1. LRFU (Least Recently/Frequently Used) Block Replacement Policy
Sang Lyul Min
Dept. of Computer Engineering
Seoul National University

2. Processor - Disk Speed Gap
Why file cache?
Processor - Disk Speed Gap
1950’s
1990’s
Processor - IBM 701
17,000 ins/sec
Disk - IBM 305 RAMAC
Density Mbits/sq. in
Average seek time ms
Processor - IBM PowerPC 603e
350,000,000 ins/sec
Disk - IBM Deskstar 5
Density - 1,319 Mbits/sq. in
Average seek time - 10 ms
x 20,000
x 600,000
x 50

3. File Cache processor main memory disk controller disks
file cache or buffer cache
disk cache

4. Operating System 101 LRU Replacement LFU Replacement heap
LRU Block
MRU Block
LFU Block
MFU Block
New reference
New reference
O(1) complexity
O(log n) complexity
O(n) complexity

5. Operating System 101 LRU LFU Advantage Disadvantage Advantage
High Adaptability
Disadvantage
Short sighted
LFU
Advantage
Long sighted
Disadvantage
Cache pollution

6. Motivation
Cache size = 20 blocks

7. Cache size = 60 blocks

8. Cache size = 100 blocks

9. Cache size = 200 blocks

10. Cache size = 300 blocks

11. Cache size = 500 blocks

12. Observation
Both recency and frequency affect the likelihood of future references
The relative impact of each is largely determined by cache size

13. Goal
A replacement algorithm that allows a flexible trade-off between recency and frequency

14. Results
LRFU (Least Recently/Frequently Used) Replacement Algorithm that
(1) subsumes both the LRU and LFU algorithms
(2) subsumes their implementations
(3) yields better performance than them

15. CRF (Combined Recency and Frequency) Value
Current time tc 1 2 3
time t1 t2 t3
Ctc(b) = F(1) + F(2) + F(3)
|| || ||
tc - t tc - t tc - t3

16. CRF (Combined Recency and Frequency) Value
Estimate of how likely a block will be referenced in the future
Every reference to a block contributes to the CRF value of the block
A reference’s contribution is determined by weighing function F(x)

17. Hints and Constraints on F(x)
should be monotonically decreasing
should subsume LRU and LFU
should allow efficient implementation

18. Conditions for LRU and LFU
LRU Condition
If F(x) satisfies the following condition, then the LRFU algorithm becomes the LRU algorithm
LFU Condition
If F(x) = c, then the LRFU algorithm becomes the LFU algorithm
current
time i
block a: x
block b: x x x x x x x x
i+1
i+2
    
i+3

19. Weighing function F(x)
F(x) = ()x
Meaning: a reference’s contribution to the target block’s CRF value is halved after every 1/ 

20. Properties of F(x) = ()x
Property 1
When  = 0, (i.e., F(x) = 1), then it becomes LFU
When  = 1, (i.e., F(x) = ()x ), then it becomes LRU
When 0 <  < 1, it is between LFU and LRU
F(x) = ()x (LRU extreme)
Spectrum
(LRU/LFU) 1
F(x) = 1 (LFU extreme)
F(x) X
current time - reference time

21. Results
LRFU (Least Recently/Frequently Used) Replacement Algorithm that
(1) subsumes both the LRU and LFU algorithms
(2) subsumes their implementations
(3) yields better performance than them

22. Difficulties of Naive Implementation
Enormous space overheads
Information about the time of every reference to each block
Enormous time overheads
Computation of the CRF value of every block at each time

23. Update of CRF value over time
= (t2 - t1)
 1
 2
 3
C t2(b) = F (1+) + F (2+) + F (3+)
= ()(1+ ) + () (2+ ) + () (3+ )
= (()1 + ()2 + ()3 ) ()
= C t1(b) x F ()

24. Properties of F(x) = ()x
Property 2
With F(x) = ()x, Ctk(b) can be computed from Ctk-1(b) as follows
Ctk(b) = Ctk-1(b) F () + F (0) ||
tk - tk-1
Implications: Only two variables are required for each block for maintaining the CRF value
One for the time of the last reference
The other for the CRF value at that time

25. Difficulties of Naive Implementation
Enormous space overheads
Information about the time of every reference to each block
Enormous time overheads
Computation of the CRF value of every block at each time

26. Properties of F(x) = ()x
Property 3
If Ct(a) > Ct(b) and neither a or b is referenced after t, then Ct'(a) > Ct'(b) for all t' > t
Why?
Ct'(a) = Ct(a) F() > Ct(b) F() = Ct'(b) (since F() > 0 )
Implications
Reordering of blocks is needed only upon a block reference
Heap data structure can be used to maintain the ordering of blocks with O(log n) time complexity

