[PPT] - Efficiently Approximating the Probability of Deadline Misses in PowerPoint Presentation

SLIDE 1

Efficiently Approximating the Probability of Deadline Misses in Real-Time Systems

Georg von der Br¨ uggen, Nico Piatkowski, Kuan-Hsun Chen, Jian-Jia Chen, and Katharina Morik

Department of Computer Science TU Dortmund University, Germany

04 July 2018

Supported by DFG, Collaborative Research Center SFB876, subproject A1.

von der Br¨ uggen et al. (TU Dortmund) 1 / 19

SLIDE 2

Rare Deadline Misses in Real-Time Systems

Usual assumption: hard real-time constraints

von der Br¨ uggen et al. (TU Dortmund) 3 / 19

SLIDE 4

Rare Deadline Misses in Real-Time Systems

Usual assumption: hard real-time constraints
Rare deadline misses often acceptable
Industrial safety standards
IEC-61508
ISO-26262

von der Br¨ uggen et al. (TU Dortmund) 3 / 19

SLIDE 5

Rare Deadline Misses in Real-Time Systems

Usual assumption: hard real-time constraints
Rare deadline misses often acceptable
Industrial safety standards
IEC-61508
ISO-26262
Soft real-time systems

von der Br¨ uggen et al. (TU Dortmund) 3 / 19

SLIDE 6

Rare Deadline Misses in Real-Time Systems

Usual assumption: hard real-time constraints
Rare deadline misses often acceptable
Industrial safety standards
IEC-61508
ISO-26262
Soft real-time systems
Important criteria: probability of deadline miss

von der Br¨ uggen et al. (TU Dortmund) 3 / 19

SLIDE 7

Rare Deadline Misses in Real-Time Systems

Usual assumption: hard real-time constraints
Rare deadline misses often acceptable
Industrial safety standards
IEC-61508
ISO-26262
Soft real-time systems
Important criteria: probability of deadline miss
Safe upper bound

von der Br¨ uggen et al. (TU Dortmund) 3 / 19

SLIDE 8

Task Model and Notation

τi(Ci, Di, Ti)

C N

i

τi

Uniprocessor, fixed priority
Sporadic tasks

von der Br¨ uggen et al. (TU Dortmund) 4 / 19

SLIDE 9

Task Model and Notation

τi(Ci, Di, Ti)

C N

i

τi

Uniprocessor, fixed priority
Sporadic tasks, implicit deadlines: Di = Ti ∀τi

von der Br¨ uggen et al. (TU Dortmund) 4 / 19

SLIDE 10

Task Model and Notation

τi((C N

i , C A i ), Di, Ti)

Normal Case

C N

i

τi

Rare, Special Case

C A

i

C A

i

τi

Uniprocessor, fixed priority
Sporadic tasks, implicit deadlines: Di = Ti ∀τi
C A

i ≥ C N i

here: C A

i = 2 · C N i

von der Br¨ uggen et al. (TU Dortmund) 4 / 19

SLIDE 11

Task Model and Notation

τi((C N

i , C A i , P(C A i ), P(C N i )), Di, Ti)

Normal Case

C N

i

τi

Rare, Special Case

C A

i

C A

i

τi

Uniprocessor, fixed priority
Sporadic tasks, implicit deadlines: Di = Ti ∀τi
C A

i ≥ C N i

here: C A

i = 2 · C N i

P(C A

i ) ≪ P(C N i )

P(C A

i ) + P(C N i ) = 1

von der Br¨ uggen et al. (TU Dortmund) 4 / 19

SLIDE 12

Task Model and Notation

τi((C N

i , C A i , P(C A i ), P(C N i )), Di, Ti)

Normal Case

C N

i

τi

Rare, Special Case

C A

i

C A

i

τi

Uniprocessor, fixed priority
Sporadic tasks, implicit deadlines: Di = Ti ∀τi
C A

i ≥ C N i

here: C A

i = 2 · C N i

P(C A

i ) ≪ P(C N i )

P(C A

i ) + P(C N i ) = 1

Probabilities independent

von der Br¨ uggen et al. (TU Dortmund) 4 / 19

SLIDE 13

Task Model and Notation

Normal Case

C N

i

τi

Rare, Special Case

C A

i

C A

i

τi

Uniprocessor, fixed priority
Sporadic tasks, implicit deadlines: Di = Ti ∀τi
C A

i ≥ C N i , here: C A i = 2 · C N i

P(C A

i ) ≪ P(C N i )

P(C A

i ) + P(C N i ) = 1

Probabilities independent

von der Br¨ uggen et al. (TU Dortmund) 4 / 19

SLIDE 14

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 15

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 16

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t)

