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Transcript of 1 Mor Harchol-Balter, CMU, Computer Sci. Alan Scheller-Wolf, CMU, Tepper Business Andrew Young,...
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Mor Harchol-Balter, CMU, Computer Sci.
Alan Scheller-Wolf, CMU, Tepper Business
Andrew Young, Morgan Stanley
Surprising results on task assignment
for high-variability workloads
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Server farm model
Q: What is a good Assignment Policy? (high-variability jobs)
incomingjobs Assignment
Policy
FIFO
FIFO
FIFO
Poisson()
Goal: Minimize mean response time: E[T]
nXE
XE
XC
n
][
][
)var(
X~ sizes job
i.i.d. general
servers
22
POLICY MATTERS!
Mor Harchol-Balter, CMU
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Good Answers
SITA (Size Interval)Split jobs based on size cutoffs.
SITA
+Isolatio
n for s
malls
+Protects against
high
variabilit
y
M/G/2
LWL (Least Work Left)Send job to host with least remaining work.
LWL
+High throughput
Mor Harchol-Balter, CMU
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Prior Work on SITA
• Supercomputing Centers [Hotovy, Schneider, O’Donnell 96] [Schroeder, Harchol-Balter 00]
• Manufacturing Centers [Buzacott, Shanthikumar 93]
• File Server Farms [Cardellini, Colajanni, Yu 01]
• Supermarkets
SITA in Practice
• [Harchol-Balter 00]• [Harchol-Balter 02]• [Thomas 08]• [Tari,Broberg,Zomaya, Baldoni 05]• [Fu, Broberg, Tari 03]
SITA variants
• [Harchol-Balter,Crovella,Murta 98]• [Bachmat, Sarfati 08]• [Sarfati 08]• [Harchol-Balter, Vesilo 10]
Optimizing SITA cutoffs
• [Broberg, Tari, Zeephongsekul 06]• [Harchol-Balter 02]• [Ciardo, Riska, Smirni 01]• [El-Taha, Maddah 06]• [Fu, Broberg, Tari 03]• [Harchol-Balter, Crovella, Murta 99]• [Oida, Shinjo 99]• [ Tari, Broberg, Zomaya, Baldoni 05]• [Thomas 08]
SITA vs. LWLSITA LWL
All conclude SITA far
superiorfor high variability
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Can’t prove anything because it’s not true!
In search of a proof of SITA’s total dominance.
OK, so not optimal, butdefinite winfor high variability.
Should at least beat all commonly used policies when variability is high enough.
Monthslater
Yearslater
SITA
Mor Harchol-Balter, CMU
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The TRUTH about SITA,under very high job size
variability
Alternative Title:
fixedXEXE
XC :][ while
][
)var(2
2
Mor Harchol-Balter, CMU
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Q: In this talk we will show ...
as C2∞a) SITA diverges & LWL diverges?b) SITA converges & LWL diverges ?c) SITA diverges & LWL converges?d) SITA converges & LWL converges?
A: All of the above
SITALWL
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Q: In this talk we will show ...
as C2∞Convergent
LWLDivergent
LWL
Convergent SITA
Divergent SITA
Looking for simple job size distributions toillustrate each.
Mor Harchol-Balter, CMU
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Results (2 server system)
p
1-p
a
b
Bimodal
Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
or
p
1-p
Exp(a)
H2
Exp(b)
depends onpa & (1-p)b
Mor Harchol-Balter, CMU
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Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
or
a
Trimodal
b
c=bm >
1
H3
Exp(a)
Exp(b)
Exp(c)
dependson m
Results (2 server system)
Mor Harchol-Balter, CMU
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Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
dependson
Results (2 server system)
Bounded Pareto()
1< < 2
Mor Harchol-Balter, CMU
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Bimodal Results
X ~
p
1-p
a
b
pa = QE[X]
(1-p)b = (1-Q)E[X]
THM: If a < 1 & b < 1 Convergent SITA
Lemma: As C2 ∞, but E[X], Q: const, a’s get little smaller QE[X]
b’s get much bigger ∞ p 1
THM: LWL always diverges.
Conv.LWL
DivergLWL
Conv.SITA
DivergSITA
dependsa & b
Mor Harchol-Balter, CMU
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Understanding LWL
Isn’t LWL always bad for high C2?
It depends …
Need 2 longs for this to be a problem!
So we need: Pr{ 2 longs } * E[T| 2 longs] ?
But shorts stuck behind longs, so E[T] ∞
Suffices to just look at E[X3/2].
LWL
Mor Harchol-Balter, CMU
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Thm:
usually) (converges LWL 1 &
while
bounded :][ If2
23
C
XE
14
Understanding LWL
LWL23
E[T] 1 & ]E[X If Thm: [Scheller-Wolf, Sigman 97], [Scheller-Wolf, Vesilo 06] (2 SERVERS)
usually) (1 spare server
C2∞
I can makeboth happen!
LWL
Mor Harchol-Balter, CMU
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Bimodal Results
THM: If a < 1 & b < 1 Convergent SITA
Conv.LWL
DivergLWL
Conv.SITA
DivergSITA
THM: LWL always diverges.
