Some ideas and open problems in real-time stochastic scheduling. Liliana CUCU, TRIO team, Nancy, France

Transcription

1 Some ideas and open problems in real-time stochastic scheduling Liliana CUCU, TRIO team, Nancy, France

2 Real-time systems Reactive systems Correct reaction Temporal constraints Gotha-- Liliana CUCU - 04/04/2008 2/26

3 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

4 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

5 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

6 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

7 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

8 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

9 Real-time systems (2) Gotha-- Liliana CUCU - 04/04/2008 3/26

10 Real-time model: τ i = (O i,c i,t i,d i ) τ 1 = (O 1,C 1,T 1,D 1 ) = (1,2,5,4) release times deadlines τ temps Gotha-- Liliana CUCU - 04/04/2008 4/26

11 Real-time model: τ i = (O i,c i,t i,d i ) τ 1 = (O 1,C 1,T 1,D 1 ) = (1,2,5,4) release times deadlines T 1 τ temps Gotha-- Liliana CUCU - 04/04/2008 4/26

12 Real-time model: τ i = (O i,c i,t i,d i ) τ 1 = (O 1,C 1,T 1,D 1 ) = (1,2,5,4) release times deadlines T 1 τ 1 response time response time temps Gotha-- Liliana CUCU - 04/04/2008 4/26

13 Why stochastic? Soft real-time constraints Uncertainness Worst-case behavior is a rare event Gotha-- Liliana CUCU - 04/04/2008 5/26

14 Where is the stochastic touch? Gotha-- Liliana CUCU - 04/04/2008 6/26

15 Where is the stochastic touch? Extracting quantitative information, i.e., obtaining distribution functions Gotha-- Liliana CUCU - 04/04/2008 6/26

16 Where is the stochastic touch? Extracting quantitative information, i.e., obtaining distribution functions and Gotha-- Liliana CUCU - 04/04/2008 6/26

17 Where is the stochastic touch? Extracting quantitative information, i.e., obtaining distribution functions and Temporal analysis of systems with at least one parameter given by a random variables Gotha-- Liliana CUCU - 04/04/2008 6/26

18 Where is the stochastic touch? Extracting quantitative information, i.e., obtaining distribution functions and Temporal analysis of systems with at least one parameter given by a random variables Gotha-- Liliana CUCU - 04/04/2008 6/26

19 Extracting quantitative information Data Distribution functions Gotha-- Liliana CUCU - 04/04/2008 7/26

20 Extracting quantitative information Data? worst-case behaviour are rare events Distribution functions Gotha-- Liliana CUCU - 04/04/2008 7/26

21 Extracting quantitative information Data? worst-case behaviour are rare events Distribution functions Joint work with N. Navet and René Schott (TRIO, Nancy) Gotha-- Liliana CUCU - 04/04/2008 7/26

22 How to estimate the average response time??? Gotha-- Liliana CUCU - 04/04/2008 8/26

23 How to estimate the average response time??? Activation model of tasks not known Gotha-- Liliana CUCU - 04/04/2008 8/26

24 How to estimate the average response time??? Activation model of tasks not known Monte-Carlo simulation Gotha-- Liliana CUCU - 04/04/2008 8/26

25 How to estimate the average response time??? Activation model of tasks not known Monte-Carlo simulation Analytical approaches Gotha-- Liliana CUCU - 04/04/2008 8/26

26 How to estimate the average response time??? Activation model of tasks not known Monte-Carlo simulation Analytical approaches Markov s, Tchebychev s, Chemoff s upper bounds Gotha-- Liliana CUCU - 04/04/2008 8/26

27 How to estimate the average response time??? Activation model of tasks not known Monte-Carlo simulation Analytical approaches Markov s, Tchebychev s, Chemoff s upper bounds Large deviation Gotha-- Liliana CUCU - 04/04/2008 8/26

28 How to estimate the average response time??? Activation model of tasks not known Monte-Carlo simulation Analytical approaches Markov s, Tchebychev s, Chemoff s upper bounds Large deviation - better suited than simulation to rare events - easily implementable - embedded in a broader analysis Gotha-- Liliana CUCU - 04/04/2008 8/26

29 Large deviation : main result M n = 1 n n R i,k k=1 mean of response times over n task instances P(M n value) R i,n Cramer s theorem : if independent identically distributed random variables P(M n G) e ninf x G I(x) G = [value, ) I(x) = sup[τx loge(e τx )] = sup τ>0 τ>0 [τx log + p k e kτ ] k= Gotha-- Liliana CUCU - 04/04/2008 9/26

30 Technical contribution Can deal with distributions given as histograms RT intervalprobability k [0, 10) 1/25 5 [10, 20) 2/25 15 [20, 30) 3/25 25 [30, 40) 10/25 35 [40, 50) 4/25 45 [50, 60) 3/25 55 [60, 70) 2/25 65 x=45 x=50 x=55 Number of successive task instances (n) Gotha-- Liliana CUCU - 04/04/ /26

