1 / 30

Improved Conditions for Bounded Tardiness under EPDF Fair Multiprocessor Scheduling

Improved Conditions for Bounded Tardiness under EPDF Fair Multiprocessor Scheduling. UmaMaheswari Devi and Jim Anderson University of North Carolina at Chapel Hill. Outline. Introduction & Motivation Overview of Pfair scheduling Contributions Summary. Context.

darcie
Télécharger la présentation

Improved Conditions for Bounded Tardiness under EPDF Fair Multiprocessor Scheduling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Improved Conditions for Bounded Tardiness under EPDF Fair Multiprocessor Scheduling UmaMaheswari Devi and Jim Anderson University of North Carolina at Chapel Hill

  2. Outline • Introduction & Motivation • Overview of Pfair scheduling • Contributions • Summary WPDRTS ‘04

  3. Context • Scheduling under Earliest-Pseudo-Deadline-First • EPDF is non-optimal Why EPDF when there exist optimal Pfair scheduling algorithms? WPDRTS ‘04

  4. Pfair Scheduling • Quantum-length subtasks are schedulable entities. • Subtasks prioritized by their deadlines. • Resolving priority ties carefully is crucial for optimality. • Optimal algorithms use non-trivial tie-breaking rules. WPDRTS ‘04

  5. Why EPDF? • Tie-breakers unnecessary or unacceptable for soft and dynamic real-time applications. • Soft real-time systems • May miss deadlines occasionally. • tardiness(Ji) = max(0, (t-d(Ji)) • Dynamic systems • Tasks leave and join • Spare capacity reallocated by changing task parameters (by “reweighting”). • Don’t use tie-breaking rules for such systems. • Just use EPDF WPDRTS ‘04

  6. Prior EPDF-based Results (Srinivasan and Anderson) • Optimal on up to 2 processors. • Optimal on M>2 processors if • The weight of each task at most 1/(M-1). • Ensures a tardiness of at most q quanta if • The weight of each task is at most q/(q+1). • Weight  ½  tardiness  1 WPDRTS ‘04

  7. Open Issues • Can tardiness under EPDF exceed one quantum? • If yes, can the weight restrictions be improved? WPDRTS ‘04

  8. Outline • Introduction & Motivation • Overview of Pfair scheduling • Contributions • Summary WPDRTS ‘04

  9. Pfair Scheduling • Introduced by Baruah et al. (’93) • Proportionate progress (uniform rate of execution) • Means for optimally scheduling recurrent real-time tasks on multiprocessors in polynomial time. • Proportionate progress with respect to: • Task weight — wt(T) • wt(T) = T.e/T.p  1 • Ideal fluid scheduler • Allocation to T in [t1,t2) = wt(T)(t2t1) WPDRTS ‘04

  10. Lag • Allocation error with respect to the ideal system • lag(T, t, S) = ideal(T, t) – actual(T, t, S) • Positive  behind, negative  ahead, zero  punctual allocation error for task T at time t total allocation to T in the ideal schedule over [0,t) total allocation to T in the schedule under consideration over [0,t) WPDRTS ‘04

  11. Pfairness 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 Example: Task T with wt(T) = 3/8 Ideal Allocation 3/8 Possible allocations to a periodic task 2 Lag 1 0 -1 -2 WPDRTS ‘04

  12. Pfairness 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 Example: Task T with wt(T) = 3/8 Ideal Allocation 3/8 Possible allocations to a periodic task 2 A schedule is Pfair iff (T, t :: 1 < lag(T, t) < 1) Lag 1 0 -1 -2 WPDRTS ‘04

  13. Subtasks • Pfairness ensured by • Scheduling subtasks • Each task is broken into a sequence of quantum-length subtasks. • Time-driven, quantum-based scheduling • Scheduling decisions at quantum boundaries. • Time is integral, measured in units of quanta. • Interval [i, i+1) – is ithquantum or time slot. • Schedules at most M subtasks (M – no. of processors). • Task migrations are allowed. WPDRTS ‘04

  14. Task Models f(T6,13) T3 T6 T5 T2 T1 T4 r(T1) d(T1) Example: Task T with wt(T) = 3/8 1 3/8 Total Ideal Allocation 0 6/8 12/8 18/8 3 30/8 36/8 42/8 6 54/8 Periodic 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 WPDRTS ‘04

  15. Task Models 30/8 36/8 42/8 6 T3 T6 T2 T5 T1 T4 Example: Task T with wt(T) = 3/8 1 3/8 Total Ideal Allocation 0 6/8 12/8 18/8 3 3 Sporadic 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 WPDRTS ‘04

  16. Task Models T2 T1 Example: Task T with wt(T) = 3/8 1 3/8 Total Ideal Allocation 0 6/8 12/8 18/8 3 30/8 36/8 42/8 6 Intra-sporadic T3 T6 T5 T4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 WPDRTS ‘04