) (as
][)1(][
)1(][
2
23
23
23
C
bXEQaXQE
bppaXE
X ~
p
1-p
a
b
pa = QE[X]
(1-p)b = (1-Q)E[X]
dependsa & b
Lemma: As C2 ∞, but E[X], Q: const,
a QE[X], b ∞, p 1
Mor Harchol-Balter, CMU
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Trimodal Results
THM: If m≤3, SITA converges If m>3, SITA diverges
Lemma: As C2 ∞, but E[X]: const, a E[X]
b ∞, c ∞ pa 1
THM: LWL always converges for <1
Conv.LWL
DivergLWL
Conv.SITA
DivergSITA
X~
b
c=bm
a
pa
pb=b-3/2
pc=c-3/2
dependson m
11][
][
23
23
23
23
23
XE
cpbpapXE cba
Mor Harchol-Balter, CMU
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Results (2 server system)
Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
p
1-p
a
b
Bimodal
a
Trimodal
b
c=bm >
1
Way more complex, because job types overlap!
“Separation in the limit”
p
1-p
Exp(a)
H2
Exp(b)
H3
Exp(a)
Exp(b)
Exp(c)
Mor Harchol-Balter, CMU
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Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
E[T] LWL
SITA
C2
E[T]
LWL
SITA
C2
E[T]
LWL
SITA
C2
18
E[T]LWL
SITA
C2
Mor Harchol-Balter, CMU
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19
Bounded Pareto(2 server system)
THM: SITA always diverges.
Lemma: As C2 ∞, but E[X], : const, k E[X]
p ∞
THM: If >3/2 and <1, then LWL converges.Else LWL diverges.
Conv.LWL
DivergLWL
Conv.SITA
DivergSITA
dependson
Extends to n>2 servers when < n-1
X~ Bounded Pareto (k,p,)
k p
1 < < 2
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Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
20
Bounded Pareto Results
= 1.6
LWL
SITA
C2
E[T]
= 1.4
LWL SITA
C2
E[T]
Why was this not noticed?
Mor Harchol-Balter, CMU
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21
Summary
Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
p
1-p
a
b
Bimodal
oror
p
1-p
Exp(a)
H2
Exp(b)
H3
Exp(a)
Exp(b)
Exp(c)
a
Trimodal
b
c=bn > 1
Bounded Pareto()
1< < 2
Mor Harchol-Balter, CMU
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22
Old Nursery Rhyme
“There once was a girl, who had a little curl
right in the middle of her forehead.
When she was good, she was very very good.
But when she was bad, she was horrid.”
When SITA is good, it is very, very good
But when it is bad, it is horrid.
Conv. SITA
Diverg. SITA
Conv. LWL
Diverg. LWL
Mor Harchol-Balter, CMU
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23
Epilogue …
Mor Harchol-Balter, CMU
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24
Where did SITA go wrong?
SITA designed to keep shorts from getting stuck behind longs. Isn’t that good?
But stringent segregation of shorts & longs can lead to underutilization of servers.
Also, for some distributions, can’t subdivide to avoid infinite variability.
Mor Harchol-Balter, CMU
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25
Must be a better way…
SITA Split jobs based on size cutoffs.
SITA+Iso
lation fo
r smalls
+Protects against
high
variabilit
yLWL Send job to host with
least remaining work.
LWL
+High throughput
M/G/2 Takenext job
Takenext job
CS Takenext small
Takenext job
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26
Must be a better way…
WIN/WIN ! Shorts have isolation from longs
And server utilization is high
WRONG!Thm: Whenever SITA diverges, CS diverges too.
Mor Harchol-Balter, CMU
CS Takenext small
Takenext job
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27
Thm: Whenever SITA diverges, CS diverges too.
SITA Split jobs by size.
SITA
PROOF: There are 2 reasons why SITA diverges under given
(1)Any way of slicing leads to
(2) There is a way of slicing away variability, but it forces 1s
2LL XEp
e.g., Pareto()
Smalls Larges
e.g., BimodalS’s
L’s
CS can’t help.
CS should help, but doesn’t.
CS Takenext small
Takenext job
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Thm: Whenever SITA diverges, CS diverges too.
SITA Split jobs by size.
SITA
PROOF: There are 2 reasons why SITA diverges under given
(1)Any way of slicing leads to
(2) There is a way of slicing away variability, but it forces 1s
2LL XEp
Pareto()
Smalls Larges
Bimodal
S’sL’s
CS can’t help.
CS should help, but doesn’t.
Small job sees L of age ~Le.Small server has been in overload for ~Le time Small sees ~Le work experiences Le delay. Delay of small ∞ as C2 ∞
CS Takenext small
Takenext job
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29
Conclusion
SITA
.
LWL
M/G/2 Takenext job
Takenext job
Maybe isolating short jobs is not the panaceafor high-variability workloads after all …
Mor Harchol-Balter, CMU
CS Takenext small
Takenext job