1. Throughput of Internally Buffered Crossbar Switch Saturday, February 20, 2016 Mingjie Lin mingjie@stanford.edu www.stanford.edu/~mingjie

2. 2 Contents 1. Motivation  High throughput performance crossbar switch  What is the impact of crosspoint buffer on throughput of crossbar switch? 2. Problem Statement and Notations  The structure of a internally buffered crossbar switch (IBCS)  Two cases: 1. With blocking, 2. Without blocking  Markov Chain model 3. Analysis approach 4. Results summary

3. 3 Background Switching Fabric IQOQ CQ

4. 4 Motivations Classic results: ~58.3%(blocking) and ~63%(non-blocking) throughput for IQ crossbar switch

5. 5 Motivations What will happen to the throughput if we add Crosspoint buffer?

6. 6 Contents 1. Motivation  High throughput performance crossbar switch  What is the impact of crosspoint buffer on throughput of crossbar switch? 2. Problem Statement and Notations  The structure of a internally buffered crossbar switch (IBCS)  Two cases: 1. With blocking, 2. Without blocking  Markov Chain model 3. Analysis approach 4. Results summary

7. 7 The structure of an internally buffered crossbar switch (IBCS) Input Traffic: i.i.d uniform Bernoulli type, independent at each input. Scheduling Algorithm (2 phases in 1 time slot): 1. Buffer In Phase: For each input queue i, each HOL packet goes to its destined crosspoint buffer cell if it is vacant. 2. Buffer Out Phase: For each output port j, randomly pick one cell from all occupied crosspoint buffer cells, and output its packet.

8. 8 Throughput Analysis 2 Cases: 1. Non-Blocking Mode 2. Blocking Mode

9. 9 Throughput Analysis 2 Cases: 1. Non-Blocking Mode 2. Blocking Mode Idea:  Using Markov Chain to model the crossbar switch behavior.

10. 10 Notation Internal buffer cell; Input queue at input port I; For any column of buffer cells, the probability of having k packets in total at time n; state transition probability of Markov chain model.

11. 11 Contents 1. Motivation  High throughput performance crossbar switch  What is the impact of crosspoint buffer on throughput of crossbar switch? 2. Problem Statement and Notations  The structure of a internally buffered crossbar switch (IBCS)  Two cases: 1. With blocking, 2. Without blocking  Markov Chain model 3. Analysis approach 4. Results summary

12. 12 Observation 1. Symmetry: a) traffic b) switching fabric structure

13. 13 Observation 1. Symmetry: a) traffic b) switching fabric structure 2. During each time slot, if there is at least 1 packet in B *,j, then there will be a packet to output

14. 14 Observation 1. Symmetry: a) traffic b) switching fabric structure 2. During each time slot, if there is at least 1 packet in B *,j, then there will be a packet to output 3. Saturation Throughput:

15. 15 Markov Chain

16. 16 Derivation

17. 17 Key Equation Total probability:

18. 18 Key Equation Total probability:

19. 19 Key Equation (cont.) N linear equations:

20. 20 IBCS without blocking Solution of transition probability:

21. 21 IBCS without blocking (cont.) Solve those N linear equations, we can compute through for any N.

22. 22 IBCS without blocking (cont.) Solve those N linear equations, we can compute through for any N. Question: what happens to throughput if N goes to infinity?

23. 23 IBCS without blocking (cont.) We know:

24. 24 IBCS without blocking (cont.) We know: when

25. 25 IBCS without blocking (cont.) Therefore:

26. 26 IBCS without blocking (cont.) Add them up: finally:

27. 27 IBCS without blocking (cont.) Which leads to:

28. 28 IBCS with blocking Markov chain model, however, state space too large to manage

29. 29 IBCS with blocking Markov chain model, however, state space too large to manage What is key difference between “with blocking” and “without blocking”?

30. 30 IBCS with blocking Markov chain model, however, state space too large to manage What is key difference between “with blocking” and “without blocking”? What is thoughput if N goes to infinity?

31. 31 IBCS with blocking (cont.)

32. 32 IBCS with blocking (cont.)

33. 33 IBCS with blocking (cont.) when Therefore: finally:

34. 34 Contents 1. Motivation  High throughput performance crossbar switch  What is the impact of crosspoint buffer on throughput of crossbar switch? 2. Problem Statement and Notations  The structure of a internally buffered crossbar switch (IBCS)  Two cases: 1. With blocking, 2. Without blocking  Markov Chain model 3. Analysis approach 4. Results summary

35. 35 Results Summary

36. 36 Results Summary (cont.)

37. 37 Results Summary (cont.) 1. Crosspoint buffer cells have a significant impact on throughput of crossbar switch Symmetry.

38. 38 Results Summary (cont.) 1. Crosspoint buffer cells have a significant impact on throughput of crossbar switch Symmetry. 2. Without crosspoint buffer, throughput will decrease while N increases, the opposite is true for ICBS.

39. 39 Results Summary (cont.) 1. Crosspoint buffer cells have a significant impact on throughput of crossbar switch Symmetry. 2. Without crosspoint buffer, throughput will decrease while N increases, the opposite is true for ICBS. 3. When N goes infinity, throughput of crossbar switch without crosspoint buffer will converge to ~63% without HOL blocking, but ICBS’s will converge to 100%.

40. 40 Results Summary (cont.) 1. Crosspoint buffer cells have a significant impact on throughput of crossbar switch Symmetry. 2. Without crosspoint buffer, throughput will decrease while N increases, the opposite is true for ICBS. 3. When N goes infinity, throughput of crossbar switch without crosspoint buffer will converge to ~63% without HOL blocking, but ICBS’s will converge to 100%. 4. When N goes infinity, throughput of crossbar switch without crosspoint buffer will converge to ~58% without HOL blocking, but ICBS’s will converge to 100%.

41. 41 Thank you!