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Schedulability Analysis of Multiprocessor
Real-Time Applications with Stochastic Task
Execution Times
Sorin Manolache, Petru Eles, Zebo Peng sorma,
petel, zebpe_at_ida.liu.se
Department of Computer and Information
ScienceLinköping University, Sweden
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Outline

• Introduction
• Problem formulation
• Application modelling
• Approximation method
• Markov chain generator construction
• Analysis
• Experimental results
• Conclusions

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Introduction
Functionality as an annotated task graph
The schedulability analysis gives the design
fitness estimate
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Motivation

• Classical schedulability analysis works on the
  worst case execution time (WCET) model
• Established analysis methods

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Applications (1)

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

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Applications (2)

• 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

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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)

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Problem Formulation (2)

• Output
• Ratio of missed deadlines per task graph
• Limitations
• Non-preemption

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3
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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)

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Approach Outline (2)
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Application Modelling (1)
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Application Modelling (2)
E
B
F
D
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Application Modelling (3)
A
E
B
C
D
F
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Approximation (1)
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Approximation (2)
a1l1
a2l2
a3l3
(1-a2)l2
(1-a1)l1
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CTMC Construction (1)
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CTMC Construction (2)
X, Y
X, Y
X
Approximation of the SMP
SMP
Approximation of X
X
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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

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Analysis Time vs. Number of Tasks
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Analysis Time vs. Number of Procs
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Growth with Number of Stages
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Accuracy
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
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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