Design and Performance Analysis of Buffers: a Constructive Approach

1. Design and Performance Analysis of Buffers:a Constructive Approach Rudolf H. Mak Dept. of Mathematics and Computer Science Technische Universiteit Eindhoven P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-mail: R.H.Mak@tue.nl

2. Baroque Buffer

3. Design issues • Functional correctness • Preservation of order • Behavioral correctness • Absence of deadlock • Performance • Cycle time, throughput • Latency • Occupancy

4. Constructive Approach • Basic building blocks • 1-place buffer • 2-way split • 2-way merge • Construction methods • Serial composition • Wagging composition • Multi-wagging

5. Functional Specification A functional specification of a system spe-cifies each output stream as a transforma-tion of (a combination of) its input streams. • Basic stream transformers • Take operator • Drop operator • Stream transformer calculus • Computation rules

6. Baroque Buffer (functional spec)

7. Sequence Functions • map the events of a system onto discrete time-slots • are used to: • establish absence of deadlock • analyze system performance • are constructed using: • a fixed set of basic sequence functions • a set of sequence function transformers

8. Baroque Buffer (sequence function)

9. Capacity and I/O-distance • The capacity of a buffer is the sum of the number of variables of each of its compo-nents • The i/o-distance of a buffer is the minimal number of variables visited by any value on its passage through the buffer

10. Cycle Time Given a sequence function s we define • The individual cycle time by • The average cycle time by

11. Latency Given a sequence function s we define • The individual latency by • The average latency by

12. Occupancy Given a sequence function s we define • The instantaneous occupancy by • The average occupancy by

13. Queuing Formula (Little) Let s be a sequence function of a buffer with finite average latency, occupancy, and cycle time. Then

14. -bounds Let X be a buffer with capacity  and i/o-distance . Then for every sequence function  for X with average cycle time , the average occupancy  is bounded by

15. Optimality Problem Given a pair of values (k, d), design the optimal buffer with capacity k and i/o-distance d, where optimal means that the occupancy can attain both bounds.

16. Optimal Buffer Families • Linear buffers • Tree-like buffers • Rectangular buffers • Square buffers

17. Rectangular Buffer (5x3)

18. Ebergen’s Square FIFO J. Ebergen, Squaring the FIFO in GasP, Async 2001

19. Ebergen’s Ring Experiment • Capacity: 74 • I/o-distance: 34 • Theoretical bounds: 17    57 • Experimental bounds: 23    63

20. Conclusions • Simple calculus to verify functional correctness • Accurate performance analysis based on sequence functions • Rectangular buffers are (, )-optimal

22. 1-place Buffer

23. 2-way Split

24. 2-way Merge

25. Serial Composition

26. Wagging Composition

27. Multi-wagging

28. Take Operator For integers  and l, 0kl, and an infinite stream of values A, we define the postfix-operator by “from A take every k-th out of l values”

29. Drop Operator For integers  and l, 0kl, and an infinite stream of values A, we define the postfix-operator by “from A drop every k-th out of l values”

30. Computation Rules • Drop-take rule • Take-take rule • Complement rule

31. Linear Buffers • Capacity  = n • I/o-distance  = n

32. Capacity  = 32n-1 2 I/o-distance  = 2n  1 Tree-like Buffers

33. Rectangular Buffers • Capacity  = mn • I/o-distance  = m + n  1

34. SF-construction 2 2,4

35. Multi-split and Multi-merge