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A MIXED MODEL FOR ESTIMATING THE PROBABILISTIC WORST CASE EXECUTION TIME Cristian MAXIM*, Adriana GOGONEL, Liliana CUCU-GROSJEAN INRIA 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

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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 1 3 7 0 1 0,7 0,2 Discrete random variable

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Probabilistic real-time systems (pRTS) Probabilities Offset WCET Period Deadline

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Probabilistic real-time systems (pRTS) Probabilities

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Probabilistic real-time systems (pRTS) 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 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

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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

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The moment generating function of random variable X is given by Moment generating function.

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