1. Q uantitative E valuation of E mbedded S ystems QUESTION DURING CLASS? Email : qees3TU@gmail.com

3. Q uantitative E valuation of E mbedded S ystems 1.Matrix equations - revisited 2.Monotonicity 3.Eigenvalues and throughput 4.Eigenvalues and the MCM

4. Q uantitative E valuation of E mbedded S ystems 1.Matrix equations - revisited 2.Monotonicity 3.Eigenvalues and throughput 4.Eigenvalues and the MCM

5. 0 ms y Cycles with a 0 execution time cause livelocks But when logging events, this is mathematically okay...

6. AB C D 1ms 2ms 4ms u y 3ms Theorem: The number of tokens on any cycle is constant! Therefore, every cycle must contain at least one token, otherwise a deadlock occurs.

9. Using induction

10. Q uantitative E valuation of E mbedded S ystems 1.Matrix equations - revisited 2.Monotonicity 3.Eigenvalues and throughput 4.Eigenvalues and the MCM

11. Theorem: (max,+) matrix addition is a monotone operator

12. Given that x(1) = 0 and for all n : u(n) ≥ 0

13. A B C 1ms 2ms u x3x3 y x1x1 x2x2 3ms

14. Time (s) Tokens Theorem: (max,+) matrix addition is a monotone operator and as a consequence, removing the input gives a best-case approximation of behavior.

15. Q uantitative E valuation of E mbedded S ystems 1.Matrix equations - revisited 2.Monotonicity 3.Eigenvalues and throughput 4.Eigenvalues and the MCM

16. In linear algebra... a pair (x, λ) is an eigenpair for a matrix A if: with denoting elementwise multiplication and assuming that

17. In (max,+) algebra... a pair (x, λ) is an eigenpair for a matrix A if: with denoting elementwise addition and assuming that

18. In linear algebra, we find for any eigenpair (x, λ) that If (x, λ) is an eigenpair then so is (αx, λ), for any scalar α

19. And so, in (max,+) algebra we find for any eigenpair (x, λ) If (x, λ) is an eigenpair then so is (α+x, λ), for any shift α

20. A B C 1ms 2ms x3x3 y x1x1 x2x2 3ms

21. for a given eigenpair (x,λ) with x≤0

22. Time (s) Tokens

23. Time (s) Tokens Theorem: The best-case throughput is always smaller than 1/λ, with λ the biggest eigenvalue of the associated matrix.

24. Q uantitative E valuation of E mbedded S ystems 1.Matrix equations - revisited 2.Monotonicity 3.Eigenvalues and throughput 4.Eigenvalues and the MCM

25. A B C 1ms 2ms x3x3 y x1x1 x2x2 3ms

26. A B C 1ms 2ms x3x3 y x1x1 x2x2 3ms where A l (k,k) is the duration of some cycle with l tokens on it hence λ = A l (k,k) /l is a cycle mean. Theorem: Every eigenvalue is the cycle mean of some cycle...

27. A B C 1ms 2ms x3x3 y x1x1 x2x2 3ms Let λ=MCM and x i represent a critical token... Now given matrix A of a dataflow graph, let us construct the following two matrices: In an entry (i,j) represents the max duration of any path from i to j minus λ for each token on that path.

28. A B C 1ms 2ms x3x3 y x1x1 x2x2 3ms Since x i represents a critical token: So for the i th column we find: And consequently: So is an eigenpair of A Theorem: The maximal cycle mean of a graph is the maximum eigenvalue of its (max,+) matrix. In an entry (i,j) represents the max duration of any path from i to j minus λ for each token on that path.

29. The maximal cycle mean of a graph is the maximum eigenvalue of its (max,+) matrix and 1/MCM is the maximal throughput. See the book by [Baccelli, Cohen, Olsder and Quadrat] for more! Like for the question “can the 1/MCM throughput actually be achieved?” Every eigenvalue is the cycle mean of some cycle... But not every cycle mean is an eigenvalue! (max,+) matrix addition is a monotone operator Thus removing the input gives a best-case approximation of behavior.

30. Simulate 6 firings

31. Give the (max,+) matrix equations

32. Calculate the MCM

33. Determine a periodic schedule for arbitrary µ

34. Plot the latency for a period µ

35. Optimize the periodic schedule for µ = 15

36. Optimize the periodic schedule for arbitrary µ

37. Plot the delayed latency for a period µ

38. Plot the minimal delay for a period µ

39. Optimize the periodic schedule for arbitrary µ