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 TU/e 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 TU/e eindhoven university of technology 4 Two Key Concepts: Arrival and Service Curve IntServ and DiffServ use the concepts of arrival curve and service curves TU/e 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) TU/e bits R2(t) 1 55.5 time t Packet model eindhoven university of technology 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 TU/e 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 TU/e 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 ( TU/e ) 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 TU/e 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 TU/e = 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 TU/e 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) TU/e 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 )] TU/e 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 )] TU/e 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 TU/e eindhoven university of technology 26 Example: Tandem of Routers R1 ⊗ T1 TU/e eindhoven university of technology 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 TU/e eindhoven university of technology 28 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)] −∞ TU/e 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 )] TU/e eindhoven university of technology 30 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 TU/e eindhoven university of technology 31 Further Reading ica1www.epfl.ch/PS_files/NetCal.htm Cheng-Shang Chang, Performance Guarantees in Communication Networks, ISBN:1852332263 TU/e eindhoven university of technology 32 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 eindhoven university of technology f (t ) ⊗ g (t ) H ( s ) = F ( s ) + G( s ) H ( s ) = F ( s ) − G( s ) TU/e = 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 TU/e 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 TU/e 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 TU/e eindhoven university of technology 36 Concatenation data data R2T2 R1T1 rate s R1 rate s R2 data data R2T2 m pe s lo R1T1 R1 TU/e 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: TU/e eindhoven university of technology 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 TU/e 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: TU/e 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 TU/e eindhoven university of technology 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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