Energy Efficient
Flow Time Scheduling
Prudence Wong
University 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
Previous
flow time + energy
work & summary of results
Algorithm
and ideas of analysis
Active Job Count (AJC)
Potential analysis
Concluding
CTAG Seminar 2008.10.30
remarks
Energy Efficient Flow Time Scheduling
Pentium
M spec (1.6 GHz)
f
1.6 GHz
0.6 GHz
IBM
Processor
V
1.484 V
0.956 V
P
24.5 W
6W
PowerPC 970FX
CTAG Seminar 2008.10. 30
Speed Scaling Functions for Flow Time Scheduling
based on Active Job Count
Energy Efficiency
For mobile devices, energy efficiency is
a major concern.
How to save energy?
Dynamic speed (voltage) scaling
Slow
To
down processor whenever possible
run at speed s, power = sα where α>1
(literature: α = 2 or 3)
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
Examples
Assume
α = 3, a job with work = 2
s=1
0
1
2
time
energy: 13 × 2 = 2
energy: 23 × 1 = 8
s=2
0
1
2
time
energy: (2/3)3 x (3/2) + 23 x 1/2
s = 2/3
&s=2
0
1
2
time
= 40/9
slower speed saves energy
same speed over same interval saves energy
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
Orthogonal Objectives
Energy
efficiency: run jobs slower
Quality
of service: run jobs faster
Examples
of QoS
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 time
CTAG 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
Fujiwara '06]
minimize F subject to a bounded E?
unfortunately,
there is no O(1)-competitive
online algorithm
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
Online Scheduling
Input:
each
Online
jobs arriving at arbitrary times
with some work to complete
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
CTAG Seminar 2008.10.30
speed vs bounded speed model
Energy Efficient Flow Time Scheduling
Example
Assume
α=2
speed
s=1
0
1
S1
Energy E Flow F
J
12x1=1
1
2
0.52x2=0.5
2
2.5
speed
J
S2
s = 0.5
J
Schedule S1 is better than S2
CTAG Seminar 2008.10.30
E+F
Energy Efficient Flow Time Scheduling
Example
Assume
0
α=2
1
2
J1
J2
E(S2) – E(S1) = -0.25
speed
s = 1.25
F(S2)-F(S1) = 0.4
S1
J1
s=1
J2
depends on overlap
speed
of span of jobs
s=1
we don't know
S2
J1
efficient optimal
offline algorithm
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
J2
r(J1) = 0
p(J1) = 1
r(J2) = 0.8
p(J2) = 1
best
schedule
flow time
inc too much
Best Speed for a Single Job
Given
a job of size p
Suppose
we run the job at speed s
= sα(p/s) = sα-1p & F = p/s
E + F = p(sα-1 + 1/s)
E
1
1 α
)
Differentiating, min E+F when s = (
α −1
When
we have n jobs (different sizes),
1
critical threshold
n
(
CTAG Seminar 2008.10.30
α −1
)α
Energy Efficient Flow Time Scheduling
Previous Work
Unit
size jobs
[Albers and Fujiwara STACS 06]
Speed function: (n/c)1/αα, c is constant
Competitive ratio 8.3e(1+Φ)α, Φ is golden ratio
Arbitrary
size
[Bansal, Pruhs, Stein SODA 07]
job selection: SJF (shortest job first)
Speed: function of remaining work
Infinite
speed model
Bounded
speed model
CTAG Seminar 2008.10.30
preemption
α = 2
α = 3
AF
>150
>400
BPS
5.236
7.940
BPS
10.472 with
max speed 1.618T
11.910 with
max speed 1.466T
Energy Efficient Flow Time Scheduling
BPS Algorithm
BPS-fractional
Speed w1/αα, w is ∑fraction of jobs remained
BPS
(use SJF)
(with speed advantage 1+εε)
Simulate BPS-F; same job; speed (1+ε
(1+ε) faster
idle
if simulated schedule runs a job we finish
J1
J2
J4
J3
speed
schedule
CTAG Seminar 2008.10.30
J1
J2
J1
J
J
3
Energy4
Efficient Flow Time Scheduling
BPS Algorithm
Somewhat
strange schedule?
continuously changing speed, sometimes lazy
required extra speed in bounded speed model
1
speed
may be lower than threshold
n α
(
)
α −1
not applicable in non-clairvoyant model, where
job size is unknown on arrival
J1
J2
J4
J3
speed
schedule
CTAG Seminar 2008.10.30
J1
J2
J1
J
J
3
Energy4
Efficient Flow Time Scheduling
Our Contribution
SRPT+AJC
(active job count)
shortest remaining
processing time
β1- & β2-comp for infinite & bounded speed
β 1=
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
Infinite speed
model
α = 2
α = 3
BPS
5.236
7.940
AJC
2.667
3.252
10.472 with max speed 1.618T
11.910 with max speed 1.466T
3.6 with max speed T
4 with max speed T
Bounded speed BPS
model
AJC
Non-clairvoyant
scheduling, batched
case: (2-1/α)-comp & 2-comp
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
SRPT-AJC (infinite speed)
SRPT:
run the job with the smallest
remaining work
run at speed n1/αα, where n is
current number of jobs
AJC:
J1
J2
J4
AJC is more stable
than BPS & can also
be used in nonclairvoyant case
J3
speed
S
J1
CTAG Seminar 2008.10.30
J2
J4
J3
Energy Efficient Flow Time Scheduling
Potential & Amortization
Let
Ga(t) & Go(t) be current F+E of
SRPT-AJC & OPT, resp.
We
show a potential function Φ(t) and
a value β s.t. following conditions hold
boundary: Φ(0) & Φ(∞
(∞) are non-negative
arrival: Φ not increase when job arrives
completion: Φ not increase when job finish
dGo(t )
running: dGa (t ) dΦ (t )
+
−β
≤0
dt
dt
dt
Implies:
CTAG Seminar 2008.10.30
SRPT-AJC is β-competitive
Energy Efficient Flow Time Scheduling
Amortized Local Competitive
Let t1, t2,
…, t3n be the time when
a job is released; online algorithm
completes a job; OPT completes a job
dGa (t ) dΦ (t )
dGo(t )
+
−β
≤0
Integrating over time
dt
dt
dt
we get
3 n +1
Ga + ∑ ∆Φ (ti ) ≤ βGo
i =1
By
arrival & completion condition
we
By
get Ga + Φ(∞
∞) – Φ(0) ≤βG
≤β o
boundary condition, Ga ≤ βGo
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
Potential Function
Let
na(q) & no(q) be # of jobs with
remaining work ≥ q in SRPT-AJC & OPT
n(q)
q1
If no more new jobs,
E+F = 2
∫
∞
0
Smallest remaining work: q1
Run one such job:
speed = n1/αα
-1/α
α
time
to
finish
=
q
n
q
1
-1/α
α
E = n (q1 n
) = n1-1/αα q1
F = n (q1 n-1/αα) = n1-1/αα q1
 n ( q ) 1−1/ α 
na ( q )

