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89-850 Communication Networks Network Calculus: Deterministic Traffic Analysis Last updated: Thursday, May 07, 2015 Prof. Amir Herzberg Dept of Computer Science, Bar Ilan University http://AmirHerzberg.com

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Network Calculus: Topics 1.Introduction Voice call processing example Network calculus Fixed-rate service and Reich’s equation Min-Plus Convolution 2.Arrival curves 3.Service curves Backlog, delay bounds, IntServ routers (Latency- Rate servers), Shapers

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Voice Call Processing & Delays Sound sampled and encoded Constant Bit Rate (CBR), e.g. 64Kbps (8000*8b) Voice activity detection (and compression?) Discard (long, e.g. >0.2s) inactive periods On-off Variable Bit Rate (VBR) Question: what link rate C is needed?

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Voice Call Processing & Delays (cont’) Sound sampled and CBR-encoded Voice activity detection, compression VBR Shaping Specifications of sent traffic Allows efficient rates, buffers Packetization (and padding if necessary)

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Voice Call Processing & Delays (cont’) Sender: Sampling and CBR-encoding Voice Activity Detection, compression VBR Shaping Packetization (and padding if necessary) Network (queues, transmission, propagation) Receiver Immediate or delayed playout?

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Voice Transport over Packet Network Deterministic vs. stochastic analysis

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Playout Delay to `Hide` Jitter Lines: actual packet delays Is output Ok? What size of buffer (backlog)?

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Playout Buffer Model Network system S: CBR input: R(t)=rt Delay variation (jitter) bounded by Δ Namely for some D>0 holds: R(t) ≦ R*(t+D+Δ) and R*(t+D-Δ) ≦ R(t) Playout buffer S’: Receive first bit at d(0), delay it by Δ Then, serve at constant rate r, when not empty if never empty after d(0), then S(t)=r(t-Δ-d(0)) +

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Playout Buffer Analysis Claim: for every t>d(0) holds 0

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Network Calculus Deterministic analysis of scheduling/ queuing / flow Uses min, max as binary operators and integrals (min-plus. max-plus algebra) Worst-case wasteful [cf. probabilistic queuing analysis – later] Applicable to many problems Voice, audio, data, … Only `traffic flow`, no processing/control Discrete or continuous time (bit or fluid model, resp.)

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Network Calculus: References and Sources Network Calculus by J. Le Boudec, P. Thiran, Springer, Aug 2001, LNCS 2050 In library Online: http://ica1www.epfl.ch/PS_files/netCalBookv4.pdf Communication Networking – an analytical approach, Kumar, Manjunath and Kuri, Elsevier, 2004 (mainly ch. 4) In (folder) in library

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Network Calculus Network Calculus: Worst-case analysis of Arrivals, Service, Buffering and Departures Used in design of QoS protocols Internet: mainly IntServ, DiffServ ATM Arrivals A(t) Departures D(t)Service Buffer X(t)

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Buffer Analysis: Fixed Rate Queue Bits sent (removed) at C bits/second (unless empty) a k : arrival of (complete) packet k L k [0,∞) : bit-length of packet k A(t), D(t): total arrivals (deliveries) till time t; A(0 - )=0 t - : just before time t X(t): queued bits at time t X(t)

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Reich’s Equation: Work- conserving schedule with output rate C t A(t) X(t) A(s) time v(t) this line has slopeC, and valueA(s) +C. (t s) att s Proof:

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Reich’s Equation: Let v be earliest time after which buffer was not empty: Hence:

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Reich’s Equation: Illustration t A(t) X(t) A(s) time b this line has slopeC, and valueA(s) +C. (t s) att s D(time)

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Reich’s Equation: Departures

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Departures and Convolutions A(s)=0 for s<0, and (t-s) + =0 for s≥t, hence… Definition: The convolution A*B of two non-decreasing functions A(t), B(t): + is: Note: compare to linear systems theory convolution: Two equivalent notations: A*B, A ⊗ B

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Convolution Operation - Visualization slopeρB(t) s t C t ρ τ t C A(t) (A * B)(t) A(τ ) + B(t τ)τ) B(t-τ) A(t)=Ct +

