Probabilistic Real-Time Systems

(1)

Probabilistic Real-Time Systems

Liliana Cucu-Grosjean, INRIA Nancy-Grand Est

(2)

Outline

•

Probabilistic Real-Time Systems

•

Probabilistic Worst-Case Execution Time

Ø  PROARTIS: measurement based approach

•

Static probabilistic timing analysis

•

Measurement based timing analysis

(3)

Model

A

periodic task

is characterized by :

release

(worst-case) execution time

(minimal) inter-arrival time

deadline

τ

i

= (

O

i

,

C

i

,

T

i

,

D

i

)

O

i

C

i

T

i

D

i

Model

A

periodic task

release

(worst-case) execution time

deadline

τ

i

= (

O

i

,

C

i

,

T

i

,

D

i

)

O

i

C

i

T

i

D

i

τ

1

= (

1

,

2

,

5

,

4

)

0 1 2 3 4 5

6

τ

1

time

Model

A

periodic task

release

(worst-case) execution time

deadline

τ

i

= (

O

i

,

C

i

,

T

i

,

D

i

)

O

_i

C

_i

T

i

D

_i

τ

1

= (

1

,

2

,

5

,

4

)

R

₁

(4)

One processor, preemptive fixed-priority

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

3

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10

One processor, preemptive fixed-priority scheduling

11 12 13 14 15 16

τ

_i

= (

O

_i

,

C

_i

,

T

_i

,

D

_i

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

₃

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

(0,1,3,3)

(3,2,6,6)

(2,1,6,6)

17 18

(5)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

3

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10

One processor, preemptive fixed-priority scheduling

11 12 13 14 15 16

τ

_i

= (

O

_i

,

C

_i

,

T

_i

,

D

_i

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

₃

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

(0,1,3,3)

(3,2,6,6)

(2,1,6,6)

17 18

(6)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

3

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10

11 12 13 14 15 16

τ

_i

= (

O

_i

,

C

_i

,

T

_i

,

D

_i

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

₃

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

(0,1,3,3)

(3,2,6,6)

(2,1,6,6)

17 18

(7)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

3

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10

11 12 13 14 15 16

τ

_i

= (

O

_i

,

C

_i

,

T

_i

,

D

_i

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

₃

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

(0,1,3,3)

(3,2,6,6)

(2,1,6,6)

17 18

(8)

One processor, preemptive fixed-priority

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

3

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10

11 12 13 14 15 16

τ

_i

= (

O

_i

,

C

_i

,

T

_i

,

D

_i

)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6)

τ

₃

= (

2

,

�

1 2

0

.

50

.

5

�

,

6

,

6

)

(0,1,3,3)

(3,2,6,6)

(2,1,6,6)

17 18

R

1

= 1

(9)

Probabilistic real-time systems

Probabilistic analysis and real-time systems

•

Uncertainties from the system and the environnement

•

Probabilistic guarantees

time

D

i

O

_i

O

_i

+

T

_i

P(

R

i

>

D

i

)

<

ε

R

i

τ

i

= (

O

i

,

C

i

,

T

i

,

D

i

)

(

O

i

,

C

i

,

T

i

,

D

i

)

X = � xk P(X =xk) �

with the notation for the discret cas

(10)

What a schedule looks like?

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ3 = (2, � 1 2 0.50.5 � ,6,6)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

(2,1,6,6)

17 18

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ₃ = (2, � 1 2 0.50.5 � ,6,6)

(

O

i

,

C

i

,

T

i

,

D

i

)

(11)

What a schedule looks like?

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ3 = (2, � 1 2 0.50.5 � ,6,6)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

(2,1,6,6)

17 18

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ₃ = (2, � 1 2 0.50.5 � ,6,6)

(

O

i

,

C

i

,

T

i

,

D

i

)

(12)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ3 = (2, � 1 2 0.50.5 � ,6,6)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

(2,1,6,6)

17 18

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ₃ = (2, � 1 2 0.50.5 � ,6,6)

(

O

i

,

C

i

,

T

i

,

D

i

)

(13)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ3 = (2, � 1 2 0.50.5 � ,6,6)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

(2,1,6,6)

17 18

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ₃ = (2, � 1 2 0.50.5 � ,6,6)

(

O

i

,

C

i

,

T

i

,

D

i

)

(14)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ3 = (2, � 1 2 0.50.5 � ,6,6)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

(2,1,6,6)

17 18

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ₃ = (2, � 1 2 0.50.5 � ,6,6)

(

O

i

,

C

i

,

T

i

,

D

i

)

(15)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ3 = (2, � 1 2 0.50.5 � ,6,6)

τ

1

= (

0

,

�

1

�

,

3

,

3

)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

(2,1,6,6)

17 18

τ

1

= (

0

,

�

1

�

,

3

,

3

)

τ2 = (3, � 2 3 4 0.50.10.4 � ,6,6) τ₃ = (2, � 1 2 0.50.5 � ,6,6)