31 Technical contribution Can deal with distributions given as histograms RT intervalprobability k [0, 10) 1/25 5 [10, 20) 2/25 15 [20, 30) 3/25 25 [30, 40) 10/25 35 [40, 50) 4/25 45 [50, 60) 3/25 55 [60, 70) 2/25 65 x=45 x=50 x=55!! Uniprocessor or multiprocessor!! Number of successive task instances (n) Gotha-- Liliana CUCU - 04/04/ /26

32 Where is the stochastic touch? Extracting quantitative information, i.e., obtaining distribution functions and Temporal analysis of systems with at least one parameter given by a random variables Gotha-- Liliana CUCU - 04/04/ /26

33 Where is the stochastic touch? Extracting quantitative information, i.e., obtaining distribution functions and Temporal analysis of systems with at least one parameter given by a random variables Gotha-- Liliana CUCU - 04/04/ /26

34 What is the model? τ i = (O i,c i,t i,d i ) Gotha-- Liliana CUCU - 04/04/ /26

35 What is the model? τ i = (O i,c i,t i,d i ) X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

36 What is the model? τ i = (O i,c i,t i,d i ) O i X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

37 What is the model? :) τ i = (O i,c i,t i,d i ) O i X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

38 What is the model? :) τ i = (O i,c i,t i,d i ) O i C i X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

39 What is the model? :) O i C i :) τ i = (O i,c i,t i,d i ) X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

40 What is the model? :) τ i = (O i,c i,t i,d i ) O i C i T i :) X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

41 What is the model? :) τ i = (O i,c i,t i,d i ) :) :) O i C i T i X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

42 What is the model? :) τ i = (O i,c i,t i,d i ) :) :) O i C i T i D i X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

43 What is the model? :) τ i = (O i,c i,t i,d i ) :) :) :( O i C i T i D i X = ( ) x k P(X = x k ) Gotha-- Liliana CUCU - 04/04/ /26

44 Response time R i = What do we want? ( ) Gotha-- Liliana CUCU - 04/04/ /26

45 Response time R i = Satisfied deadline What do we want? ( ) satis f ydeadline i = ( ) yes no Gotha-- Liliana CUCU - 04/04/ /26

46 Response time R i = Satisfied deadline What do we want? ( ) ( ) yes no satis f ydeadline i = ( ) J i = Response time jitter etc... Gotha-- Liliana CUCU - 04/04/ /26

47 Response time R i = Satisfied deadline What do we want? ( ) ( ) yes no satis f ydeadline i = ( ) J i = Response time jitter Simulations? Analytical proofs? etc... Gotha-- Liliana CUCU - 04/04/ /26

48 Response time R i = Satisfied deadline What do we want? ( ) ( ) yes no satis f ydeadline i = ( ) J i = Response time jitter Simulations? Analytical proofs? etc... Joint work with E. Tovar (Hurray, Portugal) Gotha-- Liliana CUCU - 04/04/ /26

49 Response time of a task τ i When minimal inter-arrival times are considered R i = C i + R i C j T j hp(i) j Gotha-- Liliana CUCU - 04/04/ /26

50 This time... Response time R i = C i ( k P R i C k ) ( k R N τk C k ) Gotha-- Liliana CUCU - 04/04/ /26

51 Algorithm providing a solution R i = ( ) Initial value R i 0 = 0 1 N τ k = 0, k P 1 hp(i) New arrivals are added to the response time Response time contains only changed values All values unchaged or deadline missed? NO YES SOLUTION Gotha-- Liliana CUCU - 04/04/ /26

52 Initial values R 0 = r 0 i,1 i = 0 and N = k, k R 1 hp(i) Gotha-- Liliana CUCU - 04/04/ /26

53 Iteration m - first step L j m = C i + Working random variable L m m L j C k + m N k (r 1 j ) C k k P hp(i) T k k R hp(i) r j m 1 initial value Gotha-- Liliana CUCU - 04/04/ /26

54 An example Task T C 3 τ 1 τ 2 τ 3 τ 4 τ 5 T 1 T = T 2 = = T 4 = = T L 1 4 = l = 5 1 where l 1 1 solution of equation l 1 1 = 0 C 1 + l 1 1 C C 3 +1 C 4 with r 1 0 = 0 initial value T 2 Gotha-- Liliana CUCU - 04/04/ /26

55 Iteration m - second step R m = L m ( Δ C ) i k k k R hp ( i ) Gotha-- Liliana CUCU - 04/04/ /26

56 Back to the example Task T C 3 τ 1 τ 2 τ 3 τ 4 τ 5 T 1 T = T 2 = = T 4 = = T R 2 4 = L C C 3 Gotha-- Liliana CUCU - 04/04/ /26

57 Iteration m - get ride of unchanged values L m = R m i = New R m i = Comp(R m i,l m ) = Gotha-- Liliana CUCU - 04/04/ /26

