1. University of Pittsburgh
CO 2000
Scalably Scheduling Processes with Arbitrary Speedup Curves (Better Scheduling in the Dark)
Jeff Edmonds
York University
Kirk Pruhs
University of Pittsburgh
MAPSP 2009

2. The Scheduling Problem
CO 2000
Allocate p processors to a stream of n jobs:
Shortest Remaining
Processing Time (SRPT)
Shortest Elapsed
Time First (SETF)

3. Sublinear Nondecreasing Speedup Functions
CO 2000
A set of jobs:
Each job has phases:
Each phase:
Work:
Speedup function:
Nondecreasing
Sublinear
Examples: J = f 1 ; : n g J i = 1 ; : q J q i = h W ; W q i q i

4. The Scheduling Problem
CO 2000
Allocate p processors to a stream of n jobs:
Measure of Quality:
Competitive Ratio: F ( A I ) = P n i 1 c r R t d m a x I F ( A ) O p t

5. Sublinear Nondecreasing Speedup Functions
CO 2000
Online Nonclairvoyant:
Future ?
Optimal:
All Knowing
All Powerful
Not Competitive:

6. Nonclairvoyant ­ ­ = n = ­ = F ( S E T ) O p t F ( E q u i ) n l F ( N
CO 2000 F ( S E T ) O p t = n I u O p t F ( E q u i ) = n l I F ( N o n c l a i r v y t ) O p =
[MPT]

7. Performance vs Load m a x = ­ ( n ) A I F O p t F ( A I ) F ( O p t I
CO 2000 F ( A I ) F ( O p t I ) = ( n ) m a x I F O p t A
Average Performance
Load I

8. Performance vs Load m a x = O ( 1 ) A I F ( O p t ) F ( A I ) F ( O p
CO 2000 F ( A I ) F ( O p t I ) I
Average Performance
Load = O ( 1 ) s c m a x I F ( O p t ) A

9. Performance vs Load m a x O = F ( A I ) I F ( O p t ) F ( A I ) F ( O
CO 2000 F ( A s I ) F ( O p t I ) I
Average Performance
Load c F ( A s I ) = O
Load m a x I F ( O p t )

10. Resource Augmentation
CO 2000
Nonclairvoyant:
Future ?
Extra Speed
Optimal:
All Knowing
All Powerful
Competitive:

11. Resource Augmentation
CO 2000 S E T F O p t I n u ( 1 + ) =
[KP] 2 O p t I n u E q i 2 + F ( ) 1 =
[E]
Required
Fully Parallelizable Jobs

12. Sublinear Nondecreasing Speedup Functions
CO 2000
Arrives over time
Currently Alive O p t
Opt gives all its resources to the parallelizable job and hence competes them as they arrive. The sequential jobs complete with no resources.

13. Sublinear Nondecreasing Speedup Functions
CO 2000
Arrives over time
Currently Alive S E T F s
Shortest Elapsed Time First (SETF) gives all its resources to a sequential job, wasting it.
The parallelizable jobs, getting no resources never complete. F ( S E T s ) O p t 1 = n

14. Sublinear Nondecreasing Speedup Functions
CO 2000
Arrives over time
Currently Alive `
jobs
EQUI wastes є resources,
but has є extra.
Equi spreads its resources fairly.
Most are wasted on the sequential jobs.
The parallelizable jobs don’t get enough and fall behind. E q u i 1 + F ( E q u i 1 + ) O p t = `

15. Sublinear Nondecreasing Speedup Functions
CO 2000 I n p u t O p t E q u i 2 + F ( E q u i 2 + ) O p t 1 =
Required
[E]

16. nt jobs currently alive sorted by arrival time.
CO 2000 S E T F
< L A P q u i
nt jobs currently alive sorted by arrival time.
Latest Arrival Processor Sharing O p t
1 job S E T F
But may be sequential. L A P S
jobs n t
Compromise 2 +
Too thin & needs speed
jobs E q u i n t
New result
[EP] F ( L A P S h ; 1 + i ) O p t =
Speed

17. nt jobs currently alive sorted by arrival time.
CO 2000 S E T F
< L A P q u i
nt jobs currently alive sorted by arrival time.
Latest Arrival Processor Sharing O p t
1 job S E T F F ( S E T 1 + ) O p t = L A P S
jobs n t
Compromise
jobs E q u i n t F ( E q u i 2 + ) O p t 1 =
New result
[EP] F ( L A P S h ; 1 + i ) O p t =

