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THE SYSTEM THEORY OF NETWORK CALCULUS J.-Y. Le Boudec EPFL WoNeCa, 2012 Mars 21 1

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Contents 1.Network Calculus’s System Theory and Two Simple Examples 2.More Examples 3.Time versus Space 2

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The Shaper shaper: forces output to be constrained by greedy shaper stores data in a buffer only if needed examples: constant bit rate link ( (t)=ct) ATM shaper; fluid leaky bucket controller Pb: find input/output relation 3 fresh traffic RR* shaped traffic -smooth

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A Min-Plus Model of Shaper 4 fresh traffic Rx shaped traffic -smooth

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Network Calculus’s System Theory 5

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Min-Plus Linear Operators 6

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Min-Plus Residuation Theorem 7

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Application to Shaper 8 (1) x x (2) x R (1) x x (2) x R fresh traffic RR* shaped traffic -smooth

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Variable Capacity Node node has a time varying capacity µ(t) Define M(t) = 0 t (s) ds. the output satisfies R*(t) R(t) R*(t) -R*(s) M(t) -M(s) for all s t and is “as large as possible” 9 fresh traffic (t) RR*

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Variable Capacity Node 10 fresh traffic (t) RR* R*(t) R(t) R*(t) -R*(s) M(t) -M(s) for all s t

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MORE EXAMPLES 2. 11

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A System with Loss [Chuang and Cheng 2000] node with service curve (t) and buffer of size X when buffer is full incoming data is discarded modelled by a virtual controller (not buffered) fluid model or fixed sized packets Pb: find loss ratio 12 fresh traffic R(t) buffer X Loss L(t) R’(t) R*(t)

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A System with Loss 13 fresh traffic R(t) buffer X Loss L(t) R’(t) R*(t)

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Bound on Loss Ratio 14 fresh traffic R(t) buffer X Loss L(t) R’(t) R*(t)

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Analysis of System with Loss 15 fresh traffic R(t) buffer X Loss L(t) R’(t)

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Analysis of System with Loss 16 fresh traffic R(t) buffer X Loss L(t) R’(t)

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Optimal Smoothing [L.,Verscheure 2000] Network + end-client offer a service curve to flow R’(t) Smoother delivers a flow R’(t) conforming to an arrival curve . Video stream is stored in the client buffer, read after a playback delay D. Pb: which smoothing strategy minimizes D? 17 R(t-D)R’(t)R*(t) Network Smoother ß(t) Video display B Video server R(t)

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Optimal Smoothing, System Equations 18 R(t-D)R’(t)R*(t) Network Smoother ß(t) Video display B Video server R(t)

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Optimal Smoothing, System Equations 19 R(t-D)R’(t)R*(t) Network Smoother ß(t) Video display B Video server R(t)

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Example 20 R(t-D) -50 0 50 100 150 200 250 300 350 400 450 Frame # 5432154321 * 10 6 R ( )(t-D) (t) a possible R’(t)

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Minimum Playback Delay 21

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22 t D = 435 ms 100200300400 2000 4000 6000 8000 10000 100200300400 10 20 30 40 50 60 70 t D = 102 ms

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The Perfect Battery 23

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System Equations for the Perfect Battery 24

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System Equations 25

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System Equations 26

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TIME VERSUS SPACE 3. 27

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The Residuation Theorem is a Space Method 28

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The Shaper, Time Method 29 (1) x x (2) x R (1) x x (2) x R fresh traffic RR* shaped traffic -smooth

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The Time Method for Linear Problems 30

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Perfect Battery 31

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Conclusion Min-plus and max-plus system theory contains a central result : residuation theorem ( = fixed point theorem) Establishes existence of maximum (resp. minimum) solutions and provides a representation Space and Time methods give different representations 32

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Thank You… 33

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