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

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