Network Calculus & Related Models with Applications

Network Calculus & Related Models with Applications by George I. Stassinopoulos Prof. NTUA stassin@cs.ntua.gr Berlin 14.07.06

Network Calculus & Related Models with Applications • Basic Calculus • Modelling Concepts – Traditional • Modelling Concepts – Alternatives • Applications

Basic Calculus Network Calculus ‘Deterministic Queuing Theory’ ‘A Theory for Deterministic Queuing Systems’ “for the Internet”

Network Calculus no probabilities no statistics no queuing theory non decreasing time functions flows RATEs / BUFFERs Interplay

Arrival Curve α(t)Concerns the input Places restrictions on the input’s burstiness by checking it against a ‘leaky bucket’. Input i(t) is checked againstα(t) Service Curve β(t)Concerns the system Guarantees a minimal forwarding capability for the system (‘server’). Output o(t) is bounded from below by an expression involving the input i(t) and β(t)

Arrival & Service Curves i(t) &β(t) α(t) System β(t) i(t) o(t) • GOAL: • Make output at every time as large as possible approaching the input • which is (roughly) equivalent to minimizing backlog • which is (roughly) equivalent to minimizing delay BUT: this depends on capacity of system (β) & smoothness of input (α)

Arrival & Service Curves i(t) smoothness α(t) o(t) α(t) system β(t) α(t) backlog delay t • GOAL: • Make output at every time as large as possible approaching the input • which is (roughly) equivalent to minimizing backlog • which is (roughly) equivalent to minimizing delay BUT: a theory cannot ‘make’ anything – it can only analyze so instead give BOUNDS which check or guide the design

MATHEMATICAL FRAMEWORK i(t) o(t) delay backlog t All functions are flows counted cumulatively, i.e. Nondecreasing functions of time t Starting at t=0 from 0 Very often convex (usually for β(t)) concave (usually for α(τ))

MATHEMATICAL FRAMEWORK ????? Non-causal (well known) the Ordinate (y-) value is a full address for each bit !!!! o(t) no backlog no delay i(t) delay backlog t

PHILOSOPHICAL DISGRESSION the ordinate (y-) value is a full address for each bit !!!! i(t) a mathematically trivial observation t πάντα ρεῖ Herakleitos, 6th century B.C. everything flows (against time) flow … moreover ‘you can notwash yourself in the river twice with the same water’ Herakleitos, again uniqueness

MATHEMATICAL FRAMEWORK flow f(t) t t flow as bit, packet density, traffic intensity t t flow f(t) always nondecreasing (because cumulative) but its slope (rate of increase) can be increasing - convex fdecreasing - concave f We model flows systems, via flows (service curve, response to standard inputs)

MATHEMATICAL FRAMEWORK flow f(t) t t Initial Delay does not harm Burst does not harm convexityconcavity no more a mathematical abstraction b t t T Passing a pointer (some nanosecs) can mean a new job of some Gbits still f(0)=0

Basic Goal Involving Arrival & Service Curves Exploiting the assurances on the input’s burstiness i.e. for a certain α(t) AND the guarantee on the system’s forwarding capability i.e. for a certain β(t) Derive bounds for BACKLOG, DELAY and the SMOOTHNESS of the output

f(t) Arrival Curve Concept vol b f(t) opening r Violation !!!! α(t) leaky bucket test t slope r flow f(t) must be below all functions : b t f(t’) + α(t-t’) for all t’ ≥ 0 t ‘ α constraints the growth of f ’ Arrival Curve α(t) (b,r) f(t)≤ f(t’) + α(t-t’)

Arrival Curve Concept α t Question: Give all flows being constrained by arrival curveα(t) = α t t Asw: All concave flows ‘ α constraints the growth of f ’ f(t)≤ f(t’) + α(t-t’)

f(t) Arrival Curve Concept Concave !!! vol b1 α(t) opening r1 r3 b1+b2+b3 vol b2 r2 r1 opening r2 t Arrival Curve α(t) {(bi,ri)} vol b3 opening r3 More elaborate leaky bucket test for f(t)

