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Resource augmentation and on-line scheduling on multiprocessors

Resource augmentation and on-line scheduling on multiprocessors. Phillips, Stein, Torng, and Wein. Optimal time-critical scheduling via resource augmentation . STOC (1997) . Algorithmica (to appear). Background: on-line algorithms.

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Resource augmentation and on-line scheduling on multiprocessors

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  1. Resource augmentation and on-line scheduling on multiprocessors Phillips, Stein, Torng, and Wein. Optimal time-critical scheduling via resource augmentation. STOC (1997). Algorithmica (to appear).

  2. Background: on-line algorithms • Optimization problems: given problem instance I, algorithm A obtains a value valA(I) -- goal is to maximize this value • On-line algorithmsvs an optimal off-line/ clairvoyant algorithm (OPT) • Competitive ratio of on-line algorithm A: min all I ( valA(I)/ valOPT(I) ) • Goal: Design an on-line algorithm with largest competitive ratio

  3. Background: hard-real-time scheduling • The on-line problem: • Instance I = {J1, J2, ..., Jn} of jobs • Each job Jj = (rj, pj, dj) • arrives at instant ri • needs to execute for pi units... • by a deadline at instant di • Job Ji is revealed at instant ri • Difficult to formulate as an optimization problem -- all deadlines must be met! • In uniprocessor systems, we dodged this issue • EDF/ LL are optimal algorithms (always meet all deadlines) • EDF/ LL are on-line algorithms... • ... with competitive ratio one

  4. Hard-real-time scheduling: multiprocessors • No optimal (in the EDF/LL sense) on-line algorithm exists • Must still meet all deadlines...So, give the on-line algorithm extra resources (more/ faster processors) • This paper asks: how much extra resources do EDF/ LL need, in order to meet all deadlines for sets of jobs known to be feasible on m processors? • The answers: • EDF/ LL meet all deadlines if processors are (2 - 1/m) times as fast • No on-line algorithm can meet all deadlines if processors are < 1.2 times as fast • EDF cannot always meet all deadlines if processors are (2 - 1/m - ) times as fast, for any  > 0

  5. Why we care • Our (RTS) task systems: • usually pre-specified (e.g., periodic tasks/ sporadic tasks) • “on-line”ness usually not an issue • exception: overload scheduling (later) • We’ll do feasibility analysis (does a schedule exist?) • If feasible, we’ll use the results in this paper • choose an algorithm (usually, EDF) • overallocate resources as mandated by these results • sleep well, knowing that the system performs as expected • Why choose feasibility analysis (versus schedulability analysis with chosen algorithm)? • provably competitive performance translates to approximation guarantees

  6. Model and definitions Instance I = {J1, J2, ..., Jn} of jobs Each job Jj = (rj, pj, dj) • arrives at instant ri • needs to execute for pi units... • by a deadline at instant di If I is feasible on m processors, an s-speed on-line algorithm will meet all deadlines on m processors each s times as fast (Thus, EDF is a (2 - 1/m)-speed algorithm)

  7. Digression: An example of how we’d use these results

  8. Scheduling periodic tasks - taxonomy Priorities task-level static job-level static dynamic Migration task-level fixed job-level fixed migratory bin-packing + LL (no advantage) bin-packing + EDF Baker/ Oh (RTS98) Andersson/ Jonsson Pfair scheduling Periodic task system  = {1, 2,..., n}; i = (Ti, Ci), RM EDF LL/ Pfair

  9. Remember this? (last class) RM-US(1/4) • all tasks i with (Ti/ Ci > 1/4) have highest priorities • for the remaining tasks, rate-monotonic priorities Lemma: Any task system satisfying [ (SUM j :  j : Ci /Ti)  m/4] and [ (ALL j :  j : Ci /Ti)  1/4] is successfully scheduled using RM-US(1/4) Theorem: Any task system satisfying [ (SUM j :  j : Ci /Ti)  m/4] is successfully scheduled using RM-US(1/4)

  10. A new (job-level static priority) scheduling algorithm EDF-US(1/2): • If Ci/Ti  0.5, then jobs of i get EDF priority • If Ci/Ti > 0.5, then jobs of i get highest priority • (EDF implementation: set deadline to -) Lemma: Any task system satisfying [ (SUM j :  j : Ci /Ti)  m/2] and [ (ALL j :  j : Ci /Ti)  1/2] is successfully scheduled using EDF-US(1/2) Theorem: Any task system satisfying [ (SUM j :  j : Ci /Ti)  m/2] is successfully scheduled using EDF-US(1/2)

  11. Scheduling periodic tasks w/ migration Priorities task-level static job-level static dynamic Migration task-level fixed job-level fixed migratory bin-packing + LL (no advantage) bin-packing + EDF Baker/ Oh (RTS98) Andersson/ Jonsson Pfair scheduling RM-US(1/4) EDF-US(1/4) Pfair 25% 50% 100%

  12. Back to the results in this paper...(faster processors)

  13. The big insight Definitions: • A(j,t) denotes amount of execution of job j by Algorithm A until time t • A(I,t) = [SUM: j I: A(j,t)] The crucial question: Let A be any “busy” (work-conserving) scheduling algorithm executing on m processors of speed   1. What is the smallest  such that at all times t, A(I, t)  A’(I,t) for any other algorithm A’ executing on m speed-1 processors? Lemma 2.6:  turns out to be (2 - 1/m) Use Lemma 2.6, and an individual algorithm’s scheduling rules, to draw conclusions regarding these algorithms

  14. The oh-so-important lemma 2.6 Proof: by contradiction Suppose there are time instants at which this is not true Let  = { i |  t  A(I,t) < A’(I,t) and A(i,t) < A’(i,t) } Let j be the job with the earliest release time rj in  Let to be the earliest time instant at which A(I,to) < A’(I,to) Eq (1) A(j,to) < A’(j,to) Eq (2) Lemma: Let I be an input instance, t  0 any time-instant. For any busy algorithm A using (2-1/m)-speed machines, A(I,t)  A’(I, t) for any algorithm A’ using 1-speed machines.

  15. EDF is a (2 - 1/m)-speed algorithm Instance I = {J1, J2, ..., Jn}; job Jj = (rj, pj, dj) is feasible on m procs Wlog, assume that di di+1 for all i Let Ik = {J1, J2, ..., Jk} Proof: Induction on k Base: EDF on m (1 - 2/m)-speed procs meets all deadlines for I1, .., Im IH: EDF on m (1 - 2/m)-speed procs meets all deadlines for I1, .., Ik We’re considering Ik+1. • Let Qk+1 Ik+1 denote the jobs in Ik+1 with deadlines at dk+1 • (Ik+1 \ Qk+1) is Iq for some q  k • By IH, EDF on m (1 - 2/m)-speed procs meets all deadlines for Iq • BY definition of EDF, EDF(Ik+1) is identical to EDF(Iq) on jobs of Iq; -- thus, all deadlines in Iq are met in EDF(Ik+1) • By Lemma 2.6, EDF(Ik+1,dk+1)  OPT(Ik+1, dk+1) • Since OPT meets all deadlines at dk+1, so must EDF on m (1 - 2/m)-speed procs

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