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 17

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t)

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 18

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t)

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 19

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t)

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 20

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t) for 0 < t ≤ Dk

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 21

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t) for 0 < t ≤ Dk
Probability of Deadline Miss: Φk = min0<t≤Dk P(St > t)

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 22

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t) for 0 < t ≤ Dk
Probability of Deadline Miss: Φk = min0<t≤Dk P(St > t)
Upper bound: any subset of points in (0, Dk]

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 23

Probability of Deadline Miss

τ1 = (1, 6) τ2 = (2, 9) τ3 = (2, 12) τ4 = (1, 19) 5 10 15 20

t

τ5 = (3, 22)

Looking at lowest priority task
Normally: TDA binary decision
P(St > t) for 0 < t ≤ Dk
Probability of Deadline Miss: Φk = min0<t≤Dk P(St > t)
Upper bound: any subset of points in (0, Dk]
Convolution-based approach: enumerate the state space

von der Br¨ uggen et al. (TU Dortmund) 5 / 19

SLIDE 24

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

von der Br¨ uggen et al. (TU Dortmund) 6 / 19

SLIDE 25

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

von der Br¨

uggen et al. (TU Dortmund) 6 / 19

SLIDE 26

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

8

0.72

von der Br¨

uggen et al. (TU Dortmund) 6 / 19

SLIDE 27

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

8 0.72 9 0.18

von der Br¨

uggen et al. (TU Dortmund) 6 / 19

SLIDE 28

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

8 0.72 9 0.18 10 0.08

von der Br¨

uggen et al. (TU Dortmund) 6 / 19

SLIDE 29

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

8 0.72 9 0.18 10 0.08 11 0.02

von der Br¨

uggen et al. (TU Dortmund) 6 / 19

SLIDE 30

Convolution

C1 P1 =

3 0.9 5 0.1

⊗

5 0.8 6 0.2

= C2

P2

8 0.72 9 0.18 10 0.08 11 0.02

State-of-the-art: job-wise convolution from 0 to Dk

von der Br¨ uggen et al. (TU Dortmund) 6 / 19

SLIDE 31

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

von der Br¨

uggen et al. (TU Dortmund) 7 / 19

SLIDE 32

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

t = 0

t = 8 t = 14

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 33

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

1
t = 0

t = 8 t = 14

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 34

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

1
3

0.9

5

0.1

t = 0

t = 8 t = 14

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 35

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

1
3

0.9

5

0.1

8

0.72

9

0.18

10

0.08

11

0.02

t = 0

t = 8 t = 14 P(S8 > 8) = 0.28

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 36

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

8

0.72

9

0.18

10

0.08

11

0.02

11

0.648

13

0.072

12

0.162

14

0.018

13

0.072

14

0.018

16

0.002

15

0.008

t = 0

t = 8 t = 14 P(S8 > 8) = 0.28 P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 37

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

8

0.72

9

0.18

10

0.08

11

0.02

11

0.648

13

0.072

12

0.162

14

0.018

13

0.072

14

0.018

16

0.002

15

0.008

t = 0

t = 8 t = 14 P(S8 > 8) = 0.28 P(S14 > 14) = 0.01

3 tasks → 38 jobs → 238 = 274 877 906 944

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 38

Job-Level Convolution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

8

0.72

9

0.18

10

0.08

11

0.02

11

0.648

13

0.072

12

0.162

14

0.018

13

0.072

14

0.018

16

0.002

15

0.008

13

0.144

14

0.036

t = 0

t = 8 t = 14 P(S8 > 8) = 0.28 P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 7 / 19

SLIDE 39

Performance: Job-Level Convolution

3 4 5 6 7 8 9 10 Number of Tasks 100 200 300 400 500 600 Average Analysis Runtime (seconds) Job-Level Convolution

