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
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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