alan scheller-wolf
DESCRIPTION
Dimensionality Reduction for the analysis of Cycle Stealing, Task Assignment, Priority Queueing, and Threshold Policies (PART 2). Alan Scheller-Wolf. Joint with: Mor Harchol-Balter, Taka Osogami, Adam Wierman, and Li Zhang. Affinity Scheduling. m 12. m 11. m 22. Fluid or Diffusion. - PowerPoint PPT PresentationTRANSCRIPT
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Alan Scheller-Wolf
Joint with: Mor Harchol-Balter, Taka Osogami,Adam Wierman, and Li Zhang.
Dimensionality Reductionfor the analysis of
Cycle Stealing,Task Assignment,Priority Queueing,
and Threshold Policies(PART 2)
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Affinity Scheduling
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Prior Work:Affinity Scheduling
Thresholdpolicies
Squillante, Xia, Yao and ZhangWilliams
Bell and WilliamsHarrisonHarrison and LopezSquillante, Xia, ZhangWilliams
Fluid orDiffusion
GreenSchumskyStanford and Grassman
Applications(cycle stealing)
No accurate analysis for non-limiting behavior.
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Situation 1: Self-Affinities
Optimal control policy: Cycle Stealing.
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Situation 2:Eager to Help
If server 2 overzealous, a brake is needed.
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Why? Potential Instability
Maybe server two is too eager to help: • Take too much work from server 1,leaving her idle, • Neglect own work, letting it build up.
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The Brake: T1 Policy
Asymptotically optimal, robustness concerns.We provide first easy, accurate analysis.
“Come help, but only when I call you.”
N2
N1
T1
1
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T1 Policy: Performance vs. T1
Sensitivity to T1
0
10
20
30
40
1 11 21 31T1
Res
pons
e Ti
me
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Sensitivity tor
0
20
40
60
80
100
0.85 0.9 0.95 1r
Res
pons
e Ti
me
T1=8
T1=16
T1 Performance IIT1 Policy: Performance vs. r
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N1
N2
What is the Dream? Switching Curve
Optimal?
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New Control Policy: The ADT Policy
Performs like best of T1(1) and T1(2).We propose and analyze.
“Come help when I call you.”
N1
N2
T1(1)
“If you are very busy and I am not, do not come.”“But if I really need you, you have to come.”
T1(2)
T2
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T1 Policy: Performance vs. T1
Sensitivity to T1
0
10
20
30
40
1 11 21 31T1
Res
pons
e Ti
me
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ADT Policy: Performance vs T1(1)
Sensitivity to T1(1)
0
10
20
30
40
1 11 21 31
T1(1)
Res
po
nse
Tim
e
T1ADT
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Sensitivity tor
0
20
40
60
80
100
0.85 0.9 0.95 1r
Res
po
nse
Tim
e
T1=8
T1=16
T1 Performance IIT1 Policy: Performance vs. r
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ADT Policy: Performance vs r
Sensitivity to r
0
20
40
60
80
100
0.85 0.9 0.95 1r
Res
pons
e Ti
me
T1=8
T1=16
ADT
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Goal: Mean response time per job type.
RDR and Priority Scheduling
nD-infinitechain
1D-infinitechain
HARD EASY
Priority Scheduling in M/PH/k
L
H
H
M HL
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Scaling as Single-server:
Buzen and Bondi
Aggregation intoTwo classes:
Mitrani and KingNishida
Multi-classsimpleapprox.
Two jobclasses,
exponential
Cidon and SidiFeng el atGail et alMiller
Matrix Analyticor
Gen. Functions
Aggregationor
Truncation
Two jobclasses,
exponential
Two jobclasses,hyper-
exponential
Sleptchenko et alKao and NarayananKao and WilsonKapadia et alNishidaNgo and Lee
Iterative sol tobalance
equations
Little work for > 2 classes or non-exponential.
Prior Work:Multi-Server Priority Queues
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What’s so Hard?Low
Hi
Med
Now chain grows infinitely in 3 dimensions!
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Recursive Dimensionality Reduction(RDR)
• Apply standard dimensionality reduction (DR) to two highest classes (Mor’s talk).
• Aggregate these classes -- carefully -- into single higher class. Many types of busy periods.
• Apply DR to two-class system made up of aggregated classes and third class.
• Recurse. Chain for class m used to calculate busy periods for next lower class (m+1).
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Representative Types of Busy Periods
L
H
LLL
Becomes…
Becomes…
M
H
LLLLH
H
LLLL or
M
L
LH
L
LL or
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What are these busy periods? M
M
1,00,0
M
M
3,02,0
M
M
M
M
M
M
1,10,1
M
M
3,12,1
M
M
M
M
M
1,2+0,2+
M
3,2+2,2+
M M
HH H
H HH H
H
H H H HBH BH
BH BH
Neuts[1978]
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The Low Job Chain M
M
1,00,0
0,1
HH
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The Low Job Chain M
M
5,1,05,0,0
5,0,1
HH
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The Low Job Chain
M
M
5,1,0
5,0,0
5,0,1
H
H
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The Low Job Chain
M
M
5,1,0
5,0,0
5,0,1
H
H
5,0,2H 5,1,1M5,0,2M 5,1,1H 5,2,0H 5,2,0M
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The Low Job Chain
M
M
5,1,0
5,0,0
5,0,1
H
H
5,0,2H 5,1,1M5,0,2M 5,1,1H 5,2,0H 5,2,0M
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M/M/2 Four Priority Classes: AccuracyPercent Error Higher Priority: Shorter
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95r
Class 2
Class 3
Class 4
Percent Error Higher Priority: Longer
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
r
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Higher Priority Classes: Shorter
0.0000001
0.00001
0.001
0.1
10
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95r
Re
sp
on
se
Tim
e
Class 1
Class 2
Class 3
Class 4
M/M/2 Four Priority Classes: Resp.
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M/M/2 Four Priority Classes: Perf
Higher Priority Classes: Longer
0.001
0.1
10
1000
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95r
Res
pons
e Ti
me
Class 1
Class 2
Class 3
Class 4
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Generalizations and Extensions
• Phase-type service times.
• More classes, more servers.
• Number of different busy periods grows with complexity of system (service times, servers, classes).
• RDR-A approximation for these more complex systems, within 5% error for four class problem.
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DR and RDR, future directions
We solve problems where one class depends on the other,but the dependencies can be solved sequentially (H,M,L).
What about systems that do not decouple?