Multiprocessor EDF and Deadline Monotonic Schedulability Analysis

Transcription

1 Multiprocessor EDF and Deadline Monotonic Schedulability Analysis Ted Baker Department of Computer Science Florida State University Tallahassee, FL baker

2 Overview 1. question schedulability with EDF and RM on SMP 2. key proof technique λ-busy interval 3. summary of results 4. relation to prior results

3 Dhall Effect with EDF/DM Scheduling on MP. τ 1 τ 2 τ m τ m+1 0 mx+1. τ 1 τ 2 τ m τ m+1 0 mx+1 This is only a problem when we have some tasks with very high local utilization.

4 This example shows worst-case achievable multiprocessor utilization can be as bad as 1, compared to ideal value of m. Until recently, it was used as an excuse for using partitioned scheduling on multiprocessors. However, the example depends on mixing tasks with extremely low and extremely high utilization (or ratio of compute time to deadline). Intuitively, we know we can achieve higher utilization when we have only smaller tasks. 3-1

5 Demand & Load demand of a time window = total work W that would need to be completed within the window for all the deadlines within the interval to be met load of window of length = W.

6 Schedulabity Test Based on Demand Analysis Step 1: Find a lower bound µ on the load of a window leading up to a deadline that is necessary for the deadline to be missed. Step 2: Find an upper bound β on the load of such a window that can be generated by a given task set. Step 3: If β µ the task set must be schedulable.

7 m1 Lower Bound on Load of a Problem Interval τ k is released τ k misses deadline m t t + m( x) x demand x m x, x ck load m c k

8 We call this a problem interval because it ends in a missed deadline. Since the problem job misses its deadline, the sum of the lengths of all the time intervals in which the problem job does not execute must exceed its slack time, c k. 6-1

9 Lower Bound on Load Lemma The load of the interval a missed deadline of τ k is t t W m dk 1 starting in a release of τ k and ending in c k c k

10 T i Upper Bound on Load d Ti i Ti di T i φ ε δ t t head body tail t + load ε d i 1 ci

11 The difficult part is bounding the amount of carried in work, i.e. work that is released, but not completed, before the start of the interval. If we bound this too grossly, e.g. by, we cannot get a useful result. 8-1

12 Carried-in Load Definition The carry-in of τ i at time t is the residual compute time of the last job of task τ i released before t, if any, and is denoted by the symbol ε. all m processors busy on other jobs of other tasks m y x ε φ t t t + Problem: What is the best possible upper bound on carry-in?

13 & (Mostly) Busy Interval Preceding a Missed Deadline single processor busy window: 1. demand of the interval = 100% 2. no demand is carried into the interval multiprocessor busy window: 1. demand of the interval m1!λ "$#λ 2. demand carried into the interval is bounded by c!φλ k 3. %ck λ

14 We start with the interval between a missed deadline and the corresponding release time, then extend it downward until any further extension would not be λ-busy. We call this the maximal λ-busy downward extension of a problem interval. This notion is the right generalization of busy window, for a multiprocessor. (We cannot expect 100% utilization.) 10-1

15 *., / + - ' ( Upper Bound on EDF Demand Lemma For any busy window t t) W i of τ i in the busy window is no greater than * of τk and any other task τi, the EDF demand n )max 0 ( ci,φλ where φ.nt i λ.c k, n d i 10 )di,, 2/Ti )1 if 3 di, and n.0 otherwise. T i T i T i T i φ Ti φ (n 1) T i δ =d i t t head body tail t +

16 The demand of a single task is the work that must be done by that task in the window for the deadline of the problem task to complete by its deadline. 11-1

17 9 ; : < > > > 9 87of τk the EDF load Wi Upper Bound on EDF Load Lemma For any busy window β i, where t t 6 due to τ i is at most β i i= T 1 T i= 1 T i d i T i d i λt i if λ if λ c k c k T i T i

18 This result is shown by applying the preceding bound on W i and choosing φ and n to maximize W i A. 12-1

19 F J K L C C EDF Schedulablity Test Theorem A set of periodic tasks τ 1BDC preemptive EDF scheduling if, for every task τ k, n E1 i min 1B β i where β i is as defined in the lemma above. B τn is schedulable on m processors using IGH m 1 c k c k

20 Q N N Utilization Bound Corollary A set of periodic tasks τ 1 and with deadline equal to period, is guaranteed to be schedulable on m processors MDN using preemptive EDF scheduling if n O1 i T i P m M τn with maximum individual utilization λ Rλ 1 S$Tλ This was previously shown by Goossens, Funk, Baruah, using a different proof. We have it as a consequence of the more general schedulability test, for the special case where deadline equals period. Goossens, Funk, Baruah showed that this is tight as a guaranteed utilization bound.