 ∑ i
 dq φ (q ) =  i1−1/ α  − n (q )1−1/ α n (q )
a
o
∑

 i =1

 i =1

∞
CTAG Seminar 2008.10.30
for arrival condition
Φ (t ) = β ∫ φ (q ) dq
0
Energy Efficient Flow Time Scheduling
Bounded Speed Model
AJC
speed at min {T, n1/αα}, where T is max
allowable speed
A
similar potential function but with a
different β dGa(t ) dΦ(t )
dGo(t )
dt
+
dt
−β
dt
 na ( q ) 1−1/ α 
 − na (q )1−1/ α no (q )
φ (q) =  ∑ i
 i =1

∞
Φ (t ) = β ∫ φ (q ) dq
0
CTAG Seminar 2008.10.23
Energy Efficient Flow Time Scheduling
≤0
Non-Clairvoyant Model
Online
algorithm doesn't know the job
size when it is released
1
n α
AJC*: Run at speed ( α − 1)
Run all active jobs
simultaneously (by time sharing)
RR-AJC*:
Analysis
for batched case: two steps
E+F of RR-AJC* ≤ (2-1/α
α) E+F of SJF-AJC*
SJF-AJC* is optimal
⇒ RR-AJC* is (2-1/α
α)-competitive
CTAG 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 nonclairvoyant scheduling of batched jobs
CTAG Seminar 2008.10.30
Energy Efficient Flow Time Scheduling
Future Work
scheduling of jobs
with arbitrary arrival times
Non-clairvoyant
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
CTAG Seminar 2008.10.23
Energy Efficient Flow Time Scheduling