Approximating the Optimal Replacement Algorithm Ben Juurlink

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Approximating the Optimal Replacement
AlgorithmBen Juurlink
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Plan

• Motivation
• LRU and OPT algorithms
• Related work
• TNRP algorithm
• TNRP tuning
• TNRP comparison with LRU and OPT

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Motivation

• It takes millions of cycles to serve a page fault
• Small reduction of the miss rate will pay for the
  cost of advanced replacement algorithm

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LRU Algorithm

• LRU Replaces the page that hasnt been used for
  the longest time
• The optimal algorithm (OPT) evicts the page that
  will not be used for the longest time
• LRU approximates OPT
• At times LRU is far from optimal
• 4 page frames
• 1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5
• LRUs miss rate 100
• OPTs miss rate 40
• LRU-2 replaces the page whose second to last
  access is least recent

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Related work Fiat and Rosen Algorithm

• Dynamically build weighted access graph
• An edge is created the first time the two pages
  are requested in succession
• Each time the edge is traversed, the weight is
  decreases by a constant factor
• Forgetfulness the weight of all edges
  multiplied by a constant factor every 10n pages
  accesses
• The page to be evicted on a page fault the
  furthest page from the page just accessed
• Expensive algorithm

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Time of Next Reference Predictor

• Three parameters recorded for each page
• TLAST(p) time of last reference
• STRIDE(p) of page accesses between the last
  and previous to last reference
• State
• Transient initially
• Steady when
• STRIDE(p) SD STRIDEcurr(p) STRIDE(p) SD
• (SD Stride Deviation)
• Time consecutive accesses to one page replaced
  by one

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Time of Next Reference Predictor

• On a page fault
• Replace a page with the largest expected time of
  next reference EXP-TNEXT(p)
• EXP-TNEXT(t)
• TLAST(p) STRIDE(p) if StateSteady
• current_time current_time TLAST(p) if
  StateTransient

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Time of Next Reference Predictor-Example

• Reference String 1 4 1 1 6 1 1 4 4 6 6
• SD 2
• Time 0 1 2 3 4 5 6
• Page 1 4 1 6 1 4 6
• Stride 0 0 2 0 2 4 3
• State T T S T S T T

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Time of Next Reference Predictor

• A correction is needed for EXP-TNEXT (p)
• 1,2,3,4,5,6,7,8,9,5,11,12,13,14,5,16,17,18,19,5,
• 3 pages can be kept in memory
• STRIDE(5)5
• At time 16
• 13,14 and 5 are in memory
• EXP-TNEXT(5) 15 5 20
• EXP-TNEXT(13) 16 (16 13) 19
• EXP-TNEXT(14) 16 (16 14) 18
• Page 5 will be evicted

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Time of Next Reference PredictorEXP-TNEXT (p) -
solution

• If the page hasnt been accessed before or the
  state is transient then
• EXP-TNEXT(p) current_time TF (current_time
  TLAST(p))
• TFgt1
• If the page hasnt been accessed within the
  expected time current_time gt TLAST(p) STRIDE(p)
  SD, use the above formula for EXP-TNEXT(p) and
  return its state to Transient
• In our example, take TF2
• EXP-TNEXT(5) 15 5 20
• EXP-TNEXT(13) 16 2 (16 13) 22
• EXP-TNEXT(14) 16 2 (16 14) 20
• 13 will be evicted

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Implementation Issues

• How to find a victim page
• Priority queue (priority EXP-TNEXT(p))
• Takes O(log n) to find a victim page
• But it takes O(log n) to update the queue
• Updates occur frequently
• Scan all the paged in pages on a page fault

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Experiments - Parameters

• Benchmarks from the SPEC 2000 suite
• Stride Deviation
• Investigate the distribution of STRIDEcurr
  STRIDEprev
• The distribution depends on benchmark
• SD of 5 captures over 60 of all stride values
• Time of next reference Factor
• Check the miss rate for 1lt TF lt 3 with 4 and 8
  page frames
• Performance rather independent of the exact
  values
• 4 page frames, the difference is at most 5
• 8 page frames, the difference is at most 0.8
• For some benchmarks the miss rate decreases as TF
  increases
• For other, it decreases and then increases
• Optimal value between 2 and 2.75

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Experiments Miss Rate Reduction

• TNRP incurs fewer misses than LRU for almost all
  benchmarks
• 4 page frames
• Improvement Avg 13.5, range from 6 to 28.2
• 8 page frames
• Improvement Avg 4, 20.7 for one benchmark
• 16 page frames
• Improvement less than 1 for 7 out of 9
  benchmarks
• Double the number of misses for 1 benchmark
• Performance improvement significant when of
  pages is small
• Works best in multi-tasking environment when the
  physical memory needs to be shared among several
  processes
• TNRP outperforms LRU-2 for most workloads and
  number of pages