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A CLASS OF POLICIES THAT PRIORITIZE SMALL JOBS Adam Wierman Carnegie Mellon University Pittsburgh, PA USA Web Server Use Shortest Remaining Processing Time First (SRPT) A MOTIVATIONAL EXAMPLE: Web servers Bandwidth is the bottleneck Processor Sharing (PS) Goal Minimize user

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a class of policies that prioritize small jobs adam wierman

A CLASS OF POLICIES THAT PRIORITIZE SMALL JOBSAdam Wierman

Carnegie Mellon University

Pittsburgh, PA USA

a motivational example web servers

Web

Server

Use Shortest

Remaining Processing

Time First (SRPT)

A MOTIVATIONAL EXAMPLE: Web servers

Bandwidth is the bottleneck

Processor

Sharing (PS)

Goal

Minimize user

response times

requests for files

[Harchol-Balter, Schroeder, Bansal, Agrawal, TOCS 2003]

a motivational example web servers3

PS

mean response time

?

Why SRPT?

SRPT

load

0

1

A MOTIVATIONAL EXAMPLE: Web servers

[Harchol-Balter, Schroeder, Bansal, Agrawal, TOCS 2003]

scheduling success stories across computer applications

users

CPUs

Internet

Databases

Routers

Disks

Prioritize

small jobs

Locks

SCHEDULING SUCCESS STORIES ACROSS COMPUTER APPLICATIONS

Web

Servers

…also p2p, wireless, operating systems…

Rai, Urvoy-Kelley, Biersack, Harchol-Balter, Schroeder, McWherter, Rawat, Dinda, & many others

scheduling success stories across computer applications5

SRPT

RS

PSJF

PS

LCFS

SJF

DPS

PLJF

FCFS

LJF

FSP

PLCFS

FB

LRPT

SCHEDULING SUCCESS STORIES ACROSS COMPUTER APPLICATIONS

RENEWED INTEREST IN THEORETICAL WORK

Harchol-Balter, Biersack, Ayesta, Nunez-Queija, Nuyens, Boxma, Zwart, Borst, & many others

slide6

There is a gap

between the theoretical

results and the practical

implemplentations

theory

Slowdown & Fairness are

also important

Hybrid policies outperform SRPT on these

Theory

a small set of individual, idealized

policies are analyzed

multiple metrics

are important

Practice

real policies are

hybrids

policies studied in theory are

“tweaked” for use in practice

slide8

Prioritize small jobs

BRIDGING THE GAP

Analyze classes of policiesinstead of individual policies

SRPT

RS

PSJF

PS

LCFS

SJF

DPS

PLJF

  • Classes include practical hybrids
  • Classes allow optimization of secondary metrics

FCFS

LJF

PLCFS

FSP

FB

LRPT

slide9

OUR GOAL TODAYDefine a class of policies that “prioritizes small jobs” and is bothBROADandTIGHT

behaves like SRPT

includes practical

hybrid policies

the smart class
THE SMART CLASS

SMART

PSJF

RS

SRPT

SMAll Response Times

  • Bias Property
  • Consistency Property
  • Transitivity Property

PS

LJF

PLCFS

ROS

FCFS

LRPT

two notions of small jobs
TWO NOTIONS OF “SMALL” JOBS

small original size

small remaining size

bias property
BIAS PROPERTY

If OriginalSize(a) < RemainingSize(b)

then a has priority over b

a

b

[Wierman, Harchol-Balter, Osogami 2005]

bias property13

remaining

size

0

0

original size

BIAS PROPERTY

If OriginalSize(a) < RemainingSize(b)

then a has priority over b

[Wierman, Harchol-Balter, Osogami 2005]

bias property14
BIAS PROPERTY

If OriginalSize(a) < RemainingSize(b)

then a has priority over b

lower

priority

remaining

size

?

higher

priority

0

0

original size

[Wierman, Harchol-Balter, Osogami 2005]

examples

SRPT

PSJF

!

RS

remaining

size

Partial ordering

allows time varying

policies

?

original size

EXAMPLES

If OrigSize(a) < RemSize(b)

then a has priority over b

Many others!

e.g. iRj+S

slide16

!

Partial ordering

allows time varying

policies

B

B

A

C

A

D

remaining size

*PSJF

D

C

time

slide17

!

