Push Forward: Global Fixed-Priority Scheduling of Arbitrary-Deadline - PowerPoint PPT Presentation

Push Forward: Global Fixed-Priority Scheduling of Arbitrary-Deadline Sporadic Task Systems Jian-Jia Chen, Georg von der Br¨ uggen and Niklas Ueter TU Dortmund University, Germany 06.07.2018 at ECRTS Jian-Jia Chen et al. 1 / 23

Sporadic Task Model τ i ( C i , D i , T i ) , U i = C i T i Utilization Relative Deadline τ i WCET Period Constrained Deadline: D i ≤ T i , ∀ τ i Jian-Jia Chen et al. 2 / 23

Sporadic Task Model τ i ( C i , D i , T i ) , U i = C i T i Utilization Relative Deadline τ i WCET Period Constrained Deadline: D i ≤ T i , ∀ τ i Implicit Deadline: D i = T i , ∀ τ i Jian-Jia Chen et al. 2 / 23

Sporadic Task Model τ i ( C i , D i , T i ) , U i = C i T i Utilization Relative Deadline τ i WCET Period Constrained Deadline: D i ≤ T i , ∀ τ i Implicit Deadline: D i = T i , ∀ τ i Arbitrary Deadline: otherwise. The jobs of a task must be executed in the FCFS manner. Jian-Jia Chen et al. 2 / 23

Scheduling Model • M identical multiprocessors: all the processors have the same characteristics • Global scheduling: • A job may execute on any processor • The system maintains a global ready queue • Execute the M highest-priority jobs in the ready queue • Basic Terms: EDF, FP, RM, DM • Global Scheduling: Global-EDF, Global-FP, Global-RM, Global-DM Jian-Jia Chen et al. 3 / 23

Good News for Global Scheduling For frame-based task systems, McNaughton’s wrap-around rule for P | pmtn | C max is optimal split tasks unsplit tasks D R. McNaughton. Scheduling with deadlines and loss functions. Management Science, 6:1-12, 1959. Jian-Jia Chen et al. 4 / 23

Critical Instant? • Synchronous release of higher-priority tasks and as early as possible for the following jobs • Example: M=2 and 3 tasks: (C i , D i , T i ) are τ 1 = (1 , 2 , 2) , τ 2 = (1 , 3 , 3) , τ 3 = (5 , 6 , 6) Infeasible for τ 3 . Feasible for τ 3 . Baruah in RTSS 2007. Jian-Jia Chen et al. 5 / 23

Identifying Interference for Constrained Deadlines carry-in body tail τ i τ i τ i τ i r k d k • For contrapositive, assume that a job of task τ k misses its absolute deadline at time d k with release time r k • Problem window (interval) is defined in [ r k , d k ) Jian-Jia Chen et al. 6 / 23

Necessary Condition for Deadline Misses τ k τ k τ k r k d k • demand E (∆): high-priority computation executed in the interval of length ∆. • If τ k misses its deadline at time d k , then E ( D k ) > M × ( D k − C k ) + C k Jian-Jia Chen et al. 7 / 23

Carry-In Demand • For contrapositive, assume that a job of task τ k misses its absolute deadline at time d k with release time r k . D k time r k d k Jian-Jia Chen et al. 8 / 23

Carry-In Demand • For contrapositive, assume that a job of task τ k misses its absolute deadline at time d k with release time r k . some proc. idle D k time r k t 0 d k • An innovative idea by Baruah in RTSS 2007 for Global-EDF, extended by Guan et al. in RTSS 2009 for Global-FP • Let t 0 be the earliest time instant such that the system executes jobs on M processors from t 0 to r k . • Prior to t 0 , at least one processor idles Jian-Jia Chen et al. 8 / 23

Carry-In Demand • For contrapositive, assume that a job of task τ k misses its absolute deadline at time d k with release time r k . some proc. idle D k time r k t 0 d k • An innovative idea by Baruah in RTSS 2007 for Global-EDF, extended by Guan et al. in RTSS 2009 for Global-FP • Let t 0 be the earliest time instant such that the system executes jobs on M processors from t 0 to r k . • Prior to t 0 , at least one processor idles • For constrained-deadline systems, at most M-1 carry-in jobs Jian-Jia Chen et al. 8 / 23

Carry-In Demand • For contrapositive, assume that a job of task τ k misses its absolute deadline at time d k with release time r k . some proc. idle D k time r k t 0 d k • An innovative idea by Baruah in RTSS 2007 for Global-EDF, extended by Guan et al. in RTSS 2009 for Global-FP • Let t 0 be the earliest time instant such that the system executes jobs on M processors from t 0 to r k . • Prior to t 0 , at least one processor idles • For constrained-deadline systems, at most M-1 carry-in jobs • For arbitrary-deadline systems, at most M-1 carry-in tasks and there may be multiple jobs in a carry-in task Jian-Jia Chen et al. 8 / 23

Global-FP: Arbitrary-Deadline some proc. idle ? D k time r k t 0 r k + ? D k • Essential Problems • How to define the window of interest? • How many jobs should be considered in the window of interest? Jian-Jia Chen et al. 9 / 23

