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

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

3. 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

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

5. Network Calculus’s System Theory 5

6. Min-Plus Linear Operators 6

7. Min-Plus Residuation Theorem 7

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

9. 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*

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

11. MORE EXAMPLES 2. 11

12. 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)

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

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

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

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

17. 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)

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

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

20. 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)

21. Minimum Playback Delay 21

22. 22  t  D = 435 ms 100200300400 2000 4000 6000 8000 10000 100200300400 10 20 30 40 50 60 70  t  D = 102 ms

23. The Perfect Battery 23

24. System Equations for the Perfect Battery 24

25. System Equations 25

26. System Equations 26

27. TIME VERSUS SPACE 3. 27

28. The Residuation Theorem is a Space Method 28

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

30. The Time Method for Linear Problems 30

31. Perfect Battery 31

32. 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

33. Thank You… 33