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Q uantitative E valuation of E mbedded S ystems

This exercise focuses on modeling a car manufacturing line with four assembly robots and various production units. The process includes tasks such as producing a chassis, steering installation, braking system, body, and painting, each requiring specific time commitments. We will compute the first three firings of each robot, considering safety constraints on the number of cars allowed between specific robots. The algebraic approach will help measure the traffic and efficiency of the manufacturing line using max-plus algebra while exploring linear systems theory and matrix equations.

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Q uantitative E valuation of E mbedded S ystems

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  1. Quantitative Evaluation of Embedded Systems Dataflow and Max-Plus Algebra

  2. Exercise: Model a car manufacturing line Consider a car manufacturing line consisting of... • Four assembly robots: A,B,C and D • A production unit that needs 20 minutes to produce a chassis • A production unit that needs 10 minutes to produce a steering installation • A production unit that needs 10 minutes to produce a breaking system • A production unit that needs 20 minutes to produce a body • Three painting units that each need 30 minutes to paint a body • A production unit that needs 15 minutes to produce a radio • Robot A compiles the chassis and the steering installation in 4 min. and sends it to B • Robot B adds the breaking system in 3 min. and sends it to C • Robot C adds a painted body in 5 min. and sends it to D • Robot D adds a radio in 1 min. and sends the car out of the factory • For safety reasons, there can be at most 3 ‘cars’ between A and C, and only 2 between B and D • Every robot can only deal with one of each of the assembled components at a time

  3. Answer: Model a car manufacturing line Exercise: calculate the first 3 firings of each actor 30min 20min 10min A B D C 5min 3min 4min 1min 15min Disclaimer: no actual car assembly line was studied in order to make this model. 20min 20min 10min

  4. The algebraic approach: Measuring traffic 30min 20min 10min 5min 3min 4min 1min 15min 20min 20min 10min

  5. Counters v.s. Loggers Tokens Tokens Tokens Counting tokens Logging events Time (s) Time (s) Time (s)

  6. Logging traffic A B C x2 y u x1 x3 15ms 10ms x5 x4 25ms

  7. Logging traffic A B C x2 y u x1 x3 15ms 10ms x5 x4 25ms

  8. Logging traffic A B C x2 y u 15ms 10ms x4 x’4 25ms

  9. Detour: Linear Algebra

  10. Detour: Linear algebra

  11. Detour: Linear systems theory

  12. Detour: Linear systems theory

  13. Detour: (max,+) algebra

  14. Detour: (max,+) systems theory

  15. Question: calculate this product!

  16. Question: what is now a unit matrix?

  17. Detour: Linear Systems Theory

  18. Detour: (max,+) systems theory

  19. Matrix equations A B C x2 y u 15ms 10ms x4 x’4 25ms

  20. Matrix equations A B C x2 y u 15ms 10ms x4 x’4 25ms

  21. Matrix equations A B C x2 The entries in a (max,+) algebra matrix represent the longest* token-free pathsfrom one initial token to another. * Where ‘longest’ is means ‘greatest total execution time’. y u 15ms 10ms x4 x’4 25ms

  22. Exercise: Determine the matrix equations y D C E B A F u 0 ms 1 ms 5 ms 7 ms 2 ms 15 ms

  23. y u 0 ms 1 ms 5 ms 2 7 ms D C E B A F 2 ms 15 ms 3

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