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Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks Pontus Ekberg & Wang Yi Uppsala University, Sweden ECRTS 2012
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Bounding and Shaping the Demand ofMixed-Criticality Sporadic Tasks
Pontus Ekberg & Wang Yi
Uppsala University, Sweden
ECRTS 2012
Mixed-criticality sporadic tasks
(Ci(lo) ⩽ Ci(hi))Ci(lo): WCET at low-criticalityCi(hi): WCET at high-criticality
Di: Relative deadlineTi: PeriodLi: Criticality (lo or hi)
Task τi
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 2
Mixed-criticality sporadic tasks
τ1 (L1 = lo):
τ2 (L2 = hi):
τ3 (L3 = hi):
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 3
Mixed-criticality sporadic tasks
τ1 (L1 = lo):
τ2 (L2 = hi):
τ3 (L3 = hi):
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 4
Mixed-criticality sporadic tasks
τ1 (L1 = lo):
τ2 (L2 = hi):
τ3 (L3 = hi):
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 5
Schedulability analysis
A task set τ is schedulable if
∀ℓ ⩾ 0 :∑τi∈τ
dbf(τi, ℓ) ⩽ sbf(ℓ).
Classic EDF analysis
Low-criticality mode High-criticality mode
Time
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 6
Schedulability analysis
A task set τ is schedulable if both A and B hold:
A : ∀ℓ ⩾ 0 :∑τi∈τ
dbflo(τi, ℓ) ⩽ sbflo(ℓ)
B : ∀ℓ ⩾ 0 :∑
τi∈hi(τ)
dbfhi(τi, ℓ) ⩽ sbfhi(ℓ)
Mixed-criticality EDF analysis
Low-criticality mode High-criticality mode
Time
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 7
Demand-bound functions
Half-finished jobs are carried over to high-criticality mode.
Low-criticality mode High-criticality mode
Time
Each τi behaves exactly like a standardsporadic task with WCET Ci(lo).
Use dbfs from Baruah et al., 1990!
Each τi behaves similar to a standardsporadic task with WCET Ci(hi).
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 8
Demand-bound functions
Half-finished jobs are carried over to high-criticality mode.
Low-criticality mode High-criticality mode
Time
Each τi behaves exactly like a standardsporadic task with WCET Ci(lo).
Use dbfs from Baruah et al., 1990!
Each τi behaves similar to a standardsporadic task with WCET Ci(hi).
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 9
Demand-bound functions
Half-finished jobs are carried over to high-criticality mode.
Low-criticality mode High-criticality mode
Time
Each τi behaves exactly like a standardsporadic task with WCET Ci(lo).
Use dbfs from Baruah et al., 1990!
Each τi behaves similar to a standardsporadic task with WCET Ci(hi).
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 10
Demand-bound functions
Half-finished jobs are carried over to high-criticality mode.
Low-criticality mode High-criticality mode
Time
Each τi behaves exactly like a standardsporadic task with WCET Ci(lo).
Use dbfs from Baruah et al., 1990!
Each τi behaves similar to a standardsporadic task with WCET Ci(hi).
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 11
Carry-over jobs
Half-finished jobs are carried over to high-criticality mode.
Low-criticality mode High-criticality mode
Time
To show A ∧ B, we show A ∧ (A → B).