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Convolution Operator: Properties Convolution operator * is… Commutative: A*B=B*A Associative: (A*B)*C=A*(B*C) Distributive over min function: A*min{B,C} = = min {(A*B), (A*C)} The convolution A*B is: slopeρ τ t C A(t) (A * B)(t) A(τ ) + B(t τ)τ)

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Convolutions and the δ (delay) function The convolution A*B is: Define the delay function as: Since A(t) is non-decreasing: B(t)≥0 is causal if B(t)=0 for t<0; If A, B are causal then…

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Network Calculus: Topics 1.Introduction 2.Arrival curves Arrival curve and min-plus convolution Leaky bucket Good arrival curves are sub-additive Minimal arrival curve and min-plus deconvolution 3.Service curves Backlog, delay bounds, IntServ routers (Latency- Rate servers), Shapers

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Cumulative flows Cumulative flow R(t) F, t real (or integer) F = { x(t) | x(t) is non decreasing and x(t) = 0 for t < 0 } Models: time t bits 12567 time t bits 155.5 R 1 (t)R 2 (t) Fluid model (continuous)Packet model (left continuous)

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Arrival Curves Flow R has arrival curve (envelope) iff ∀ 0 s t: R(t) -R(s) (t-s) and (x<0)(α(x)=0) Example 1: affine arrival curve r,b (t) = r,b (t) = rt+b for t>0, r,b≥0 (Example 2, stair arrival curve kv T, : hidden next foil) time bits b slope r R(t)

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(t) = kv T, (t) = k (t+ )/T with T = period, = tolerance, k = constant packet size Characterizes flows that are periodic stream of packets of same size k (cells), that suffers a variable delay <= time bits k kv T, (t) = k (t+ )/T T- 2T- 3T- 2k 3k 4k Example 2: stair arrival curve kv T, o All packets of size k. Then R conforms to =kv T, R conforms to = r,b with r = k/T and b = k( +T)/T

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Leaky Bucket Conforming Arrivals Leaky Bucket (LB) conformance test: Parameters: rate r, bucket size b Initially: bucket 0 (empty) Upon packet arrival: If bucket+|packet| ≦ b then {bucket+=|packet|; mark packet `conforming`} else mark packet `non-conforming’; Every 1/r seconds: if bucket>0 then bucket--; R(t) r x(t) b R(t)

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Affine Arrivals are LB-Conforming Let R* be leaky-bucket conforming arrivals R* has affine arrival curve (t) = r,b (t) = rt+b for t>0, r,b≥0 If R conforms to (has envelope) (t), then R*=R All packets of R are conformant R(t) r,b r x(t) b R(t) slope r b

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GCRA (T, ) All packets (cells) of flow R are of the same size k Arrival time of nth = A n Theoretical arrival just after nth arrival is n = max(A n, n-1 ) + T If A n+1 >= n – then cell is conformant, otherwise not Example: GCRA (10,2) n123345 n-1 01121213141 A n 11116202938 ccncccnc Equivalences: R conforms to GCRA (T, ) R conforms to staircase arrival curve = kv T, R conforms to leaky bucket (r = k/T, b = k( +T)/T) R conforms to affine arrival curve = r,b

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Min-Plus Convolution Flow R has envelope (arrival curve) iff ∀ 0 s t: R(t) -R(s) (t-s) and (x<0)(α(x)=0) The min-plus convolution of R andαis: Flow R has arrival curve α,iff R ≦ R ⊗ α R has arrival curve all 0 s t, R(t) -R(s) (t-s) R(t) R(s) + (t-s) for all 0 s t R(t) inf u { R(u) + (t-u) } R R

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Min-plus convolution (cont’) Definition (f g) (t) = inf u { f(t-u) + g(u) } t f(t) g(t) (f g)(t)

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Enveloped Arrivals Process A(t) has envelope E if for every t, s (s t) holds A(t)-A(s) E(t-s) A has envelope E if and only if A A*E Envelopes are non-decreasing, non-negative If A, E are causal (A(t)=0 for t<0, E δ 0 ) then: A*E A* δ 0 = A A*E A=A*E If arrivals with envelope E pass via element with lower service curve S, define `max delay` as:

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Some properties of min-plus convolution (f g) F is associative is commutative Neutral element: 0 : f 0 = f ( 0 (t) = 0 for t = 0 and 0 (t) = for t > 0) is distributive with respect to min ( is isotone: f f’ and g g’ f g f’ g’ Functions passing through origin (f(0) = g(0) = 0): f g f g Concave functions passing through the origin: f g = f g

