1. Deadline Miss Rates of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
{sorma, petel,
Department of Computer and Information Science Linköping University, Sweden

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

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

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
Solutions based on approximation
1. Execution time PDF approximation
2. Independence assumption among various random variables

6. Execution Time PDF Approximation

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

8. Application modelling (1)B C F D

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

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

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

12. CTMC construction (2)
The global generator of the Markov chain becomes then
M is expressed in terms of small matrices and can be generated 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 Stages 2 3 4 5 Relative error 8.7% 4.1% 1.04% 0.4%
Accuracy vs analysis complexity compared to the exact approach
Stages 2 3 4 5
Relative error
8.7%
4.1%
1.04%
0.4%

17. Independence Assumption-Based Approximation

18. 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
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 

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

20. Load-arrival time independenceB C A B C
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