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Towards optimal priority assignments for real-time tasks with probabilistic arrivals and probabilistic execution times. Dorin MAXIM INRIA Nancy Grand Est. Model of the Probabilistic Real-Time System. n independent tasks with independent jobs

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Dorin MAXIM INRIA Nancy Grand Est

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## Towards optimal priority assignments for real-time tasks with probabilistic arrivals and probabilistic execution times

Dorin MAXIM

INRIA Nancy Grand Est

RTSOPS, PISA, ITALY 10/07/2012

### Model of the Probabilistic Real-Time System

• n independent tasks with independent jobs

• a task τi is characterized by τi = (Ti, Ci, Di),

period

• probabilistic execution time

• single processor, synchronous, preemptive, fixed priorities

The goal: Assigning priorities to tasks so that each task meets certain conditions referring to its timing failures.

RTSOPS, PISA, ITALY 10/07/2012 2/7

### Probabilistic Period Probabilistic ET

MIT

WCET

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Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

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Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T1,0 = 1

0 1 2 3 4

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T1,0 = 1

T2,0 = 2

0 1 2 3 4

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T1,0 = 1

T2,0 = 2

0 1 2 3 4

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T1,0 = 1

Probability of occurrence = 0.42

T2,0 = 2

0 1 2 3 4

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Example

Task-set τ = {τ1, τ2} with:

τ1=;

τ2=

Probability of occurrence =

3.24 * 10-6

T2,0 = 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RTSOPS, PISA, ITALY 10/07/2012 4/7

Open problems

RTSOPS, PISA, ITALY 10/07/2012 5/7

Open problems

• Algorithm for computing the response time distribution of different jobs of the given tasks.

RTSOPS, PISA, ITALY 10/07/2012 5/7

Open problems

• Algorithm for computing the response time distribution of different jobs of the given tasks.

• Priority assignment so that each task meets certain conditions referring to its timing failures.

RTSOPS, PISA, ITALY 10/07/2012 5/7

Open problems

• Algorithm for computing the response time distribution of different jobs of the given tasks.

• Priority assignment so that each task meets certain conditions referring to its timing failures.

• Study interval.

RTSOPS, PISA, ITALY 10/07/2012 5/7

Intuitions and counter-intuitions

Rate Monotonic is NOT optimal for probabilistic systems:

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Intuitions and counter-intuitions

Rate Monotonic is NOT optimal for probabilistic systems:

RM does not take into account the probabilistic character of the tasks

RTSOPS, PISA, ITALY 10/07/2012 6/7

Intuitions and counter-intuitions

Rate Monotonic is NOT optimal for probabilistic systems:

RM does not take into account the probabilistic character of the tasks

RM considers tasks periods, which here are random variables that may not be comparable

RTSOPS, PISA, ITALY 10/07/2012 6/7

Intuitions and counter-intuitions

Rate Monotonic is NOT optimal for probabilistic systems:

RM does not take into account the probabilistic character of the tasks

RM considers tasks periods, which here are random variables that may not be comparable

RM was proved not optimal for tasks with deterministic arrivals and probabilistic executions times

RTSOPS, PISA, ITALY 10/07/2012 6/7

Thank you!

RTSOPS, PISA, ITALY 10/07/2012 7/7