optimal real-time scheduling for uniform multiprocessors

National Taiwan UniversityDepartment of Computer Science and Information Engineering

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Optimal Real-Time Scheduling for Uniform Multiprocessors

薛智文助理教授台灣大學資訊工程學系資訊網路暨多媒體研究所[email protected]

Shih-Ying Chen and Chih-Wen Hsueh, "Optimal Dynamic-priority Real-Time Scheduling Algorithms for Uniform Multiprocessors," Proc. 29th IEEE Real-Time Systems Symposium, 147-156, Barcelona, Spain, Dec. 2008.

2011/3/2

mailto:[email protected]

National Taiwan UniversityDepartment of Computer Science and Information Engineering /46

Task1

Task2

Taskn

…

(periodically)

Looking for simple condition to be feasible or schedulable.

Scheduling

2

Processor2

…

Processorm

Processor1


Kwang S. Hong and Joseph Y-T. Leung, ON-LINE SCHEDULING OF REAL-TIME TASKS, Real-Time Systems Symposium,1988


OutlineIntroduction

Multiprocessor EnvironmentsMotivationContributionDefinitions and Assumptions

Model and T-Ler PlanesScheduling AlgorithmsConclusion and Future WorkQ&A

4


Multiprocessor EnvironmentsIdentical

All the processors are identical.Uniform

Execution time is dependent on the running processor.Scheduling algorithms for it might be adapted on

asymmetric multi-core platform (AMP).Unrelated

Execution time is independent on the running processor.

5


P1

T1T2Tn …

(periodically)

RM and EDF are optimal.

Uniprocessor Scheduling

6

C. L. Liu and J. Layland. Scheduling algorithms for multiprogramming in a hard real-time environment. Journal of the ACM, 10(1):46–61, 1973.


P1

T1 P2T2Tn …

…

(periodically)

LLREF is optimal, with preemption and migration.

Identical Multiprocessor Scheduling

Pm

7

Hyeonjoong Cho, Binoy Ravindran, and E. Douglas Jensen. An optimal real-time scheduling algorithm for multiprocessors. RTSS, pages 101–110, Oct. 2006. (LLREF)

R.R. Muntz and E.G. Coffman. Optimal preemptive scheduling on two-processor systems. IEEE Transactions on Computers, (11):1014–1020, Nov. 1969. (level algorithm)


P1

T1T2Tn ……

(periodically)

Uniform Multiprocessor Scheduling

Pm

8

P2

Jane W.S. Liu and Ai-Tsung Yang, Optimal scheduling of independent tasks on heterogeneous computing systems, ACM '74 Proceedings of the 1974 annual conference, 1974

Well…


P1

T1

P2

T2Tn ……

(periodically)

PG or PCG is optimal, with preemption and migration.

Uniform Multiprocessor Scheduling (cont.)

Pm

9

Edward C. Horvath, Shui Lam, and Ravi Sethi. A level algorithm for preemptive scheduling. Journal of the ACM, 24(1):32–43, January 1977.Shih-Ying Chen and Chih-Wen Hsueh, "Optimal Dynamic-priority Real-Time Scheduling Algorithms for Uniform Multiprocessors," Proc. 29th IEEE Real-Time Systems Symposium, 147-156, Barcelona, Spain, Dec. 2008. (PC, PCG)


Why “Uniform”?

processor 1 with computing capacity = 2

processor 2 with computing capacity = 1

time0 1 2 3 4 5task

0 1 2 3 4 5task

task with execution requirement = 4

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T1 (8, 10)T2 (7, 10)

One task can only run on one processor at a time.

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Task migration might be easier on multicore platform

P1

P2

…

Pm

Sharedmemory

12


MotivationA simpler way to describe the execution behavior

of tasks and processors.No existing optimal scheduling algorithm in terms

of feasibility condition.

13


Guest OS

VCPU VCPU

Guest OS

VCPU

…

…

PCPU PCPU PCPU …

App App

Hypervisor

Scheduling

OS Scheduling on Hypervisor

14


Contribution

A novel description plane: T-Ler planeTwo optimal dynamic-priority on-line scheduling

algorithms.

