Energy Efficient Deadline Scheduling in TwoProcessor Systems

Tak-Wah Lam1 Lap-Kei Lee1 Isaac K. K. To2 Prudence W. H.Wong2

1Department of Computer ScienceUniversity of Hong Kong

2Department of Computer ScienceUniversity of Liverpool

CTAG seminar 2008 MarchBased on ISAAC 2007 presentation

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 1 / 27

Problem and contribution

Outline

1 Problem and contributionSpeed scalingOnline real-time schedulingKnown bounds and our contribution

2 Main ideas in algorithm and proofReview of previous algorithmsNew algorithm: Slow-SR

3 Open Problems and Summary

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 2 / 27

Problem and contribution Speed scaling

Saving energy via Speed scaling

Experience using laptops: Running slower saves energy.Processors can dynamically slow down (usually in 0.5ms).Why slow is good? Transistors are capacitor-like. . .

When state not changing, no current⇒ no energy used.To switch state, a current flows to (dis)charge the base oftransistors, which expend energy.

� Transistors also leaks, making them somewhat resistor-like. Let’s ignore such details.

Running slower reduces frequency f of state changes.Nearly useless: you just spend energy later.� Cooling requirement is reduced, though.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 3 / 27

Problem and contribution Speed scaling

Saving energy via Speed scaling

Experience using laptops: Running slower saves energy.Processors can dynamically slow down (usually in 0.5ms).Why slow is good? Transistors are capacitor-like. . .

When state not changing, no current⇒ no energy used.To switch state, a current flows to (dis)charge the base oftransistors, which expend energy.

� Transistors also leaks, making them somewhat resistor-like. Let’s ignore such details.

Running slower reduces frequency f of state changes.Nearly useless: you just spend energy later.� Cooling requirement is reduced, though.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 3 / 27

Problem and contribution Speed scaling

Saving energy via Speed scaling

Experience using laptops: Running slower saves energy.Processors can dynamically slow down (usually in 0.5ms).Why slow is good? Transistors are capacitor-like. . .

When state not changing, no current⇒ no energy used.To switch state, a current flows to (dis)charge the base oftransistors, which expend energy.

� Transistors also leaks, making them somewhat resistor-like. Let’s ignore such details.

Running slower reduces frequency f of state changes.Nearly useless: you just spend energy later.� Cooling requirement is reduced, though.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 3 / 27

Problem and contribution Speed scaling

But it does help!

When frequency is lowered: voltage V required for reliableoperation is reduced.Power consumption (should) varies as V 2f !E.g., Pentium M spec (1.6GHz)

f V P P/f1.6 GHz 1.484V 24.5W 15.3nW0.6 GHz 0.956V 6W 10nW

Looks like P/f varies as V instead of V 2.

Ideally: If we assume f and V required are proportional, powervaries as f 3, energy varies as f 2.� Not quite the case: f varies as (V − Vt )

2/V , typical Vt of 0.5V ruins the day. Still, a

reasonable approximation, especially if Vt can be lowered.

Our model: Processor speed adjustable. When running atspeed s, spend power sα (α is around 2 or 3).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 4 / 27

Problem and contribution Speed scaling

But it does help!

When frequency is lowered: voltage V required for reliableoperation is reduced.Power consumption (should) varies as V 2f !E.g., Pentium M spec (1.6GHz)

f V P P/f1.6 GHz 1.484V 24.5W 15.3nW0.6 GHz 0.956V 6W 10nW

Looks like P/f varies as V instead of V 2.

Ideally: If we assume f and V required are proportional, powervaries as f 3, energy varies as f 2.� Not quite the case: f varies as (V − Vt )

2/V , typical Vt of 0.5V ruins the day. Still, a

reasonable approximation, especially if Vt can be lowered.

Our model: Processor speed adjustable. When running atspeed s, spend power sα (α is around 2 or 3).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 4 / 27

Problem and contribution Speed scaling

But it does help!