58 One entire iteration (3) of our example The periodic higher tasks are giving a response time: Ι = The random higher tasks are giving a response time: R 3 n,0 = where Ι F * ( * F (20) C ) (20) = = , 0.08 Gotha-- Liliana CUCU - 04/04/ /26

59 One entire iteration (3) of our example The periodic higher tasks are giving a response time: Ι = The random higher tasks are giving a response time: R 3 n,0 = where Ι F * ( * F (20) C ) (20) = = , 0.08 Gotha-- Liliana CUCU - 04/04/ /26

60 Some precautions when we think stochastic... Gotha-- Liliana CUCU - 04/04/ /26

61 Some precautions when we think stochastic... Analysis able to give an answer in the deterministic case and to allow mixing hard and soft real-time constraints Gotha-- Liliana CUCU - 04/04/ /26

62 Some precautions when we think stochastic... Analysis able to give an answer in the deterministic case and to allow mixing hard and soft real-time constraints Robustness based on large deviations Gotha-- Liliana CUCU - 04/04/ /26

63 Some precautions when we think stochastic... Analysis able to give an answer in the deterministic case and to allow mixing hard and soft real-time constraints Robustness based on large deviations Next step? Gotha-- Liliana CUCU - 04/04/ /26

64 How to validate stochastic? Gotha-- Liliana CUCU - 04/04/ /26

65 How to validate stochastic? Initial condition: deterministic case Gotha-- Liliana CUCU - 04/04/ /26

66 How to validate stochastic? Initial condition: deterministic case Robustness condtion: worst-case behavior of the algorithms is rare Gotha-- Liliana CUCU - 04/04/ /26

67 How to validate stochastic? Initial condition: deterministic case Robustness condtion: worst-case behavior of the algorithms is rare Worst-case condition: insure worst-case when mixing hard and soft Gotha-- Liliana CUCU - 04/04/ /26

68 Some references A. Burns, G. Bernat, I. Broster M. Pinedo S. Edgar, A. Burns N. Navet, L. Cucu, R. Schott J.L. Diaz, D.F. Garcìa, K. Kim, C.-G. Lee, L. Lo Bello, J.M. Lopez, S.L. Min, O. Mirabella L. Cucu, E. Tovar A probabilistic framework for schedulability analysis Offline deterministic scheduling, stochastic scheduling and online deterministic scheduling: a comparative overview Statistical analysis of WCET for scheduling Probabilistic estimation of response times through large deviations Stochastic analysis of real-time systems A framework for response times analysis of fixed-priority tasks with stochastic inter-arrival times Gotha-- Liliana CUCU - 04/04/ /26

69 Some references A. Burns, G. Bernat, I. Broster M. Pinedo S. Edgar, A. Burns N. Navet, L. Cucu, R. Schott J.L. Diaz, D.F. Garcìa, K. Kim, C.-G. Lee, L. Lo Bello, J.M. Lopez, S.L. Min, O. Mirabella L. Cucu, E. Tovar A probabilistic framework for schedulability analysis Offline deterministic scheduling, stochastic scheduling and online deterministic scheduling: a comparative overview Statistical analysis of WCET for scheduling Probabilistic estimation of response times through large deviations Stochastic analysis of real-time systems A framework for response times analysis of fixed-priority tasks with stochastic inter-arrival times Gotha-- Liliana CUCU - 04/04/ /26

70 Some references A. Burns, G. Bernat, I. Broster M. Pinedo S. Edgar, A. Burns N. Navet, L. Cucu, R. Schott J.L. Diaz, D.F. Garcìa, K. Kim, C.-G. Lee, L. Lo Bello, J.M. Lopez, S.L. Min, O. Mirabella L. Cucu, E. Tovar A probabilistic framework for schedulability analysis Offline deterministic scheduling, stochastic scheduling and online deterministic scheduling: a comparative overview Statistical analysis of WCET for scheduling Probabilistic estimation of response times through large deviations Stochastic analysis of real-time systems A framework for response times analysis of fixed-priority tasks with stochastic inter-arrival times Gotha-- Liliana CUCU - 04/04/ /26

71 Some references A. Burns, G. Bernat, I. Broster M. Pinedo S. Edgar, A. Burns N. Navet, L. Cucu, R. Schott J.L. Diaz, D.F. Garcìa, K. Kim, C.-G. Lee, L. Lo Bello, J.M. Lopez, S.L. Min, O. Mirabella L. Cucu, E. Tovar A probabilistic framework for schedulability analysis Offline deterministic scheduling, stochastic scheduling and online deterministic scheduling: a comparative overview Statistical analysis of WCET for scheduling Probabilistic estimation of response times through large deviations Stochastic analysis of real-time systems A framework for response times analysis of fixed-priority tasks with stochastic inter-arrival times Gotha-- Liliana CUCU - 04/04/ /26

72 Thank you for your attention Open problems in stochastic real-time scheduling : introduction of dependent random variables Gotha-- Liliana CUCU - 04/04/2008