Partial ordering

allows time varying

policies

B

A

?

B

A

D

remaining size

*PSJF

*PSJF

D

C

*SRPT

time

slide18

!

Partial ordering

allows time varying

policies

  • Use a parameterized policy setthat is (nearly) dense in SMART,e.g. iRj + S
  • Search (i,j) space for policy thatoptimizes secondary objectives,e.g. fairness and predictability

ONLINE MULTI-OBJECTIVE SCHEDULING USING SMART

the bias property isn t enough
THE BIAS PROPERTYISN’T ENOUGH

CONSISTENCY

TRANSITIVITY

remaining

size

?

orig. size

[Wierman, Harchol-Balter, Osogami 2005]

the bias property isn t enough20
THE BIAS PROPERTYISN’T ENOUGH

CONSISTENCY

If a is served ahead of

b then a will always have priority over b

+

TRANSITIVITY

If an arriving job b preempts c, then until b leaves, every arriving job a with original size smaller than b has priority over c.

remaining

size

?

at most 1 has

higher priority

orig. size

[Wierman, Harchol-Balter, Osogami 2005]

slide21

DOING THE “SMART” THING

lower

priority

SMAll Response Times

remaining

size

  • Bias Property
  • Consistency Property
  • Transitivity Property

at most 1 job with higher priority

higher

priority

0

0

original size

slide22

?

Which policies

are notSMART?

SMART

PSJF

RS

SRPT

FB

SJF

PS

LJF

PLCFS

ROS

FCFS

LRPT

Blind policies

Non-preemptive policies

slide23

OUR GOAL TODAYDefine a class of policies that “prioritizes small jobs” and is bothBROADandTIGHT

behaves like SRPT

includes practical

hybrid policies

analysis setting

M/GI/1 preempt-resume queue

ANALYSIS SETTING:

E[T]SMART

APPROACH:

Bound T(x)SMART

Pr(TSMART>y)

t x result plot for e t x

Residence time

Residence time

Waiting time

Waiting time

T(x) RESULT, plot for E[T(x)]

CONDITIONAL RESPONSE TIME UNDER SMART POLICIES

Theorem: Under the M/GI/1, for all SMART policies P,

Response time

for a job of size x

[Wierman, Harchol-Balter, Osogami 2005]

t x result plot for e t x26

SRPT

SMART

PSJF

original size

T(x) RESULT, plot for E[T(x)]

CONDITIONAL RESPONSE TIME UNDER SMART POLICIES

Theorem: Under the M/GI/1, for all SMART policies P,

Picture “proof”: Waiting time

remaining

size

?

t x result plot for e t x27

SRPT

PSJF

T(x) RESULT, plot for E[T(x)]

CONDITIONAL RESPONSE TIME UNDER SMART POLICIES

Theorem: Under the M/GI/1, for all SMART policies P,

Picture “proof”: Residence time

SMART

remaining

size

No higher

priority jobs

?

?

original size

analysis setting28

M/GI/1 preempt-resume queue

ANALYSIS SETTING:

E[T]SMART

APPROACH:

Bound T(x)SMART

Pr(TSMART>y)

slide29

Theorem: In the M/GI/1,

mean response time

PS

E[T]

!

These bounds

are tight

SMART

load, ρ

0

1

SMART POLICIES ARE 2-COMPETITIVE

[Wierman, Harchol-Balter, Osogami 2005]

slide30

Theorem: In the M/GI/1,

!

These bounds

are tight

SMART POLICIES ARE 2-COMPETITIVE

Consider the M/D/1

SRPT does FCFS (only in M/D/1). So as ρ1

PLCFS is in SMART (only in M/D/1)

As ρ1, E[T]PLCFS  2 E[T]SRPT

slide31

Theorem: In the M/GI/1,

?

Tight under

Deterministic

Tight under Pareto

(as ρ1)

What is the heavy

traffic behavior?

SMART POLICIES ARE 2-COMPETITIVE

With work, the bounds can be rewritten as

analysis setting32

M/GI/1 preempt-resume queue

ANALYSIS SETTING:

E[T]SMART

APPROACH:

Bound T(x)SMART

Pr(TSMART>y)

tail behavior of smart

λ1

λ2

λN

NB

Many sources

Large buffer

!