When frequency is lowered: voltage V required for reliableoperation is reduced.Power consumption (should) varies as V 2f !E.g., Pentium M spec (1.6GHz)

f V P P/f1.6 GHz 1.484V 24.5W 15.3nW0.6 GHz 0.956V 6W 10nW

Looks like P/f varies as V instead of V 2.

Ideally: If we assume f and V required are proportional, powervaries as f 3, energy varies as f 2.� Not quite the case: f varies as (V − Vt )

2/V , typical Vt of 0.5V ruins the day. Still, a

reasonable approximation, especially if Vt can be lowered.

Our model: Processor speed adjustable. When running atspeed s, spend power sα (α is around 2 or 3).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 4 / 27

Problem and contribution Online real-time scheduling

Real-time scheduling

Who decides to go slow? One possibility: schedulers.Our focus: firm deadline scheduling

Input is a sequence of jobs.Each job has some work to complete by processor(s).Each job has a deadline.

Online real-time scheduling:Learn everything about a job when it is released.The algorithm decides the job and speed to run now.

We want to reduce energy consumption. . .But we don’t want to give up too much performance—we measurethat by throughput of jobs completed by deadline.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 5 / 27

Problem and contribution Online real-time scheduling

Real-time scheduling

Who decides to go slow? One possibility: schedulers.Our focus: firm deadline scheduling

Input is a sequence of jobs.Each job has some work to complete by processor(s).Each job has a deadline.

Online real-time scheduling:Learn everything about a job when it is released.The algorithm decides the job and speed to run now.

We want to reduce energy consumption. . .But we don’t want to give up too much performance—we measurethat by throughput of jobs completed by deadline.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 5 / 27

Problem and contribution Online real-time scheduling

Real-time scheduling

Who decides to go slow? One possibility: schedulers.Our focus: firm deadline scheduling

Input is a sequence of jobs.Each job has some work to complete by processor(s).Each job has a deadline.

Online real-time scheduling:Learn everything about a job when it is released.The algorithm decides the job and speed to run now.

We want to reduce energy consumption. . .But we don’t want to give up too much performance—we measurethat by throughput of jobs completed by deadline.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 5 / 27

Problem and contribution Online real-time scheduling

Example schedule

Here α = 2. . .

j2j1

Jobs

S1

Time

Speed 1

2× 12 + 2× 0.52 = 2.5

Energy

S2

Time

Speed 0.75

4× 0.752 = 2.25

Speed 0.5

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 6 / 27

Problem and contribution Online real-time scheduling

Infinite speed model

Original model“Infinite speed model” [Yao, Demers, Shenker; FOCS 1995]

s is arbitrary non-negative number.Can be as large as the algorithm want.

Requirements of algorithms:All jobs can (and must) be completed.Want to be competitive in energy.

Easier for theoretical work, but not very realistic.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 7 / 27

Problem and contribution Online real-time scheduling

Infinite speed model

Original model“Infinite speed model” [Yao, Demers, Shenker; FOCS 1995]

s is arbitrary non-negative number.Can be as large as the algorithm want.

Requirements of algorithms:All jobs can (and must) be completed.Want to be competitive in energy.

Easier for theoretical work, but not very realistic.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 7 / 27

Problem and contribution Online real-time scheduling

Bounded speed model

Real processors cannot run at arbitrary speed.“Bounded speed model” [Chan, Chan, Lam, Lee, Mak, Wong; SODA 2007]

s ≤ T for some speed bound T (it suffices to consider just theT = 1 case).But now: Some jobs may need to be abandoned.

Requirements of algorithms:Find the “best” schedule:

The one which maximizes throughput.And among those, the one which minimizes energy.

Want to be competitive in both throughput and energy against thisparticular schedule.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 8 / 27

Problem and contribution Online real-time scheduling

Bounded speed model

Real processors cannot run at arbitrary speed.“Bounded speed model” [Chan, Chan, Lam, Lee, Mak, Wong; SODA 2007]

s ≤ T for some speed bound T (it suffices to consider just theT = 1 case).But now: Some jobs may need to be abandoned.

