Scalably Scheduling Processes with
Arbitrary Speedup Curves
(Better Scheduling in the Dark)
Jeff Edmonds
York University
Kirk Pruhs
University of Pittsburgh
MAPSP 2009
The Scheduling Problem
• Allocate p processors to a stream of n jobs:
Shortest Remaining
Processing Time
(SRPT)
Shortest Elapsed
Time First (SETF)
Sublinear Nondecreasing
Speedup JFunctions
= f J ;:::;J g
• A set of jobs:
• Each job has phases:
• Each phase:
– Work:
– Speedup function:
• Nondecreasing
• Sublinear
• Examples:
1
Ji =
Jq =
i
Wq
¡
i
q
i
n
J 1 ; : : : ; J qi
i
i
hW q ; ¡ q i
i
i
®
The Scheduling Problem
• Allocate p processors to a stream of n jobs:
• Measure of Quality:P
F (A(I )) =
n (c
i= 1 i
• Competitive Ratio:F ( A ( I ) )
max
I F ( O pt ( I ) )
¡ ri ) =
R
t
n t dt
Sublinear Nondecreasing
Speedup Functions
Online
Nonclairvoyant:
Optimal:
Not Competitive:
Future
All Knowing
All Powerful
?
Nonclairvoyant
I nput
Opt
F ( SE T F )
F ( O pt )
SE T F
=
(n)
I nput
E qui
Opt
F ( E qu i )
F ( O pt )
F ( N on cl ai r voy an t )
F ( O pt )
=
=
(
n
ln n
)
p
( n)
[MPT]
Performance vs Load
Average
Performance
F (A(I ))
Load
maxI
F (Opt(I ))
I
F (A(I ))
F ( O pt ( I ) )
=
(n)
Performance vs Load
Average
Performance
F (A(I ))
F (Opt(I ))
s
c
Load
maxI
I
F (A(I ))
F ( O pt ( I ) )
= O(1)
Performance vs Load
Average
Performance
F (Opt(I ))
F (A s (I ))
c
Load
maxI
I
F (A s (I ))
F ( O pt ( I ) )
= O(c)
Resource Augmentation
Nonclairvoyant:
Future
Extra Speed
Optimal:
Competitive:
All Knowing
All Powerful
?
Resource Augmentation
1+ 2²
I nput
Opt
F ( SE T F 1 + ² )
F ( O pt 1 )
SE T F
= £ ( 1)
²
I nput
E qui 2+ ²
Opt
F ( E qu i 2 + ² )
F ( O pt 1 )
= £ ( 1)
Required
²
Fully Parallelizable Jobs
[KP]
[E]
Sublinear Nondecreasing
Speedup Functions
Arrives
over time
Currently
Alive
Opt gives all its resources to the
parallelizable job and hence
Opt
competes them as they arrive.
The sequential jobs complete
with no resources.
Sublinear Nondecreasing
Speedup Functions
Arrives
over time
Currently
Alive
Shortest Elapsed Time First (SETF)
gives all its resources to a sequential
job, wasting it.
The parallelizable jobs, getting no
resources never complete.
SE T F s
F ( SE T F s )
F ( O pt 1 )
=
(n)
Sublinear Nondecreasing
Speedup Functions
Arrives
over time
Currently
Alive
`
²
jobs
Equi spreads its resources fairly.
Most are wasted on the sequential jobs.
The parallelizable jobs don’t get enough
and fall behind.
`
jobs
EQUI wastes є resources,
has є extra.
E qui 1+but
²
F ( E qu i 1 + ² )
F ( O pt 1 )
=
` =² + `
`
=
1
²
Sublinear Nondecreasing
Speedup Functions
I nput
E qui 2+ ²
Opt
F ( E qu i 2 + ² )
F ( O pt 1 )
= £ ( 1)
Required
²
[E]
nt jobs currently alive
sorted by arrival time.
Latest Arrival Processor Sharing
Opt
SE T F
1 job But may be sequential.
L AP S ¯n t
E qui
New result
[EP]
F ( L A P Sh¯ ; 1 + ¯ +
F ( O pt 1 )
² ;i
)
nt
jobs
= £(
Speed
Compromise
Too thin
& needs
2+speed
²
jobs
1
¯²
)
nt jobs currently alive
sorted by arrival time.
Latest Arrival Processor Sharing
Opt
SE T F
1 job
L AP S ¯n t
nt
E qui
F ( SE T F 1+ ² )
F ( O pt 1 )
New result
[EP]
= £(
1
0²
¯ ¼0
Compromise
jobs
¯= 1
jobs
)
F ( L A P Sh¯ ; 1 + ¯ + ² ; i ) =
F ( O pt 1 )
F ( E qu i 2+ ² ) = £ ( 1 )
F ( O pt 1 )
1²
£(
1
¯²
)
Backwards Quantifies
Desired result:
Obtained:
New result
[E STOC09?]
