Multiresolution Resource Behavior Queries Using Wavelets

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Multi-resolution Resource Behavior Queries Using
Wavelets

• Jason Skicewicz
• Peter A. Dinda
• Jennifer M. Schopf
• Northwestern University

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The Tension
Video App
Sensor
Fine-grain measurement

Resource-appropriate measurement
Grid App
Resource Signal (periodic sampling) Example
host load
Course-grain measurement
3
Video Scheduling
Video App
Sensor
Fine-grain measurements needed
4
Grid Scheduling
Grid App
Sensor
Coarse-grain measurements sufficient
5
Interval Averages
Application
Sensor
Average over interval
Average over interval
Ideal Result
Adequate Result
6
Contributions / Outline

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

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Schematic Representation of Query Model
Application
Sensor

x
x
Lower bandwidth used
Measurements at fs samples/second
Desired rate at fq samples/second
The desired rate signal is an estimate error x
x

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Application
Sensor
Query
Stream Error
x
t
t
?
?q
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Application
Sensor
Query
Average CI
tnowinowD
(inow-N1)D
x
t
t
Application gets average over this interval
Application wants average over this interval
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Contributions / Outline

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

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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
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Multi-resolution Views
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High Level View of a 4-level Wavelet Decomposition
Sensor
Level 0
Wavelet Transform
Level 1
Wavelet Coefficients
Level 2
Level 3

• Resource Signal is decomposed into levels
• Samples at each level are at a different rate
• Each level captures different frequency content
• Corresponding inverse transform

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4-level Wavelet DecompositionTime-frequency
Localization
Level
Frequency
0
0 fs/16
fs/16 fs/8
1
fs/8 fs/4
2
fs/4 fs/2
3
xn
0 fs/2
?
fs1/?
time
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Example Decomposition of Host Load
Lossless representation of resource signal
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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, n2k
• Levels, M lg(n) 1
• Circular convolution over block, O(n)

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Proposed System
Application
Sensor
Network
Stream
Interval
Level 0
Level 0
Wavelet Transform
Inverse Wavelet Transform
Level L
Level M-1
Level M
Application receives levels based on its needs
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Multi-resolution Views Using 14 Levels
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Wavelet Compression Gains, 14 Levels
Typical appropriate number of levels for host
load, error lt 20
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Contributions / Outline

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

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Offline Analysis System
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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

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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

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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
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Streaming Queries, Relative Error Variance
Fewer than 1 of coefficients, error lt 20
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Streaming Queries, Error Histogram at Level 6
Errors follow a near-Gaussian distribution
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Interval Queries, Error Variance
Error variance approaches zero as interval
increases
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Interval Queries, Error Histograms at Level 5
Distributions not always Gaussian
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Contributions / Outline

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

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Block By Block System Delay
M Levels
Wavelet Transform
Inverse Wavelet Transform

xn
xrn

Block
Block
n samples in block
n samples in block
Sample Acquisitions
Wavelet transform
Inverse transform
time
Samples delayed by block size
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Streaming System Delay, Example with Length 4
Wavelets (D4), 4 Levels
Level 0
Length 22
Length 22
Level 1
Length 22
Length 22
xn
xrn-d
Level 2
Length 10
Length 10
Delay K1
Level 3
Length 4
Length 4
Delay K2
High levels delayed waiting for low frequency
computations, output delayed by high order filter
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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!
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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

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Conclusions

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

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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

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Contact Information

• Webpage
• http//www.cs.northwestern.edu/jskitz
• Email address
• jskitz_at_cs.northwestern.edu
• Load traces and tools
• http//www.cs.northwestern.edu/pdinda/LoadTraces
• Matlab scripts
• Available by request (jskitz_at_cs.northwestern.edu)

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Frequency Information Vs. Rate
Input Signal, xn
Decomposition

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

Q Why is this true?
A The Nyquist Criterion- sampling theory
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Wavelet Transform, 1 Stage
LPF, HPF FIR filters
xn
yn
hn
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Increasing Stages, Mallats Tree Algorithm
xn
Stages can be arbitrarily increased
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Frequency Response
HPF
LPF

• Filters must be even order for PR
• Other special properties to retain PR
• The filters are order N8 (D8 wavelet)

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Reconstruction From the Wavelet Coefficients, 1
Stage
Upsampler
LPF, HPF time reversed filters, same response
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Reconstruction From Multiple Stages, The Inverse
Wavelet Transform
Reconstructed signal is exactly the resource
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Q How are the number of levels determined?
Answers

• Determined by accuracy constraints
• Determined by what levels are available
• Determined by the rate (fq) at which measurements
  are requested

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Example, Choosing Levels
Solution
L 2
fq fs / 6
M 4 levels
Equation Satisfied!
Levels 0, 1 and 2 coefficients returned
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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
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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
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Streaming Queries, Energy in Auto-covariance
Error becomes uncorrelated as levels added
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Interval Queries, Error Mean (32 seconds)
Error mean is zero at 8 levels, 3 of coefficients
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Interval Queries, Error Mean (512 seconds 8½
minutes)
As interval increases, need fewer levels