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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

  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

  2. Table of Contents Motivation and Problem Definition Job-Level Convolution Task-Level Convolution Runtime Improvement Evaluation von der Br¨ uggen et al. (TU Dortmund) 2 / 19

  3. Rare Deadline Misses in Real-Time Systems • Usual assumption: hard real-time constraints von der Br¨ uggen et al. (TU Dortmund) 3 / 19

  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

  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

  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

  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

  8. Task Model and Notation τ i ( C i , D i , T i ) C N τ i i • Uniprocessor, fixed priority • Sporadic tasks von der Br¨ uggen et al. (TU Dortmund) 4 / 19

  9. Task Model and Notation τ i ( C i , D i , T i ) C N τ i i • Uniprocessor, fixed priority • Sporadic tasks, implicit deadlines: D i = T i ∀ τ i von der Br¨ uggen et al. (TU Dortmund) 4 / 19

  10. Task Model and Notation τ i (( C N i , C A i ) , D i , T i ) Normal Case C N τ i i Rare, Special Case C A C A τ i i i • Uniprocessor, fixed priority • Sporadic tasks, implicit deadlines: D i = T i ∀ τ i • C A i ≥ C N here: C A i = 2 · C N i i von der Br¨ uggen et al. (TU Dortmund) 4 / 19

  11. Task Model and Notation τ i (( C N i , C A i , P ( C A i ) , P ( C N i )) , D i , T i ) Normal Case C N τ i i Rare, Special Case C A C A τ i i i • Uniprocessor, fixed priority • Sporadic tasks, implicit deadlines: D i = T i ∀ τ i • C A i ≥ C N here: C A i = 2 · C N i 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

  12. Task Model and Notation τ i (( C N i , C A i , P ( C A i ) , P ( C N i )) , D i , T i ) Normal Case C N τ i i Rare, Special Case C A C A τ i i i • Uniprocessor, fixed priority • Sporadic tasks, implicit deadlines: D i = T i ∀ τ i • C A i ≥ C N here: C A i = 2 · C N i 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

  13. Task Model and Notation Normal Case C N τ i i Rare, Special Case C A C A τ i i i • Uniprocessor, fixed priority • Sporadic tasks, implicit deadlines: D i = T i ∀ τ i • 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

  14. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  15. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  16. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  17. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  18. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  19. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  20. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) for 0 < t ≤ D k von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  21. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) for 0 < t ≤ D k • Probability of Deadline Miss: Φ k = min 0 < t ≤ D k P ( S t > t ) von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  22. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) for 0 < t ≤ D k • Probability of Deadline Miss: Φ k = min 0 < t ≤ D k P ( S t > t ) • Upper bound: any subset of points in (0 , D k ] von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  23. Probability of Deadline Miss τ 1 = (1 , 6) τ 2 = (2 , 9) τ 3 = (2 , 12) τ 4 = (1 , 19) τ 5 = (3 , 22) t 0 5 10 15 20 • Looking at lowest priority task • Normally: TDA binary decision • P ( S t > t ) for 0 < t ≤ D k • Probability of Deadline Miss: Φ k = min 0 < t ≤ D k P ( S t > t ) • Upper bound: any subset of points in (0 , D k ] • Convolution-based approach: enumerate the state space von der Br¨ uggen et al. (TU Dortmund) 5 / 19

  24. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  25. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 � � von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  26. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 � 8 � 0 . 72 von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  27. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 � 8 � 9 0 . 72 0 . 18 von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  28. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 � 8 � 9 10 0 . 72 0 . 18 0 . 08 von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  29. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 � 8 � 9 10 11 0 . 72 0 . 18 0 . 08 0 . 02 von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  30. Convolution � 3 � 5 � � C 1 5 6 = C 2 P 1 = ⊗ 0 . 9 0 . 1 0 . 8 0 . 2 P 2 � 8 � 9 10 11 0 . 72 0 . 18 0 . 08 0 . 02 • State-of-the-art: job-wise convolution from 0 to D k von der Br¨ uggen et al. (TU Dortmund) 6 / 19

  31. Job-Level Convolution τ 1 � 3 C 1 5 � P 1 = 0 . 9 0 . 1 D 1 = T 1 = 8 τ 2 � 5 C 2 6 � P 2 = 0 . 8 0 . 2 D 2 = T 2 = 14 von der Br¨ uggen et al. (TU Dortmund) 7 / 19

  32. Job-Level Convolution t = 0 t = 8 t = 14 τ 1 � 3 C 1 5 � P 1 = 0 . 9 0 . 1 D 1 = T 1 = 8 τ 2 � 5 C 2 6 � P 2 = 0 . 8 0 . 2 D 2 = T 2 = 14 von der Br¨ uggen et al. (TU Dortmund) 7 / 19

  33. Job-Level Convolution t = 0 t = 8 t = 14 � 3 � 5 τ 1 5 � 6 � � 3 0 . 9 0 . 1 0 . 8 0 . 2 C 1 5 � P 1 = 0 . 9 0 . 1 D 1 = T 1 = 8 τ 2 � 5 C 2 6 � P 2 = 0 . 8 0 . 2 D 2 = T 2 = 14 � 0 � 1 von der Br¨ uggen et al. (TU Dortmund) 7 / 19

  34. Job-Level Convolution t = 0 t = 8 t = 14 � 3 � 5 τ 1 5 � 6 � � 3 0 . 9 0 . 1 0 . 8 0 . 2 C 1 5 � P 1 = 0 . 9 0 . 1 D 1 = T 1 = 8 � 3 � 0 . 9 τ 2 � 5 C 2 6 � P 2 = 0 . 8 0 . 2 D 2 = T 2 = 14 � 0 � 1 � 5 � 0 . 1 von der Br¨ uggen et al. (TU Dortmund) 7 / 19

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