Probabilistic real-time scheduling. Liliana CUCU-GROSJEAN. TRIO team, INRIA Nancy-Grand Est

Transcription

1 Probabilistic real-time scheduling Liliana CUCU-GROSJEAN TRIO team, INRIA Nancy-Grand Est

2 Outline What is a probabilistic real-time system? Relation between pwcet and pet Response time analysis Optimal fixed-priority scheduling algorithms How do we get the distributions?

3 What is a probabilistic real-time system? τ i (O i,c i,t i, D i ) (O i,c i,t i, D i ) C i =! # " $ & % Fixed-priority scheduling Preemption allowed 1 processor P( R i > D ) i < ε

4 What is a probabilistic real-time system? (2) (O i, C i,t i, D i ) C 2 =! # " $ & %

5 Probabilistic execution time (pet) The pet of the job of a task describes the probability that the execution time of the job is equal to a given value C j i = the execution time of the j th job of The execution time of j th job of the i th task has a probability of 0.7 of being 2

6 Probabilistic Worst Case Execution Time (pwcet) The pwcet of a task describes the probability that the worst-case execution time of the task is equal to a given value C 2 = The worst-case execution time of the i th task has a probability of 0.75 of being 2

7 Relation between pwcet and pet A safe pwcet is an upper bound on the pets pets C i, 8 C i C j i, 8j. ays under the

8 Outline What is a probabilistic real-time system? Relation between pwcet and pet Response time analysis Optimal fixed-priority scheduling algorithms How do we get the distributions?

9 Reponse time analysis [Diaz02] It requires probabilistical independence R i j = B i (λ i, j ) I(λ i, j ) C i j If pwcets are used, then the random variables are probabilistically independent If pets are used, then the random variables may be dependent

10 Convolution It requires probabilistical independence! # " $! & # 0 4 % " $ & = %! # " $ & % R i j = B i (λ i, j ) I(λ i, j ) C i j

11 Backlog The sum of the remaining execution times of all jobs that didn t complete their execution at λ i, j Adding the new arrival at the existing backlog Shrinking the obtained random variable R i j = B i (λ i, j ) I(λ i, j ) C i j

12 An (historical) example f W P t 3/9 f C 1/3 1/3 1/3 1/9 2/9 1/18 3/18 1/18 1/ w r hat job is shown in the box) f W P f t C 6/27 11/545/27 10/27 f W P t 0 3/27 5/54 2/27 1/27 1/54 1/54 1/54 1/ t 0 t w 11/54 5/27 5/54 2/27 1/54 1/54 1/54 1/ w

13 Outline What is a probabilistic real-time system? Relation between pwcet and pet Response time analysis Optimal fixed-priority scheduling algorithms How do we get the distributions?

14 Optimal (task) fixed-priority scheduler Rate Monotic is not optimal ## τ 1 = % 1& & ## 3 4 & & % (,2,2,40%( τ $ $ 1 2 = %% (,6,6,30%( ' ' $ $ ' '

15 Optimal (task) fixed-priority scheduler (2) A feasible task fixed-priority assignement ## τ 1 = % 1& & ## 3 4 & & % (,2,2,40%( τ $ $ 1 2 = %% (,6,6,30%( ' ' $ $ ' '

16 Optimal (task) fixed-priority scheduler (3) Theorem (Maxim,2011) The order of higher priority tasks does not have any impact on the probability of missing the deadline of a task Audsley reasoning may be proposed

17 Outline What is a probabilistic real-time system? Relation between pwcet and pet Response time analysis Optimal fixed-priority scheduling algorithms How do we get the distributions?

18 18 How do we get the distributions? pwcet estimate may be obtained by applying Extreme Value Theory (EVT) Central limit theory for tails Used previously: flood estimation, wear estimation, finance F X (x) 1 CDF pwcet X 2 < X 1 X 3 < X 1 X 3 6< X 2 X 2 6< X 3 O 1 O j X 1 X 2 X 3 x

19 19 Extreme Value Theory (MBPTA) The input of EVT are the observations of the target system EVT requires I.i.d. hypotheses for the observations Good understanding of hardware platform, compiler and operating system Generalized Extreme Value based on block maxima

20 MBPTA* Steps of applying Extreme Value Theory (EVT) Observations Sampling EVT More runs Fitting Comparison Observations EVT Convergence criteria Tail extension Convergence CRPS EVT Fitting I.i.d tests passed/not passed passed/not passed *ECRTS 2012

21 Conclusion What is a probabilistic real-time system? Relation between pwcet and pet Response time analysis Optimal fixed-priority scheduling algorithms How do we get the distributions?

22 Thank you for your attention

23 7th Junior Researcher Workshop on Real-Time Computing (JRWRTC 2013) Important dates Submission deadline: Sept. 8 th, 2013 Notification of acceptance: Sept. 15 th, 2013 Final manuscript due: Sept. 25 th, 2013 Conference: Oct. 16 th -18 th, 2013