Arrival Curve Concept Concave !!! Convex subgraph same f(t)tested 3x (against 3 hyperplanes) h3 r3 α(t) r2 h2 r1 h1 t Arrival Curve α(t) {(bi,ri)} vol b1 vol b2 vol b3 A convex set as a intersection of linear half spaces defined by hyperplanes r1 r2 r3 …and another elaborate leaky bucket test for f(t)

f(t) = f(t’) + f (t-t’) The flow coincides with the arrival curve by which it is constrainedf(t) = α(t) f(t) = α(t) The flow f(t) fills to the limit all test buckets determined by α(t) =f(t) f(t)is greedy some other f constrained by α t The flow f(t) is sub-additive and f(0)=0 f(t+s) ≤ f(t) + f(s) for all t,s ≥ 0 α is a ‘good’ function α = αα α = αα α = α(sub-additive closure)

Service Curve Concept i o System R t T output is above lower envelope of: Service Curve β(t) i(t’) + β(t-t’) for all t’ ≥ 0 t

i o System c < R i = ct R output is above lower envelope of: t T t ct’+ β(t-t’) for all t’ ≥ 0 Service Curve β(t) i = ct c > R above !!! BACKLOG ??

i o i = ct System c > R R output is above lower envelope of: t T t ct’+ β(t-t’) for all t’ ≥ 0 Service Curve β(t) Actual Output of System is Server always busy

i o System c < R i = ct R output is above lower envelope of: t T t ct’+ β(t-t’) for all t’ ≥ 0 Service Curve β(t) Actual Output of System is slope c cT/(R-c)

Service Curve Concept i o System R t T output is above lower envelope of: Service Curve β(t) t i(t’) + β(t-t’) for all t’ ≥ 0 However: Beware of busy / idle periods in the system (‘server’) ! for t0 beginning of last idle period o(t) ≥i(t0) + β(t-t0) simpler !!

Service Curve Concept i o System i(t) !!!! R t t0 T Service Curve β(t) Explain: for constant rate server and t0beginning of last idle period……. o(t) ≥i(t0) + β(t-t0) Draw answer above !!!!

Arrival vs. Service Curve An arrival curve concerns the growth property of a flow (usually an input) (remember the bucket(s) test) α(t) A service curve concerns the ability of a system to sustain an output as long as there is backlog. A service curve concerns a particular system’s response to any input β(t) i o System

CONCAVE !! Service Curve i o System i(t) … and then levels off !! A low latency server! t t0 Service Curve β(t) Responds massively to initially accumulated traffic serving it at the highest rate… The input must be steep enough in order to provide volume to ‘be served’

CONCAVE !! Service Curve i o System i(t) backlog A low latency server! t Service Curve β(t) … in this case the server is not able to sustain the input rate and the backlog grows beyond any bound

CONVEX !! Service Curve i o System i(t) Theory says: o(t) is above red line It actually meets i(t) high throughput server! t Service Curve β(t) … in this case the server has initially slow response not caring for the delay caused – eventually it clears the backlog

i o System i T δ(t) T output is above lower envelope of: t T t i(t’)+ δ(t-t’) for all t’ ≥ 0 Service Curve But also: o(t) ≥ i(t0) + δ(t-t0), t0 idle point Every t is a t0 !!!! - Infinite rate server CONVEX !! Service Curve Pure (lossless) Delay

CONVEX !! Service Curves δ(t) pure delay t high throughput server! T rate latency server t slope R delay T

from Arrival Curve Concept f(t)≤ f(t’) + α(t-t’) ‘ α constraints f ’ Set t’ -> s f -> g α-> f Generalization (fg)(t) = inf { f(t-s) + g(s) / 0 ≤ s ≤ t } Min-Plus Convolution between f & g

from Arrival Curve Concept f(t)≤ f(t’) + α(t-t’) Flow f is constrained by arrival curve α for all t’ ≤ t is written now f(t) ≤ (f α)(t) f is bounded from above by the Min-Plus Convolution between f & α

from Arrival Curve Concept f(t)≤ f(t’) + α(t-t’) …. when is a concave flow constrained by itself, taken as arrival curve ? for all t’ ≤ t Concave flow t

from Arrival Curve Concept α(t – t2) f(t)≤ f(t’) + α(t-t’) α(t – t1) for all t’ ≤ t α(t) For f = a & concave t t1 t2 α(t) ≤ (αα)(t) Trivial, but useful !!! A concave α is always constrained by ‘arrival curve’ α

Min-Plus Convolution Interpretation Go back on f by s and replace by a piece from the beginning of g f(t-s) g(s) Do that for the s giving min (inf) value s s t (f g)(t) = inf { f(t-s) + g(s) } 0 ≤ s ≤ t