27. Optimized Implementation
Blocks that can compete with a currently referenced block

28. Optimized Implementation
Reference to a new block
Reference to a block in the heap
Reference to a block in the linked list
linked list
heap
linked list
heap
linked list
heap
1. replaced
referenced
block
2. promoted
2. demoted
1. demoted
3. new block
3. heap
restored
4. heap restored
referenced block
1. heap restored

29. Question
What is the maximum number of blocks that can potentially compete with a currently referenced block?

30. time block a: x block b: x x x x x • • • dthreshold dthreshold +1
• • •
block a: x
block b: x x x x x
time
• • • + F(d threshold +2) + F(d threshold +1) + F(d threshold ) < F(0)
dthreshold
dthreshold +1
dthreshold +2
current

31. Properties of F(x) = ()x
Property 4 :
log  (1- ()) 
dthreshold =
When   0
When   1
= 
= 1
Archi & Network LAB
Seoul National University

32. Optimized implementation (Cont’d)
linked list
heap
(single element)
LRU
extreme
LFU extreme
linked list (null)
O(log n)
O(1)
Archi & Network LAB
Seoul National University

33. Results
LRFU (Least Recently/Frequently Used) Replacement Algorithm that
(1) subsumes both the LRU and LFU algorithms
(2) subsumes their implementations
(3) yields better performance than them

34. Correlated References

35. LRFU with correlated references
Masking function Gc(x)
C'tk(b), CRF value when correlated references are considered, can be derived from C'tk-1(b)
C'tk(b) = F(tk - tk) F(tk - ti )*Gc(ti+1 - ti )
= F( tk - tk-1) * [F(0) * Gc( tk - tk-1) + C'tk-1(b) - F(0)] + F(0)
Archi & Network LAB
Seoul National University

36. Trace-driven simulation
Sprite client trace
Collection of block references from a Sprite client
contains 203,808 references to 4,822 unique blocks
DB2 trace
Collection of block references from a DB2 installation
Contains 500,000 references to 75,514 unique blocks
Archi & Network LAB
Seoul National University

37. Effects of  on the performance
Hit Rate
(a) Sprite client  X
(b) DB2 
Hit Rate X
Archi & Network LAB
Seoul National University

38. Combined effects of  and correlated period
Hit Rate
Correlated
Period 
(a) Sprite client
Hit Rate
Correlated
Period 
(b) DB2
Archi & Network LAB
Seoul National University

39. Previous works FBR (Frequency-Based Replacement) algorithm
Introduces correlated reference concept
LRU-K algorithm
Replaces blocks based on time of the K’th-to-last non-correlated references
Discriminates well the frequently and infrequently used blocks
Problems
Ignores the K-1 references
linear space complexity to keep the last K reference times
2Q and sLRU algorithms
Use two queues or two segments
Move only the hot blocks to the main part of the disk cache
Work very well for “used-only-once” blocks
Archi & Network LAB
Seoul National University

40. Comparison of the LRFU policy with other policies
Hit Rate
Cache Size (# of blocks)
(a) Sprite client
Hit Rate
Cache Size (# of blocks)
(b) DB2
Archi & Network LAB
Seoul National University

41. Implementation of the LRFU algorithm
Buffer cache of the FreeBSD 3.0 operating system
Benchmark: SPEC SDET benchmark
Simulates a multi-programming environment
consists of concurrent shell scripts each with about 150 UNIX commands
gives results in scripts / hour
Archi & Network LAB
Seoul National University

42. SDET benchmark results
Hit rate
SDET Throughput
(scripts/ hour)
Hit Rate 
Archi & Network LAB
Seoul National University

43. Conclusions
LRFU (Least Recently/Frequently Used) Replacement Algorithm that
(1) subsumes both the LRU and LFU algorithms
(2) subsumes their implementations
(3) yields better performance than them

44. Future Research Dynamic version of the LRFU algorithm
LRFU algorithm for heterogeneous workloads
File requests vs. VM requests
Disk block requests vs. Parity block requests (RAID)
Requests to different files (index files, data files)

45. People REAL PEOPLE (Graduate students) Lee, Donghee Choi, Jongmoo
Kim, Jong-Hun
Guides
(Professors)
Noh, Sam H.
Min, Sang Lyul
Cho, Yookun
Kim, Chong Sang

46. Adaptive LRFU policy
Adjust  periodically depending on the evolution of workload
Use the LRU policy as the reference model to quantify how good (or bad) the locality of the workload has been
Algorithm of the Adaptive LRFU policy
if ( > )
 value for period i+1 is updated in the same direction
else
the direction is reversed
Archi & Network LAB
Seoul National University

47. Results of the Adaptive LRFU
Client Workstation 54
DB2
Archi & Network LAB
Seoul National University