Requirements of algorithms:Find the “best” schedule:

The one which maximizes throughput.And among those, the one which minimizes energy.

Want to be competitive in both throughput and energy against thisparticular schedule.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 8 / 27

Problem and contribution Online real-time scheduling

Reasonable requirement?

Example. . .If the best algorithm completes 100 units of work, and to completeso much work requires at least 100 units of energy. . .To be 4-competitive in both work and energy: an algorithm need atleast 25 units of work with at most 400 units of energy.

Sounds unfair?Fix energy and maximize throughput: impossible.Can we compare against optimal energy schedule that completesjust as much work as online (perhaps 25)?No idea at all.� Optimal might require much less than 100 (or even 25) units of energy!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 9 / 27

Problem and contribution Online real-time scheduling

Reasonable requirement?

Example. . .If the best algorithm completes 100 units of work, and to completeso much work requires at least 100 units of energy. . .To be 4-competitive in both work and energy: an algorithm need atleast 25 units of work with at most 400 units of energy.

Sounds unfair?Fix energy and maximize throughput: impossible.Can we compare against optimal energy schedule that completesjust as much work as online (perhaps 25)?No idea at all.� Optimal might require much less than 100 (or even 25) units of energy!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 9 / 27

Problem and contribution Online real-time scheduling

Reasonable requirement?

Example. . .If the best algorithm completes 100 units of work, and to completeso much work requires at least 100 units of energy. . .To be 4-competitive in both work and energy: an algorithm need atleast 25 units of work with at most 400 units of energy.

Sounds unfair?Fix energy and maximize throughput: impossible.Can we compare against optimal energy schedule that completesjust as much work as online (perhaps 25)?No idea at all.� Optimal might require much less than 100 (or even 25) units of energy!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 9 / 27

Problem and contribution Known bounds and our contribution

Single processor scheduling algorithms

Infinite speed (complete all jobs)Optimal Available (OA): Energy αα-competitive [Bansal, Kimbrel, Pruhs;FOCS 2004].BKP: Energy 2( α

α−1 )αeα-competitive [Bansal, Kimbrel, Pruhs; FOCS2004].No algorithm is better than eα-competitive when α is large [ditto].qOA: When α = 3, Energy 6.7-competitive [Bansal, Chan, Pruhs; 2008].

Bounded speed:FSA-OAT: Throughput 14-competitive; Energy(αα + 4αα2)-competitive [Chan, Chan, Lam, Lee, Mak, Wong; SODA 2007].Slow-D: Throughput 4-competitive; Energy (αα + 4αα2)-competitive[Bansal, Chan, Lam, Lee; 2008].� 4-competitive is the best we can hope for!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 10 / 27

Problem and contribution Known bounds and our contribution

Single processor scheduling algorithms

Infinite speed (complete all jobs)Optimal Available (OA): Energy αα-competitive [Bansal, Kimbrel, Pruhs;FOCS 2004].BKP: Energy 2( α

α−1 )αeα-competitive [Bansal, Kimbrel, Pruhs; FOCS2004].No algorithm is better than eα-competitive when α is large [ditto].qOA: When α = 3, Energy 6.7-competitive [Bansal, Chan, Pruhs; 2008].

Bounded speed:FSA-OAT: Throughput 14-competitive; Energy(αα + 4αα2)-competitive [Chan, Chan, Lam, Lee, Mak, Wong; SODA 2007].Slow-D: Throughput 4-competitive; Energy (αα + 4αα2)-competitive[Bansal, Chan, Lam, Lee; 2008].� 4-competitive is the best we can hope for!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 10 / 27

Problem and contribution Known bounds and our contribution

Single processor scheduling algorithms

Infinite speed (complete all jobs)Optimal Available (OA): Energy αα-competitive [Bansal, Kimbrel, Pruhs;FOCS 2004].BKP: Energy 2( α

α−1 )αeα-competitive [Bansal, Kimbrel, Pruhs; FOCS2004].No algorithm is better than eα-competitive when α is large [ditto].qOA: When α = 3, Energy 6.7-competitive [Bansal, Chan, Pruhs; 2008].

