
packetization
and
computation
time
9000
CPL
CPL/nu
UPP
8000
Swaping
between
function
classes
7000
Experiment
Conclusion
6000
51
102
M. Boyer (ONERA,France)
153
204
255
306
357
408
459
510
561
An efficient ans simple class
612
663
714
765
816
867
918
969
WEED - Nov. 2011
19 / 22
Experiment values
An efficient
ans simple
class
Method
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Computation time
Min gain
Max gain
Av. gain
Min gain on 1000 biggest
Max gain on 1000 biggest
Av. gain on 1000 biggest
CPL
(float)
0.9 s
-
CPL/b.νT ,τ
(float)
1.1 s
0%
7.8%
2.49%
0.8%
4.4%
2.9%
UPP
(rat)
7.2 s
0.15%
15.2%
5.92%
2.0%
11.9%
8.3%
Experiment
Conclusion
M. Boyer (ONERA,France)
Gain correlation: 0.785
An efficient ans simple class
WEED - Nov. 2011
20 / 22
Outline
An efficient
ans simple
class
M. Boyer
1 Network calculus
Network
calculus
2 Shaping, packetization and computation time
Shaping,
packetization
and
computation
time
3 Swaping between function classes
Swaping
between
function
classes
Experiment
4 Experiment
5 Conclusion
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
21 / 22
Conclusion
An efficient
ans simple
class
M. Boyer
Network
calculus
Shaping,
packetization
and
computation
time
Swaping
between
function
classes
Two critical aspects: shaping and packetization
Two existing methods:
fluid: bad packetization, quick computation
stair-case: good packetization, longer computation
Contribution: trade-off tightness/computation time
Perspective: use in optimisation loop
quick computation in first iterations
longer computation to finalise
Experiment
Conclusion
M. Boyer (ONERA,France)
An efficient ans simple class
WEED - Nov. 2011
22 / 22