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Recap from last time -I

Recap from last time -I. Given a system of periodic tasks :  = { 1 ,  2 ,...  n };  i = (T i , C i ) Schedule using static priorities (of the first kind ) Two approaches: partitioning Dhall (1977), Dhall & Liu (1979): min #-procs for given 

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Recap from last time -I

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  1. Recap from last time -I • Given a system of periodic tasks:  = {1, 2,... n}; i = (Ti, Ci) • Schedule using static priorities (of the first kind) • Two approaches: • partitioning • Dhall (1977), Dhall & Liu (1979): min #-procs for given  • Baker & Oh (1998): utilization bound for fixed m • non-partitioning • (John generalizes...) • In both approaches, feasibility-determination is NP-H in the SS (Leung & Whitehead -- from bin-packing) • The two approaches are incomparable

  2. Recap from last time - II • RM may have arbitrarily low utilization (the “Dhall effect”) • An upper bound on the achievable utilization of any static-priority scheme... Question:Is this bound tight? (Prove for m=2!) I.e., Given with [ (SUM j :  j : Ci /Ti)  4/3], prove that there is a static priority-assignment for which results in all deadlines being met.

  3. A detour: Not-quite-static priorities Question:Construct a similar upper bound for priority-assignment schemes of type “2”. (Is this tight? For n=2?) I.e., Given with [ (SUM j :  j : Ci /Ti)  3/2], prove that there is a static priority-assignment for which results in all deadlines being met.

  4. This paper -I • Obs 1 & 2: increasing period may reduce feasibility • (reason: parallelism of processor left over by higher-pri tasks increases) • Obs 3: Critical instant not easily identified • Obs 4: Response time of a task depends upon relative priorities of higher-priority tasks • ==> the Audsley technique of priority assignment cannot be used • Theorem 1: A sufficient condition for feasibility • idea of the proof • possible problems?

  5. Recap • Phil’s example of EDF anomaly • John’s generalization of partitioning/ non-partitioning • Shelby -- all about bin-packing 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) Jim wants to know... This paper Pfair scheduling

  6. This paper -I • Obs 1 & 2: increasing period may reduce feasibility • (reason: parallelism of processor left over by higher-pri tasks increases) • Obs 3: Critical instant not easily identified • Obs 4: Response time of a task depends upon relative priorities of higher-priority tasks • ==> the Audsley technique of priority assignment cannot be used • Theorem 1: A sufficient condition for feasibility • idea of the proof • possible problems?

  7. This paper -II “Circumventing Dhall’s effect” [Dhall’s effect: 1 = 2 = ... = m = (2, 2); m+1 = (1+ , 1) ] • Would like m+1 to have higher priority: Least slack assignment of priorities? • doesn’t quite work • TkC priority assignment: • choose a constant k • for each i = (Ti, Ci), priority-number of i := Ti - k  Ci • lowest priority-number gets highest priority • Seems a reasonable idea, but...

  8. This paper -III Deep thoughts about TkC: (priority-number of i := Ti - k  Ci) k=0: RM • not good: Dhall’s effect • k very large: assign priorities according to Ci’s • not good: 1=2= (100,1); 3=4= (10000,100) is infeasible on 2 procs • k somewhere in between...

  9. This paper -IV Why I don’t like TkC • Where’s the simple idea? • Conjecture is that (m+1) tasks is the worst-case... • goes against uniprocessor experience To disprove: (as opposed to not believe) • find a counterexample to the utilization bound • (likely easiest for m=2 -- static least-slack)

  10. How would we approach this problem? • Special cases (e.g., harmonic task sets) • Different kinds of priority schemes • priority-number of i = f(i) • relative priorities of two tasks depends upon only the two tasks • must examine all tasks prior to assigning priorities • Implications to on-line admission control

  11. Tractable special cases? Harmonic task sets • Result: Critical instant is easily identified • Result: Priority detemination remains NP-H in the SS • (since the Leung/Whitehead proof had all periods equal) Question:What about fixed number of processors? • (provably NP-H, but in the ordinary sense, for m=2)

  12. Open problems • Is the upper bound on achievable utilization tight? • Is the type-2 priority bound tight? • Any results on harmonic task sets? • [Uniprocessors:] Think deep thoughts about type 2 vs type 3 priority-assignment

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