Bounded speed:FSA-OAT: Throughput 14-competitive; Energy(αα + 4αα2)-competitive [Chan, Chan, Lam, Lee, Mak, Wong; SODA 2007].Slow-D: Throughput 4-competitive; Energy (αα + 4αα2)-competitive[Bansal, Chan, Lam, Lee; 2008].� 4-competitive is the best we can hope for!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 10 / 27

Problem and contribution Known bounds and our contribution

Multiple processors algorithms

Infinite speed (complete all jobs)Algorithm known only when jobs are known to have “agreeabledeadline”, i.e., same ordering as release time.Energy is 16ααα-competitive. [Albers, Fujiwara; SPAA 2007]

Bounded speedNo known result.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 11 / 27

Problem and contribution Known bounds and our contribution

Our Results in ISAAC 07

Focus on simpler case: 2 processors only.Infinite speed model:

Trivial algorithm: use only one processor, (2α−1αα)-competitive.Bounded speed model:

Trivial algorithm: throughput 8-competitive, energy(2α−1αα + 22α−1α2)-competitiveSlow-SR: modify Slow-D using ideas of the 2-processor algorithmSafe-Risky [Koren, PhD Thesis 1993].Throughput: 3-competitive (lower bound of 2-competitive known).� Safe-Risky is 2-competitive in throughput when migration is allowed, 3-competitivewhen migration is not allowed, no guarantee on energy.Energy: (2ααα + 22αα2)-competitive.Need migration.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 12 / 27

Problem and contribution Known bounds and our contribution

Our Results in ISAAC 07

Focus on simpler case: 2 processors only.Infinite speed model:

Trivial algorithm: use only one processor, (2α−1αα)-competitive.Bounded speed model:

Trivial algorithm: throughput 8-competitive, energy(2α−1αα + 22α−1α2)-competitiveSlow-SR: modify Slow-D using ideas of the 2-processor algorithmSafe-Risky [Koren, PhD Thesis 1993].Throughput: 3-competitive (lower bound of 2-competitive known).� Safe-Risky is 2-competitive in throughput when migration is allowed, 3-competitivewhen migration is not allowed, no guarantee on energy.Energy: (2ααα + 22αα2)-competitive.Need migration.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 12 / 27

Main ideas in algorithm and proof

Outline

1 Problem and contributionSpeed scalingOnline real-time schedulingKnown bounds and our contribution

2 Main ideas in algorithm and proofReview of previous algorithmsNew algorithm: Slow-SR

3 Open Problems and Summary

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 13 / 27

Main ideas in algorithm and proof Review of previous algorithms

OA

Single processor algorithm, Infinite speed modelOA algorithm

At any time, there is a minimum speed, at which we can stillcomplete all jobs by keep running at that speed.Just use that speed to run the earliest deadline job.

A very conservative algorithm: never use more speed than enough.� This makes many analysis much easier.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 14 / 27

Main ideas in algorithm and proof Review of previous algorithms

Example

j2j1

j3Jobs

OA Load

TimeOPT Load

Time

4.75

3.75

Energy(α = 2)

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 15 / 27

Main ideas in algorithm and proof Review of previous algorithms

Slow-D (Finite speed model)

Run a simulated copy of OA.Since OA runs in infinite model, it meets all deadlines, but may runat speed > 1.Slow-D imitates OA whenever OA speed ≤ 1 (slow time).Problems with OA speed > 1 (fast time): must give up some jobs.

Definition (Slow-down time tslow)The future time when speed fall back to at most 1.

