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

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

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

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

Network Calculus’s System Theory 5

Min-Plus Linear Operators 6

Min-Plus Residuation Theorem 7

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

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*

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

MORE EXAMPLES 2. 11

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)

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

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

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

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

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)

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

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

Example 20 R(t-D) Frame # * 10 6 R    (    )(t-D)    (t) a possible R’(t)

Minimum Playback Delay 21

22  t  D = 435 ms  t  D = 102 ms

The Perfect Battery 23

System Equations for the Perfect Battery 24

System Equations 25

System Equations 26

TIME VERSUS SPACE 3. 27

The Residuation Theorem is a Space Method 28

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

The Time Method for Linear Problems 30

Perfect Battery 31

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

Thank You… 33