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Weina Wang. UIUC . . CMU. Performance 2018: Joint work with Mor Harchol-Balter, Haotian Jiang, Alan Scheller-Wolf, and R. Srikant. Solving Limited Fork-Join Model:. Independent or Not?. Model. ID: 757811

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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?

Slide2Model

...

...

1

2

3

4

5

10

4

“job with 3 tasks”

random

1

Slide3Model

...

...

1

2

3

4

5

10

4

2

Slide4Model

...

...

1

2

3

4

5

10

4

3

Slide5Model

...

...

1

2

3

4

5

10

4

4

Slide6Model

...

...

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 done

Slide7Goal: What is job delay?

...

...

n

6

Slide8“Limited fork-join model”

...

...

n

7

Slide9...

...

n

task delay

vs.

job delay

intensively studied

well-known hard problem:

fork-join: tight analysis only when

n

= 2

8

Slide10Our result:first tight characterization ofjob delay

in a large system9

Slide11Job delay: “max of task delays”

Analyzing job delay

...

...

n

2

1

3

10

?

k

=

n

Equal

if independent

Slide12k

= nINDPT?

Job delay: “max of task delays”

Analyzing job delay

...

...

n

11

?

NO!

Slide13k

= nINDPT?

k

= 1

INDPT?

Job delay: “max of task delays”

Analyzing job delay

...

...

n

1

12

?

NO!

YES

!

Slide14k

= 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

!

←large

Slide15k

= 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

!

Slide16k

= 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

!

Slide17Analyzing 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

17

Slide19Result: 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

18

Slide20Proof 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

19

Slide21ORIG

time

time

INDPT

Proof sketch

Take a detour

20

Slide22Proof sketch

ORIG

time

time

INDPT

...

...

1

2

3

4

5

n

k

(

n

)

tasks

first

k

(

n

)

queues

empty

empty

21

Slide23Proof 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

)

?

22

Slide24Proof 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

?

23

Slide25Proof 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

?

24

Slide26Proof sketch

ORIG

time

time

INDPT

Pr

(some job strikes

“≥ 2 queues within

k

(n)

”

in

)

?

25

Slide27Proof sketch

ORIG

time

time

INDPT

?

Best choice:

26

Slide28Proof sketch

ORIG

time

time

INDPT

when

,

27

Slide29Proof sketch

ORIG

time

time

INDPT

“small”

✓

“small”

✓

“small”

✓

“small”

✓

when

,

28

Slide30Result: Delay asymptotics for k(n)

Theorem For

, as

n

⟶ ∞,

queue lengths ⟶

k

(

n

)

-wise independent

job delay ⟶

max of

k(n)

independent task delays

✓

29

Slide31Summary

: 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

30

Slide32k

= 1

?

Asymptotically independent as

n

⟶ ∞

?

INDPT

Not ASYM INDPT

ASYM INDPT

?

?

Job Delay

Tail Probability (CCDF)

Job Delay

Tail Probability (CCDF)

31

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