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A CLASS OF POLICIES THAT PRIORITIZE SMALL JOBS Adam Wierman

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### A CLASS OF POLICIES THAT PRIORITIZE SMALL JOBSAdam Wierman

Carnegie Mellon University

Pittsburgh, PA USA

Server

Use Shortest

Remaining Processing

Time First (SRPT)

A MOTIVATIONAL EXAMPLE: Web serversBandwidth is the bottleneck

Processor

Sharing (PS)

Goal

Minimize user

response times

requests for files

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

mean response time

?

Why SRPT?

SRPT

load

0

1

A MOTIVATIONAL EXAMPLE: Web servers[Harchol-Balter, Schroeder, Bansal, Agrawal, TOCS 2003]

CPUs

Internet

Databases

Routers

Disks

Prioritize

small jobs

Locks

SCHEDULING SUCCESS STORIES ACROSS COMPUTER APPLICATIONSWeb

Servers

…also p2p, wireless, operating systems…

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

RS

PSJF

PS

LCFS

SJF

DPS

PLJF

FCFS

LJF

FSP

PLCFS

FB

LRPT

SCHEDULING SUCCESS STORIES ACROSS COMPUTER APPLICATIONSRENEWED INTEREST IN THEORETICAL WORK

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

also important

Hybrid policies outperform SRPT on these

Theorya 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

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

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

behaves like SRPT

includes practical

hybrid policies

THE SMART CLASS

SMART

PSJF

RS

SRPT

SMAll Response Times

- Bias Property
- Consistency Property
- Transitivity Property

PS

LJF

PLCFS

ROS

FCFS

LRPT

BIAS PROPERTY

If OriginalSize(a) < RemainingSize(b)

then a has priority over b

a

b

[Wierman, Harchol-Balter, Osogami 2005]

size

0

0

original size

BIAS PROPERTYIf OriginalSize(a) < RemainingSize(b)

then a has priority over b

[Wierman, Harchol-Balter, Osogami 2005]

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]

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 PROPERTYISN’T ENOUGH

CONSISTENCY

TRANSITIVITY

remaining

size

?

orig. size

[Wierman, Harchol-Balter, Osogami 2005]

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]

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

Which policies

are notSMART?

SMART

PSJF

RS

SRPT

FB

SJF

PS

LJF

PLCFS

ROS

FCFS

LRPT

Blind policies

Non-preemptive policies

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

behaves like SRPT

includes practical

hybrid policies

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]

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

?

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

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]

!

These bounds

are tight

SMART POLICIES ARE 2-COMPETITIVE

Consider the M/D/1

SRPT does FCFS (only in M/D/1). So as ρ1

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

As ρ1, E[T]PLCFS 2 E[T]SRPT

?

Tight under

Deterministic

Tight under Pareto

(as ρ1)

What is the heavy

traffic behavior?

SMART POLICIES ARE 2-COMPETITIVE

With work, the bounds can be rewritten as

Nμ

λ2

λN

NB

Many sources

Large buffer

!

SMART policies

are asymptotically

equivalent in both

TAIL BEHAVIOR OF SMARTPr(T>y)is difficult to study directly

so it is typically studied asymptotically

[Nuyens, Wierman, Zwart 2007]

[Yang, Wierman, Shakkattai,

Harchol-Balter 2006]

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…

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

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

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

OUR NEW GOALBROADENtheSMART class while keeping itTIGHT

2

k

E[T]P ≤ E[T]SRPT

include policies

that use job size estimates

?

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...

OUR NEW GOALBROADENtheSMART class while keeping itTIGHT

k

E[T]P ≤ E[T]SRPT

include policies

that use inexact job sizes

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

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

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

Nμ

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

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

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

heuristics and techniques

A NEW APPROACH

Analyze classes of policies instead of individual policies

Results apply to polices that are

implemented in practice

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

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]

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 THEMESgoing beyond individual policies

SCHEDULING CLASSIFICATIONS

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 THEMESgoing beyond the M/GI/1

BROADER MODELS

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

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