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Example: rate latency function T (delay function) R,T (rate-latency function) latency T R,T (t) Slope (rate) R delay T T (t) R (t)=Rt Rate R R (delay function)

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Sub-Additive Functions Recall cumulative flow functions A function f F is sub-additive iff: If flow has arrival curve f F, then it also has sub-additive arrival curve In particular, the `sub-additive closure` Focus on sub-additive arrival curves Mostly, `good` functions: sub-additive and f(0)=0

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Affine Arrivals are Sub-Additive affine arrival curve: r,b (t) = rt+b for t>0, r,b≥0 Is sub-additive (and `good`): r,b (s+t) = r(s+t)+b ≦ ≦ rs+b+rt+b = r,b (s)+ r,b (t) time bits b slope r

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Sub-additivity and arrival curves T k 2T 3T 2k 3k 4k 4T T k 2T 3T 2k 3k 4k 4T Flow R has arrival curve , the sub-additive closure, is arrival curve What is the sub-additive closure ? Use min-plus convolution

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Sub-additive Closure The sub-additive closure of an arrival curve, is a (sub-additive) arrival curve f F, f(0)=0 f ≥ f ⊗ f ≥ 0 Repeat… sequence of smaller and smaller functions: is sub-additive with (`good`) is sub-additive with f (0) = 0 = f f = f f

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Examples time bits T R,T (t)+K’ R K’ time bits T R,T (t)+K’’ R K’’ 2T 2K’ R,T (t)+K’) (2) 2T R,T (t)+K’ ’ R,T (t)+K’

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Combining Two Leaky Buckets time bits b M slope r slope p time bits b slope r IntServ Standard Traffic Specifications: T-Spec 4-tuple: (p,M, r, b) Peak rate p (link speed), packet size M `Store and forward`: entire packet `received at once’ Burst size b, long-term average rate r (u) = min (pu+M, ru+b) = (pu+M) (ru+b) Exercise: show (u) is sub-additive (and `good`)

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Combining Two Arrival Curves (1) Generalize: Flow R has arrival curves ’ ∀ 0 s t: R(t) -R(s) (t-s), R(t) -R(s) ’(t-s) Clearly: R has arrival curve ’ E.g., two affine arrival curves (leaky bucket) But sometimes there is a tighter curve!

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Combining Arrival Constraints (2) Flow R has arrival curves ’ Clearly: R has arrival curve ’ Example: t)= r,b (t), ’(t)= v T, (t) Pick r=1, b=0, T=2, =0… Is ’ sub-additive? time bits slope 1 1,0 (t) time bits 1 V 2,0 (t) [stair function] 2 3 4 46 ’ ’(2.5)=2 ’( 2)=1 ’( 0.5)=0.5

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Combining with Sub-Additive Closure Flow R has arrival curves ’ also ’ Also arrival curves: ⊗ ’ and The sub-additive closure of ’ If ’ are `good`, then ⊗ ’= time bits slope 1 1,0 (t) time bits 1 V 2,0 (t) [stair function] 2 3 4 46 ’

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Minimal arrival curve Assume we only know arrival rate R (empirically) How can we compute the (minimal) arrival curve ? The answer uses min-plus deconvolution R

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Min-plus deconvolution Ø Definition (f Ø g) (t) = sup u { f(t+u) - g(u) } t f(t) g(t) (f g)(t)

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Properties of min-plus deconvolution (f g) F in general (f f) F (f f) is sub-additive with (f f) (0) = 0 (f g) h = f (g h) Duality with : f g h f g h

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Minimal arrival curve The minimal arrival curve of flow R is R Ø R . Proof: It is an arrival curve because R(t) – R(s) = R((t-s)+s) - R(s) sup u { R((t-s)+u) - R(u) } = (R Ø R) (t-s) l If ’ is another arrival curve for flow R, then R R ’ [use duality rule: f g h f g h ] R Ø R ’. ∎

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Contents 1.Introduction 2.Arrival curves 3.Service curves, backlog, delay bounds Service curve: definition Backlog and delay bounds Shapers

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Goal of Service Curve and GR node define an abstract node model independent of a specific type of scheduler applies to real routers, which are not a single scheduler, but a complex interconnection of delay and scheduling elements applies to nodes that are not work-conserving

