A Generalized View on Beneficial Task Sortings for Partitioned RMS - - PowerPoint PPT Presentation

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

Dirk Müller, Matthias Werner Operating Systems Group Chemnitz University of Technology GERMANY

RTSOPS 2011: 2nd International Real-Time Scheduling Open Problems Seminar

Porto, Portugal, July 5th, 2011 Co-located with ECRTS 2011

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Partitioned RMS

• Partitioned Scheduling as resort to multiple unipro-

cessor 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 RMS complicated
• TDA is exact, but pseudo-polynomial
• Sufficient tests

pessimistic

Allocation Heuristic (e.g. NF, FF, BF) Admission Control (e.g. TDA, LL, HB, R-BOUND, Burchard) RMS RMS RMS RMS Online stream of tasks

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• Offline situation: permutation of tasks in order to in-

crease success rates of allocation

• Re-use of established online schedulers
• Restriction of input space to sorted instances
• Most famous: Decreasing Utilization (DU)

known from bin packing as e.g. FFD (First Fit Decreasing)

Sorting as Preprocessing

Allocation Heuristic (e.g. NF, FF, BF) Admission Control (e.g. TDA, LL, HB, R-BOUND, Burchard) RMS RMS RMS RMS Online stream of tasks Presorting (e.g. DU) Task set (offline)

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Problem Statement: Sorting Criteria

• DU is optimal for EDF as FFD is for pure bin packing
• Is there an optimal sorting sequence for the prepro-

cessing step in partitioned RMS heuristics?

• Systematization and generalization
• One-stage
• Decreasing utilization
• Increasing (transformed) period
• Increasing S-value with
• Two-stage
• 1. decreasing utilization, 2. increasing S-values

– RMGT (Burchard et al. 1995) – RMMatching (Rothvoß 2009) – k-RMM (Karrenbauer & Rothvoß 2010)

S=log2 p−⌊log2 p⌋

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A Common Root of RMST and R-BOUND-MP

• RMST (Burchard et al. 1995) vs.

R-BOUND-MP (Lauzac et al. 1998)

• RMST with increasing S-values
• R-BOUND-MP transforms with ScaleTaskSet to same-
• rder-of-magnitude periods, then increasing periods
• Logarithmization to base 2 is strictly increasing

=> order-preserving operation

• max. period as a constant yielding an index shift

Si=log2 pi−⌊log2 pi⌋

pi' =pi 2⌊log 2( p max/ pi)⌋ log2 pi' =log2 pi+⌊log2 pmax−log2 pi⌋

pmax

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Example

• Ties are broken by index
• Results differ just by an index shift
• Circular relationship suggests that there is no out-

standing starting point => consider all index offsets in the general case

1 2 3 4 5 6 7 8 9 10 7 16 21 32 48 64 66 75 96 100 56 64 84 64 96 64 66 75 96 100 rank 1 2 7 3 8 4 5 6 9 10 0.807 0.000 0.392 0.000 0.585 0.000 0.044 0.229 0.585 0.644 rank 10 1 6 2 7 3 4 5 8 9

i pi pi' Si pi' Si pi' =Si mod 10+1

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Summary

• Sorting as preprocessing for partitioned RMS is a key

step for increasing success ratio when scheduling

• Decreasing utilization is optimal for partitioned EDF

and a good starting point for partitioned RMS, but not

• ptimal since it ignores period compatibility completely
• Previously thought of independent approaches RMST

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