  17. Task Models Theorem(Baruah et al., Anderson & Srinivasan): A Pfair schedule exists for a GIS system  on M Processors iff . . …. T1 Example: Task T with wt(T) = 3/8 1 3/8 Total Ideal Allocation 0 6/8 12/8 18/8 3 30/8 36/8 42/8 Generalized Intra-sporadic T3 T6 T5 T4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 WPDRTS ‘04

  18. Pfair-based Scheduling Algorithms • PF (Baruah et al.), PD (Baruah et al.), PD2(Anderson and Srinivasan) • Prioritize subtasks on an earliest-pseudo-deadline-first basis. • Ties among subtaskscannot be resolved arbitrarily. • Differ in tie-breaking rules. • BF (Zhu et al.) • For reduced context switches. • PDQ (QRFair) (Anderson et al.) • For efficient and nearly fair reallocation of spare capacity. WPDRTS ‘04

  19. Outline • Introduction & Motivation • Background on Pfair scheduling • Contributions • Summary WPDRTS ‘04

  20. Results of This Paper • Tardiness under EPDF may exceed three quanta. • EPDF ensures a tardiness of at most q quanta if the weight of each task is at most (q+1)/(q+2). • Weight  2/3  tardiness  1 • Without any restrictions, tardiness under EPDF is at most WPDRTS ‘04

  21. Counterexample • Tardiness under EPDF may exceed 1 quantum. • 10 processors • 13 tasks • 4 tasks of weight 1/2 • 3 tasks of weight 3/4 • 6 tasks of weight 23/24 • A subtask can miss its deadline by two at time 48. • Examples of task systems with tardiness of 3 and 4 are provided in the paper. WPDRTS ‘04

  22. System Lag – LAG • Difference between the total allocation that  receives over [0, t) in the ideal and actual schedules. • Simply, the sum of the lags of individual tasks. • LAG(, t) =  lag(T, t) T WPDRTS ‘04

  23. Proof Sketch (Tardiness of 1) • Theorem to prove: • EPDF ensures a tardiness of at most one quantum for a GIS task system  on M processors if the following hold. • (T   :: wt(T)  2/3) • ( Twt(T)  M) • Proof by contradiction. WPDRTS ‘04

  24. Proof Sketch (Tardiness of 1) t t+1 LAG = 0 LAG  M+1 • Let td be the earliest time that a subtask in a  misses its deadline by 2 under EPDF. h idle processors Setup similar to that of Srinivasan and Anderson At most M-h tasks are scheduled at t Lags of tasks not scheduled at t is 0 or less at t+1 misses deadline by 2 Suffices to determine the lags at t+1 of the M-h tasks scheduled … … 0 1 td LAG < M+1 LAG  M+1 WPDRTS ‘04

  25. Proof Sketch (Tardiness of 1) Recall: T3 lag(T, t+1) < 0 T2 T1 Task T scheduled at t h idle processors CASE 1: d(Ti) > t+1 … … 0 1 t t+1 td LAG < M+1 LAG  M+1 LAG = 0 LAG  M+1 WPDRTS ‘04

  26. Proof Sketch (Tardiness of 1) Recall: T3 lag(T, t+1) = 0 T2 lag(T, t+1) < wt(T) T1 Task T scheduled at t h idle processors CASE 2: d(Ti) = t+1 … … 0 1 t t+1 td LAG < M+1 LAG  M+1 LAG = 0 LAG  M+1 WPDRTS ‘04

  27. Proof Sketch (Tardiness of 1) Recall: T3 lag(T, t+1) = wt(T) lag(T, t+1) = 1 T2 T1 Task T scheduled at t h idle processors CASE 3(a): d(Ti) = t … … 0 1 t t+1 td LAG < M+1 LAG  M+1 LAG = 0 LAG  M+1 WPDRTS ‘04

  28. Proof Sketch (Tardiness of 1) Recall: T3 lag(T, t+1) = 1+f(Ti+2,t) Ti+2 Ti+1 Ti T2 T1 Task T scheduled at t h idle processors CASE 3(b): d(Ti) = t If the weight of each task is at most 2/3, then the sum of the lags of the tasks scheduled at t is less than M+1. f(Ti+2, t) is maximized if Ti is the first subtask of a job. However, we show that the number of such subtasks is bounded! … … 0 1 t t+1 td LAG < M+1 LAG  M+1 LAG = 0 LAG  M+1 WPDRTS ‘04

  29. Other Results • The same proof is generalized for a tardiness of q. • Can be used to show that tardiness of an arbitrary feasible task set is less than maximum of the execution times over all the tasks. Can the weight restrictions be improved?? WPDRTS ‘04

  30. Summary • Showed that tardiness under EPDF can exceed 1 quantum. • Improved per-task weight restrictions for bounded tardiness under EPDF. • Other related work: • Supporting multiple tardiness classes • Bounds on schedulable utilization of EPDF WPDRTS ‘04

More Related