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An efficient and simple class of functions to M. Boyer model - - PowerPoint PPT Presentation

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An efficient and simple class An efficient and simple class of functions to M. Boyer model arrival curve of packetised flows Network calculus Marc Boyer, J orn Migge, Nicolas Navet Shaping, packetization and computation time Swaping

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SLIDE 1

An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

An efficient and simple class of functions to model arrival curve of packetised flows

Marc Boyer, J¨

  • rn Migge, Nicolas Navet

RTSS/WCTT Workshop

  • Nov. 29th, 2011
  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 1 / 26

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SLIDE 2

An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Outline

1 Network calculus 2 Shaping, packetization and computation time 3 Swaping between function classes 4 Experiment 5 Conclusion

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 2 / 26

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SLIDE 3

An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Outline

1 Network calculus 2 Shaping, packetization and computation time 3 Swaping between function classes 4 Experiment 5 Conclusion

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 3 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

What is Network Calculus ?

A theory designed to compute guaranteed bounds on delays. With a strong mathematical background: (min,+) algebra

Basic object: non-decreasing, non-negative functions F = {f : R+ → R+ x < y = ⇒ f (x) ≤ f (y)} Three basic operations: the convolution ∗, deconvolution ⊘, the sub-additive closure f ∗. (f ∗ g)(t) = inf

0≤u≤t(f (t − u) + g(u))

(1) (f ⊘ g)(t) = sup

0≤u

(f (t + u) − g(u)) (2) f ∗ = δ0 ∧ f ∧ (f ∗ f ) ∧ (f ∗ f ∗ f ) ∧ · · · (3)

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 4 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Network calculus overview

Two basic objects: Flow:

modelling: R ∈ F = {R+ → R+, non-decreasing} semantics: R(t), cumulative amount of data up to t

Server:

modelling: S ∈ F × F: R

S

− → R′ = ⇒ R′ ≤ R semantics: relation between some input and some output, no loss, output comes after input (R′(t) ≤ R(t))

delay: d(R, S) ≤ max

R

S

− →R′ h(R, R′) h(R, R′) : horizontal deviation

R R’ h(R,R’) v(R,R’) t d(t) b(t)

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 5 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Contract modelling

Flow contract: arrival curve α R ≺ α ⇐ ⇒ ∀t, ∆ ∈ R+R(t + ∆) − R(t) ≤ α(∆) ⇐ ⇒ R ≤ R ∗ α Server contract: service curve

simple service of curve β R

S

− → R′ ⇐ ⇒ R′ ≥ R ∗ β strict service of curve β for all backlogged period [t, t + ∆[ (i.e.∀x ∈ [t, t + ∆[: R′(x) < R(x)): R′(t + ∆) − R′(t) ≥ β(∆)

Results: R

S

− → R′, R ≺ α, S has service curve β: R′ ≺ α ⊘ β d(R, S) ≤ h(α, β)

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 6 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Outline

1 Network calculus 2 Shaping, packetization and computation time 3 Swaping between function classes 4 Experiment 5 Conclusion

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 7 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Shaping on links

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

Impact: up to 40% in industrial system

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 8 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Shaping and network calculus

Kb ms Shaping Group sum

Let S be a server, with shaping curve σ, then, the output is constrained by σ. If the output is constrained by α′, it is by α′ ∧ σ. R

S

− → R′ = ⇒ R′ ≺ σ R ≺ α, S β = ⇒ R′ ≺ σ ∧ (α ⊘ β)

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 9 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Modelling a packetized flow

Common example: sporadic flow inter emission “period”: T frame size (fixed or max): b Two modelling: fluid (“token bucket”): affine function, continuous packetized: stair-case functions, discontinuous

Kb ms Packet Fluid Frame Size T

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 10 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Fluid modelling: the virtual burst problem

Jitter “shifts” the arrival curve: if jitter < period: instantaneous burst unchanged in fluid modelling: creation of virtual burst = ⇒ increase bounds

Packet Fluid Kb Frame Size ms

Virtual Burst

T − Jitter

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 11 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Putting all together

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 and simple class WCTT - Nov. 2011 12 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Outline

1 Network calculus 2 Shaping, packetization and computation time 3 Swaping between function classes 4 Experiment 5 Conclusion

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 13 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Getting the better of each class

Classes strengths/weaknesses: jitter effect: stair-case class summing (“grouping”): CPL class shaping: CPL class Idea: keeping stair-case for individual flow constraint converting into CPL when summing and shaping

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 14 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

From stair-case to CPL

t b T − τ νT,τ γ

b T−τ ,b

γ b

T ,b(1+τ/T)

Figure: CPL overapproximation of a stair-case function

cpl(bνT,τ) = γ

b nT−τ ,nb ∧ γ b T ,b(1+τ/T)

(4)

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 15 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Algorithm adaptation

Adaptation: replace

F k

i ∈F αk

i by F k

i ∈F cpl(αk

i )

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 16 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Outline

1 Network calculus 2 Shaping, packetization and computation time 3 Swaping between function classes 4 Experiment 5 Conclusion

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 17 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Testbed configuration

industrial (Thales) configuration 104 nodes 8 switches 974 multicast flows 6501 end-to-end bounds

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 18 / 26

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An efficient and simple class

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Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Comparing methods

326 652 978 1304 1630 1956 2282 2608 2934 3260 3586 3912 4238 4564 4890 5216 5542 5868 6194 2000 4000 6000 8000 10000 12000 CPL CPL/nu UPP

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 19 / 26

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An efficient and simple class

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Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Zoom on worst delays

51 102 153 204 255 306 357 408 459 510 561 612 663 714 765 816 867 918 969 6000 7000 8000 9000 10000 11000 CPL CPL/nu UPP

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 20 / 26

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An efficient and simple class

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Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Pessimism evaluation

Method comparison: based on upper bound (UBm) Best comparison: based on pessism pessm = UBm − WCTT Worst case unknown (WCTT) Delay lower bound: trajectorial based approach (LB) LB ≤ WCTT ≤ UBm pessm ≤ UBm − LB

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 21 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Uppers and lower bounds

326 652 978 1304 1630 1956 2282 2608 2934 3260 3586 3912 4238 4564 4890 5216 5542 5868 6194 2000 4000 6000 8000 10000 12000 Icc Bucket Shaped stairs Upp stairs Unfavorable Upp Stairs Pessimism Bound

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 22 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Pessimism bounding, per method

326 652 978 1304 1630 1956 2282 2608 2934 3260 3586 3912 4238 4564 4890 5216 5542 5868 6194 500 1000 1500 2000 2500

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 23 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Experiment values

Method CPL CPL/b.νT,τ UPP (float) (float) (rat) Computation time 0.9 s 1.1 s 7.2 s Min gain

  • 0%

0.15% Max gain

  • 7.8%

15.2%

  • Av. gain
  • 2.49%

5.92% Min gain on 1000 biggest

  • 0.8%

2.0% Max gain on 1000 biggest

  • 4.4%

11.9%

  • Av. gain on 1000 biggest
  • 2.9%

8.3% Gain correlation: 0.785

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 24 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Outline

1 Network calculus 2 Shaping, packetization and computation time 3 Swaping between function classes 4 Experiment 5 Conclusion

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 25 / 26

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An efficient and simple class

  • M. Boyer

Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion

Conclusion

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 Don’t use CPL, use CPL/b.νT,τ

simple to implement low computation time over-head significant bound improvement

Perspective: use in optimisation loop

quick computation in first iterations longer computation to finalise

  • M. Boyer (ONERA,France)

An efficient and simple class WCTT - Nov. 2011 26 / 26