1. Recap from last time -I qGiven a system of periodic tasks:  = {  1,  2,...  n };  i = (T i, C i ) qSchedule 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...) qIn both approaches, feasibility-determination is NP-H in the SS (Leung & Whitehead -- from bin-packing) qThe two approaches are incomparable

2. Recap from last time - II qRM may have arbitrarily low utilization (the “Dhall effect”) qAn 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  : C i /T i )  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  : C i /T i )  3/2], prove that there is a static priority-assignment for  which results in all deadlines being met.

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

5. Recap qPhil’s example of EDF anomaly qJohn’s generalization of partitioning/ non-partitioning qShelby -- all about bin-packing Priorities task-level static job-level static dynamic Migration task-level fixed job-level fixed migratory Baker/ Oh (RTS98) Pfair scheduling This paper Jim wants to know... bin-packing + LL (no advantage) bin-packing + EDF

6. This paper -I qObs 1 & 2: increasing period may reduce feasibility –(reason: parallelism of processor left over by higher-pri tasks increases) qObs 3: Critical instant not easily identified qObs 4: Response time of a task depends upon relative priorities of higher-priority tasks –==> the Audsley technique of priority assignment cannot be used qTheorem 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) ] qWould like  m+1 to have higher priority: Least slack assignment of priorities? –doesn’t quite work qTkC priority assignment: –choose a constant k –for each  i = (T i, C i ), priority-number of  i := T i - k  C i –lowest priority-number gets highest priority qSeems a reasonable idea, but...

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

9. This paper -IV Why I don’t like TkC qWhere’s the simple idea? qConjecture 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? qSpecial cases (e.g., harmonic task sets) qDifferent 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 qImplications to on-line admission control

11. Tractable special cases? Harmonic task sets qResult: Critical instant is easily identified qResult: 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 qIs the upper bound on achievable utilization tight? qIs the type-2 priority bound tight? qAny results on harmonic task sets? q[Uniprocessors:] Think deep thoughts about type 2 vs type 3 priority-assignment