Global-FP: Arbitrary-Deadline some proc. idle ? D k time r k t 0 r k + ? D k • Essential Problems • How to define the window of interest? • How many jobs should be considered in the window of interest? • Existing Results • Baker (RTSJ 2006): downward extension • Baruah and Fisher (OPODIS 2007, ICDCN 2008): to-be-detailed later • Guan et al. (RTSS 2009): M-1 carry-in jobs (flawed) • Sun et al. (RTCSA 2014): complex carry-in workload functions • Huang and Chen (RTNS 2015): a more precise quantification for the number of carry-in jobs Jian-Jia Chen et al. 9 / 23

Resource Augmentation Speedup (resource augmentation) factor ρ ( ρ ≥ 1): ⇒ If the task set (system) is schedulable (feasible), Algorithm A also returns a schedulable (feasible) answer when speeding up the system by a factor ρ . Jian-Jia Chen et al. 10 / 23

Resource Augmentation Speedup (resource augmentation) factor ρ ( ρ ≥ 1): ⇒ If the task set (system) is schedulable (feasible), Algorithm A also returns a schedulable (feasible) answer when speeding up the system by a factor ρ . ⇐ If Algorithm A does not return a schedulable (feasible) answer, the system is also unschedulable (infeasible) when slowing down by a factor ρ , i.e., at speed 1 /ρ . Jian-Jia Chen et al. 10 / 23

Resource Augmentation of Global DM implicit deadlines constrained deadlines arbitrary deadlines 2( M − 1) 2.668 (Lundberg 3 − 1 / M (Baruch et al. 12 M 2 − 8 M +1 ≤ 3 . 73 √ 4 M − 1 − upper bounds 2002) 2011) (Baruah and Fisher 2007) 2.823 (Chen et al. 3 − 1 / M (Chen et al. 2016) 2016) 2.668 (Lundberg 2.668 (Lundberg 2002) 2.668 (Lundberg 2002) lower bounds 2002) Jian-Jia Chen et al. 11 / 23

Resource Augmentation of Global DM implicit deadlines constrained deadlines arbitrary deadlines 2( M − 1) 2.668 (Lundberg 3 − 1 / M (Baruch et al. 12 M 2 − 8 M +1 ≤ 3 . 73 √ 4 M − 1 − upper bounds 2002) 2011) (Baruah and Fisher 2007) 3 − 1 2.823 (Chen et al. 3 − 1 / M (Chen et al. M 2016) 2016) 2.668 (Lundberg 2.668 (Lundberg 2002) 2.668 (Lundberg 2002) lower bounds 2002) 3 3 − M +1 Jian-Jia Chen et al. 11 / 23

Outline Introduction Some Details Conclusion Jian-Jia Chen et al. 12 / 23

Push Forward (Basic Idea from Baruah and Fisher) ≥ ( ℓ − 1) T k + D k = D ′ deadline miss k time t 0 t a t d • τ k is continuously active from t a to t d with deadline miss at t d Jian-Jia Chen et al. 13 / 23

Push Forward (Basic Idea from Baruah and Fisher) ≥ ( ℓ − 1) T k + D k = D ′ deadline miss k time t 0 t a t d • τ k is continuously active from t a to t d with deadline miss at t d • t 0 is the smallest value of t ≤ t a such that Ω( t , t d ) ≥ µ k • E ( t , t d ): the amount of workload (sum of the execution times) of the higher-priority jobs, i.e., from τ 1 , τ 2 , . . . , τ k − 1 , executed in the time interval [ t , t d ) • C ∗ k : amount of time that task τ k is executed from t a to t d • ℓ -th job of task τ k misses its deadline, i.e., C ∗ k < ℓ C k = C ′ k • Ω( t , t d ) is C ∗ k + E ( t , t d ) t d − t Jian-Jia Chen et al. 13 / 23

Push Forward (Basic Idea from Baruah and Fisher) ∆ ≥ ( ℓ − 1) T k + D k = D ′ deadline miss k time t 0 t a t d • τ k is continuously active from t a to t d with deadline miss at t d • t 0 is the smallest value of t ≤ t a such that Ω( t , t d ) ≥ µ k • ∆ is t d − t 0 • time instant t i is the arrival time of a higher-priority carry-in task τ i if τ i is continuously active in time interval [ t i , t 0 + ε ], where t i < t 0 and ε > 0 is an arbitrarily small number Jian-Jia Chen et al. 13 / 23

Push Forward (Basic Idea from Baruah and Fisher) ∆ ≥ ( ℓ − 1) T k + D k = D ′ τ i is active deadline miss k time t i t 0 t a t d • τ k is continuously active from t a to t d with deadline miss at t d • t 0 is the smallest value of t ≤ t a such that Ω( t , t d ) ≥ µ k • ∆ is t d − t 0 • time instant t i is the arrival time of a higher-priority carry-in task τ i if τ i is continuously active in time interval [ t i , t 0 + ε ], where t i < t 0 and ε > 0 is an arbitrarily small number Jian-Jia Chen et al. 13 / 23

Properties of Pushing Forward Suppose that µ k = M − ( M − 1) ρ for a certain ρ with 1 ≥ ρ ≥ C ′ k . k D ′ Lemma If τ k misses its deadline at t d , for any ρ with 1 ≥ ρ ≥ C ′ k , the time k D ′ t 0 always exists with Ω( t 0 , t d ) ≥ µ k and t 0 ≤ t a . Jian-Jia Chen et al. 14 / 23