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Service Curve System S has (minimal/lower) service curve , if F, and for any arrival process A(t), the delivery process D(t) satisfies: D≥ A ⊗ Assuming system is full (some backlog) S A(t)D(t) t s A(s) A( ) D( ) (t)

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Service Curve System S has (minimal/lower) service curve , if F, and for any arrival process A(t), the delivery process D(t) satisfies: D≥ A ⊗ Assuming system is full (some backlog) If is continuous, for all t exists some s ≦ t such that D(t) - A(s) (t-s) [see hidden foil] (t) S A(t)D(t) t s A(s) A( ) D( )

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Equivalence of Service Curve Defs Equivalence of min-plus convolution and rates definitions Exists some 0 s t : y(t) - x(s) (t-s) y(t) x(s) + (t-s) for some 0 s t y(t) inf u { x(u) + (t-u) } y x

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Constant rate server (t)=ct Proof: take s = beginning of busy period: y(t) – y(s) = c (t-s) and y(s) = x(s) -> y(t) – x(s) = c (t-s) buffer s t0 ct t

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Rate-latency service curve Standard model for Internet (IntServ) router: (lower) service curve S(t)= Rt + ⊗ δ T E.g.: fixed rate, but (max) packet size L T=L/R in this case T bits R seconds

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WFQ Scheduler is a Latency Rate Server Notations: Bound on length of packets GPS departure time of k th packet (any queue) WFQ departure time of same packet WFQ Delay Theorem [4.3 in KMK]: GPS deliveries for non-empty queue j : Hence GPS, WFQ have lower service curves: Latency Rate Server

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Example 1: Constant-Rate Server Infinite buffer, output rate r `Fluid model` service curve S(t)=rt + (Lower) service curve S(t)=(rt + -L)=rt + * δ L / r A(t)D(t) S(t) S(t) r sloper sloper L r

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Example 2: Coder and Constant-Rate Server Coder output bursts (can be < L bits), rate r Packetizer outputs packets: rate r, max delay L /r D p (t)≥A(t-L /r), i.e. lower service curve is D(t)=(D p *(r t + )) ≥ (A*δ L/r )*(r t + ) Latency rate servers: rate r, delay (latency) d Lower service curve S(t)=δ d *(rt + )=r(t-d) + Typical `Per-Hop Behavior` service specifications A(t)D(t) D p (t) r Coder Packetizer L

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The service curve of a Greedy shaper is its shaping curve x(t) Shaper y(t) If is sub-additive and y(t) = (x (t). The service curve of a shaper is thus

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Fixed-delay node: service curve T seconds T x y 0 T T (t) Function T t

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Min-plus deconvolution Ø Definition (f Ø g) (t) = sup u { f(t+u) - g(u) } t f(t) g(t) (f g)(t)

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Tight Bound on Backlog If flow has arrival curve and node offers service curve then the backlog is at most: sup ( (s) - (s)) = ( Ø )(0) = v( , ) Proof: simple application of definitions of curves δ(s) v(,)v(,) s Recall (f Ø g) (t) = sup u { f(t+u) - g(u) }

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Tight Bound on Delay If flow has arrival curve and node has service curve then delay h( , )=sup s { δ (s)}, δ (s)= inf{ τ≥0: (s) ≦ (s+τ)} Proof: omitted. δ(s) v(,)v(,) s

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Theorem: Delay of Enveloped Arrivals Given causal arrivals A with envelope E into element with lower service curve S(t), the departures D satisfy: Namely, delay is at most Where: Proof: Since S is lower service curve: D≥A*S By definition of : Hence: A=A*E, since E is envelope of (causal) A Claim follows (from associativity of convolution *)

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Leaky Bucket via IntServ Router Flow with LB-arrivals, A(t)=b+rT Latency-rate server (e.g. router): B(t)=R(t-T) + If r ≤ R: buffer ≦ b+rT, delay ≦ T+b/R Burstiness of flow increased by rT. If r > R, the bounds are infinite. data Arrivals A(t)=b+rt b T b+rT T+b/R Latency-rate Service curve: B(t)=R(t-T) +

