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

2. WPDRTS ‘042 Outline Introduction & Motivation Overview of Pfair scheduling Contributions Summary

3. WPDRTS ‘043 Context Scheduling under Earliest-Pseudo-Deadline- First EPDF is non-optimal Why EPDF when there exist optimal Pfair scheduling algorithms?

4. WPDRTS ‘044 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.

5. WPDRTS ‘045 Why EPDF? Tie-breakers unnecessary or unacceptable for soft and dynamic real-time applications. Soft real-time systems –May miss deadlines occasionally. –tardiness(J i ) = max(0, (t-d(J i )) 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

6. WPDRTS ‘046 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

7. WPDRTS ‘047 Open Issues Can tardiness under EPDF exceed one quantum? If yes, can the weight restrictions be improved?

8. WPDRTS ‘048 Outline Introduction & Motivation Overview of Pfair scheduling Contributions Summary

9. WPDRTS ‘049 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 [t 1,t 2 ) = wt(T)  (t 2  t 1 )

10. WPDRTS ‘0410 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)

11. WPDRTS ‘0411 Pfairness Example: Task T with wt(T) = 3/8 3/8 1 012345678910111213141516 0 1 2 -2 Ideal Allocation Possible allocations to a periodic task Lag

12. WPDRTS ‘0412 Pfairness Example: Task T with wt(T) = 3/8 3/8 1 012345678910111213141516 0 1 2 -2 Ideal Allocation Possible allocations to a periodic task Lag A schedule is Pfair iff (  T, t ::  1 < lag(T, t) < 1)

13. WPDRTS ‘0413 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 i th quantum or time slot. Schedules at most M subtasks (M – no. of processors). Task migrations are allowed.

14. WPDRTS ‘0414 Task Models Example: Task T with wt(T) = 3/8 3/8 1 012345678910111213141516 06/812/818/8330/836/842/86 T1T1 T2T2 T3T3 T4T4 T5T5 T6T6 1718 54/8 Periodic Total Ideal Allocation r(T 1 )d(T 1 ) f(T 6,13)

15. WPDRTS ‘0415 Task Models Example: Task T with wt(T) = 3/8 0123456789101112131415161718 Sporadic Total Ideal Allocation 3/8 1 06/812/818/8330/836/842/863 T1T1 T2T2 T3T3 T4T4 T5T5 T6T6

16. WPDRTS ‘0416 Task Models Example: Task T with wt(T) = 3/8 3/8 1 012345678910111213141516 06/812/818/8330/836/842/86 1718 Intra-sporadic Total Ideal Allocation T1T1 T2T2 T3T3 T4T4 T5T5 T6T6

17. WPDRTS ‘0417 Task Models Example: Task T with wt(T) = 3/8 3/8 1 012345678910111213141516 06/812/818/8330/836/842/8 1718 Generalized Intra-sporadic Total Ideal Allocation T1T1 T3T3 T4T4 T5T5 T6T6 Theorem (Baruah et al., Anderson & Srinivasan): A Pfair schedule exists for a GIS system  on M Processors iff.. …. Theorem (Baruah et al., Anderson & Srinivasan): A Pfair schedule exists for a GIS system  on M Processors iff.. ….

18. WPDRTS ‘0418 Pfair-based Scheduling Algorithms PF (Baruah et al.), PD (Baruah et al.), PD 2 (Anderson and Srinivasan) –Prioritize subtasks on an earliest-pseudo-deadline-first basis. –Ties among subtasks cannot be resolved arbitrarily. –Differ in tie-breaking rules. BF (Zhu et al.) –For reduced context switches. PD Q (QRFair) (Anderson et al.) –For efficient and nearly fair reallocation of spare capacity.

19. WPDRTS ‘0419 Outline Introduction & Motivation Background on Pfair scheduling Contributions Summary

20. WPDRTS ‘0420 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

21. WPDRTS ‘0421 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.

22. WPDRTS ‘0422 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 

23. WPDRTS ‘0423 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.

24. WPDRTS ‘0424 Proof Sketch (Tardiness of 1) Let t d be the earliest time that a subtask in a  misses its deadline by 2 under EPDF. 01 … tdtd LAG  M+1LAG = 0 tt+1 LAG < M+1LAG  M+1 h idle processors At most M-h tasks are scheduled at t Lags of tasks not scheduled at t is 0 or less at t+1 Suffices to determine the lags at t+1 of the M-h tasks scheduled Setup similar to that of Srinivasan and Anderson … misses deadline by 2

25. WPDRTS ‘0425 Proof Sketch (Tardiness of 1) 01 tdtd tt+1 h idle processors Task T scheduled at t lag(T, t+1) < 0 CASE 1: d(T i ) > t+1 T1T1 T2T2 T3T3 Recall: … LAG  M+1LAG = 0 LAG < M+1LAG  M+1 …

26. WPDRTS ‘0426 Proof Sketch (Tardiness of 1) 01 tdtd tt+1 lag(T, t+1) = 0 lag(T, t+1) < wt(T) CASE 2: d(T i ) = t+1 … LAG  M+1LAG = 0 LAG < M+1LAG  M+1 … h idle processors Task T scheduled at t T1T1 T2T2 T3T3 Recall:

27. WPDRTS ‘0427 T1T1 T2T2 T3T3 Recall: Proof Sketch (Tardiness of 1) 01 tdtd tt+1 lag(T, t+1) = 1 lag(T, t+1) = wt(T) CASE 3(a): d(T i ) = t … LAG  M+1LAG = 0 LAG < M+1LAG  M+1 … h idle processors Task T scheduled at t

28. WPDRTS ‘0428 Proof Sketch (Tardiness of 1) 01 tdtd tt+1 TiTi T i+1 T i+2 f(T i+2, t) is maximized if T i is the first subtask of a job. However, we show that the number of such subtasks is bounded! 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. CASE 3(b): d(T i ) = t … LAG  M+1LAG = 0 LAG < M+1LAG  M+1 … h idle processors Task T scheduled at t lag(T, t+1) = 1+f(T i+2,t) T1T1 T2T2 T3T3 Recall:

29. WPDRTS ‘0429 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??

30. WPDRTS ‘0430 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