1. Multi-resolution Resource Behavior Queries Using Wavelets Jason Skicewicz Peter A. Dinda Jennifer M. Schopf Northwestern University

2. 2 The Tension Sensor Video App Course-grain measurement Resource- appropriate measurement Fine-grain measurement Grid App … Resource Signal (periodic sampling) Example: host load

3. 3 Video Scheduling Sensor Video App Fine-grain measurements needed

4. 4 Grid Scheduling Grid App Sensor Coarse-grain measurements sufficient

5. 5 Interval Averages Sensor Application Ideal ResultAdequate Result Average over interval

6. 6 Contributions / Outline Application-sensor tension Query model to address tension Wavelets as basis for query model Promising early results Delay conundrum

7. 7 Schematic Representation of Query Model ApplicationSensor Measurements at f s samples/second Desired rate at f q samples/second Lower bandwidth used The desired rate signal is an estimate error = x – x ^ x x ^

8. 8 Query Stream + Error Application x t Δ t ΔqΔq Sensor

9. 9 Query Average + CI ApplicationSensor x tt Application gets average over this interval t now =i now  (i now  N+1)  Application wants average over this interval

10. 10 Contributions / Outline Application-sensor tension Query model to address tension Wavelets as basis for query model Promising early results Delay conundrum

11. 11 Wavelets As Basis for Query Model Natural time/frequency decomposition Provides a multi-resolution view of a resource Well known mathematical tool Invented in the ’80s, hot in ‘90s and today Linear complexity Non-stationarity, other “normal” behaviors acceptable Burrus, Gopinath, Gao, intro to wavelets and wavelet transforms: A primer Analytic enabler Prediction on different resolutions Compression of measurement streams … Queries over wavelet domain representation of signal

12. 12 Multi-resolution Views

13. 13 High Level View of a 4-level Wavelet Decomposition Resource Signal is decomposed into levels Samples at each level are at a different rate Each level captures different frequency content Corresponding inverse transform Wavelet Transform Level 1 Level 0 Level 3 Level 2 Wavelet Coefficients Sensor

14. 14 4-level Wavelet Decomposition Time-frequency Localization Level 0 1 2 3 x[n] time Frequency [0 f s /2] [f s /4 f s /2] [f s /8 f s /4] [f s /16 f s /8] [0 f s /16] Δ f s =1/Δ

15. 15 Example Decomposition of Host Load Lossless representation of resource signal

16. 16 Computing Wavelet Coefficients Streaming operation –Number of levels, M, chosen arbitrarily –Amortized work per sample: O(1) –O(n) for n samples Block by block operation –Block of samples, n=2 k –Levels, M = lg(n) + 1 –Circular convolution over block, O(n)

17. 17 Proposed System Wavelet Transform Level 0 Sensor Inverse Wavelet Transform Application Level M-1 Level M Level 0 Level L Network Application receives levels based on its needs StreamInterval

18. 18 Multi-resolution Views Using 14 Levels

19. 19 Wavelet Compression Gains, 14 Levels Typical appropriate number of levels for host load, error < 20%

20. 20 Contributions / Outline Application-sensor tension Query model to address tension Wavelets as basis for query model Promising early results Delay conundrum

21. 21 Offline Analysis System

22. 22 Load Traces DEC Unix 5 second exponential average –1 Hz sample rate –Traces collected in August 1997 AXP0-PSC – Interactive machine with high load AXP7-PSC – Batch machine Sahara-CMU – Large-memory compute server Themis-CMU – Desktop workstation Windows 2000 percentage of CPU –1Hz sample rate –Trace collected in May 2001 Tlab-03-NU – Desktop, teaching lab machine

23. 23 Testcases Stream Queries One million samples per trace Interval Queries 2, 8, 32, 128, 512, 2048, 8192 second intervals 1000 randomized queries per interval length per trace

24. 24 Performance Evaluation Streaming queries metrics –Error variance –Error histograms –Error mean –Energy in error auto-covariance Interval query metrics –Error variance –Error histograms –Error mean Error mean ~ 0 for all evaluations

25. 25 Streaming Queries, Relative Error Variance Fewer than 1% of coefficients, error < 20%

26. 26 Streaming Queries, Error Histogram at Level 6 Errors follow a near-Gaussian distribution

27. 27 Interval Queries, Error Variance Error variance approaches zero as interval increases

28. 28 Interval Queries, Error Histograms at Level 5 Distributions not always Gaussian

29. 29 Contributions / Outline Application-sensor tension Query model to address tension Wavelets as basis for query model Promising early results Delay conundrum

