a mixed model for estimating the probabilistic worst case execution time
DESCRIPTION
A mixed model FOR ESTIMATING THE PROBABILISTIC WORST CASE EXECUTION TIME. Cristian MAXIM* , Adriana GOGONEL, Liliana CUCU-GROSJEAN INRIA Paris-Rocquencourt, France *Airbus, Toulouse. - PowerPoint PPT PresentationTRANSCRIPT
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A MIXED MODEL FOR ESTIMATING THE PROBABILISTIC WORST CASE
EXECUTION TIME
Cristian MAXIM*, Adriana GOGONEL, Liliana CUCU-GROSJEANINRIA Paris-Rocquencourt, France
*Airbus, Toulouse
Open problems in real-time computing April 4th, 2014, ULB, Brussels, Belgium
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Summary
• About probabilities
• Measurement-based probabilistic time analysis (MBPTA)
• Genetic algorithms
• Our mixed model
WHY MBPTA NEEDS to be IMPROVED?
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Probabilities
• What is a distribution function?
• What is a probabilistic real time system?
• Central limit theorem
• Extreme value theory
• Independence and identical distribution (i.i.d.)
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What is a probability distribution function?
• A function that gives the probability of a random variable to be equal to a given value
• Continuos random variable Probability density function (pdf)
Probabilities
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What is a probability distribution function?
• A function that gives the probability of a random variable to be equal to a given value
• Discrete random variable Probability mass function (pmf)
Probabilities
𝒞𝑖=( 1 3 70.2 0.5 0.3)
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Cumulative distribution function (cdf)• It describes the probability that a real-valued random
variable X with a given probability distribution will be found at a value less than or equal to x
Probabilities
Continuous random variable Discrete random variable
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Probabilistic real-time systems (pRTS)
• pRTS – a real time system with at least one of the parameters represented as a random variable
• Model of real time system:
Probabilities
task (task set)
Offset
WCET
Period
Deadline
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Probabilistic real-time systems (pRTS)
• One parameter described by a random variable:
• • Most known• Studied by Diaz, Cucu and others.
• • Practical example: two cars backing up
• •
Probabilities
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Probabilistic real-time systems (pRTS)• Example:
Probabilities
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Central Limit Theorem (CLT)• Lehoczky [1992, 1995], Tia [1995], Broster [2002]
• It states that the sample mean is aproximatively a Gaussian distribution, given a sufficiently large sample. (gaussian distribution = normal distribution)
Probabilities
Tail
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Extreme value theory (EVT)• Estimates the probability of occurrence of extreme events, when their
distribution function is unknown, based on sequences of observations. • If the distribution of rescaled maxima converges, then the limit G(x) is one
of the three following types:
Probabilities
Gumbel pdf
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Independence and identical distribution (i.i.d.)
• In order to use EVT or CLT, the input data for these techniques has to be:
• Independent
• Identical distributed
Probabilities
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Probabilistic Worst Case Execution Time (pWCET)
• The pWCET is an upper bound on the execution times of all possible jobs of the task
Probabilities
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Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
- Tested to be i.i.d.
- A fair amount of observation is needed
- The input data should vary
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Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
Block maxima technique
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Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
Finding the parameters for the Gumble distribution• Location - μ• Scale - β• Shape -α
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Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
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Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
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Measurement-based probabilistic timing analysis (MBPTA)
• The MBPTA ensures safeness (tight and pessimistic bound on WCET) with respect to the input data
How we build representative input data with respect to the WCET?
MBPTA
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Genetic Algorithms
Genetic Algorithms
• Belong to the larger class of evolutionary algorithms
• Used in optimization problems in order to get better solutions
• In our case – we use it to get a large and diversified number of inputs in order to access all paths of a program
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Genetic Algorithms
Genetic Algorithms
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A mixed model for estimating the probabilistic worst case execution time
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Conclusion
• Experiments needed
• Verification of i.i.d. for both inputs and execution times
• Is there any corelation between the inputs and the execution times?
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Thank you for your attention