1. Linköping University, IDA, ESLAB
Schedulability Analysis of Real-Time Systems with Stochastic Task Execution Times
Sorin Manolache
Linköping University, IDA, ESLAB

2. System-Level Design Process
Informal specification ok ok
Modelling
System model
Estimation
Functional simulation
Formal verification
Simulation
Analysis
Mapping
System architecture
Architecture selection P1 P2
Scheduling
Mapped and scheduled model
Lower levels of design

3. Real-Time Systems Soft RTS Hard RTS
Occasionally missing a deadline is not desired but accepted
Analysis
Can be based on other execution time models
Provides a feasibility degree
Focus of this thesis
Hard RTS
Missing a deadline is unaccaptable
Analysis
Is based on the WCET
Provides yes-no answers
Established methods

4. Outline Stochastic execution time model Contribution
Problem formulation
Exact solution
Approximate solution
Extensions
Conclusions and future work

5. Task Execution Time Variability
Application characteristics (data dependent loops and branches)
Architectural factors (pipeline hazards, cache misses)
External factors (network load)
Insufficient knowledge
Alternative Models:
Average
Interval
Stochastic

6. Why not WCET?
Soft real-time applications (missing a deadline is acceptable)
WCET becomes pessimistic
Leads to processor under-utilization
fast
WCET
probability density
computation time
slow
WCET
computation time

7. Related Work
L. Abeni and G. Butazzo, “Integrating Multimedia Applications in Hard Real-Time Systems”, 1998
J. Kim and K.G. Shin, “Execution Time Analysis of Communicating Tasks in Distributed Systems”, 1996
A. Kalavade, P. Moghe, “A Tool for Performance Estimation for Networked Embedded Systems”, 1998
J. Lehoczky, “Real Time Queueing Systems”, 1996
T. Tia et al., “Probabilistic Performance Guarantee for Real-Time Tasks with Varying Computation Times”, 1995
T. Zhou et al., “A Probabilistic Performance Metric for Real-Time System Design”, 1999

8. Limitation of Previous Work
Monoprocessor systems
Particular classes of task execution time probability density functions (exponential)
Discrete sets of possible execution times
Particular scheduling policies (FIFO, fixed priority)
Restricted application classes (independent tasks)
Analysis applicable under particular circumstances (heavy traffic)

9. Contribution
An exact method for schedulability analysis, efficiently applicable but no limited to monoprocessor systems
An approximate method for schedulability analysis, trading analysis efficiency for result accuracy
Both methods are applicable on systems with as unrestricted assumptions as possible
Experiments and method-specific extensions

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

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

12. Outline Stochastic execution time model Contribution
Problem formulation
Exact solution
Approximate solution
Extensions
Conclusions and future work

13. Analysis Method
Relies on the analysis of the underlying stochastic process
A state of the process should capture enough information to be able to generate the next states and to compute the corresponding transition probabilities

14. Group as many states as possible in equivalent states
Naive Stochastic Process 3 5
A, 0, {B}
B, t0, {}
B, tk, {A}
B, tk+1, {A}
B, t1, {}
Number of next states equals the number of possible execution times (infinitely many)
Group as many states as possible in equivalent states

15. PMIs A PMI is delimited by the arrival times and deadlines
The sorting of the tasks according to their priorities is unique inside of a PMI t0 t1 tk
tk+1 3 5 6 9 10 12 15
A, 0, {B}
B, tk, {A}
B, tk+1, {A}
B, t0, {}
B, t1, {}
B, [0, 3), {}
B, [3, 5), {A}

16. Stochastic Process s1 execA s2 s3 s6 s4 s5 execB 3 5 3 A, [0, 3), {B}
A, [0, 3), {B} s1
execA
B, [0, 3), {} s2
B, [3, 5), {A} s3
A, [5, 6), {B} s6
-, [0, 3), {} s4
A, [3, 5), {} s5 5 3
execB

17. Analysis (1)
[0, 3)
[3, 5)
[5, 6)
[6, 9)
[9, 10)
[10, 12)
[12, 15)

18. Influence of the Number of Tasks

19. Sliding Window Size

20. Influence of the Data Dependencies

21. Influence of the Instantiations

22. Outline Stochastic execution time model Contribution
Problem formulation
Exact solution
Approximate solution
Extensions
Conclusions and future work

23. Approximate the ETPDFs by functions of exponential distributions
Limitations of the Exact Solution
The number of states increases dramatically in the case of multiprocessor systems
It has to perform as many convolutions as there exist states in the stochastic process (time)
It has to store as many probability distributions as there exist states in the sliding window (memory)
Approximate the ETPDFs by functions of exponential distributions
A much larger Markov chain is obtained, but it requires less resources to solve

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

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

26. Application Modelling (2)B C F D

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

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

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

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

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

32. Construction of the CTMC
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

33. Analysis Time vs. Number of Tasks

34. Analysis Time vs. Number of Procs

35. Growth with Number of Stages

36. Accuracy Stages 2 3 4 5 Relative error 8.7% 4.1% 1.04% 0.4%
Accuracy vs analysis complexity compared to an exact approach presented in previous work
Stages 2 3 4 5
Relative error
8.7%
4.1%
1.04%
0.4%

37. Individual Task Periods (1)
360
120 A 2 15 9 B 4 C 6 G 3 H 5 J 9 D 12 I 15 E 60 F 24

38. Stochastic process size
Individual Task Periods (2)
Tasks
Stochastic process size
Increase
Identical
Individual 12
922.14
56.20% 15
32.17% 18
99.57% 21
16.60% 24
78.23% 27
52.52%

39. Deadlines < Periods (1)
Deadlines shorter than periods lead to an increase in the number of PMIs 3 5 6 9 10 12 15 2 3 4 5 6 8 9 10 11 12 14 15

40. Stochastic process size
Deadlines < Periods (2)
Tasks
Stochastic process size
Increase
d=p
d<p 12
63.93% 15
22.14% 18
42.73% 21
106.46% 24
39.33% 27
14.76%

41. Stochastic process size
Rejection vs. Discarding (1)
Tasks
Stochastic process size
Increase
Discarding
Rejection 12
42.07 15
121.60 18
73.85 21
99.68 24
80.15 27
95.04

42. Stochastic process size
Rejection vs. Discarding (2)
Tasks
Stochastic process size
Increase
Discarding
Rejection 12
1.16 15
2.23 18
7.06 21
4.13 24
4.17 27
4.30

43. Conclusions
Exact solution for the schedulability analysis. Mainly applicable to monoprocessors systems
Approximation approach to performance analysis of multiprocessor real-time applications
Larger scale applications can be analysed due to the PMI and sliding window approache (exact solution) and due to an efficient scheme to store the underlying stochastic process (approximate solution)
Provides the possibility to trade-off analysis speed and memory demand with analysis accuracy

44. Future Work
Better support for design space exploration (more performance indicators for diagnosis)
More efficient extraction of the performance indicators (exploiting symetries at the application and modelling level)
Relaxation of the assumptions (inspecting different mapping possibilities)

45. Analysis (2)
[0, 2)
[2, 4)
[4, 6)
[6, 8)

46. Stochastic Process execA execB z2 z3 s1 s2 s3 z2*execB s6 s4 s5 3 5 35 3 3 5
execA
execB 3 z2 3 z3
A, [0, 3), {B} s1
B, [0, 3), {} s2
B, [3, 5), {A} s3 3 5 8
z2*execB
A, [5, 6), {B} s6
-, [0, 3), {} s4
A, [3, 5), {} s5