30. 30 Block By Block System Delay Wavelet Transform Inverse Wavelet Transform Block x[n]x r [n] … M Levels n samples in block n samples in block Sample Acquisitions Wavelet transform Inverse transform time Samples delayed by block size ^

31. 31 Streaming System Delay, Example with Length 4 Wavelets (D4), 4 Levels High levels delayed waiting for low frequency computations, output delayed by high order filter x[n] Length 22 Length 10 Length 4 Length 22 Length 10 Length 4 x r [n-d] Delay K1 Delay K2 Level 0 Level 1 Level 2 Level 3

32. 32 Delay Conclusions System implementation Delay must be taken into account Prediction may help reduce streaming delay Application scheduling Fine-grain apps more sensitive to delay Coarse-grain apps less sensitive to delay Suggestions? We are working on a solution!

33. 33 Related Work Database queries over wavelet coefficients –Shahabi, et al [SSDBM 2000] –Chakrabarti, et al [VLDB 2000] –Vitter, et al [CIKM ‘98, SIGMOD ‘99] Network traffic analysis and modeling –Ribeiro, et al [IEEE INFOCOM 2000] –Riedi, et al [IEEE DSPCS ’99] –Feldman, et al [SIGCOMM ’98] Wavelet theory –Daubechies [Ten Lectures on Wavelets ‘92, SIAM] –Mallat [IEEE Trans. on Pattern Analysis and Machine Intelligence, ’89]

34. 34 Conclusions Application-sensor tension Query model to address tension Wavelets as basis for query model Promising early results Delay conundrum

35. 35 Future Work Wavelets are an enabler of other techniques –Prediction over wavelet coefficients Possibility of better results Can reduce system delay –Further compression through processing –Adaptive decompositions based on resource Looking at other resource streams RPS implementation

36. 36 Contact Information Webpage http://www.cs.northwestern.edu/~jskitz Email address jskitz@cs.northwestern.edu Load traces and tools http://www.cs.northwestern.edu/~pdinda/LoadTraces Matlab scripts Available by request (jskitz@cs.northwestern.edu)

37. 37 Frequency Information Vs. Rate Input Signal, x[n]Decomposition f(Hz) f s /2 f(Hz) f s /2 f s /8 f s /4 f s /16 0123 Levels Frequency information retained = f s /2 Measurement rate, f s Q: Why is this true? A: The Nyquist Criterion- sampling theory

38. 38 Wavelet Transform, 1 Stage Level 0 y l [n] 2 2 Level 1 HPF LPF y h [n] x[n] LPF, HPF FIR filters h[n] x[n] y[n] Downsampler 2 y[n] c[k],for all k

39. 39 Increasing Stages, Mallat’s Tree Algorithm x[n] Level 0 Level 1 HPF LPF HPF LPF HPF LPF Level M-1 Level M Stages can be arbitrarily increased

40. 40 Frequency Response Filters must be even order for PR Other special properties to retain PR The filters are order N=8 (D8 wavelet) HPFLPF

41. 41 Reconstruction From the Wavelet Coefficients, 1 Stage Upsampler LPF, HPF time reversed filters, same response y[n] = c[k] 2 Level 0 2 Level 1 HPF LPF x r [n] 2 + ^

42. 42 Reconstruction From Multiple Stages, The Inverse Wavelet Transform Level 0Level 1 x r [n] HPF + LPF HPF + LPF HPF + LPF Level M-1 Level M ^ Reconstructed signal is exactly the resource

43. 43 Determined by accuracy constraints Determined by what levels are available Determined by the rate (f q ) at which measurements are requested: Q: How are the number of levels determined? Answers:

44. 44 Example, Choosing Levels f(Hz) f s /2 f s /8 f s /4 f s /16 0123 Levels M = 4 levels f q = f s / 6 Solution: L = 2: Equation Satisfied! Levels 0, 1 and 2 coefficients returned

45. 45 Streaming Query Tradeoffs Measurement rate, f q high –Lower error variance –Higher communication costs Measurement rate, f q low –Higher error variance –Very low communication costs Wavelet approach yields accuracy at low rates

46. 46 Interval Query Tradeoffs Interval length N long –Less dynamic rate –Tighter confidence intervals Interval length N short –More dynamic rate –Wider confidence intervals Rate, f q high –Shorter interval length –Tighter confidence intervals Rate, f q low –Longer interval length –Wider confidence intervals Confidence interval (c) provides flexibility

47. 47 Streaming Queries, Energy in Auto-covariance Error becomes uncorrelated as levels added

48. 48 Interval Queries, Error Mean (32 seconds) Error mean is zero at 8 levels, 3% of coefficients

49. 49 Interval Queries, Error Mean (512 seconds ~ 8½ minutes) As interval increases, need fewer levels