Analyzing Fixed-Priority Global Multiprocessor

Scheduling

Analyzing Fixed-Priority Global Multiprocessor

Scheduling

Seminar Multiprocessor Scheduling

May 22 2006

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Scheduling

Content

Problem Definition

Fixed-Priority Scheduling• Rate-monotonic Scheduling (RMS)• Dhall’s effect• Bound on utilization bounds

RM-US[US-LIMIT]• US-LIMIT = 0,33• US-LIMIT = 0,37482

Summary

Problem DefinitionProblem Definition

- Partition- Priorities- Definitions

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Scheduling

Partioned versus global scheduling

Partioned• A task is assigned ones to only one processor

Global• Task migration is permittedAssumption: no penalty associated with task migrationOne task can use more the one processorBut only one at any given time

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Scheduling

Priority driven scheduling algorithms

Preemptive• Tasks with low priority will be interrupted by incoming tasks

with higher priority

Static• Priorities for task are assigned ones

Dynamic• Priorities might change during runtime

vs. Non-Preemprive

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Scheduling

Problem definition

n independent tasks• Task ti (1 ≤ i ≤ n)

• Execution requirement Ci

• Period Ti

• Utilization Ui = Ci / Ti

multiprocessor with m identical processors .

A task set is schedulable if all tasks meet their deadlines (end of period) in all periods

/iUS U m

Fixed-Priority SchedulingFixed-Priority Scheduling

- Rate-monotonic Scheduling (RMS)- Dhall’s effect- Bound on utilization bounds

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Scheduling

Rate-monotonic Scheduling

Algorithm• Priorities assigned inversely proportional to their periods

Proved Theorems [3]:• A critical instant for any task occurs whenever the task is

requested simultaneously with requests for all higher priority tasks.

• If a feasible priority assignment exists for some task set, the rate-monotonic priority assignment is feasible for the task set.

• For a set of m tasks with fixed priority order the least upper bound to processor utilization factor is U = m(21/m – 1)

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Scheduling

t3: C3=4 T3=5 U3=80%

t2: C2=1 T2=4 U2=25%

Dhall’s Effect

t1: C1=1 T1=4 U1=25%

m = 2, US = (25%+25%+80%)/2 = 65%

misses deadline

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Scheduling

tn: Cn=4 Tn=5 Un=80%

tn-1: Cn-1=1 Tn-1=4 Un-1=25%

t2: C2=1 T2=4 U2=25%

t1: C1=1 T1=4 U1=25%

Dhall’s Effect

m = n-1, US = ((n-1)*25% + 80%)/m = 25%

.

.

.

misses deadline

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Scheduling

tn: Cn=8 Tn=9 Un=88%

tn-1: Cn-1=1 Tn-1=8 Un-1=12,5%

t2: C2=1 T2=8 U2=12,5%

t1: C1=1 T1=8 U1=12,5%

Dhall’s Effect

m=n-1, US=12,5%

.

.

.

misses deadline

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Scheduling

Dhall’s Effect

t1: C1=1 T1=x U2=0%

m=n-1, US0% with infinitely large x

tn-1: Cn-1=1 Tn-1=x Un-1=0%

tn: Cn=x Tn=x+1 Un=100%

t1: C1=1 T1=x U1=0%

.

.

.

misses deadline

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Scheduling

t4: C4=2 T4=4 U4-=50%

t2: C2=2 T2=4 U2=50%

t1: C1=2 T1=4 U1=50%

t3: C3=2 T3=4 U2=50%

Bound on utilization bounds

“The utilization guarantee bound for any static-priority multiprocessor scheduling algorithm (partitioned or global) cannot be higher than 1/2 of the capacity of the multiprocessor platform.” [2]

t4: C4=1 T4=4 U4=25%

t2: C2=1 T2=4 U2=25%

t1: C1=1 T1=4 U1=25%

t3: C3=1 T3=4 U2=25%

m = 3

: 0,5 mit 1 1

( ) 0,5i i it C L e T L i m

m L US

RM-US[US-LIMIT]RM-US[US-LIMIT]

- Idea - Example- Finding optimal US-LIMIT

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Scheduling

Idea

Task ti with Ui ≤ US-LIMIT• Lower priorities than tasks with Ui > US-LIMIT

• Priorities assigned according to the rate monotonic priority assignment scheme

Task ti with Ui > US-LIMIT• Highest priority

Proved Limit [2]: US-LIMIT 3 21US-LIMIT mit 3

mm

m

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Scheduling

Example

Processors m = 3 Scheduling Algorithm RM-US[0,429] Task set:

1 1

2 2

3 3

4 4

1,4 0,25

3,5 0,60

2,7 0,29

7,8 0,88

t U

t U

t U

t U

2 3 1 3

5

m

t0

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Scheduling

Finding optimal US-LIMIT

Select a task set such that the task set is not schedulable and such that there is no other unschedulable task set with a lower US (called extremal task set).

• m high prior tasks form block interference to a lower prior task (tn)

• All tasks are released first at the same time

• All m task are released two times within Tn

• m is infinitely large

• Ci = ∂ is infinitely small (1 ≤ i ≤ x)

• Ti = Tn - i∂ (1 ≤ i ≤ m)

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Scheduling

Finding optimal US-LIMIT (2)

Definition:

Following Situation:

11yUS LIMIT y

1,1 1,2 1,

1 , 1

, 1

1

1 , 0

,

n n

i

i i i m

C y T y

C i n

y x

n mx

T T T i i x

m x

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Scheduling

1

1 mit 1mx

n i

n ii

C CUS n mxmT m T

Finding optimal US-LIMIT (3)

Calculating US-LIMIT1

1

1nn i

n ii

C CUS T T m

11

xn

n i

CUS mT i

/

01

y

n

n i

CUS mT i

0

11

y

n

n

CUS dtmT t

ln 1n

n

CUS ymT

1 1 ln 11 1

1 0 ln 1 mit 1

y y yy m y

y y my

0,45473321765

0,37482252818

y

US

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Scheduling

Being able to schedule all task sets?

Three exhaustive cases

• All tasks have Ui ≤ US-LIMIT

• All tasks have Ui > US-LIMIT

• Some tasks have Ui > US-LIMIT and some have Ui ≤ US-LIMiT

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Scheduling

Comparison and Conclusion

[1]

Thank you for your attentionThank you for your attention

- References- Questions?

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Scheduling

References

[1] Lars Lundberg, "Analyzing Fixed-Priority Global Multiprocessor Scheduling“, rtas, p. 145,  Eighth IEEE Real-Time and Embedded Technology and Applications Symposium (RTAS'02),  2002.

[2] B. Andersson, S. Baruah, and J. Jonsson, “Static-priority scheduling on multiprocessors”, in Proc. of the IEEE Real-Time Systems Symposium (RTSS'01), Dec. 3-6, 2001, London, pp. 193 - 202.

[3] C. L. Liu and J. W. Layland, “Scheduling algorithms for multiprogramming in a hard real-time environment”, Journal of the ACM, 20(1): 46-61, Jan. 1973.