Min-Plus Convolution Interpretation (fg)(t) f(t-s) g(s) s s t (fg)(t) = inf { f(t-s) + g(s) } 0 ≤ s ≤ t

Min-Plus Convolution Properties • (fg)(t) = inf { f(t-s) + g(s) } • ≤ f(t) set s = 0 • ≤ g(t) set s = t • = (gf)(t) set t’ = t-s (commutativity) • = min {f(t), g(t) } if f, g linear 0 ≤ s ≤ t

Min-Plus Convolution Properties (fg)(t) = inf { f(t-s) + g(s) / 0 ≤ s ≤ t Take f and g convex !!! f g Slope non decreasing equal slope Slopes Rule f g Take successive slopes in increasing order and put these in succession !!!

Modelling Concepts – Traditional Berlin 14.07.06

i = ct i o System β(t) convex ‘File Server’ output is above lower envelope of: t concave β(t) t i(t’) + β(t-t’) for all t’ ≥ 0 Low Latency i = ct t Service Curve β(τ)

i α(s) i(t) output is above lower envelope Backlog o(t) ≥ inf {i(t-s)+ β(s) / 0 ≤ s ≤t} i(t) – o(t) ≤ i(t) - inf { i(t-s)+ β(s) } 0 ≤ s ≤ t i(t) – o(t) ≤ sup { i(t) - i(t-s)- β(s) } BOUNDS 0 ≤ s ≤ t α & β combined! i(t) – o(t) ≤ sup { α(s) – β(s)} 0 ≤ s ≤ t

i(t) – o(t) ≤ sup { α(s) – β(s)} 0 ≤ s ≤ t o(t) ≥ i(t) β(s) 1 Input / output relationship two out of few results justifying the term ‘calculus’ β 2 β1 β2 Systems in series β= β1β2

Backlog A system offering a service curve β and fed with a flow constrained by arrival curve α has backlog at t bounded by sup { α(s) – β(s) / 0 ≥ s ≤ t} α BOUNDS β sup over t t backlog increasing with time !!! !!! BOUND for …………. uncertainty of our theory while working far into the future i(t) – o(t) ≤ sup { α(s) – β(s) / 0 ≥ s ≤ t}

Backlog & Delay vertical horizontal i(t) BOUNDS o(t) backlog the t1-th bit at the input delay t2-t1 is the delay at time t1 appears here at the output t t1 t2

i(t) α(t) o(t) β(t) t τ s + τ Delay d(t) BOUNDS δ(s) = inf {τ≥ 0 / α(s) ≤β(s+τ) } virtual delay of hypothetical system, input α(t(), output β(t) find h(α,β) sup (over s) of all values δ(s) For an input, constrained by αand traversing a system offering β d(t) ≤ h(α,β)

Output Smoothness i o System BOUNDS Input i is constrained by arrival curveα System offers service curveβ then Output o is constrained by ‘arrival curve’ α β min-plus deconvolution

What is min-plus deconvolution ? … in some sense the opposite of min-plus convolution ? min-plus deconvolution (fg)(t) = sup { f(t + u) - g(u) } u ≥ 0 MIN-PLUS DECONVOLUTION How much can f grow over g after a time interval t ? min-plus convolution (fg)(t) = inf { f(t - s) + g(s) } 0 ≤ s ≤ t

Output Smoothness i o Output o is constrained by ‘arrival curve’ αβ System with α (αβ)(t) = sup {α(t + u) - β(u) } BOUNDS u ≥ 0 sup { α(u) – β(u)} β (αβ)(0) = backlog 0 ≤ u u giving sup for t=0 easily explainable

Output Smoothness i o Output o is constrained by ‘arrival curve’ αβ System with α (αβ)(t) = sup {α(t + u) - β(u) } BOUNDS u ≥ 0 β (αβ)(0) = backlog (αβ)(t) …… for increasing t Compare what arrives t secs ahead with what the system can output ‘arrival curve’ of output backlog

Backlog bound sup { α(s) – β(s) } 0 ≤ s ≤ t Delay bound d(t) ≤ h(α,β) Output has ‘arrival curve’αβ BOUNDS RECAP min-plus deconvolution …. i.e. the defining role of arrival & service curve !!!

Network Calculus & Related Models with Applications