1

Pruhs, Woeginger, Uthaisombut 2004

Qos Objective: Minimize total flow time Flow time fi of a job i is completion time Ci – ri

Power Objective: constraint that at most E energy is used

We make the simplifying assumptions that all jobs have the same (unit) amount of work Optimal job selection policy?

2

Pruhs, Woeginger, Uthaisombut 2004

Qos Objective: Minimize total flow time Flow time fi of a job i is completion time Ci – ri

Power Objective: constraint that at most E energy is used

We make the simplifying assumptions that all jobs have the same (unit) amount of work In this case the optimal job selection policy is

First Come First Served. We thus focus on speed setting policy.

wlog assume, r1 ≤ r2 ≤ … ≤ rn

Warm up Exercise: n unit jobs released at time 0

How much energy to devote to each job?

3

Warm up Exercise: n unit jobs released at time 0

How much energy to devote to each job? Min Σ_i (n-i+1)/s_i

Subject to Σ_i s_i^2 <= E

Either by Lagrange multipliers or intuitive reasoning, s_i ≈ (n_i+1)^(1/3)

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Convex Program with Release Times

Min Σ_i (C_i – r_i) Subject to

Σ_i (C_i – max(r_{i}, C_{i-1})^2 <= E C_i > C_{i-1}

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KKT Optimality Conditions(2)

Consider a strictly-feasible convex differentiable program

A sufficient condition for a solution x to be optimal is theexistence of Lagrange multipliers λi such that

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KKT Optimality Conditions

Total energy of E is used Ci < ri+1 implies ρi = ρn

Ci > ri+1 implies ρi = ρi+1 + ρn

Ci = ri+1 implies ρn ≤ ρi ≤ ρi+1 + ρn Example:

Pn = p7

2pn3pnpnpn

p2

P2 + pn

r3r2 r6 r7

Offline Algorithm

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9

KKT Optimality Conditions

Algorithmic Difficulties: This doesn’t tell us the value of ρn

Solution: Binary search Don’t know the value of pi when Ci = ri+1

Solution: Can calculate since you know interval when job runs

Don’t know if Ci < ri+1, Ci = ri+1, or Ci > ri+1

Easy for high energy E, Ci < ri+1

Solution: Trace out optimal schedules as E decreases

p2

p2 + pn

r3r2

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Algorithmic Evolution

< <

= <

> =

> >

High Energy

Low Energy

> <

r2 r3

Configurations

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Intuition

Intuitively as you lose energy, should jobs run faster or slower?

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Intuition

Intuitively as you lose energy, jobs should run slower, but this intuition is false

Example: Higher energy: p1=2p3 and p2 = p3

Lower energy: p1 = 3p3 and p2 = 2 p3

p1/p2 decreases and job 2 speeds up as we lose energy

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What Goes Wrong With Arbitrary Work Jobs

≤ ≤

= ≤

≥ ≤

Arbitrary length

Open Question: What is thecomplexity of finding optimal

flow time schedules whenjobs have arbitrary work?

Optimal scheuduleis not a continuous

function of energy E

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Theorem: There is no O(1)-competitive online algorithm for the bounded energy problem Proof Idea: How much energy do you give the

first job that arrives?

If it is not an Ω(E) then you are not O(1)-competitive

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Energy/Flow Trade-Off Problem Definition [AF06]

Job i has release date ri and work yi

Optimize total flow + ρ * energy used Natural interpretation: User specifies an

energy amount ρ that he is willing to spend to get a unit improvement in responsee.g. If the user is willing to spend 1 ergs

of energy for a 3 microsecond improvement in response, then ρ=3.

wlog, ρ=1.

One job example

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Natural Policies

Job Selection?

Speed Scaling?

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Natural Policies

Job Selection: SRPT

Speed Scaling: Power = Number of unfinished jobs Increase in energy objective = increase in flow

objective

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Bansal, Chan, Pruhs, SODA 2009

We consider allowing an arbitrary power function P(s) Only require something like P is piece-wise smooth, e.g.

