Energy Efficient Flow Time Schedulingcgi.csc.liv.ac.uk/~ctag/seminars/prudence_ajc2008.pdf ·...

Energy Efficient Flow Time Scheduling

Prudence WongPrudence WongUniversity of Liverpool

CTAG Seminar

Joint work with Tak-Wah LAM, Lap-Kei LEE (U of Hong Kong)Isaac K.K. TO (Liverpool)

Appeared in ESA 2008

Outline�Minimizing flow time + energy

�Previous work & summary of results

�Algorithm and ideas of analysis� Active Job Count (AJC)� Active Job Count (AJC)

� Potential analysis

�Concluding remarks

CTAG Seminar 2008.10.30 Energy Efficient Flow Time Scheduling

Processor�Pentium M spec (1.6 GHz)

�IBM PowerPC 970FX

f V P

1.6 GHz 1.484 V 24.5 W

0.6 GHz 0.956 V 6 W

CTAG Seminar 2008.10. 30Speed Scaling Functions for Flow Time Scheduling

based on Active Job Count

Energy EfficiencyFor mobile devices, energy efficiency is

a major concern.

How to save energy? Dynamic speed (voltage) scaling

�Slow down processor whenever possible

�To run at speed s, power = sα where α>1(literature: α = 2 or 3)


Examples�Assume α = 3, a job with work = 2

time0 1 2

time

energy: 13 × 2 = 2

energy: 23 × 1 = 8

s = 1

s = 2


time0 1 2

time0 1 2

energy: (2/3)3 x (3/2) + 23 x 1/2

= 40/9

� slower speed saves energy� same speed over same interval saves energy

s = 2/3 & s = 2

Orthogonal Objectives�Energy efficiency: run jobs slower

�Quality of service: run jobs faster

�Examples of QoS� deadline feasibility, flow time, etc� deadline feasibility, flow time, etc

�In previous talks: deadline feasibility� a job has an arrival time a, deadline d,

amount of work w

� slowest possible speed w/(d-a)

�This time: flow timeCTAG Seminar 2008.10.30 Energy Efficient Flow Time Scheduling

Flow time�Flow time: time elapsed from release to

complete

�We want: small flow time, small energy

�Objective: minimize F+E [Albers and �Objective: minimize F+E [Albers and Fujiwara '06]

� minimize F subject to a bounded E?�unfortunately, there is no O(1)-competitive

online algorithm


Online Scheduling�Input: jobs arriving at arbitrary times

�each with some work to complete

�Online scheduling� learn about a job only when it is released

� decide which job to run and at what speed

�Competitive analysis� worst case ratio of F+E of online algorithm

to that of optimal algorithm

�Infinite speed vs bounded speed modelCTAG Seminar 2008.10.30 Energy Efficient Flow Time Scheduling

Example�Assume α = 2

J

speed

J0 1

speed

Energy E Flow F E+F

12x1=1 1 2S1

s = 1


J

s = 0.5

speed

J 0.52x2=0.5 2 2.5S2

Schedule S1 is better than S2

Example�Assume α = 2

J1

J2

0 1 2

speed

best s = 1.25 s = 1

r(J1) = 0 p(J1) = 1

r(J2) = 0.8 p(J2) = 1

E(S2) – E(S1) = -0.25

F(S2)-F(S1) = 0.4


J1

best scheduleJ2

s = 1.25 s = 1

speed

J1 J2

s = 1 flow time inc too much

S1

S2

F(S2)-F(S1) = 0.4

� depends on overlap of span of jobs

� we don't know efficient optimal offline algorithm

Best Speed for a Single Job�Given a job of size p

�Suppose we run the job at speed s

�E = sα(p/s) = sαααα-1p & F = p/sE + F = p(sαααα-1 + 1/s)E + F = p(sαααα-1 + 1/s)

�Differentiating, min E+F when s =

�When we have n jobs (different sizes), critical threshold


α

α

1

)1

1(

−

α

α

1

)1

(−

n

Previous Work�Unit size jobs [Albers and Fujiwara STACS 06]

� Speed function: (n/c)1/αααα, c is constant

� Competitive ratio 8.3e(1+Φ)αααα, Φ is golden ratio

�Arbitrary size [Bansal, Pruhs, Stein SODA 07]Arbitrary size � job selection: SJF (shortest job first)

� Speed: function of remaining work


αααα = 2 αααα = 3

Infinite speed model

AF >150 >400

BPS 5.236 7.940

Bounded speed model

BPS 10.472 with max speed 1.618T

11.910 withmax speed 1.466T

preemption

BPS Algorithm�BPS-fractional (use SJF)

� Speed w1/αααα, w is ∑fraction of jobs remained

�BPS (with speed advantage 1+εεεε)� Simulate BPS-F; same job; speed (1+εεεε) faster


Simulate BPS-F; same job; speed (1+ ) faster� idle if simulated schedule runs a job we finish

CTAG Seminar 2008.10.30

J1J2

J3J4

speed

scheduleJ1 J2 J1 J4 J3

BPS Algorithm�Somewhat strange schedule?