21 `_X Ti Y Z [ X U V Upper Bound on DM Demand Lemma For any busy window t tw of τ k and any task τ i, the DM demand W i of τ i in the busy window is no greater than n Ymax 0 V ci Yφλ λ\ where n\ c k_, ^] δ i W1, φ\nt i Wδi Y, δ\ci i if iak, and δi \\di if i k.

22 Essentially the same analysis, using the λ-busy window, can be applied to fixed-priority scheduling, including deadline monotonic. 15-1

23 Upper Bound on DM Demand (Pictures) T i Ti T i Ti d i... φ Ti φ (n 1) T i δ = t t head body tail t + T k T k T k T k c c c k k... c k k φ T k φ (n 1) T k δ = t t head body tail t +

24 There are two cases because (only) τ k must have its absolute deadline at the end of the interval. The other tasks can have absolute deadlines after that point, so long as their relative deadlines are b dk. 16-1

25 l m k i j f c d f e e h i n n m f o e i n Upper bound on DM load Lemma For any busy window at most t t e of τ k the DM load W i g due to τi, i hk, is β i T i T i 1 1 T i δ i T i δ i λt i if λ if λ c k c k T i T i where δ i ici for i hk, and δk idk.

26 u DM Schedulability Test Theorem A set of periodic tasks is schedulable on m processors using preemptive deadline-monotonic scheduling if, for every task τ k, kp1 β i q1 i m r s 1t where β i is as defined in the lemma above. c k

27 } } y x RM Utilization Bound A set of periodic tasks with maximum individual utilization λ and deadline equal to period, is guaranteed to be schedulable on m processors, m rate monotonic scheduling if n iw1 T i m 2 1zλ {$ λ v 2, using preemptive This improves on the result of Baruah and Goossens that a set of tasks, all with deadline equal to period, is guaranteed to be schedulable on m processors using RM scheduling if T for i~1 i D 3 n and n iw 1 T 3. i x 1 x m

28 ƒ Tightness of RM Utilization Bound There exist task sets with maximum individual task utilization λ that are not feasible with preemptive RM scheduling on m processors and have utilization arbitrarily close to λ mln 2 1 λ

29 We conjecture that this may be the actual guaranteed RM utilization bound, though we have not been able to prove it yet. 20-1

30 utilization bound lower bound upper bound λ m 25 30

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32 ratio of upper to lower bound ratio λ m 25 30

33 The figures show how much of a gap there is between our lower bound on the minimum achievable RM utilization and our upper bound. The first is a direct plot of the upper and lower bounds. The second showns the ratio of the upper bound to the lower bound. 20-2

34 Other Papers on Global MP EDF and RM Schedulability 1. A. Srinivasan, S. Baruah, Deadline-based scheduling of periodic task systems on multiprocessors, Information Processing Letters 84 (2002) x 2. B. Andersson, S. Baruah and J. Jonsson, Static-priority scheduling on multiprocessors, Proceedings of the IEEE Real-Time Systems Symposium London, England (December 2001). 3. B. Andersson and J. Jonsson, Fixed-priority preemptive multiprocessor scheduling: to partition or not to partition, Proceedings of the International Conference on Real-Time Computing Systems, Cheju Island, South Korea (December 2000). 4. S. Baruah and Joel Goossens, Rate-monotonic scheduling on uniform multiprocessors, UNC-CS TR02-025, University of North Carolina Department of Computer Science (May 2002).

35 5. S. Baruah, J. Gherke and C.G. Plaxton, Fast scheduling of periodic tasks on multiple resources, Proceedings of the 9th International Parallel Processing Symposium (April 1995) S. Funk, J. Goossens and S. Baruah, On-line scheduling on uniform multiprocessors, Proceedings of the IEEE Real-Time Systems Symposium, IEEE Computer Society Press (December 2001). 7. J. Goossens, S. Funk and S. Baruah, Priority-driven scheduling of periodic task systems on multiprocessors, technical report UNC-CS TR01-024, University of North Carolina Computer Science Department, Real Time Systems, Kluwer, (to appear). 8. A. Srinivasan and S. Baruah, Deadline-based scheduling of periodic task systems on multiprocessors, Information Processing Letters 84 (2002) C.A. Phillips, C. Stein, E. Torng, J Wein, Optimal time-critical scheduling via resource augmentation, Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing (El Paso, Texas, 1997)