Presentation of the problem The heuristic Results and conclusion Line search method for solving a non-preemptive strictly periodic scheduling problem Cl´ ement Pira and Christian Artigues MOGISA Team, LAAS-CNRS, Toulouse, France August 29, 2013, Gent, Belgium Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 1 / 18
Presentation of the problem The heuristic Results and conclusion Context ◮ P processors distributed on the plane ◮ N periodic tasks to be executed (measures of sensors, etc.). ⇒ A multiprocessor periodic scheduling problem. ◮ non-preemptive : processing in “one shot” ◮ multiperiodic : each task has its own period ◮ strictly periodic : no jitter Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 2 / 18
Presentation of the problem The heuristic Results and conclusion Context ◮ P processors distributed on the plane ◮ N periodic tasks to be executed (measures of sensors, etc.). ⇒ A multiprocessor periodic scheduling problem. ◮ non-preemptive : processing in “one shot” ◮ multiperiodic : each task has its own period ◮ strictly periodic : no jitter Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 2 / 18
Presentation of the problem The heuristic Results and conclusion Context ◮ P processors distributed on the plane ◮ N periodic tasks to be executed (measures of sensors, etc.). ⇒ A multiprocessor periodic scheduling problem. ◮ non-preemptive : processing in “one shot” ◮ multiperiodic : each task has its own period ◮ strictly periodic : no jitter Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 2 / 18
Presentation of the problem The heuristic Results and conclusion Context ◮ P processors distributed on the plane ◮ N periodic tasks to be executed (measures of sensors, etc.). ⇒ A multiprocessor periodic scheduling problem. ◮ non-preemptive : processing in “one shot” ◮ multiperiodic : each task has its own period ◮ strictly periodic : no jitter Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 2 / 18
Presentation of the problem The heuristic Results and conclusion Context ◮ P processors distributed on the plane ◮ N periodic tasks to be executed (measures of sensors, etc.). ⇒ A multiprocessor periodic scheduling problem. ◮ non-preemptive : processing in “one shot” ◮ multiperiodic : each task has its own period ◮ strictly periodic : no jitter Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 2 / 18
Presentation of the problem The heuristic Results and conclusion Context ◮ P processors distributed on the plane ◮ N periodic tasks to be executed (measures of sensors, etc.). ⇒ A multiprocessor periodic scheduling problem. ◮ non-preemptive : processing in “one shot” ◮ multiperiodic : each task has its own period ◮ strictly periodic : no jitter Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 2 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion The uniprocessor problem ◮ Problem defined in (Korst, 1991), (Eisenbrand et al , 2010), (Al Sheikh et al , 2012). ◮ Each periodic task i is defined by : ◮ A processing time p i . ◮ A fixed period T i (multiperiodic). ◮ A reference offset t i induces a set of occurrences : t i + T i Z (strictly periodic). ⇒ To determine the reference offsets ( t i ) such that... ... no two tasks on the same processor overlap. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 3 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion Constraints between two tasks ◮ We generalize processing times p i by latency delays l i , j ◮ Constraint to express : whenever an occurrence of j follows an occurrence of i , a latency delay l i , j should be respected ⇒ The smallest positive difference between an occurrence of j and an occurrence of i should be greater than l i , j : ( t j − t i ) mod gcd ( T i , T j ) ≥ l i , j Hint : let’s define the set of all the possibles differences ( t j + T j Z ) − ( t i + T i Z ) = ( t j − t i ) + gcd ( T i , T j ) Z The modulo is the smallest positive representative of this set. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 4 / 18
Presentation of the problem The uniprocessor problem The heuristic The multiprocessor problem Results and conclusion Constraints between two tasks ◮ We generalize processing times p i by latency delays l i , j ◮ Constraint to express : whenever an occurrence of j follows an occurrence of i , a latency delay l i , j should be respected ⇒ The smallest positive difference between an occurrence of j and an occurrence of i should be greater than l i , j : ( t j − t i ) mod gcd ( T i , T j ) ≥ l i , j Hint : let’s define the set of all the possibles differences ( t j + T j Z ) − ( t i + T i Z ) = ( t j − t i ) + gcd ( T i , T j ) Z The modulo is the smallest positive representative of this set. Pira, Artigues Line search method and periodic scheduling August 29, 2013, Gent 4 / 18
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