15


Definitions and Assumptions

Assumptions :Uniform multiprocessorsTask is periodicDeadline is equal to the end of period.Full migration

Basic definitionsOn-line schedulingDynamic-priority schedulingPreemptive scheduling

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Definitions and Assumptions (cont.)Processors :

Processor Pi with computing capacity siAssume si in decreasing order :

Tasks : Task Ti with utilization ui where

Assume ui in decreasing order :01 21 nuuu

01 21 nsss

periodtrequiremen execution

iu

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Feasibility Conditions (Funk)

S. Funk, J. Goossens, and S. Baruah. “On-line scheduling on uniform multiprocessors,” RTSS, Dec. 2001.

Set Processors Tasks

1

2

n

1s 1u

21 ss 21 uu

nsss 21 nuuu 21

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OutlineIntroductionModel and T-Ler Plane

P-fair and T-L PlaneT-L PlanesT-Ler Planes

Scheduling AlgorithmsConclusion and Future WorkQ&A

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P-fair and T-L PlaneProportionate Fairness ( P - fair 1)

Each task is scheduled resources in proportion to its utilization.

Time and Local Remaining Execution Time Plane ( T-L plane 2)Show that scheduling for multiprocessors can be

viewed as repeatedly occurring T-L planes.

1 S. Baruah, N. Cohen, C. Plaxton, and D. Varvel, “Proportionate progress: A notion of fairness in resource allocation,” Algorithmica, June 1996.2 H. Cho, B. Ravindran, and E. D. Jensen. “An optimal real-time scheduling algorithm for multiprocessors,” RTSS, Oct. 2006.

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P-fair:Each task is scheduled resources in proportion to its utilization.

time

remaining execution tim

e 0

Task= (4, 8), (execution_time, period)

84

4

fluid schedule

21


local remaining execution time: utilization * ( t2 – t1 )

period period

time

period period period

t1 t2

T1

T2


e

P-fair and T-L Plane (cont.)

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T-L Plane:Time and Local Remaining Execution Time Plane

local remaining execution tim

e

timet2t1

T1

T2

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local remaining execution time: utilization * ( t2 – t1 )

period period

time

period period period

t1 t2

T1

T2


e

T-L Planes …

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T-Ler Planes

time

remaining execution requirem

ent 0

Task= (4, 8), (execution_requirement, period)

84

4

fluid schedule

Time and Local Remaining Execution Requirement Plane ( T-Ler Plane)

25


local remaining execution requirement: utilization * ( t2 – t1 )

period

time

period

t1 t2

T1

T2


ent

T-Ler Planes (cont.)

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Processor Boundary

local remaining execution requirem

ent

time0

Processor 1

Processor 2

Processor 3

… Processor n

Speed 1 0.6 0.4 … 0.11P

2P

3P

nP

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Rescheduling Events


ent

time0 Event B (Bottom hitting event)

Event C (Ceiling hitting event)

Event F (Floor hitting event)

1P

2P

3P

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OutlineIntroductionModel and T-Ler PlanesScheduling Algorithms

Precaution Greedy Scheduling AlgorithmPrecaution Cut Greedy Scheduling AlgorithmProof of Optimality

Conclusion and Future WorkQ&A

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Framework

2P 1P1T time0 84

time


ent

0 84

4

local local

timeT1

T2


entlocal rem

aining execution requirem

ent

time

1T

T-Ler plane

local

local

2T

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Why P and G? GreedyPrecaution

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enttime0

1P

2P

3P



ent time

Precaution Greedy Scheduling Algorithm

The times of rescheduling is unpredictable.1T

2T

3T

1P

2P

0

32


time


ent

Precaution Cut Greedy Scheduling Algorithm

Give an upper bound of “n” rescheduling in a T-Ler plane.

1T

2T

3T

1P

2P

0

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PG vs. PCGWhen any event occurs, PG and PCG will reschedule.PCG reduce the times of rescheduling dramatically.

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ent time0

Task Order

?