(

O

i

,

C

i

,

T

i

,

D

i

)

(16)

Outline

•

Static probabilistic timing analysis

•

(17)

PROARTIS: Probabilistically

Analysable Real-Time Systems

(18)

(19)

Outline

•

(20)

SPTA*

•

Static Probabilistic Timing Analysis

•

The pWCET estimate is obtaining by convolving the ETPs

of each instruction

{(2, 3, 8, 11),

0.2 0.7 0.05 0.05

2 3 8 11

(0.2 , 0.7, 0.05, 0.05)}

(21)

SPTA/2: convolution and requirements

•

Static Probabilistic Timing Analysis

•

Requirements of SPTA

Ø  The convolution requires independent random variables

Ø  The ETPs must be defined by a single distribution

Independent and identically distributed

random variables

1

2

0.7 0.3

!

"

#

$

%

& '

7

1

!

"

#

$

%

&

=

1

+

7

2

+

7

1

(

0.7 1

(

0.3

!

"

#

$

%

&

=

8

9

0.7 0.3

!

"

#

$

%

&

(22)

I.i.d. random variables

•

Independent random variables

Ø  Random variables that describe events which are not related

Ø  “Event A: my laptop died” and “Event B: the beamer does not like

my laptop”

•

Identically distributed random variables

Ø  Random variables that have the same distribution function

Ø  “Event A: arrival of a client in a bank” and “Event B: arrival of a car

at a gas station”

(23)

•

Data must be independent

Ø

Hypothesis testing with

Wald-Wolfowitz

test

§  A parameter P is obtained to quantify how often consecutive

execution times are higher/lower than the median

§  If the data are random, P should fall in the central part of a

Gaussian distribution

–  Threshold (0.05) is chosen based on experience/common practice. P

should fall in the central 95% part of the distribution

Required properties: Independence

(24)

•

Data must be identically distributed

Ø

Kolmogorov-Smirnov

test

§  Two subsets of data randomly obtained from the trace

§  Distributions compared. A parameter P obtained based on the

maximum distance among both data sets

§  If P is above a given threshold, data are identically distributed

–  Threshold (0.05) is chosen based on experience/common practice

Required properties: Identically distributed

Maximum distance

Whole distribution

(25)

SPTA: I.i.d. random variables

•

Static Probabilistic Timing Analysis

•

Ensured at the level of processor instructions

•

Example: randomisation at cache level, evict on access

§  Pathological eviction patterns are bounded pessimistically (N is the

number of cache entries, K is the number of memory accesses between two consecutive accesses to the same cache entry)

Ø  Some numbers

(26)

Outline

•

(27)

•

Measurement Based Probabilistic Timing Analysis

•

The pWCET estimate is obtained by applying Extreme

Value Theory

(28)

Luxembourg June 19th, 2012

•

Measurement Based Probabilistic Timing Analysis

•

First sound utilisation of EVT to pWCET estimating

Ø  The hypothesis of independence and identical distribution is

checked before any EVT utilisation

§  I.i.d. hypothesis is ensured at the level of processor instructions

Ø  Block maxima applied with a proper definition of minimum number

of runs

Ø  Utilisation of a (proven) correct statistical test to check that data

belong to the Gumbel domain

MBPTA/2

(29)

Required properties: Gumbel

•

Data must fit a Gumbel distribution

Ø

exponential tail

test

§  A parameter P is obtained for the actual data

§  A confidence interval CI is obtained for the actual data

assuming it fits a Gumbel distribution

§  If P belongs to CI then the data fit a Gumbel distribution

Execution times 6372 3728 6321 8328 8231 9827 : : + Gumbel Parameter P Confidence interval CI

(30)

June 19th, 2012

MBPTA/3

•

Steps of applying EVT (single-path programs)

Ø  Observations Ø  Grouping Ø  Fitting Ø  Comparison Ø  Tail extension

•

Convergence

Ø  Continuous rank probability score Observa- tions I.i.d tests Gumbel Fitting (ET)

passed/not passed passed/not passed

CRPS Convergence criteria More runs EVT EVT

(31)

MBPTA/4

•

Measurement Based Probabilistic Timing Analysis

•

Multi-path programs

Ø  Independent and identical distributions

§  Identical distributions ensured by the convergence process

Ø  Minimum number of observations

§  Each path must be observed a minimum number of times and it is

ensured by the convergence process

Ø  Path coverage

(32)

Conclusions and future work

•

SPTA offers a tight pWCET estimate, but impractical for

complex programs

•

MBPTA is a sound utilisation of EVT, but limited by the

path coverage

Ø  Proof of the limited pessimism of EVT application

•

Proposition of hybrid approaches

•

Extending the probabilistic techniques proposed for the

single-core case to the multi-core case

Ø  Identify the existing probabilistic techniques for multi-core

Ø  Ensure the requirements of the PROARTIS single-core