1

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)

2

Affinity Scheduling

3

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.

4

Situation 1: Self-Affinities

Optimal control policy: Cycle Stealing.

5

Situation 2:Eager to Help

If server 2 overzealous, a brake is needed.

6

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.

7

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

8

T1 Policy: Performance vs. T1

Sensitivity to T1

0

10

20

30

40

1 11 21 31T1

Res

pons

e Ti

me

9

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

10

N1

N2

What is the Dream? Switching Curve

Optimal?

11

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

12

T1 Policy: Performance vs. T1

Sensitivity to T1

0

10

20

30

40

1 11 21 31T1

Res

pons

e Ti

me

13

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

14

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

15

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

16

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

17

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

18

What’s so Hard?Low

Hi

Med

Now chain grows infinitely in 3 dimensions!

19

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).

20

Representative Types of Busy Periods

L

H

LLL

Becomes…

Becomes…

M

H

LLLLH

H

LLLL or

M

L

LH

L

LL or

21

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]

22

The Low Job Chain M

M

1,00,0

0,1

HH

23

The Low Job Chain M

M

5,1,05,0,0

5,0,1

HH

24

The Low Job Chain

M

M

5,1,0

5,0,0

5,0,1

H

H

25

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

26

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

27

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

28

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.

29

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

30

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.

31

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?