Multi-resolution Resource Behavior Queries Using Wavelets

Multi-resolution Resource Behavior Queries Using

Wavelets

Jason Skicewicz

Peter A. Dinda

Jennifer M. Schopf

Northwestern University

2

The Tension

Sensor

Video App

Network

Course-grain measurement

Resource-appropriate

measurement

Fine-grain measurement

Grid App

…

Resource Signal (periodic sampling)Example: host load

3

Video Scheduling

Sensor

Network

Video App

Fine-grain measurements needed

4

Grid Scheduling

Network

Grid AppSensor

Coarse-grain measurements sufficient

5

Interval Averages

Sensor

Network

Application

Ideal Result Adequate Result

Average over interval

Average over interval

6

Contributions / Outline

• Application-sensor tension

• Query model to address tension

• Wavelets as basis for query model

• Promising early results

• Delay conundrum

7

Schematic Representation of Query Model

Network

ApplicationSensor

Measurements atfs samples/second

Desired rate atfq samples/second

Lower bandwidthused

The desired rate signal is an estimateerror = x – x^

xx ^

8

Query

Stream + Error

Application

2ˆ,ˆ eiq xfyStreamQuer

/1sf qqf /1

x

tΔ t

x̂

Δq

sq ff

Sensor

9

Query

Average + CI

Application

NavgNavgNavgfcNeryIntervalQu hlq ,,,,

Sensor

x

t t

x̂

Application gets average over this interval

tnow=inow(inowN+1)

Application wants average over this interval

10

Contributions / Outline

• Application-sensor tension

• Query model to address tension

• Wavelets as basis for query model

• Promising early results

• Delay conundrum

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

Multi-resolution Views

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

WaveletTransform

Level 1

Level 0

Level 3

Level 2Wavelet

Coefficients

Sensor

14

4-level Wavelet DecompositionTime-frequency Localization

Level

0

1

2

3

x[n]

time

Frequency

[0 fs/2]

[fs/4 fs/2]

[fs/8 fs/4]

[fs/16 fs/8]

[0 fs/16]

Δ fs=1/Δ

15

Example Decomposition of Host Load

Lossless representation of resource signal

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=2k

– Levels, M = lg(n) + 1– Circular convolution over block, O(n)

17

Proposed System

WaveletTransform

Level 0

Sensor

InverseWavelet

Transform

Application

Level M-1

Level M

Level 0

Level L

Network

Application receives levels based on its needs

Stream Interval

18

Multi-resolution Views Using 14 Levels

19

Wavelet Compression Gains, 14 Levels

Typical appropriate number of levels for host load, error < 20%

20

Contributions / Outline

• Application-sensor tension

• Query model to address tension

• Wavelets as basis for query model

• Promising early results

• Delay conundrum

21

Offline Analysis System

Seg. 1 Seg. 2 Seg. PWavelet

Transform

Host LoadTraces

8192 SecondSegments

knx ,

knx ,ˆkne ,

14 Levels

knc ,

ChooseSubset of

Levels

L LevelsL Levels

knx ,

+-

Reconstructionknx ,ˆ

Streaming Error Interval Error

+-

Ne

NxE

Interval

2,8,32,...,8192

Average NxE ˆ

NxE ˆ

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

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

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

Streaming Queries, Relative Error Variance

Fewer than 1% of coefficients, error < 20%

26

Streaming Queries, Error Histogram at Level 6

Errors follow a near-Gaussian distribution

27

Interval Queries, Error Variance

Error variance approaches zero as interval increases

28

Interval Queries, Error Histograms at Level 5

Distributions not always Gaussian

29

Contributions / Outline

• Application-sensor tension

• Query model to address tension

• Wavelets as basis for query model

• Promising early results

• Delay conundrum

30

Block By Block System Delay

WaveletTransform

InverseWavelet

TransformBlock Block

x[n] xr[n]…

M Levels

n samplesin block

n samplesin block

Sample AcquisitionsWavelet transformInverse transform

time

Samples delayed by block size

^

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 22

Length 10

Length 4

Length 22

Length 22

Length 10

Length 4

xr[n-d]

Delay K1

Delay K2

Level 0

Level 1

Level 2

Level 3

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

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

Conclusions

• Application-sensor tension

• Query model to address tension

• Wavelets as basis for query model

• Promising early results

• Delay conundrum

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

Contact Information

• Webpage• http://www.cs.northwestern.edu/~jskitz

• Email address• [email protected]

• Load traces and tools• http://www.cs.northwestern.edu/~pdinda/LoadTraces

• Matlab scripts• Available by request ([email protected])

37

Frequency Information Vs. Rate

Input Signal, x[n] Decomposition

f(Hz) fs/2 f(Hz) fs/2 fs/8 fs/4 fs/16

0 1 2 3

Levels

• Frequency information retained = fs/2 • Measurement rate, fs

Q: Why is this true?A: The Nyquist Criterion- sampling theory

38

Wavelet Transform, 1 StageLevel 0 yl[n]

2

2Level 1

HPF

LPF

yh[n]

x[n]

LPF, HPF FIR filters

h[n] x[n] y[n]

N

k

knxkhny0

Downsampler

2 y[n] c[k]

kykc 2 ,for all k

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

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

Reconstruction From the Wavelet Coefficients, 1 Stage

Upsampler

LPF, HPF time reversed filters, same response

0

2nc

else ,

2 of multiplen , y[n] = c[k] 2

Level 0

2Level 1

HPF

LPF

xr[n]2

+^

42

Reconstruction From Multiple Stages, The Inverse Wavelet Transform

Level 0

Level 1

xr[n]

HPF

+ LPF

HPF

+ LPF

HPF

+ LPF

Level M-1

Level M

^

Reconstructed signal is exactly the resource

43

• Determined by accuracy constraints

• Determined by what levels are available• Determined by the rate (fq) at which

measurements are requested:

Q: How are the number of levels determined?

Answers:

LMs

qLMs f

ff

22 1

44

Example, Choosing Levels

f(Hz) fs/2 fs/8 fs/4 fs/16

0 1 2 3

Levels

M = 4 levels fq = fs / 6

Lss

Ls fff

414 262

Solution:

23 262sss fff

L = 2:

Equation Satisfied!

Levels 0, 1 and 2 coefficients returned

45

Streaming Query Tradeoffs

• Measurement rate, fq high– Lower error variance– Higher communication costs

• Measurement rate, fq low– Higher error variance– Very low communication costs

Wavelet approach yields accuracy at low rates

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, fq high– Shorter interval length– Tighter confidence

intervals

• Rate, fq low– Longer interval length– Wider confidence

intervals

Confidence interval (c) provides flexibility

47

Streaming Queries, Energy in Auto-covariance

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 3 5 7 9 11 13

0.001 0.01 0.1 1

Level

Compression and Peak Frequency

Error becomes uncorrelated as levels added

48

Interval Queries, Error Mean (32 seconds)

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

1 3 5 7 9 11 13

0.001 0.01 0.1 1

Level

Compression and Peak Frequency

Error mean is zero at 8 levels, 3% of coefficients

49

Interval Queries, Error Mean (512 seconds ~ 8½ minutes)

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

1 3 5 7 9 11 13

0.001 0.01 0.1 1

Level

Compression and Peak Frequency

As interval increases, need fewer levels