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Sorin Manolache , Petru Eles, Zebo Peng {sorma, petel, zebpe}@ida.liu.se

Schedulability Analysis of Multiprocessor Real -Time Applications with Stochastic Task Execution Times. Sorin Manolache , Petru Eles, Zebo Peng {sorma, petel, zebpe}@ida.liu.se. Department of Computer and Information Science Linköping U niversit y, Sweden. Outline. Introduction

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Sorin Manolache , Petru Eles, Zebo Peng {sorma, petel, zebpe}@ida.liu.se

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  1. Schedulability Analysis of Multiprocessor Real-Time 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. Outline Introduction Problem formulation Application modelling Approximation method Markov chain generator construction Analysis Experimental results Conclusions

  3. Scheduling Partitioning Mapping Allocation P2 P1 No Mapped and scheduled tasks on the allocated processors Fit? Introduction Functionality as an annotated task graph The schedulability analysis gives the design fitness estimate

  4. Motivation “Classical” schedulability analysis works on the worst case execution time (WCET) model Established analysis methods

  5. Applications (1) Soft real-time applications (missing a deadline could be acceptable) WCET becomes pessimistic Leads to processor under-utilization

  6. Applications (2) Early design phases, early estimations for future design guidance • Alternative Models: • Average • Interval • Stochastic

  7. Sources of Variability Application characteristics (data dependent loops and branches) Architectural factors (pipeline hazards, cache misses) External factors (network load) Insufficient knowledge

  8. probab probab execution time execution time Problem Formulation (1) Input: • Set of task graphs, periodic tasks, deadlines equal periods, statically mapped • Set of execution times probability density functions (continuous) • Scheduling policy • Deadlines less than or equal to the periods • Designer controlled rejection (discarding)

  9. Problem Formulation (2) Output: • Ratio of missed deadlines per task graph Limitations: • Non-preemption 15% 3%

  10. Approximate the ETPDFs by Coxian distributions A much larger Markov chain is obtained, but it is easier to solve Approach Outline (1) The application with stochastic task execution times can be regarded as a system with random character • The solution can be obtained by constructing and analysing the underlying stochastic process • Very difficult to solve in the case of arbitrary task execution time PDFs (ETPDFs)

  11. Approximation Modelling CTMC constr. Analysis Coxian distribs Task graphs GSPN CTMC Results Approach Outline (2)

  12. Approximation Modelling CTMC constr. Analysis Coxian distribs Task graphs GSPN CTMC Results Application Modelling (1)

  13. C A Application Modelling (2) E B F D

  14. A C F D B E Firing delay equals execution time probab firing delay Application Modelling (3) A E B C D F

  15. Approximation Modelling CTMC constr. Analysis Coxian distribs Task graphs GSPN CTMC Results Approximation (1)

  16. Approximation (2) a1l1 a2l2 a3l3 (1-a1)l1 (1-a2)l2

  17. Approximation Modelling CTMC constr. Analysis Coxian distribs Task graphs GSPN CTMC Results CTMC Construction (1)

  18. CTMC Construction (2) X, Y X, Y X SMP Approximation of the SMP Approximation of X X

  19. Construction of the CTMC The global generator of the Markov chain becomes then M is expressed in terms ofsmallmatrices and can begenerated on the fly– memory savings

  20. Analysis Time vs. Number of Tasks

  21. Analysis Time vs. Number of Procs

  22. Growth with Number of Stages

  23. Accuracy Accuracy vs analysis complexity compared to an exact approach presented in previous work

  24. Conclusions Approximation approach to performance analysis of multiprocessor real-time applications with stochastic execution times Larger scale applications can be analysed due to an efficient scheme to store the underlying stochastic process Provides the possibility to trade-off analysis speed and memory demand with analysis accuracy

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