Deadline Miss Rates of Applications with Stochastic Task Execution Times

1. Deadline Miss Rates of Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng {sorma, petel, zebpe}@ida.liu.se Department of Computer and Information ScienceLinköping University, Sweden

2. Affordable hardware <5% missed deadlines Probability Task execution time Motivation Expensive hardware 0% missed deadlines Probability Task execution time

3. 2s • Task graphs • Task periods • Task execution time probability density functions 2s 4s • Task and task graph deadlines 6s miss 4% • Mapping of tasks to processors and messages to buses • Deadline miss ratio thresholds 10s miss 2% miss 10% 10s Probability Task execution time Problem formulation, input • Message transmission time probability density functions

4. Problem formulation, output miss 0% miss 2% • Deadline miss ratios per task and task graph miss 0% miss 2% miss 5% miss 0% miss 7% miss 3% miss 3%

5. Solutions based on approximation Outline • For monoprocessor systems, we found an exact solution based on concurrent construction and analysis of the underlying generalized semi-Markov process[Manolache et al. “Memory and Time-Efficient Schedulability Analysis of Task Graphs with Stochastic Execution Time”, ECRTS-01] • The solution is theoretically applicable to multiprocessor systems, but practically to only very small ones, because of complexity 1. Execution time PDF approximation 2. Independence assumption among various random variables

6. Execution Time PDF Approximation

7. Approximation Modelling CTMC constr. Analysis Coxian distribs Task graphs GSPN CTMC Results Coxian approximation-based

8. C A Application modelling (1) E B F D

9. A C F D B E Firing delay equals execution time probab firing delay Application modelling (2) A E B C D F

10. Approximation Modelling CTMC constr. Analysis Coxian distribs Task graphs GSPN CTMC Results Coxian approximation-based

11. Approximation of the GSMP CTMC construction (1) X, Y X, Y X GSMP Approximation of X X

12. CTMC construction (2) The global generator of the Markov chain becomes then M is expressed in terms ofsmallmatrices and can begenerated on the fly– memory savings

13. Analysis time vs. number of tasks

14. Analysis time vs. number of procs

15. Growth with number of stages

16. Accuracy Accuracy vs analysis complexity compared to the exact approach

17. Independence Assumption-Based Approximation

18.  • Analysis complexity is reduced by two means: • Task start and finish times are approximated with discrete values • Two types of dependencies between some random variables are neglected Independence assumption-based • Faster and approximate analysis for multiprocessor systems [ICCAD 2002] • However it is still too slow to be plugged into an optimization loop

19. A Z Z Y Y X X Independence of predecessors Z Y X P(X>max(Y, Z)) = P(X>Y)  P(X>Z)

20. Load-arrival time independence A C B A C B Time P(LC(t)) = P(LC(t)|AC<t)

21. Approximation effects

22. Experimental results 22

23. Conclusions • Two approaches for obtaining approximations of deadline miss ratios • Based on the approximation of the ETPDF by Coxian distributions • Efficient scheme to store the underlying stochastic process and to construct it on the fly • Based on independence assumptions among random variables • Both approaches provide the possibility to trade analysis speed and memory demand for analysis accuracy