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Leaky Bucket via WFQ Flow with LB-arrivals, A(t)=b+rt Latency-rate server (e.g. router): S(t)=R(t-T) + If r ≤ R: buffer ≦ b+rT, delay ≦ T+b/R Burstiness of flow increased by rT. If r > R, the bounds are infinite. For WFQ: data Arrivals A(t)=b+rt b T B+rT T+b/R Latency-rate Service curve: S(t)=R(t-T) + Details: Network Calculus lecture (or [KMK/BT]

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Service Curves: not just lower… Delay often bounded by service curves: S(t) is a (lower/minimal) service curve if D≥A*S S(t) is an upper service curve if D A*S S(t) is a service curve if D=A*S S(t),S(t): non-negative, non-decreasing, causal

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The composition theorem Theorem: the concatenation of two network elements each offering (lower) service curve i offers the (lower) service curve 1 2 Holds for lower, upper service curvers

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Cascade of Service Elements Traffic passes thru several `elements` Each has upper and lower service curves Composing: Composite lower service curve: Similarly, composite upper service curve: A(t) D 1 (t)D 2 (t) S 1 ( t ) D n (t) S 1 (t) S 2 (t) S 2 (t) S n (t) S n (t)

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Cascade of Latency Rate Servers Latency rate servers: rate r, delay (latency) d Lower service curve of k th server: S k (t)= r k (t-d k ) + A(t) D 1 (t)D 2 (t) S 1 ( t ) D n (t) S 1 (t) S 2 (t) S 2 (t) S n (t) S n (t)

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R1R2 T2 = T1 Example: Cascade of Two Latency-Rate Servers (IntServ Routers) R1 T2T1+T2

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Pay Bursts Only Once D1D1 D2D2 D D b /R + T 1 + T 2 end to end delay bound is less!! D b /R + T 1 + T 2 end to end delay bound is less!! D 1 +D 2 (2b + rT 1 )/ R + T 1 + T 2 Recall, LB via router: delay ≦ T+b/R b out =b in +RT

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Regulators (aka Shapers) How to ensure envelope E? Regulator: departures envelope is E (any arrivals) A regulator with envelope E satisfies: Arrivals A(t), Departures D(t), and D(t) A(t) Departures envelope is E (any arrivals) For all s t holds D(t)-D(s) E(t-s) D D*E

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Regulators (Shapers) Lemma Def: A regulator with envelope E satisfies: Arrivals A(t), Departures D(t), and D(t) A(t) Departures envelope is E (any arrivals) For all s t holds D(t)-D(s) E(t-s) D D*E Lemma: Proof:

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Leaky Bucket Regulator / Shaper D(t)-D(s) σ+ρ(t-s) Such process is (σ,ρ)-regulated, or (σ,ρ)-process D has envelope E(t)= σ+ρt for t≥0, E(t)=0 for t<0 Or use notation: E(t)= (σ+ρt)I [t≥0] D(0)=0 for t<0 D, E are causal D=D*E Called leaky-bucket (LB) regulator / shaper σ ρ A(t) D(t) X 1 (t) X 2 (t) Source buffer Token bucket

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Leaky Bucket Shaper Operation [fluid model - ignore packetization] LB becomes empty buffer becomes empty buffer nonempty amount of data in the buffer departure process time σ LB full LB nonempty amount of tokens in LB Bucket is full

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LB Shaper: Properties, Operation Denote peak (input=output) rate by R D(t)-D(s) σ+ρ(t-s) Bucket size σ, average rate ρ Max burst (at rate R ): σR/(R-ρ) LB becomes empty buffer becomes empty buffer nonempty amount of data in the buffer time σ LB nonempty σ

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LB Shaper Theorem: D=A*E Lemma: Claim: Consider any time t. If source buffer at t is empty then: Source buffer non-empty at t bucket empty at t Let v(t) be last time before t bucket was full At t all tokens were used for departures, so: A(t): arrivals to LB shaper with envelope E(t)= σ+ρt for t>0, E(t)=0 for t 0

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Cascade of Leaky Buckets Shapers Theorem: D 2 =D 1 =A*E Associativity: D 2 =A*(E*E) E*E E, since E is causal (E(t)=0 for t<0) E E*E Holds for any subadditive E [for all s t holds E(t) E(s)+E(t-s)] E.g. for Leaky Bucket… E=E*E D 2 =A*E=D 1 A(t) D 1 (t)D 2 (t) E(t)= =(σ+ρt)I [t≥0] E(t)

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