Definition (Fast jobs)Jobs with deadline at or before tslow.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 16 / 27

Main ideas in algorithm and proof Review of previous algorithms

Jobs life-cycle in Slow-D

JobRelease

Slow Job

CompletedAdmittedFast Job

WaitingFast Job

Abandoned

Deadline> tslow

tslow increasesto > deadline

Active fast jobs may be admitted, waiting or abandoned. Slow-D run theearliest deadline admitted fast job at full speed.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 17 / 27

Main ideas in algorithm and proof Review of previous algorithms

Jobs life-cycle in Slow-D

JobRelease

Slow Job

CompletedAdmittedFast Job

WaitingFast Job

Abandoned

Deadline> tslow

Feasible

NotFeasible

tslow increasesto > deadline

Add job as admitted jobs if resulting set of admitted jobs is feasible.Otherwise it waits.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 17 / 27

Main ideas in algorithm and proof Review of previous algorithms

Jobs life-cycle in Slow-D

JobRelease

Slow Job

CompletedAdmittedFast Job

WaitingFast Job

Abandoned

Deadline> tslow

Feasible

NotFeasible

tslow increasesto > deadline

LST;large

LST;small

Accept anotherLST job

When a waiting fast job has its last chance (LST), Slow-D must abandoneither that or some admitted fast jobs.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 17 / 27

Main ideas in algorithm and proof Review of previous algorithms

Slow-D example run

j2

j1

j3Jobs

OA

Load

Time

j4

Slow-DTime

Load

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 18 / 27

Main ideas in algorithm and proof Review of previous algorithms

Slow-D example run

j2

j1

j3Jobs

OA

Load

Time

j4

Slow-DTime

Load

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 18 / 27

Main ideas in algorithm and proof Review of previous algorithms

Slow-D example run

j2

j1

j3Jobs

OA

Load

Time

j4

Slow-DTime

Load

Give up j3!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 18 / 27

Main ideas in algorithm and proof Review of previous algorithms

Slow-D example run

j2

j1

j3Jobs

OA

Load

Time

j4

Slow-DTime

Load

Want j4, abandon j2!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 18 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR

How the extra processor help? To have a cake and eat it too.Slow-SR algorithm (Changes from Slow-D)

When a waiting fast job has LST. . .Rather than abandoning admitted jobs, we run the fast job aboutto fail in the extra processor: Risky Processor (RP).If there is already another job running there, abandon the smallerone.Job in RP migrates to the “Slow-D Processor” (SP) if there is nomore fast job in it.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 19 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR

How the extra processor help? To have a cake and eat it too.Slow-SR algorithm (Changes from Slow-D)

When a waiting fast job has LST. . .Rather than abandoning admitted jobs, we run the fast job aboutto fail in the extra processor: Risky Processor (RP).If there is already another job running there, abandon the smallerone.Job in RP migrates to the “Slow-D Processor” (SP) if there is nomore fast job in it.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 19 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

From Slow-D to Slow-SR

Slow-D. . .

JobRelease

Slow Job

CompletedAdmittedFast Job

WaitingFast Job

Abandoned

Deadline> tslow

Feasible

NotFeasible

tslow increasesto > deadline

LST;large

LST;small

Accept anotherLST job

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 20 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

From Slow-D to Slow-SR

Slow-SR. . .

JobRelease

Slow Job(SP)

CompletedAdmittedFast Job (SP)

UrgentFast Job (RP)Waiting

Fast JobAbandoned

Deadline> tslow

Feasible

NotFeasible

LST: normal

LST; smallerthan running

A larger jobhas LST

No acceptedjobs in SP

tslow increasesto > deadline

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 20 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Our example

j2

j1

j3Jobs

OA

Load

Time

j4

Time

Load

RP

SP

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 21 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Our example

j2

j1

j3Jobs

OA

Load

Time

j4

Time

Load

SP

RP

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 21 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Our example

j2

j1

j3Jobs

OA

Load

Time

j4

Time

Load

SP

RP

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 21 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Our example

j2

j1

j3Jobs

OA

Load

Time

j4

Time

Load

SP

RPGive up j3: j4 better!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 21 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Our example

j2

j1

j3Jobs

OA

Load

Time

j4

Time

Load

SP

RP Migrate!

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 21 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Energy guarantee

For the trivial algorithm (throw away 1 processor and run Slow-D):Boring extension of the Slow-D techniques, taking into account thatoptimal can now use 2 processors.This shows (single processor) OA clamped at speed-1 is(2α−1αα + 22α−1α2)-competitive.