Restricting to the interesting cases
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 12
Carry-over jobs
t t+ Di
Release of τi Absolute deadline
Switch to high-criticality mode
Remaining schedulingwindow Ci(hi)−Ci(lo)
… Time
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 13
Carry-over jobs
t t+ Di
Release of τi Absolute deadline
Switch to high-criticality mode
Remaining schedulingwindow
Ci(hi)−Ci(lo)
… Time
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 14
Carry-over jobs
t t+ Di
Release of τi Absolute deadline
Switch to high-criticality mode
Remaining schedulingwindow
Ci(hi)−Ci(lo)
… Time
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 15
Carry-over jobs
t t+ Di
Release of τi Absolute deadline
Switch to high-criticality mode
Remaining schedulingwindow Ci(hi)−Ci(lo)
… Time
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 16
Adjusting the demand of carry-over jobs
t t+ Di(lo) t+ Di(hi)
Release of τi
Deadlines in low- and high-criticality mode
… Time
… Time
Switch to high-criticality mode
… Time
Switch to high-criticality mode
Remaining scheduling window
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 17
Adjusting the demand of carry-over jobs
t t+ Di(lo) t+ Di(hi)
Release of τi
Deadlines in low- and high-criticality mode
… Time
… Time
Switch to high-criticality mode
… Time
Switch to high-criticality mode
Remaining scheduling window
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 18
Adjusting the demand of carry-over jobs
t t+ Di(lo) t+ Di(hi)
Release of τi
Deadlines in low- and high-criticality mode
… Time… Time
Switch to high-criticality mode
… Time
Switch to high-criticality mode
Remaining scheduling window
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 19
Adjusting the demand of carry-over jobs
t t+ Di(lo) t+ Di(hi)
Release of τi
Deadlines in low- and high-criticality mode
… Time… Time
Switch to high-criticality mode
… Time
Switch to high-criticality mode
Remaining scheduling window
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 20
Adjusting the demand of carry-over jobs
t t+ Di(lo) t+ Di(hi)
Release of τi
Deadlines in low- and high-criticality mode
… Time… Time
Switch to high-criticality mode
… Time
Switch to high-criticality mode
Remaining scheduling window
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 21
Demand-bound functions for high-criticality mode
0 5 10 15 20 25 30Time interval length (`)
0
5
10
15
20
25
30
Dem
and
dbfHI(τi, `)
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 22
Demand-bound functions for high-criticality mode
0 5 10 15 20 25 30Time interval length (`)
0
5
10
15
20
25
30
Dem
and
dbfHI(τi, `)
dbfLO(τi, `)
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 23
The effect of the low-criticality relative deadline
If Di(lo) is decreased by δ ∈ Z, then
dbflo(τi, ℓ) ; dbflo(τi, ℓ+ δ)
dbfhi(τi, ℓ) ; dbfhi(τi, ℓ− δ)
Shifting lemma
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 24
The effect of the low-criticality relative deadline
0 5 10 15 20 25 30Time interval length (`)
0
5
10
15
20
25
30
Dem
and
δδ
dbfHI(τi, `)
dbfLO(τi, `)
dbfHI(τi, `), Di(LO) decreased by δ
dbfLO(τi, `), Di(LO) decreased by δ
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 25
Shaping the demand of the task set
A task set τ is schedulable if both A and B hold:
A : ∀ℓ ⩾ 0 :∑τi∈τ
dbflo(τi, ℓ) ⩽ sbflo(ℓ)
B : ∀ℓ ⩾ 0 :∑
τi∈hi(τ)
dbfhi(τi, ℓ) ⩽ sbfhi(ℓ)
Mixed-criticality EDF analysis
Is there a valid assignment of Di(lo)s to each high-criticalitytask τi such that both A and B hold?
A constraint satisfaction problem
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 26
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 27
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 28
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 29
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 30
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
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90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 31
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 32
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 33
Shaping the demand of the task set
0 10 20 30 40 50 60 70 80 90 100Time interval length (`)
0
10
20
30
40
50
60
70
80
90
100
Dem
and
∑dbfHI∑dbfLO
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 34
Evaluation
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Average utilization
0
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40
50
60
70
80
90
100Accep
tanc
eratio(%
)
OurOCBP-prioAMC-maxVestalEDF-VDOCBP-loadNaive
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 35
Evaluation
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Probability of high-criticality
0
10
20
30
40
50
60
70
80
90
100Weigh
tedacceptan
ceratio(%
)
OurOCBP-prioAMC-maxVestalEDF-VDOCBP-loadNaive
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 36
Evaluation
1 3 5 7 9 11 13 15 17 19Maximum ratio of high- to low-criticality WCET
0
10
20
30
40
50
60
70
80
90
100Weigh
tedacceptan
ceratio(%
)
OurOCBP-prioAMC-maxVestalEDF-VDOCBP-loadNaive
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 37
Conclusions
1 Demand-bound functions are useful also for mixed-criticality systems.
2 The particulars of mixed-criticality demand-bound functions allowus to easily shape the demand to the supply of the platform.
3 Experiments indicate that this approach performs well.
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 38
Questions?
Pontus Ekberg Bounding and Shaping the Demand of Mixed-Criticality Sporadic Tasks 39