SMART policies

are asymptotically

equivalent in both

TAIL BEHAVIOR OF SMART

Pr(T>y)is difficult to study directly

so it is typically studied asymptotically

[Nuyens, Wierman, Zwart 2007]

[Yang, Wierman, Shakkattai,

Harchol-Balter 2006]

slide34

OUR GOAL TODAYDefine a class of policies that “prioritizes small jobs” and is bothBROADandTIGHT

  • behaves like SRPT:
  • E[T]
  • Pr(T>y)

includes practical

hybrid policies

But…

theory35
Theory

a small set of individual, idealized

policies are analyzed

multiple metrics

are important

implementation

restrictions

Practice

real policies are

hybrids

policies studied in theory are

“tweaked” for use in practice

theory36
Theory

a small set of individual, idealized

policies are analyzed

implementationsonly have job

size estimates

implementation

restrictions

Practice

implementations

only use 7-10

priority levels

policies studied in theory are

“tweaked” for use in practice

slide37

?

What is the impact

of inexact job size

information?

no k such that

use a SMART

policy

know exact

job sizes

1

?

have an estimate

of job sizes

prioritize jobs with

small estimated sizes

2

know only the

distribution of

job sizes

if sizes are variable use FB

otherwise use FCFS

3

slide38

OUR NEW GOALBROADENtheSMART class while keeping itTIGHT

2

k

E[T]P ≤ E[T]SRPT

include policies

that use job size estimates

slide39

rem.

size

?

orig. size

rem.

size

?

Weaken the

Bias Property

orig. size

smart

remaining

size

ε(x)

?

?

original size

original size

SMART

SMARTε

If OrigSize(a) = x and

ε(x) < RemSize(b)

then a has priority over b

If OrigSize(a) < RemSize(b)

then a has priority over b

slide41

ε(x)

?

original size

SMARTε

EXAMPLES

If OrigSize(a) = x and

ε(x) < RemSize(b)

then a has priority over b

How do you get back SMART?

ε(x) = x

How can you characterize job

size estimates?

ε(x) = (1+σ) x

You can also capture policies with

a finite number of priority levels...

slide42

OUR NEW GOALBROADENtheSMART class while keeping itTIGHT

k

E[T]P ≤ E[T]SRPT

include policies

that use inexact job sizes

analysis setting43

M/GI/1 preempt-resume queue

ANALYSIS SETTING:

E[T]SMART

APPROACH:

Bound T(x)SMART

Pr(TSMART>y)

slide44

ε(x)

rem.

size

?

!

Under SMART

σ=δ=0, so

E[T] ≤ 2E[T]SRPT

orig. size

!

x

(1+σ) = (1-δ)1-α

under Pareto(α)

distributions

SMARTε POLICIES ARE NEAR OPTIMAL

Theorem:

In an M/GI/1 under SMARTεpolicy P

σbounds the SIZE of larger

jobs that get higher priority

δbounds the LOAD of larger jobs that get higher priority

slide45

?

What is the effect

of inexact job

size information?

Theorem:

In an M/GI/1 with Pareto(α) job sizes

and SMARTεpolicy P

where for all x,

20

E[T] / E[T]SRPT

10

2

0

200%

100%

estimate accuracy (σ)

INTERPRETING THE THEOREM

Take α=1.1

slide46

?

What is the effect

of inexact job

size information?

Theorem:

In an M/GI/1 with Pareto(α) job sizes

and SMARTεpolicy P

where for all x,

PS

INTERPRETING THE THEOREM

mean response time

SMARTε

SRPT

load

0

1

analysis setting47

M/GI/1 preempt-resume queue

ANALYSIS SETTING:

E[T]SMART

APPROACH:

Bound T(x)SMART

Pr(TSMART>y)

slide48

λ1

λ2

λN

NB

Many sources

Large buffer

!

SMARTε policies

are asymptotically

equivalent for unbounded

service distributions

TAIL BEHAVIOR OF SMARTε POLICIES

Pr(T>y)is difficult to study directly

so it is typically it is studied asymptotically

slide50

!