1T

2T

nT

t35


Task Order (cont.)


ent time

1T

2T

0

1T

2T

event occurs

36


OutlineIntroductionModel and T-Ler PlanesScheduling Algorithms

Precaution Greedy Scheduling AlgorithmPrecaution Cut Greedy Scheduling AlgorithmProof of Optimality

Conclusion and Future WorkQ&A

37


IFF Feasibility Conditions (Funk)

S. Funk, J. Goossens, and S. Baruah. “On-line scheduling on uniform multiprocessors,” RTSS, Dec. 2001.

Set Processors Tasks

1

2

n

1s 1u

21 ss 21 uu

nsss 21 nuuu 21

Feasibility Condition violated

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0

local remaining execution

requirement

time

Proving PG is Feasible

feasibility condition is violated

s1 ≥ u’1

s1 + s2 = u’1 + u’2

s1 + s2 + s3 ≥ u’1 + u’2 + u’3

feasibility condition is violated

1T

3T

2T

s1 ≥ u1

s1 + s2 ≥ u1 + u2

s1 + s2 + s3 ≥ u1 + u2 + u3

initial condition

1T 2T

3T

( T’1 , T’2 ) could be ( T1 ,T3 ) or ( T2 ,T3 )

1P

3P

2P

When any event occurs, PG will reschedule.Here we want to show any event occurs earlier

than feasibility condition is violated.If u2 > s2

T1 or T2 will invoke event before feasibility condition is violated

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0

local remaining execution

requirement

time

Proving PG is Feasible (cont.)

feasibility condition violate

s1 ≥ u’1

s1 + s2 = u’1 + u’2

s1 + s2 + s3 ≥ u’1 + u’2 + u’3

feasibility condition violate1T

3T2T

s1 ≥ u1

s1 + s2 ≥ u1 + u2

s1 + s2 + s3 ≥ u1 + u2 + u3

initial condition

1T 2T

3T

( T’1 , T’2 ) could be ( T1 ,T3 ) or ( T2 ,T3 )

1P

3P

2P

If u2 < s2

T3 will invoke event before feasibility condition is violated

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Proving PG is Optimal

In the beginning, set of tasks and processors is feasible.

PG reschedules at any evens and is still feasible.

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Proving PCG is Feasible

When any event occurs, set of tasks and processors is feasible by PCG.

Remove the task and the processor that invoke event.When event B occurs :

u’n = 0When event C occurs :

s1 = u’1 When event F occurs :

si = u’j i = jsi = u’j i > jsi = u’j i < j

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Proving PCG is Feasible (cont.)

We want to derive :Processors Tasks

s1 ≥ u’1

s1 + s2 ≥ u’1 + u’3 (s2 ≥ s4 = u’2 ≥ u’3)s1 + s2 + s3 ≥ u’1 + u’3 + u’4 (s3 ≥ s4 =

u’2 ≥ u’4)s1 + s2 + s3 + s5 ≥ u’1 + u’3 + u’4 + u’5

When event F occurs, s4 = u’2

Processors Tasks

s1 ≥ u’1

s1 + s2 ≥ u’1 + u’2

s1 + s2 + s3 ≥ u’1 + u’2 + u’3

s1 + s2 + s3 + s4 ≥ u’1 + u’2 + u’3 + u’4

s1 + s2 + s3 + s4 + s5

≥ u’1 + u’2 + u’3 + u’4 + u’5

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Proving PCG is OptimalIn the beginning, the set of tasks and processors

is feasible.PCG reschedules at any event and is still

feasible.

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ConclusionWe presented T-Ler plane which is easier for reasoning

about the execution behavior of tasks and processors.Two optimal on-line scheduling algorithms

Precaution Greedy scheduling algorithmPrecaution Cut Greedy scheduling algorithmPCG reduces the times of rescheduling dramatically.

Our result might be applicable to current asymmetric multicore platform.

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Future WorkVerifying the performance of PG and PCG.Implementing PG and PCG scheduling algorithms on

asymmetric multicore platform.Reduce the migration times of PCG.Extend PCG to globally periodical.

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Q&A

Thanks for your attention!!

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optimal real-time scheduling for uniform multiprocessors

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