von der Br¨ uggen et al. (TU Dortmund) 8 / 19

SLIDE 40

Considering Time Points Individually

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

8

0.72

9

0.18

10

0.08

11

0.02

11

0.648

13

0.072

12

0.162

14

0.018

13

0.072

14

0.018

16

0.002

15

0.008

13

0.144

14

0.036

t = 14

P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 9 / 19

SLIDE 41

Considering Time Points Individually

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

8

0.72

9

0.18

10

0.08

11

0.02

11

0.648

12

0.162

14

0.018

13

0.072

14

0.018

16

0.002

15

0.008

13

0.072

13

0.072

13

0.144

14

0.036

t = 14

P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 9 / 19

SLIDE 42

Considering Time Points Individually

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

t = 14

P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 9 / 19

SLIDE 43

Considering Time Points Individually

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

10

0.01

8

0.09

8

0.09

t = 14

P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 9 / 19

SLIDE 44

Considering Time Points Individually

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

10

0.01

8

0.18

t = 14

P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 9 / 19

SLIDE 45

Considering Time Points Individually

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 14

τ2

C2 P2 =

5

0.8 6 0.2

3

0.9 5 0.1

5

0.8 6 0.2

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

10

0.01

8

0.18

11

0.648

12

0.162

13

0.144

14

0.036

15

0.008

16

0.002

t = 14

P(S14 > 14) = 0.01

von der Br¨ uggen et al. (TU Dortmund) 9 / 19

SLIDE 46

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 47

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 48

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 49

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 50

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243 0.027

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 51

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243 0.027 0.001

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 52

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243 0.027 0.001

Set of equivalence classes

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 53

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243 0.027 0.001

Set of equivalence classes
For one equivalence class:
Execution time
Number of paths
Probability of each path

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 54

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243 0.027 0.001

Set of equivalence classes
For one equivalence class:
Execution time
Number of paths
Probability of each path
Multinomial distribution
Probability of each path:

Pi(1)ℓi,1 ·Pi(2)ℓi,2 ·...·Pi(h)ℓi,h

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 55

Equivalence Classes and Multinomial Distribution

D1 = T1 = 8

τ1

C1 P1 =

3

0.9 5 0.1

1
3

0.9

5

0.1

6

0.81

8

0.09

8

0.09

10

0.01

9

0.729

11

0.081

11

0.081

13

0.009

11

0.081

11

0.081

13

0.009

15

0.001

13

0.009

t = 19

# C A

i

jobs 1 2 3 Total Ci 9 11 13 15 Probability 0.729 0.243 0.027 0.001

Set of equivalence classes
For one equivalence class:
Execution time
Number of paths
Probability of each path
Multinomial distribution
Probability of each path:

Pi(1)ℓi,1 ·Pi(2)ℓi,2 ·...·Pi(h)ℓi,h

Number of paths:

ρi,t! ℓi,1!ℓi,2!...ℓi,h!

von der Br¨ uggen et al. (TU Dortmund) 10 / 19

SLIDE 56

Task-Level Convolution

D1 = T1 = 15

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

von der Br¨ uggen et al. (TU Dortmund) 11 / 19

SLIDE 57

Task-Level Convolution

D1 = T1 = 15

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D1 = T1 = 15

τ1

C1 P1 =

6

0.81 8 0.18 10 0.01

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

von der Br¨ uggen et al. (TU Dortmund) 11 / 19

SLIDE 58

Task-Level Convolution

D1 = T1 = 15

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D1 = T1 = 15

τ1

C1 P1 =

6

0.81 8 0.18 10 0.01

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24
1
von der Br¨

uggen et al. (TU Dortmund) 11 / 19

SLIDE 59

Task-Level Convolution

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24
1
6

0.81

8

0.18

10

0.01

von der Br¨

uggen et al. (TU Dortmund) 11 / 19

SLIDE 60

Task-Level Convolution

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24
1
6

0.81

8

0.18

10

0.01

12

0.648

16

0.162

14

0.144

18

0.036

16

0.008

20

0.002

von der Br¨

uggen et al. (TU Dortmund) 11 / 19

SLIDE 61

Task-Level Convolution

t = 24

1
6

0.81

8

0.18

10

0.01

12

0.648

16

0.162

14

0.144

18

0.036

16

0.008

20

0.002

17

0.4536

19

0.1944

21

0.1134

23

0.0486

19

0.1008

21

0.0432

23

0.0252

25

0.0108

21

0.0056

23

0.0024

25

0.0014

27

0.0006

von der Br¨

uggen et al. (TU Dortmund) 11 / 19

SLIDE 62

Task-Level Convolution

t = 24

1
6

0.81

8

0.18

10

0.01

12

0.648

16

0.162

14

0.144

18

0.036

16

0.008

20

0.002

17

0.4536

19

0.1944

21

0.1134

23

0.0486

19

0.1008

21

0.0432

23

0.0252

25

0.0108

21

0.0056

23

0.0024

25

0.0014

27

0.0006

von der Br¨

uggen et al. (TU Dortmund) 11 / 19

SLIDE 63

Performance: Task-Level Convolution

3 4 5 6 7 8 9 10 Number of Tasks 100 200 300 400 500 600 Average Analysis Runtime (seconds) Job-Level Convolution Task-Level Convolution