New result
[EP]
9Alg 8²
F ( A l g1+ ² )
F ( O pt 1 )
=
8² 9Alg
F ( A l g1+ ² )
F ( O pt 1 )
= £(
F ( A l g1+ ² )
F ( O pt 1 )
= ! (1)
¯=
1²
2
8Alg 9²
²=
1¯
2
F ( L A P Sh¯ ; 1 + ¯ +
F ( O pt 1 )
²i
’
)
= £(
1
² O ( 1)
1
¯²
)
’
1
²2
)
Performance vs LLoad
Threshold
2 [0; 1]
F (Opt L (I )) = 1
Defn:
A set of jobs has load
if
i.e. can be optimally handled with speed L.
Defn:
F ¯ (L ) = max wit h load
I
L
F (L AP Sh¯ ;1i (I ))
Equi (β=1)
has the best performance,
but it only can handle half load.
L=
1
2
Small β can handleL = 1¡ ¯
almost full load
but its performance
1.
degrades with ¯
L
Proof Sketch
F ( L A P Sh¯ ; 1 + ¯ +
F ( O pt 1 )
² ;i
)
= £(
1
¯²
)
Proof Sketch
• In the worst cast inputs, each phase is either
sequential or parallelizable.
LAPS
LAPS
Proof Sketch
Potential Function
•
•
•
•
•
Define a potential function Φt.
It says how much debt Laps has in the bank.
Φ0 = Φfinal = 0.
Φt does not increase as jobs arrive or complete.
At other dF
times,
( L A P S)
dF ( O pt )
d©
dt
+
dt
· c
• Result follows by integrating
dt
F (L AP S) + ©f i n al ¡ ©0 · cF (Opt)
Potential Function
nt jobs currently alive
sorted by arrival time.
Coefficient:
Parallelizable work
done by Opt
not by LAPS:
1 2 3 …
x 1x 2x 3
…
©= °
xnnt n0t+1
t
P
Job arrives:
i 2 [n t ]
d©
dt
i ¢max(x i ; 0)
= 0
Potential Function
n jobs currently alive
sorted by arrival time.
Coefficient:
Parallelizable work
done by Opt
not by LAPS:
x11x22x33 …
…
©= °
P
xnnt nt-1
i i+1i
0
i 2 [n t ]
d©
Job completes: dt
t
i ¢max(x i ; 0)
· 0
Potential Function
n jobs currently alive
sorted by arrival time.
Coefficient:
Parallelizable work
done by Opt
not by LAPS:
x11x22x33 …
xnnt
…
©= °
Opt works:
P
i 2 [n t ]
d©
dt
i ¢max(x i ; 0)
· ° ¢n t ¢1
t
Opt
Potential Function
n jobs currently alive
sorted by arrival time.
Coefficient:
Parallelizable work
done by Opt
not by LAPS:
LAPS works:
d©
dt
(1 ¡ ¯)n t
x ( 1¡ ¯ ) n
x11x22x33 …
…
©= °
· °
xnnt
t
P
i 2 [n t ]
t
L AP S
i ¢max(x i ; 0)
P
i 2 [( 1¡ ¯ ) n t ;n t ¡ `bt ]
`bt =
i ¢¡
( 1+ ¯ + ² )
¯nt
# of jobs
• sequential under LAPS
xi · 0
• LAPS is a head, i.e.
Function
dF (Potential
L A P S) + d©
· c dF ( O pt )
dt
dt
dt
nt
· £(
# jobs alive
under Laps
1
¯²
)
b̀ · N
t
# jobs alive
under Opt
resulting
competitive ratio
A page of math later,
and the proof is done.
Opt works:
LAPS works:
d©
dt
· ° ¢n t ¢1
P
+°
i 2 [( 1¡ ¯ ) n t ;n t ¡ `bt ]
`bt =
i ¢¡
( 1+ ¯ + ² )
¯nt
# of jobs
• sequential under LAPS
xi · 0
• LAPS is a head, i.e.
Conclusions
nt jobs currently alive
sorted by arrival time.
Latest Arrival Processor Sharing
L AP S ¯n t
[EP] Resource
F ( LAugmentation:
APS
)
h¯ ; 1 + ¯ + ² ; i
F ( O pt 1 )
[
Work in Progress
jobs
= £(
1
¯²
)
]: Suboptimal
Load
Threashold
F
(
A
l g ) = ! (1)
8Alg 9²
1+ ²
F ( O pt 1 )
Other Models
Same Techniques
•Broadcast:
Many page requests
serviced simultaneously
[EP:SODA02, EP:SODA03]
•TCP:
Add Incr & Mult Decr
~ EQUI
[EDD:PAA03, E:Latin 04]
•Speed Scaling:
Each algorithm can dynamically choose its speed s,
but it must pay for it with energy P(s) = sα
[CELLSP:STACS09,CEP:STACS09]
Thank you