PowerP(s)

Speed s

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Our Results

Main Theorem: SRPT + variation of natural speed scaling algorithm is 3-competitive for the objective of flow+energy for an arbitrary power function P(s) and for arbitrary work jobs

Later improved to 2-competitive by Lachlan L. H. Andrew, Adam Wierman and Ao Tang.

Second Theorem: HDF + variation of natural speed scaling algorithm is 2-competitive for the objective of fractional weighted flow+energy for essentially any power function P(s) and for arbitrary work and weight jobs Probably not possible to get such a result for integral

weighted flow since to be competitive for weighted flow requires resource/speed augmentation [BC09]

Equation for Amortized Local Competitiveness Argument

Pon(t) + Non(t) + dΦ(t)/dt ≤ c [ Popt(t) + Nopt(t) ] P(t) is the power at time t N(t) is the unfinished jobs at time t on is the online algorithm opt is the adversary/optimal Φ(t) is the potential function c is the competitive ratio

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Initial Equation: Pon + Non + dΦ/dt ≤ 2 [ Popt + Nopt ]

Since Pon = Non, 2Pon + dΦ/dt ≤ 2 [ Popt + Nopt ]

Guess potential Φ is a function of N = Non – Nopt so job arrivals do not affect potential function

Worst case is when Nopt = 0, or equivalently when N = Non

By the chain rule dΦ/dt=dΦ/dN dN/dt. Note dN/dt=(sopt – son), 2Pon + dΦ/dN * (sopt – son) ≤ 2 * Popt

Solving for Φ gives dΦ/dN ≤ 2 [ Popt – Pon] /(sopt – son)

The term [ Popt – Pon] /(sopt – son) looks like the slope of P(s) around the point s=son.

So set dΦ/dN = 2 dP(son)/ds = 2 dP(P-1(N))/ds

Or equivalently, Φ = 2∫0

N dP(P-1(N))/ds dy

Derivation of Potential Function for Unit Work Jobs

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PowerP

Speed s

Chan, Edmonds, Pruhs 2009

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1. Nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

2. Speed scaling on a uniprocessor

SPAA 2009: Speed scaling and

nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

We define and address one algorithmic multiprocessor power management

problem

Outline of the Talk

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1. Nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

Speed-up Curves

Each portion of work/code has a speed up function that specifies how fast work is processed as a function of the number of processors assigned.

25

Ratework is

processed

Number of Processors

1

Parallel speed-up curve

Sequential speed-up curve

Arbitrary speed-up curve

Definition: Nonclairvoyant means that the scheduling algorithm doesn’t know the speed-up curves or the work of the jobs

Question: What is the most natural scheduling algorithm if one doesn’t know the speed-up curves?

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Definition: Nonclairvoyant means that the scheduling algorithm doesn’t know the speed-up curves or the work of the jobs

Question: What is the most natural scheduling algorithm if one doesn’t know the speed-up curves?

Answer: Equipartition (round-robin, processor sharing) which assigns an equal number of processors to each job

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Theorem [E99]: Equipartition is 2+εspeed O(1)-competitive The average waiting time using Equipartition at

at most a constant factor larger than the optimal schedule on processors that are slightly slower than ½ as fast

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Question: What else can a nonclairvoyant algorithm do besides sharing the processor equally among the jobs that might be better?

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Question: What else can a nonclairvoyant algorithm do besides sharing the processor equally among the jobs that might be better?

Answer: Late Arrival Processor Sharing (LAPS) which shares the processors equally among the latest arriving δn jobs

Theorem [EP09]: LAPS is 1+εspeed O(1)-competitive Note that 1+εspeed is required to be

competitive even if the online algorithm knows the speed-up curves and the work of each job

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Analysis of Equipartition and LAPS

Key Lemma: The worst case is if all work is parallel or sequential

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Outline of the Talk

32

1. Nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

2. Speed scaling on a uniprocessor

SPAA 2009: Speed scaling and

nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

Outline of the Talk

33

2. Speed scaling on a uniprocessor

Second most natural nonclairvoyant online algorithm Job selection: LAPS

Recall LAPS shares the processing power equally among the latest arrive constant fraction of the jobs

