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Analyzing Fixed-Priority Global Multiprocessor Scheduling. Seminar Multiprocessor Scheduling May 22 2006 Tobias Queck. 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 - PowerPoint PPT Presentation
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Analyzing Fixed-Priority Global Multiprocessor
Scheduling
Analyzing Fixed-Priority Global Multiprocessor
Scheduling
Seminar Multiprocessor Scheduling
May 22 2006
Tobias Queck
2Tobias QueckMultiprocessor
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
6Tobias QueckMultiprocessor
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
8Tobias QueckMultiprocessor
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)
9Tobias QueckMultiprocessor
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
10Tobias QueckMultiprocessor
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
11Tobias QueckMultiprocessor
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
12Tobias QueckMultiprocessor
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
13Tobias QueckMultiprocessor
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
16Tobias QueckMultiprocessor
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
19Tobias QueckMultiprocessor
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
20Tobias QueckMultiprocessor
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.