von der Br¨ uggen et al. (TU Dortmund) 12 / 19

SLIDE 64

State Space Pruning

t = 24

1
6

0.81

8

0.18

10

0.01

12

0.648

16

0.162

14

0.144

18

0.036

16

0.008

20

0.002

17

0.4536

19

0.1944

21

0.1134

23

0.0486

19

0.1008

21

0.0432

23

0.0252

25

0.0108

21

0.0056

23

0.0024

25

0.0014

27

0.0006

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 65

State Space Pruning

t = 24

1
6

0.81

8

0.18

10

0.01

12

0.648

16

0.162

14

0.144

18

0.036

16

0.008

20

0.002

17

0.4536

19

0.1944

21

0.1134

23

0.0486

19

0.1008

21

0.0432

23

0.0252

25

0.0108

21

0.0056

23

0.0024

25

0.0014

27

0.0006

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 66

State Space Pruning

t = 24

1
6

0.81

8

0.18

10

0.01

12

0.648

16

0.162

14

0.144

18

0.036

16

0.008

20

0.002

17

0.4536

19

0.1944

21

0.1134

23

0.0486

19

0.1008

21

0.0432

23

0.0252

25

0.0108

21

0.0056

23

0.0024

25

0.0014

27

0.0006

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 67

State Space Pruning

D1 = T1 = 15

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D1 = T1 = 15

τ1

C1 P1 =

6

0.81 8 0.18 10 0.01

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24
1
von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 68

State Space Pruning

D1 = T1 = 15

τ1

C1 P1 =

3

0.9 5 0.1

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D1 = T1 = 15

τ1

C1 P1 =

6

0.81 8 0.18 10 0.01

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

min = 6 + 5 = 11 max = 10 + 7 = 17 min = 5 max = 7

1
von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 69

State Space Pruning

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

min = 6 + 5 = 11 max = 10 + 7 = 17

1
6

0.81

8

0.18

10

0.01

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 70

State Space Pruning

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D2 = T2 = 30

τ2

C2 P2 =

6

0.8 10 0.2

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

min = 6 + 5 = 11 max = 10 + 7 = 17

1
6

0.81

8

0.18

10

0.01

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 71

State Space Pruning

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

min = 6 + 5 = 11 max = 10 + 7 = 17 min = 5 max = 7

1
6

0.81

8

0.18

10

0.01

14

0.144

18

0.036

16

0.008

20

0.002

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 72

State Space Pruning

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

D3 = T3 = 24

τ3

C3 P3 =

5

0.7 7 0.3

t = 24

min = 6 + 5 = 11 max = 10 + 7 = 17 min = 5 max = 7

1
6

0.81

8

0.18

10

0.01

14

0.144

18

0.036

16

0.008

20

0.002

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 73

State Space Pruning

t = 24 min = 6 + 5 = 11 max = 10 + 7 = 17 min = 5 max = 7

1
6

0.81

8

0.18

10

0.01

14

0.144

18

0.036

16

0.008

20

0.002

23

0.0252

25

0.0108

von der Br¨

uggen et al. (TU Dortmund) 13 / 19

SLIDE 74

Performance: Job-Level Convolution with Pruning

3 4 5 6 7 8 9 10 Number of Tasks 100 200 300 400 500 600 Average Analysis Runtime (seconds) Job-Level Convolution Task-Level Convolution Pruning

von der Br¨ uggen et al. (TU Dortmund) 14 / 19

SLIDE 75

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

12

1.9·10−07

13

1.6·10−09

14

6.1·10−12

# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 76

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

12

1.9·10−07

13

1.6·10−09

14

6.1·10−12

# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 77

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

12

1.9·10−07

13

1.6·10−09

14

6.1·10−12

# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 78

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12 # C A

i

jobs 1 2 3 4 Total Ci 7 8 9 10 11 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 79

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12 # C A

i

jobs 1 2 3 4 5, 6, or 7 Total Ci 7 8 9 10 11 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 80

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

14
# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12 # C A

i

jobs 1 2 3 4 5, 6, or 7 Total Ci 7 8 9 10 11 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 81

Union of Equivalence Classes

1
7

0.84

8

0.15

9

0.012

10

4.9·10−04

11

1.3·10−05

14

1.916061·10−07

# C A

i

jobs 1 2 3 4 5 6 7 Total Ci 7 8 9 10 11 12 13 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.9 · 10−07 1.6 · 10−09 6.1 · 10−12 # C A

i

jobs 1 2 3 4 5, 6, or 7 Total Ci 7 8 9 10 11 14 Probability 0.84 0.15 0.012 4.9 · 10−04 1.3 · 10−05 1.916061 · 10−07

von der Br¨ uggen et al. (TU Dortmund) 15 / 19

SLIDE 82

Evaluation: Setup

Utilization: 70%
Periods: UUniFast, 10ms-1000ms
Pi(A) = 0.025, Pi(N) = 0.975
For 5-20 tasks: 20 task sets
For 25-35 tasks: 5 task sets

