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Memory and Time-Efficient Schedulability Analysis
of Task Sets with Stochastic Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
Department of Computer and Information
ScienceLinköpings universitet
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Outline

• Introduction
• Task model and problem formulation
• Analysis method
• Experimental results
• Conclusions and future work

3
Introduction
Functionality as an annotated task graph
The schedulability analysis gives the design
fitness estimate
Mapped and scheduled tasks on the allocated
processors
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Motivation

• Classical schedulability analysis works on the
  WCET model
• Established analysis methods

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Applications

• Soft real-time applications (missing a deadline
  is acceptable)
• WCET becomes pessimistic
• Leads to processor under-utilization

• Early design phases, early estimations for future
  design guidance

• Alternative Models
• Average
• Interval
• Stochastic

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Sources of Variability

• Application characteristics (data dependent loops
  and branches)
• Architectural factors (pipeline hazards, cache
  misses)
• External factors (network load)
• Insufficient knowledge

7
Related Work

• L. Abeni and G. Butazzo, Integrating Multimedia
  Applications in Hard Real-Time Systems, 1998
• A. Atlas and A. Bestavros, Stochastic Rate
  Monotonic Scheduling, 1998
• 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

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Outline

• Introduction
• Task model and problem formulation
• Analysis method
• Experimental results
• Conclusions and future work

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Problem Formulation

• Input
• Set of task graphs
• Set of execution time probability distribution
  functions (continuous)
• Scheduling policy
• Output
• Ratio of missed deadlines per task or per task
  graph
• Limitations
• Discarding, non-pre-emption

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Task Model
360
120
2
15
9
6
4
5
3
9
12
15
60
24
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Outline

• Introduction
• Task model and problem formulation
• Analysis method
• Experimental results
• Conclusions and future work

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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

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PMIs
0
5
3
14
PMIs
0
5
3
6
9
10
12
15

• 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

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Stochastic Process
0
5
3
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Analysis
0, 3)
3, 5)
5, 6)
6, 9)
9, 10)
10, 12)
12, 15)
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Outline

• Introduction
• Task model and problem formulation
• Analysis method
• Experimental results
• Conclusions and future work

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Experimental Results
Influence of number of tasks on the process size
155000
110000
Number of process states
65000
20000
10
11
12
13
14
15
16
17
18
19
Tasks
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Experimental Results
Influence of dependency degree on the process size
1000000
100000
Number of process states
10000
1000
0
1
2
3
4
5
6
7
8
9
Dependency degree
20
Experimental Results
Influence of the period LCM on the process size
1800000
1200000
Number of process states
600000
0
2500
4000
5500
1000
Least common multiple
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Conclusions

• Schedulability analysis of set of tasks with
  stochastic execution times
• Construction and analysis of the process at the
  same time ? sliding window size between 16 to 172
  times smaller than the total number of process
  states
• Future work extension for multiprocessor case