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Job Delay Analysis in Data Centers Job Delay Analysis in Data Centers

Job Delay Analysis in Data Centers - PowerPoint Presentation

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Job Delay Analysis in Data Centers - PPT Presentation

Weina Wang UIUC CMU Performance 2018 Joint work with Mor HarcholBalter Haotian Jiang Alan SchellerWolf and R Srikant Solving Limited ForkJoin Model Independent or Not Model ID: 757811

indpt job time delay job indpt delay time task max sketch asymptotically tasks orig proof queues independent delays

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Slide1

Job Delay Analysis in Data Centers

Weina WangUIUC  CMUPerformance 2018: Joint work with Mor Harchol-Balter, Haotian Jiang, Alan Scheller-Wolf, and R. Srikant

Solving Limited Fork-Join Model:Independent or Not?Slide2

Model

...

...

1

2

3

4

5

10

4

“job with 3 tasks”

random

1Slide3

Model

...

...

1

2

3

4

5

10

4

2Slide4

Model

...

...

1

2

3

4

5

10

4

3Slide5

Model

...

...

1

2

3

4

5

10

4

4Slide6

Model

...

...

1

2

3

4

5

n

Job arrival: Poisson process

Each job has

k

tasks (

k

n

)

random

5

Job not done until

all

its tasks are doneSlide7

Goal: What is job delay?

...

...

n

 

6Slide8

“Limited fork-join model”

...

...

n

7Slide9

...

...

n

task delay

vs.

job delay

intensively studied

well-known hard problem:

fork-join: tight analysis only when

n

= 2

8Slide10

Our result:first tight characterization ofjob delay

in a large system9Slide11

Job delay: “max of task delays”

 

Analyzing job delay

...

...

n

2

1

3

 

 

 

 

 

 

10

 

?

k

=

n

Equal

if independentSlide12

k

= nINDPT?

Job delay: “max of task delays”

 

Analyzing job delay

...

...

n

 

 

 

11

 

?

NO!Slide13

k

= nINDPT?

k

= 1

INDPT?

Job delay: “max of task delays”

 

Analyzing job delay

...

...

n

1

 

 

12

 

?

NO!

YES

!Slide14

k

= nINDPT?

k

= 3

INDPT?

k

= 1

INDPT?

Job delay: “max of task delays”

 

Analyzing job delay

...

...

n

2

1

3

 

 

 

 

 

 

13

 

?

NO!

YES

!

asymptotically

YES

!

←largeSlide15

k

= nINDPT?

k

= log

n

INDPT?

k

= 3

INDPT?

k

= 1

INDPT?

Job delay: “max of task delays”

 

Analyzing job delay

...

...

n

 

 

 

14

 

?

NO!

YES

!

asymptotically

YES

!

←large

asymptotically

YES

!Slide16

k

= nINDPT?

k

= o (

n

1/4

)

INDPT?

k

= log

n

INDPT?

k

= 3

INDPT?

k

= 1

INDPT?

Job delay: “max of task delays”

 

Analyzing job delay

...

...

n

 

 

 

15

 

?

NO!

YES

!

asymptotically

YES

!

←large

asymptotically

YES

!

asymptotically

YES

!Slide17

Analyzing job delay

...

...

n

 

 

 

16

 

?

k

=

n

INDPT?

k

= o (√

n

)

INDPT?

k

= o (

n

1/4

)

INDPT?

k

= log

n

INDPT?

k

= 3

INDPT?

k

= 1

INDPT?

NO!

YES

!

asymptotically

YES

!

asymptotically

YES

!

asymptotically

YES

!

UNKNOWN!

Job delay: “max of task delays”

 Slide18

: job delay converges to max of independent task delays

 

Summary

...

...

1

2

3

4

5

n

Job arrival: Poisson process

Each job has

k

tasks

random

17Slide19

Result: Asymptotic independence

Theorem For

, as

n

⟶ ∞,

queue lengths ⟶

k

(

n

)

-wise independent

job delay ⟶

max of

k(n) independent task delays

 

prove this

Each job has

k

tasks

...

...

1

2

3

4

5

n

# servers:

n

Load

of each queue

 

Job arrival: Poisson

 

 

 

 

 

 

 

18Slide20

Proof sketch

Want to show:this distance as

 

...

...

1

2

3

4

5

n

k

(

n

)

tasks

first

k

(

n

)

queues

steady state

ORIG

time

 

 

time

 

 

INDPT

19Slide21

ORIG

time

 

 

time

 

 

INDPT

Proof sketch

Take a detour

20Slide22

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

...

...

1

2

3

4

5

n

k

(

n

)

tasks

first

k

(

n

)

queues

empty

empty

21Slide23

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

k

(

n

)

tasks

...

...

1

2

3

4

5

n

first

k

(

n

)

queues

1s

1s

Pr

(some job strikes

“≥ 2 queues within

k

(n)

in

1s

)

?

22Slide24

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

10s

10s

Pr

(some job strikes

“≥ 2 queues within

k

(n)

in

10s

)

k

(

n

)

tasks

...

...

1

2

3

4

5

n

first

k

(

n

)

queues

?

23Slide25

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

 

Pr

(some job strikes

“≥ 2 queues within

k

(n)

in

)

 

 

k

(

n

)

tasks

...

...

1

2

3

4

5

n

first

k

(

n

)

queues

?

24Slide26

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

 

Pr

(some job strikes

“≥ 2 queues within

k

(n)

in

)

 

 

?

 

25Slide27

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

 

 

?

 

 

Best choice:

 

26Slide28

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

 

 

 

 

when

,

 

27Slide29

Proof sketch

ORIG

time

 

 

time

 

 

INDPT

 

 

“small”

“small”

“small”

“small”

when

,

 

28Slide30

Result: Delay asymptotics for k(n)

Theorem For

, as

n

⟶ ∞,

queue lengths ⟶

k

(

n

)

-wise independent

job delay ⟶

max of

k(n)

independent task delays 

29Slide31

Summary

: job delay converges to max of independent task delays

Finite

k

and

n

(

k

n

)

: independence gives an upper bound 

...

...

1

2

3

4

5

n

Job arrival: Poisson process

Each job has

tasks

 

random

30Slide32

k

= 1

 

?

 

Asymptotically independent as

n

⟶ ∞

?

INDPT

Not ASYM INDPT

ASYM INDPT

?

?

 

Job Delay

Tail Probability (CCDF)

 

Job Delay

Tail Probability (CCDF)

 

31