For Slow-SR:We can easily show that RP works only during fast time.Also, RP only works when SP is working.So Slow-SR cannot use more than twice as much energy.⇒ (2ααα + 22αα2)-competitive.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 22 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Energy guarantee

For the trivial algorithm (throw away 1 processor and run Slow-D):Boring extension of the Slow-D techniques, taking into account thatoptimal can now use 2 processors.This shows (single processor) OA clamped at speed-1 is(2α−1αα + 22α−1α2)-competitive.

For Slow-SR:We can easily show that RP works only during fast time.Also, RP only works when SP is working.So Slow-SR cannot use more than twice as much energy.⇒ (2ααα + 22αα2)-competitive.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 22 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Throughput guarantee

Split input jobs into two sets. . .Ls: Those which are slow jobs when released.Lf : Those which are fast jobs when released.

Optimal algorithm:Ls: Might complete all.Lf : Might complete 2F work, where F is length of fast time.� These jobs can only run during fast time.So at most |Ls|+ 2F throughput.

Slow-SR:Ls: Complete all.Lf : No guarantee! (Can reject nearly everything.)But. . . all fast jobs ever admitted is guaranteed to complete.Can guarantee at least F work during fast time.At least max{|Ls|,F} throughput.

This means 3-competitive.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 23 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Throughput guarantee

Split input jobs into two sets. . .Ls: Those which are slow jobs when released.Lf : Those which are fast jobs when released.

Optimal algorithm:Ls: Might complete all.Lf : Might complete 2F work, where F is length of fast time.� These jobs can only run during fast time.So at most |Ls|+ 2F throughput.

Slow-SR:Ls: Complete all.Lf : No guarantee! (Can reject nearly everything.)But. . . all fast jobs ever admitted is guaranteed to complete.Can guarantee at least F work during fast time.At least max{|Ls|,F} throughput.

This means 3-competitive.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 23 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Throughput guarantee

Split input jobs into two sets. . .Ls: Those which are slow jobs when released.Lf : Those which are fast jobs when released.

Optimal algorithm:Ls: Might complete all.Lf : Might complete 2F work, where F is length of fast time.� These jobs can only run during fast time.So at most |Ls|+ 2F throughput.

Slow-SR:Ls: Complete all.Lf : No guarantee! (Can reject nearly everything.)But. . . all fast jobs ever admitted is guaranteed to complete.Can guarantee at least F work during fast time.At least max{|Ls|,F} throughput.

This means 3-competitive.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 23 / 27

Main ideas in algorithm and proof New algorithm: Slow-SR

Slow-SR: Throughput guarantee

Split input jobs into two sets. . .Ls: Those which are slow jobs when released.Lf : Those which are fast jobs when released.

Optimal algorithm:Ls: Might complete all.Lf : Might complete 2F work, where F is length of fast time.� These jobs can only run during fast time.So at most |Ls|+ 2F throughput.

Slow-SR:Ls: Complete all.Lf : No guarantee! (Can reject nearly everything.)But. . . all fast jobs ever admitted is guaranteed to complete.Can guarantee at least F work during fast time.At least max{|Ls|,F} throughput.

This means 3-competitive.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 23 / 27

Open Problems and Summary

Outline

1 Problem and contributionSpeed scalingOnline real-time schedulingKnown bounds and our contribution

2 Main ideas in algorithm and proofReview of previous algorithmsNew algorithm: Slow-SR

3 Open Problems and Summary

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 24 / 27

Open Problems and Summary

Open problems

Can we do better than 3-competitive?� Safe-Risky is 2-competitive instead.

Could Slow-SR be much better actually? We are really wasting alot by that silly max!

We can denote “work processed during slow time” as ps. ThenSlow-SR has at least ps + F throughput.F could mostly be slow jobs, optimal might complete them earlyand find another 2F fast-job work to run in fast time.But it really cannot happen that ps = 0. By guaranteeing a ratiobetween ps and F , it is promising to get better competitive ratio.� My conjecture: around 2.31.

