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Q uantitative E valuation of E mbedded S ystems. QUESTION DURING CLASS? Email : qees3TU@gmail.com. FAIL!. Thank you, Dimitrios Chronopoulos !. Q uantitative E valuation of E mbedded S ystems. Matrix equations - revisited Monotonicity Eigenvalues and throughput
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Quantitative Evaluation of Embedded Systems QUESTION DURING CLASS? Email : qees3TU@gmail.com
FAIL! Thank you, DimitriosChronopoulos!
Quantitative Evaluation of Embedded Systems Matrix equations - revisited Monotonicity Eigenvalues and throughput Eigenvalues and the MCM
Quantitative Evaluation of Embedded Systems Matrix equations - revisited Monotonicity Eigenvalues and throughput Eigenvalues and the MCM
Recall! Cycles with a 0 execution time cause livelocks But when logging events, this is mathematically okay... y 0 ms
Recall! A B Theorem: The number of tokens on any cycle is constant! Therefore, every cycle must contain at least one token, otherwise a deadlock occurs. y u C D 2ms 1ms 3ms 4ms
Solving the (max,+) matrix equations Using induction
Quantitative Evaluation of Embedded Systems Matrix equations - revisited Monotonicity Eigenvalues and throughput Eigenvalues and the MCM
Monotonicity Theorem: (max,+) matrix addition is a monotone operator
Monotonicity Given that x(1) = 0 and for all n : u(n) ≥ 0
Best-case approximation… A x1 B C x3 y u 2ms 1ms x2 3ms
Best-case approximation Tokens Theorem: (max,+) matrix addition is a monotone operator and as a consequence, removing the input gives a best-case approximation of behavior. Time (s)
Quantitative Evaluation of Embedded Systems Matrix equations - revisited Monotonicity Eigenvalues and throughput Eigenvalues and the MCM
Eigenvalues In linear algebra... a pair (x,λ) is an eigenpair for a matrix A if: with denoting elementwise multiplication and assuming that
Eigenvalues In (max,+) algebra... a pair (x,λ) is an eigenpair for a matrix A if: with denoting elementwise addition and assuming that
Eigenvalues 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
Eigenvalues 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
Find an eigenpair… A x1 B C x3 y 2ms 1ms x2 3ms
Eigenvalues for a given eigenpair (x,λ) with x≤0
Eigenvalues Tokens Time (s)
Eigenvalues Tokens Theorem: The best-case throughput is always smaller than 1/λ, with λthe biggest eigenvalue of the associated matrix. Time (s)
Quantitative Evaluation of Embedded Systems Matrix equations - revisited Monotonicity Eigenvalues and throughput Eigenvalues and the MCM
Eigenvalues and the MCM A x1 B C x3 y 2ms 1ms x2 3ms
Eigenvalues are cycle means A x1 B C Theorem: Every eigenvalue is the cycle mean of some cycle... x3 y 2ms 1ms x2 where Al(k,k) is the duration of some cycle with l tokens on it hence λ = Al(k,k)/l is a cycle mean. 3ms
The MCM is an eigenvalue Let λ=MCM and xi represent a critical token... A x1 B Now given matrix A of a dataflow graph,let us construct the following two matrices: C x3 y 2ms 1ms x2 3ms In an entry (i,j) representsthe max duration of any path from i to j minus λ for each token on that path.
The MCM is an eigenvalue Since xi represents a critical token: A x1 B So for the ithcolumn we find: Theorem: The maximal cycle mean of a graphis the maximum eigenvalue of its (max,+) matrix. C x3 y And consequently: 2ms 1ms x2 3ms In an entry (i,j) representsthe max duration of any path from i to j minus λ for each token on that path. So is an eigenpair of A
Summary: (max,+) matrix addition is a monotone operator Thus removing the input gives a best-case approximation of behavior. Every eigenvalue is the cycle mean of some cycle... But not every cycle mean is an eigenvalue! The maximal cycle mean of a graphis 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?”
Give the (max,+) matrix equations
