A mixed model FOR ESTIMATING THE PROBABILISTIC WORST CASE EXECUTION TIME
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Transcript of A mixed model FOR ESTIMATING THE PROBABILISTIC WORST CASE EXECUTION TIME
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
Summary
• About probabilities
• Measurement-based probabilistic time analysis (MBPTA)
• Genetic algorithms
• Our mixed model
WHY MBPTA NEEDS to be IMPROVED?
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.)
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
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)
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
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
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
Probabilistic real-time systems (pRTS)• Example:
Probabilities
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
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
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
Probabilistic Worst Case Execution Time (pWCET)
• The pWCET is an upper bound on the execution times of all possible jobs of the task
Probabilities
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
Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
Block maxima technique
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
Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
Measurement-based probabilistic timing analysis (MBPTA)• Steps of applying EVT (single-path programs)
Observations
Grouping
Fitting
Comparison
Tail extension
MBPTA
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
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
Genetic Algorithms
Genetic Algorithms
A mixed model for estimating the probabilistic worst case execution time
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?
Thank you for your attention