1 Pruning: multinomial-based task-level convolution with pruning 2 Unify: pruning and union of equivalence classes (max error 10−6) 3 Approx: only considering Dk and last release times 4 Analytical approach with Chernoff bounds (Chen and Chen) 5 Analytical approach with Hoeffding’s inequality (this paper) 6 Analytical approach with Bernstein inequalities (this paper)

von der Br¨ uggen et al. (TU Dortmund) 16 / 19

SLIDE 83

Evaluation: Runtime

5 10 15 20 25 30 35 Number of Tasks - Runtime Comparison 10−2 10−1 100 101 102 103 104 Average Runtime (sec)

Pruning Unify Approx Chernoff Hoeffding Bernstein von der Br¨ uggen et al. (TU Dortmund) 17 / 19

SLIDE 84

Evaluation: Precision

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

5 10 15 20 25 30 35 Number of Tasks - Runtime Comparison 10−2 10−1 100 101 102 103 104 Average Runtime (sec)

Pruning Unify Approx Chernoff Hoeffding Bernstein

Set 1 Set 2 Set 3 Set 4 Set 5 Sets with 15 Tasks - Approximation Quality 10−6 10−5 10−4 10−3 10−2 10−1 Calculated Probability

von der Br¨ uggen et al. (TU Dortmund) 18 / 19

SLIDE 85

Conclusion

Probability of deadline miss important in system design
For multiple execution times: no binary schedulability decision
Job-level convolution not scalable (not more than 10 tasks)
Idea: Task-level convolution
Multinomial distribution
Pruning improves runtime without precision loss
Union improves runtime with bounded precision loss
With pruning: approach scalable
75 tasks: average 621.6 sec per time point
100 tasks: average 791.1 sec per time point
Easy parallelization
Analytical bounds: Hoeffding’s and Bernstein inequality
Precision roughly proportional to runtime

von der Br¨ uggen et al. (TU Dortmund) 19 / 19

SLIDE 86

Conclusion

Probability of deadline miss important in system design
For multiple execution times: no binary schedulability decision
Job-level convolution not scalable (not more than 10 tasks)
Idea: Task-level convolution
Multinomial distribution
Pruning improves runtime without precision loss
Union improves runtime with bounded precision loss
With pruning: approach scalable
75 tasks: average 621.6 sec per time point
100 tasks: average 791.1 sec per time point
Easy parallelization
Analytical bounds: Hoeffding’s and Bernstein inequality
Precision roughly proportional to runtime

Thank You!

von der Br¨ uggen et al. (TU Dortmund) 19 / 19

Efficiently Approximating the Probability of Deadline Misses in Real-Time Systems

Georg von der Br¨ uggen, Nico Piatkowski, Kuan-Hsun Chen, Jian-Jia Chen, and Katharina Morik

Department of Computer Science TU Dortmund University, Germany

04 July 2018

Table of Contents

Motivation and Problem Definition Job-Level Convolution Task-Level Convolution Runtime Improvement Evaluation

Rare Deadline Misses in Real-Time Systems

Rare Deadline Misses in Real-Time Systems

Rare Deadline Misses in Real-Time Systems

Rare Deadline Misses in Real-Time Systems

Rare Deadline Misses in Real-Time Systems

Task Model and Notation

τi(Ci, Di, Ti)

Task Model and Notation

τi(Ci, Di, Ti)

Task Model and Notation

τi((C N

Normal Case

Rare, Special Case

here: C A

Task Model and Notation

τi((C N

Normal Case

Rare, Special Case

here: C A

Task Model and Notation

τi((C N

Normal Case

Rare, Special Case

here: C A

Task Model and Notation

Normal Case

Rare, Special Case

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Probability of Deadline Miss

Convolution

C1 P1 =

3

0.9 5 0.1

5

0.8 6 0.2

P2

Convolution

C1 P1 =

3

0.9 5 0.1

5

0.8 6 0.2

P2

Convolution

C1 P1 =

3

0.9 5 0.1

5

0.8 6 0.2

P2

8

0.72

Convolution

C1 P1 =

3

0.9 5 0.1

5

0.8 6 0.2

P2

8

0.72 9 0.18

Convolution

C1 P1 =

3

0.9 5 0.1

5