Can we have 2-competitive algorithms?Slow-SR is definitely not one (we have bad examples).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 25 / 27

Open Problems and Summary

Open problems

Can we do better than 3-competitive?� Safe-Risky is 2-competitive instead.

Could Slow-SR be much better actually? We are really wasting alot by that silly max!

We can denote “work processed during slow time” as ps. ThenSlow-SR has at least ps + F throughput.F could mostly be slow jobs, optimal might complete them earlyand find another 2F fast-job work to run in fast time.But it really cannot happen that ps = 0. By guaranteeing a ratiobetween ps and F , it is promising to get better competitive ratio.� My conjecture: around 2.31.

Can we have 2-competitive algorithms?Slow-SR is definitely not one (we have bad examples).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 25 / 27

Open Problems and Summary

Open problems

Can we do better than 3-competitive?� Safe-Risky is 2-competitive instead.

Could Slow-SR be much better actually? We are really wasting alot by that silly max!

We can denote “work processed during slow time” as ps. ThenSlow-SR has at least ps + F throughput.F could mostly be slow jobs, optimal might complete them earlyand find another 2F fast-job work to run in fast time.But it really cannot happen that ps = 0. By guaranteeing a ratiobetween ps and F , it is promising to get better competitive ratio.� My conjecture: around 2.31.

Can we have 2-competitive algorithms?Slow-SR is definitely not one (we have bad examples).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 25 / 27

Open Problems and Summary

Open problems

Can we do better than 3-competitive?� Safe-Risky is 2-competitive instead.

Could Slow-SR be much better actually? We are really wasting alot by that silly max!

We can denote “work processed during slow time” as ps. ThenSlow-SR has at least ps + F throughput.F could mostly be slow jobs, optimal might complete them earlyand find another 2F fast-job work to run in fast time.But it really cannot happen that ps = 0. By guaranteeing a ratiobetween ps and F , it is promising to get better competitive ratio.� My conjecture: around 2.31.

Can we have 2-competitive algorithms?Slow-SR is definitely not one (we have bad examples).

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 25 / 27

Open Problems and Summary

Open problems (cont’d)

Can we avoid migration?� For 3-competitive throughput, a variant of Safe-Risky is non-migratory

Classical solution: replace migration with processor renaming.Difficulty: slow jobs may have been partially executed and thenbecome fast jobs.If processor it had executed is renamed as risky: might not be ableto run at all!

More random thoughtsCan we lower the energy requirement? E.g., base on one of thenewer algorithms?Can we make algorithms that work for more processors?Can we incorporate static energy consumption and sleep states?

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 26 / 27

Open Problems and Summary

Open problems (cont’d)

Can we avoid migration?� For 3-competitive throughput, a variant of Safe-Risky is non-migratory

Classical solution: replace migration with processor renaming.Difficulty: slow jobs may have been partially executed and thenbecome fast jobs.If processor it had executed is renamed as risky: might not be ableto run at all!

More random thoughtsCan we lower the energy requirement? E.g., base on one of thenewer algorithms?Can we make algorithms that work for more processors?Can we incorporate static energy consumption and sleep states?

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 26 / 27

Open Problems and Summary

Open problems (cont’d)

Can we avoid migration?� For 3-competitive throughput, a variant of Safe-Risky is non-migratory

Classical solution: replace migration with processor renaming.Difficulty: slow jobs may have been partially executed and thenbecome fast jobs.If processor it had executed is renamed as risky: might not be ableto run at all!

More random thoughtsCan we lower the energy requirement? E.g., base on one of thenewer algorithms?Can we make algorithms that work for more processors?Can we incorporate static energy consumption and sleep states?

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 26 / 27

Open Problems and Summary

Summary

New migratory algorithm Slow-SR for 2-processor systems.Throughput 3-competitive, energy (2ααα + 4αα2)-competitive.Whole bunch of open problems.

Lam, Lee, To, Wong (HKU, UoL) 2-processor energy efficient scheduling CTAGS 2008 27 / 27