1. 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 Science Linköping University, Sweden

2. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 2 Outline nIntroduction nProblem formulation nApplication modelling nApproximation method nMarkov chain generator construction nAnalysis nExperimental results nConclusions

3. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 3 Introduction Mapped and scheduled tasks on the allocated processors The schedulability analysis gives the design fitness estimate Scheduling AllocationP1P1 P2P2 Partitioning Mapping Functionality as an annotated task graph Fit? No

4. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 4 Motivation n“Classical” schedulability analysis works on the worst case execution time (WCET) model nEstablished analysis methods

5. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 5 Applications (1) nSoft real-time applications (missing a deadline could be acceptable) nWCET becomes pessimistic nLeads to processor under-utilization

6. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 6 Applications (2) nEarly design phases, early estimations for future design guidance nAlternative Models: nAverage nInterval nStochastic

7. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 7 nApplication characteristics (data dependent loops and branches) nArchitectural factors (pipeline hazards, cache misses) nExternal factors (network load) nInsufficient knowledge Sources of Variability

8. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 8 Problem Formulation (1) Input: nSet of task graphs, periodic tasks, deadlines equal periods, statically mapped nSet of execution times probability density functions (continuous) nScheduling policy nDeadlines less than or equal to the periods nDesigner controlled rejection (discarding) execution time probab

9. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 9 Problem Formulation (2) Output: nRatio of missed deadlines per task graph Limitations: nNon-preemption 15%3%

10. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 10 Approach Outline (1) The application with stochastic task execution times can be regarded as a system with random character nThe solution can be obtained by constructing and analysing the underlying stochastic process nVery difficult to solve in the case of arbitrary task execution time PDFs (ETPDFs) Approximate the ETPDFs by Coxian distributions A much larger Markov chain is obtained, but it is easier to solve

11. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 11 Approach Outline (2) Approximation Coxian distribsTask graphsGSPNCTMCResults ModellingCTMC constr.Analysis

12. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 12 Application Modelling (1) Approximation Coxian distribs Task graphsGSPN CTMCResults Modelling CTMC constr.Analysis

13. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 13 Application Modelling (2) B C D A E F

14. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 14 C Application Modelling (3) AFDBE firing delay probab Firing delay equals execution time A BC D E F

15. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 15 Approximation (1) Approximation Coxian distribs Task graphsGSPNCTMCResults ModellingCTMC constr.Analysis

16. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 16 Approximation (2)  1 1  2 2  3 3  2 ) 2  1 ) 1

17. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 17 CTMC Construction (1) Approximation Coxian distribsTask graphsGSPN CTMC Results Modelling CTMC constr. Analysis

18. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 18 CTMC Construction (2) X X, Y X Approximation of the SMPSMP Approximation of X

19. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 19 Construction of the CTMC nThe global generator of the Markov chain becomes then nM is expressed in terms of small matrices and can be generated on the fly – memory savings

20. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 20 Analysis Time vs. Number of Tasks

21. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 21 Analysis Time vs. Number of Procs

22. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 22 Growth with Number of Stages

23. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 23 Accuracy Accuracy vs analysis complexity compared to an exact approach presented in previous work Stages2345 Relative error8.7%4.1%1.04%0.4%

24. Schedulability Analysis of Multiprocessor Real-Time Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden 24 Conclusions nApproximation approach to performance analysis of multiprocessor real-time applications with stochastic execution times nLarger scale applications can be analysed due to an efficient scheme to store the underlying stochastic process nProvides the possibility to trade-off analysis speed and memory demand with analysis accuracy