A Generalized View on Beneficial Task Sortings for Partitioned RMS Task Allocation on Multiprocessors

Transcription

1 RTSOPS 2011: 2 nd International Real-Time Scheduling Open Problems Seminar Porto, Portugal, July 5th, 2011 Co-located with ECRTS 2011 A Generalized View on Beneficial Task Sortings for Partitioned Task Allocation on Multiprocessors Dirk Müller, Matthias Werner Operating Systems Group Chemnitz University of Technology GERMANY

2 2/7 Partitioned Partitioned Scheduling as resort to multiple uniprocessor scheduling problems (easy) and allocation Allocation closely related to Bin Packing, NP-hard Use of online heuristics like Next Fit, First Fit, Best Fit Schedulability test ( Fit ) for complicated TDA is exact, but pseudo-polynomial Sufficient tests pessimistic Online stream of tasks Allocation Heuristic (e.g. NF, FF, BF) Admission Control (e.g. TDA, LL, HB, R-BOUND, Burchard)

3 3/7 Sorting as Preprocessing Offline situation: permutation of tasks in order to increase success rates of allocation Re-use of established online schedulers Restriction of input space to sorted instances Most famous: Decreasing Utilization (DU) Task set (offline) Presorting (e.g. DU) known from bin packing as e.g. FFD (First Fit Decreasing) Online stream of tasks Allocation Heuristic (e.g. NF, FF, BF) Admission Control (e.g. TDA, LL, HB, R-BOUND, Burchard)

4 4/7 Problem Statement: Sorting Criteria DU is optimal for EDF as FFD is for pure bin packing Is there an optimal sorting sequence for the preprocessing sten partitioned heuristics? Systematization and generalization One-stage Decreasing utilization Increasing (transformed) period Increasing S-value with S =log 2 p log 2 p Two-stage 1. decreasing utilization, 2. increasing S-values RMGT (Burchard et al. 1995) RMMatching (Rothvoß 2009) k-rmm (Karrenbauer & Rothvoß 2010)

5 5/7 A Common Root of T and R-BOUND-MP T (Burchard et al. 1995) vs. R-BOUND-MP (Lauzac et al. 1998) T with increasing S-values S i =log 2 log 2 R-BOUND-MP transforms with ScaleTaskSet to sameorder-of-magnitude periods, then increasing periods ' = 2 log 2 ( p max / ) Logarithmization to base 2 is strictly increasing => order-preserving operation log 2 ' =log 2 + log 2 p max log 2 p max max. period as a constant yielding an index shift

6 6/7 Example i ' ' S i S i rank rank Ties are broken by index ' =S i mod 10+1 Results differ just by an index shift Circular relationship suggests that there is no outstanding starting point => consider all index offsets in the general case

7 7/7 Summary Sorting as preprocessing for partitioned is a key step for increasing success ratio when scheduling Decreasing utilization is optimal for partitioned EDF and a good starting point for partitioned, but not optimal since it ignores period compatibility completely Previously thought of independent approaches T and R-BOUND-MP have a common root concerning sorting sequence (S-values describe period similarity) Shows potential for generalizations Recent results suggest superiority of two-stage approaches with 1. decreasing utilization and 2. increasing S-values