� continuously changing speed, sometimes lazy

� required extra speed in bounded speed model�speed may be lower than threshold

� not applicable in non-clairvoyant model, where

α

α

1

)1

(−

n


� not applicable in non-clairvoyant model, where job size is unknown on arrival


J1J2

J3J4

speed

scheduleJ1 J2 J1 J4 J3

Our Contribution�SRPT+AJC (active job count)

� β1- & β2-comp for infinite & bounded speed�β1= 2/(1- (α-1)/αα/(α-1)) → 2α/ln α

�β2= 2(α+1)/(α-(α-1)/(α+1)1/(α-1)) → 2α/ln α

�N.B.: µεγ1, µεγ2 → 2(α/ln α)2

shortest remaining processing time


�N.B.: µεγ1, µεγ2 → 2(α/ln α)

�Non-clairvoyant scheduling, batched case: (2-1/α)-comp & 2-comp

αααα = 2 αααα = 3

Infinite speed model

BPS 5.236 7.940

AJC 2.667 3.252

Bounded speed model

BPS 10.472 with max speed 1.618T 11.910 with max speed 1.466T

AJC 3.6 with max speed T 4 with max speed T

SRPT-AJC (infinite speed)�SRPT: run the job with the smallest

remaining work

�AJC: run at speed n1/αααα, where n is current number of jobs


J1J2

J3J4

J1 J2 J4 J3

speed

S

AJC is more stable than BPS & can also

be used in non-clairvoyant case

Potential & Amortization�Let Ga(t) & Go(t) be current F+E of

SRPT-AJC & OPT, resp.

�We show a potential function ΦΦΦΦ(t) and a value ββββ s.t. following conditions hold

boundary: (0) & ( ) are non-negative� boundary: ΦΦΦΦ(0) & ΦΦΦΦ(∞∞∞∞) are non-negative

� arrival: ΦΦΦΦ not increase when job arrives

� completion: ΦΦΦΦ not increase when job finish

� running:

�Implies: SRPT-AJC is β-competitiveCTAG Seminar 2008.10.30 Energy Efficient Flow Time Scheduling

0dt

)(d

dt

)(d

dt

)(d≤−

Φ+

tGttG oaβ

Amortized Local Competitive�Let t1, t2, …, t3n be the time when

� a job is released; online algorithm completes a job; OPT completes a job

�Integrating over timewe get

0dt

)(d

dt

)(d

dt

)(d≤−

Φ+

tGttG oaβ

� we get

�By arrival & completion condition� we get Ga + ΦΦΦΦ(∞∞∞∞) – ΦΦΦΦ(0) ≤β≤β≤β≤βGo

�By boundary condition, Ga ≤≤≤≤ ββββGoCTAG Seminar 2008.10.30 Energy Efficient Flow Time Scheduling

dtdtdt

o

n

i

ia GtG β≤∆Φ+∑+

=

13

1

)(

Potential Function�Let na(q) & no(q) be # of jobs with

remaining work ≥ q in SRPT-AJC & OPTn(q) Smallest remaining work: q1

Run one such job:� speed = n1/αααα

� time to finish = q n-1/αααα


qq1

� time to finish = q1 n-1/αααα

� E = n (q1 n-1/αααα) = n1-1/αααα q1

� F = n (q1 n-1/αααα) = n1-1/αααα q1If no more new jobs,

E+F = qiqn

i

d20

)(

1

/11

∫ ∑∞

=

−

α)()()(

/11)(

1

/11qnqniq oa

qn

i

a

ααφ −

=

− −

= ∑

qqt d)()(0∫∞

=Φ φβfor arrival condition

Bounded Speed Model�AJC

� speed at min {T, n1/αααα}, where T is max allowable speed

�A similar potential function but with a different βdifferent β


0dt

)(d

dt

)(d

dt

)(d≤−

Φ+

tGttG oaβ

)()()(/11

)(

1

/11qnqniq oa

qn

i

a

ααφ −

=

− −

= ∑

qqt d)()(0∫∞

=Φ φβ

Non-Clairvoyant Model�Online algorithm doesn't know the job

size when it is released

�AJC*: Run at speed

�RR-AJC*: Run all active jobs

α

α

1

)1

(−

n

�RR-AJC*: Run all active jobs simultaneously (by time sharing)

�Analysis for batched case: two steps� E+F of RR-AJC* ≤ (2-1/αααα) E+F of SJF-AJC*

� SJF-AJC* is optimal

⇒RR-AJC* is (2-1/αααα)-competitiveCTAG Seminar 2008.10.23 Energy Efficient Flow Time Scheduling

Summary�We proposed a more stable speed

function AJC

�Combining AJC with SRPT improves the competitive ratios of BPS


�Combining with RR (round robin) gives constant competitiveness for non-clairvoyant scheduling of batched jobs


Future Work�Non-clairvoyant scheduling of jobs

with arbitrary arrival times

�Energy efficient multi-processor flow time scheduling

�Save energy by scaling speed and let processor go to a sleep state

�Other quality of service

�Other practical considerations� switching overhead


Date post:	19-Oct-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Energy Efficient Flow Time Schedulingcgi.csc.liv.ac.uk/~ctag/seminars/prudence_ajc2008.pdf ·...

Documents