Speed scaling: Power = number of unfinished jobs

Theorem [CELLMP09]: The above algorithm is O(1)-competitive for total flow time plus energy Proof: amortized local competitiveness argument

34

Outline of the Talk

35

1. Nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

2. Speed scaling on a uniprocessor

SPAA 2009: Speed scaling and

nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

Outline of the Talk

36

SPAA 2009: Speed scaling and

nonclairvoyant scheduling of jobs with

arbitrary speed-up curves on a

multiprocessor

Problem we address in SPAA 2009 paper

Nonclairvoyantly scheduling and speed scaling on a multiprocessor so as to minimize total flow time plus energy

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Schedule

Job 1

Height =speed

Processor 1

Processor 2

Warm-up Problem

Question: If you are running a single parallel job on m processors, what should the speed s be?

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Warm-up Problem

Question: If you are running a single parallel job on m processors, what should the speed s be?

Answer: To optimize flow+energy you should equate

flow and energy, since x+y = Θ(max(x, y)) Thus you want the rate of increase of flow

(which is the number of unfinished jobs) to equal the rate of increase in energy (which is power P).

Therefore total power = mP=1 P=1/m or equivalently s = 1/m1/3 since P = s3

Note that the total power used is independent of the number of machines

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Impossibility Theorem: There is no algorithm who competitiveness scales reasonably with the number of processors

Proof: Consider an instance of a single job that is either parallel or sequential

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Impossibility Theorem: There is no algorithm who competitiveness scales reasonably with the number of processors

Proof: Consider an instance of a single job that is either parallel or sequential If you run the job on few processors, then either the flow

or energy must be much higher than optimal in the case that the job is parallel

If you run the job on many processors, then either the flow or the energy must be much higher than optimal in the case that the job is parallel

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Question: If you have only one job that you know is either parallel or sequential, how would you schedule it?

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Question: If you have only one job that you know is either parallel or sequential, how would you schedule it?

Answer: Run one copy at speed 1 on one processor and

one copy at speed 1/m1/3 on the rest of the processors

This is O(1)-competitive This is allowable if the job has no side effects

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Impossibility Theorem: There is no algorithm whose competitiveness scales reasonably with the number of processors if jobs can have side effects

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Candidate algorithm for 1 job instance: Run one copy on 2i processors at power 1/2i

This candidate algorithm works if all portions of the job has a single speed up curve

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Candidate algorithm for 1 job instance: Run one copy on 2i processors at power 1/2i

Impossibility Theorem: The candidate algorithm, and no other possible nonclairvoyant algorithm, has a competitive ratio that scales reasonably with the number of processors Proof: Consider a jobs where the speed-up

curves of the different portions of the jobs change over time

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Question: If you have only one job where different portions have different speed up curves, how would you schedule it?

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Question: If you have only one job where different portions have different speed up curves, how would you schedule it?

Answer: Run one copy on 2i processors at power 1/2i

checkpointing constantly This is O(log m) competitive

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Impossibility Theorem: There is no algorithm who competitiveness scales reasonably with the number of processors if jobs can have side effects or can not be checkpointed

But we can get a positive result if jobs don’t have side effects and we can checkpoint cheaply

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Main Theorem: If jobs have no side effects and are checkpointable then there a nonclairvoyant scheduling and speed scaling algorithm that is O(log m) competitive for the objective of flow + energy

Corollary: O(1)-competitiveness is possible for clairvoyant online algorithms

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Main Theorem: If jobs have no side effects and are checkpointable then there a nonclairvoyant scheduling and speed scaling algorithm that is O(log m) competitive for the objective of flow + energy

Algorithm Description: Scheduling: LAPS, plus run copies at a faster

rate on fewer processors in case the jobs are not parallel, checkpointing constantly

Speed Scaling: Natural algorithm that equates power and number of unfinished jobs

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Algorithm Analysis

Key Lemma: The worst case is if every speed up curve is parallel up to some number of processors and then is sequential

Contrast this to the fixed speed processor case where the worst case is if all work is either parallel or sequential

The rest of the analysis is a reduction to the single processor case

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Ratework is

processed

Number of Processors

Speed up curve