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

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 between function classes Experiment Conclusion RTSS/WCTT Workshop Nov. 29th, 2011 M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 1 / 26

Outline An efficient and simple class M. Boyer 1 Network calculus Network 2 Shaping, packetization and computation time calculus Shaping, packetization and 3 Swaping between function classes computation time Swaping 4 Experiment between function classes Experiment 5 Conclusion Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 2 / 26

Outline An efficient and simple class M. Boyer 1 Network calculus Network 2 Shaping, packetization and computation time calculus Shaping, packetization and 3 Swaping between function classes computation time Swaping 4 Experiment between function classes Experiment 5 Conclusion Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 3 / 26

What is Network Calculus ? An efficient and simple A theory designed to compute guaranteed bounds on class delays. M. Boyer With a strong mathematical background: (min,+) algebra Basic object: non-decreasing, non-negative functions Network calculus F = { f : R + → R + x < y = ⇒ f ( x ) ≤ f ( y ) } Shaping, packetization and computation Three basic operations: the convolution ∗ , deconvolution time ⊘ , the sub-additive closure f ∗ . Swaping between function ( f ∗ g )( t ) = 0 ≤ u ≤ t ( f ( t − u ) + g ( u )) inf (1) classes Experiment ( f ⊘ g )( t ) = sup ( f ( t + u ) − g ( u )) (2) Conclusion 0 ≤ u f ∗ = δ 0 ∧ f ∧ ( f ∗ f ) ∧ ( f ∗ f ∗ f ) ∧ · · · (3) M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 4 / 26

Network calculus overview An efficient Two basic objects: and simple class Flow: M. Boyer modelling: R ∈ F = { R + → R + , non-decreasing } semantics: R ( t ), cumulative amount of data up to t Network Server: calculus → R ′ = ⇒ R ′ ≤ R S modelling: S ∈ F × F : R − Shaping, packetization semantics: relation between some input and some output, and no loss, output comes after input ( R ′ ( t ) ≤ R ( t )) computation time delay: Swaping between function R d(t) classes b(t) R’ v(R,R’) Experiment h ( R , R ′ ) d ( R , S ) ≤ max Conclusion S − → R ′ R h ( R , R ′ ) : horizontal deviation h(R,R’) t M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 5 / 26

Contract modelling An efficient Flow contract: arrival curve α and simple class ⇒ ∀ t , ∆ ∈ R + R ( t + ∆) − R ( t ) ≤ α (∆) R ≺ α ⇐ M. Boyer ⇐ ⇒ R ≤ R ∗ α Network Server contract: service curve calculus simple service of curve β Shaping, → R ′ ⇐ ⇒ R ′ ≥ R ∗ β S packetization R − and computation strict service of curve β time for all backlogged period [ t , t + ∆[ Swaping ( i.e. ∀ x ∈ [ t , t + ∆[: R ′ ( x ) < R ( x )): between function R ′ ( t + ∆) − R ′ ( t ) ≥ β (∆) classes S → R ′ , R ≺ α , S has service curve β : Results: R − Experiment Conclusion R ′ ≺ α ⊘ β d ( R , S ) ≤ h ( α, β ) M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 6 / 26

Outline An efficient and simple class M. Boyer 1 Network calculus Network 2 Shaping, packetization and computation time calculus Shaping, packetization and 3 Swaping between function classes computation time Swaping 4 Experiment between function classes Experiment 5 Conclusion Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 7 / 26

Shaping on links An efficient and simple A link is shared by a set of flows: what is the throughput of class this set ? M. Boyer Principle: whatever the applicative throughput is, is it limited by the links capacity Network calculus Also known has: Shaping, Serialisation: the frames of the different flows can not be packetization and sent at the same time computation time Grouping: computes per-group throughput, not per-flow Swaping Interest: considering long term rate ρ and instantaneous between function burst b classes applicative flows: small ρ , big b Experiment link: big ρ , null b Conclusion Impact: up to 40% in industrial system M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 8 / 26

Shaping and network calculus An efficient and simple Kb class Shaping M. Boyer Group sum Network calculus ms Shaping, packetization and Let S be a server, with shaping curve σ , then, the output is computation time constrained by σ . Swaping If the output is constrained by α ′ , it is by α ′ ∧ σ . between function classes Experiment → R ′ = ⇒ R ′ ≺ σ S R − Conclusion ⇒ R ′ ≺ σ ∧ ( α ⊘ β ) R ≺ α, S � β = M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 9 / 26

Modelling a packetized flow Common example: sporadic flow An efficient and simple class inter emission “period”: T M. Boyer frame size (fixed or max): b Two modelling: Network calculus fluid (“token bucket”): affine function, continuous Shaping, packetized: stair-case functions, discontinuous packetization and computation time Kb Swaping between function classes Fluid Experiment Frame Packet Conclusion Size ms T M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 10 / 26

Fluid modelling: the virtual burst problem An efficient and simple Jitter “shifts” the arrival curve: class M. Boyer if jitter < period: instantaneous burst unchanged in fluid modelling: creation of virtual burst = ⇒ increase bounds Network calculus Shaping, packetization and Kb computation time Swaping between Fluid Virtual function Burst classes Frame Packet Size Experiment Conclusion ms T − Jitter M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 11 / 26

Putting all together An efficient and simple class M. Boyer Network calculus Shaping, packetization and computation time Swaping fluid + shaping: concave piecewise linear function (CPL) between function Efficient min, max, sum classes Implementation in floating points Experiment stair-case modelling: general class (UPP) Conclusion Complex min, max, sum Implementation in exact rationals ( Q ) M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 12 / 26

Outline An efficient and simple class M. Boyer 1 Network calculus Network 2 Shaping, packetization and computation time calculus Shaping, packetization and 3 Swaping between function classes computation time Swaping 4 Experiment between function classes Experiment 5 Conclusion Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 13 / 26

Getting the better of each class An efficient and simple class M. Boyer Classes strengths/weaknesses: jitter effect: stair-case class Network calculus summing (“grouping”): CPL class Shaping, packetization shaping: CPL class and computation Idea: time Swaping keeping stair-case for individual flow constraint between function converting into CPL when summing and shaping classes Experiment Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 14 / 26

From stair-case to CPL An efficient and simple class γ b M. Boyer γ T , b (1+ τ/ T ) b T − τ , b ν T ,τ Network calculus b Shaping, T − τ packetization and t computation time Swaping Figure: CPL overapproximation of a stair-case function between function classes Experiment Conclusion cpl ( b ν T ,τ ) = γ nT − τ , nb ∧ γ b (4) b T , b (1+ τ/ T ) M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 15 / 26

Algorithm adaptation An efficient and simple class M. Boyer Network calculus Shaping, packetization and computation time Swaping between function classes Experiment Conclusion i ∈F α k i ∈F cpl ( α k Adaptation: replace � i by � i ) F k F k M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 16 / 26

Outline An efficient and simple class M. Boyer 1 Network calculus Network 2 Shaping, packetization and computation time calculus Shaping, packetization and 3 Swaping between function classes computation time Swaping 4 Experiment between function classes Experiment 5 Conclusion Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 17 / 26

Testbed configuration An efficient and simple class M. Boyer industrial (Thales) configuration Network calculus 104 nodes Shaping, packetization 8 switches and computation 974 multicast flows time Swaping 6501 end-to-end bounds between function classes Experiment Conclusion M. Boyer (ONERA,France) An efficient and simple class WCTT - Nov. 2011 18 / 26