Other forms of ε(x)

are also useful

SMARTε

when

PSJF

RS

E[T]≤ k E[T]SRPT

SRPT

Pr(T>x)~ Pr(TSRPT>x)

PS

LJF

FCFS

Show that SMARTε

policies behave like SRPT

Formalize the heuristic of

“prioritizing small jobs”

  • Bias Property
  • Consistency Property
  • Transitivity Property
slide51

RS

SRPT

DPS

LRPT

FSP

PSJF

PS

SJF

LCFS

PLJF

FCFS

LJF

PLCFS

FB

Remaining size based

Age based

Prioritize small jobs

(SMART)

Non-preememptive

Preemptive

size based

Time Sharing

Prioritize large jobs

(FOOLISH)

A NEW APPROACH

Analyze classes of policies instead of individual policies

slide52

Isolates impact of scheduling

heuristics and techniques

A NEW APPROACH

Analyze classes of policies instead of individual policies

Results apply to polices that are

implemented in practice

slide53

A. Wierman, and M. Harchol-Balter. “Classifying scheduling policies with respect to unfairness in an M/GI/1.” Sigmetrics 2003.

  • A. Wierman, M. Harchol-Balter, and T. Osogami. ”Nearly insensitive bounds on SMART scheduling.” Sigmetrics 2005.
  • A. Wierman and M. Harchol-Balter. ”Classifying scheduling policieswith respect to higher moments of conditional response time.” Sigmetrics 2005.
  • C. Woo Yang, A. Wierman, S. Shakkottai, M. Harchol-Balter. “Tail asymptotics for policies favoring small jobs in a many-sources regime.” Sigmetrics 2006.
  • M. Nuyens, A. Wierman, B. Zwart. “Preventing large sojourn times using SMART scheduling.” Operations Res. 2007.
  • A. Wierman. “On the effect of inexact size information in size based policies.” MAMA 2006.

A CLASS OF POLICIES THAT PRIORITIZE SMALL JOBS Adam Wiermanhttp://www.cs.cmu.edu/~acwacw@cs.cmu.edu

my research themes
MY RESEARCH THEMES

going beyond individual policies

SCHEDULING CLASSIFICATIONS

How can we use classifications

to bridge the gap between theory

and practice?

[Sigmetrics 03] [Sigmetrics 05] [Sigmetrics 06]

[OR 06]

my research themes55

going beyond mean response time

DIVERSE METRICS

Can a policy be both fair and

efficient?

...What is “fair” anyway?

[Perf Eval 02] [Sigmetrics 03]

[Sigmetrics 05]

How predictable are response

times under common policies?

[Sigmetrics 05] [OR 06]

MY RESEARCH THEMES

going beyond individual policies

SCHEDULING CLASSIFICATIONS

my research themes56

going beyond individual policies

SCHEDULING CLASSIFICATIONS

going beyond mean response time

DIVERSE METRICS

Can a policy be both fair and

efficient?

...If so, when?

...What is “fair” anyway?

[Perf Eval 02] [Sigmetrics 03, 05]

How does having multiple servers

change the effect of scheduling?

[QUESTA 05] [Perf Eval 06]

How do the effects of scheduling

differ in open/closed systems?

...Is the “real world” open or closed?

[NSDI 06]

How predictable are response

times under common policies?

...Is the mean a good measure

of the distribution?

[Sigmetrics 05] [QUESTA 06]

How do different scheduling

techniques / heuristics react to

customer abandonment?

[Work in progress]

MY RESEARCH THEMES

going beyond the M/GI/1

BROADER MODELS

slide57

A. Wierman, and M. Harchol-Balter. “Classifying scheduling policies with respect to unfairness in an M/GI/1.” Sigmetrics 2003.

  • A. Wierman, M. Harchol-Balter, and T. Osogami. ”Nearly insensitive bounds on SMART scheduling.” Sigmetrics 2005.
  • A. Wierman and M. Harchol-Balter. ”Classifying scheduling policieswith respect to higher moments of conditional response time.” Sigmetrics 2005.
  • C. Woo Yang, A. Wierman, S. Shakkottai, M. Harchol-Balter. “Tail asymptotics for policies favoring small jobs in a many-sources regime.” Sigmetrics 2006.
  • M. Nuyens, A. Wierman, B. Zwart. “Preventing large sojourn times using SMART scheduling.” Operations Res. 2007.
  • A. Wierman. “On the effect of inexact size information in size based policies.” MAMA 2006.

A CLASS OF POLICIES THAT PRIORITIZE SMALL JOBS Adam Wiermanhttp://www.cs.cmu.edu/~acwacw@cs.cmu.edu