18. Backwards Quantifies 9 A l g 8 ² = 8 ² 9 A l g = £ 8 A l g 9 ² = ! = £
CO 2000
Desired result: 9 A l g 8 F ( 1 + ) O p t =
Obtained: 8 9 A l g F ( 1 + ) O p t = 2 = 1 2
New result
[E STOC09?] 8 A l g 9 F ( 1 + ) O p t = ! = 1 2
New result
[EP] F ( L A P S h ; 1 + i ) O p t = ’

19. Performance vs Load Threshold
CO 2000
Defn: A set of jobs has load if
i.e. can be optimally handled with speed L. L 2 [ ; 1 ] F ( O p t L I ) = 1
Defn: F ( L ) = m a x I w i t h l o d A P S ; 1
Equi (β=1)
has the best performance,
but it only can handle half load. L = 1 2
Small β can handle almost full load L = 1 1 .
but its performance degrades with L

20. Proof Sketch
CO 2000 F ( L A P S h ; 1 + i ) O p t =

21. Proof Sketch
CO 2000
In the worst cast inputs, each phase is either sequential or parallelizable.
LAPS
LAPS

22. Proof Sketch
CO 2000

23. Potential Function + · c F ( L A P S ) + © ¡ · c O p t
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 times,
Result follows by integrating d F ( L A P S ) t + c O p F ( L A P S ) + f i n a l c O p t

24. Potential Function = © = ° P ¢ m a x ( ; ) d © t i 2 [ n ] x
CO 2000
nt jobs currently alive sorted by arrival time.
Job arrives: d t =
nt+1
Coefficient: 1 2 3 … nt
Parallelizable work done by Opt not by LAPS: x 1 2 3 … n t = P i 2 [ n t ] m a x ( ; )

25. Potential Function · © = ° P ¢ m a x ( ; ) d © t i 2 [ n ] x x x x x x
CO 2000
n jobs currently alive sorted by arrival time. i
Coefficient: 1 2 3 …
i+1 d t
nt-1 i nt
Parallelizable work done by Opt not by LAPS: … x 1 x 2 x 3 x n t x n t x n t = P i 2 [ n t ] m a x ( ; )
Job completes:

26. Potential Function · ° ¢ n 1 © = ° P ¢ m a x ( ; ) © = ° P ¢ m a x ( ;
CO 2000
n jobs currently alive sorted by arrival time.
Coefficient: 1
Opt works: O p t d n 1 2 3 … nt
Parallelizable work done by Opt not by LAPS: … x 1 x 2 x 3 x n t x n t x n t = P i 2 [ n t ] m a x ( ; ) = P i 2 [ n t ] m a x ( ; ) = P i 2 [ n t ] m a x ( ; )

27. Potential Function · ° P ¢ © = ° P ¢ m a x ( ; ) © = ° P ¢ m a x ( ; )
CO 2000
n jobs currently alive sorted by arrival time.
Coefficient: 1 2 3 … nt L A P S ( 1 ) n t
Parallelizable work done by Opt not by LAPS: … x 1 x 2 x 3 x n t x n t x n t x ( 1 ) n t = P i 2 [ n t ] m a x ( ; ) = P i 2 [ n t ] m a x ( ; ) = P i 2 [ n t ] m a x ( ; )
LAPS works: d t P i 2 [ ( 1 ) n ; b ` ] +
# of jobs
sequential under LAPS
LAPS is a head, i.e. x =

28. Potential Function b ` £ ( ) n · · N · ° ¢ n 1 + P + · c b ` = d F ( L
CO 2000 d F ( L A P S ) t + c O p b ` ( 1 ) n t N t
# jobs alive under Laps
resulting competitive ratio
# jobs alive under Opt
A page of math later, and the proof is done.
LAPS works:
Opt works: d t n t 1 + P i 2 [ ( ) ; b ` ]
# of jobs
sequential under LAPS
LAPS is a head, i.e. x i b ` t =

29. Conclusions = £ 8 A l g 9 ² = ! [EP] Resource Augmentation:
nt jobs currently alive sorted by arrival time. L A P S
jobs n t
Latest Arrival Processor Sharing
[EP] Resource Augmentation: F ( L A P S h ; 1 + i ) O p t =
[Work in Progress]: Suboptimal Load Threashold 8 A l g 9 F ( 1 + ) O p t = !

30. Other Models Same Techniques
CO 2000
Other Models Same Techniques
Broadcast: Many page requests [EP:SODA02, EP:SODA03] serviced simultaneously
TCP: Add Incr & Mult Decr ~ EQUI
Speed Scaling:
Each algorithm can dynamically choose its speed s, but it must pay for it with energy P(s) = sα
[EDD:PAA03, E:Latin 04]
[CELLSP:STACS09,CEP:STACS09]

31. CO 2000
Thank you