1. The document discusses block replacement algorithms like FIFO and optimal and compares their performance on sample reference strings. 2. It shows an example calculation of hits and misses using FIFO replacement on a reference string with 3 blocks. FIFO had 5 hits and 15 misses, for a hit ratio of 25%. 3. The document introduces Belady's Anomaly, where increasing the number of blocks can paradoxically increase misses for some algorithms like FIFO. It provides an example reference string where FIFO has more misses with 4 blocks than with 3 blocks.

1. Block Replacement Algorithms
FIFO & Belady's Anomaly
Bindu Agarwalla
1 Tuesday, March 16, 2021

2. Block Replacement
• A replacement policy determines, out of all possible candidates,
which one should be evicted from the cache upon arrival of a new
piece of data.
• We evaluate an algorithm by running it on a particular string of
memory references and computing the number of block misses.
The string of memory references is called a reference string.
• Reference strings can be generated.
– Artifically (By using a random-number generator).
– We can trace a given system and record the address of each memory reference.
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3. Block Replacement Algorithms
• First-in, First-out (FIFO).
• Optimal.
• Least Recently used (LRU).
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For each algorithm, we will calculate
The number of Hits.
The number of misses.
Hit Ratio (The number of Hits / Total number of attempts)
The algorithm, that gives the more number of hits, will also increase the
system performance by reducing the effective access time.

4. Block Replacement Algorithms
Assumptions:
1. Three blocks are given to the process.
2. Initially all three blocks are empty.
• First-in, First-out (FIFO): When a block must be replaced, the
oldest is chosen.
Implementation:
By using a FIFO Queue: We replace the block at the head of
the queue. When a block is brought into memory we insert it
at the tail of the queue.
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5. First in First Out (FIFO)
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5
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
Blocks : 3 Nos

6. First in First Out (FIFO)..
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6
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
7
*

7. First in First Out (FIFO)..
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7
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
0
7 7
* *

8. First in First Out (FIFO)..
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8
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1
0 0
7 7 7
* * *

9. First in First Out (FIFO)..
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9
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1
0 0 0
7 7 7 2
* * * *

10. First in First Out (FIFO)..
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10
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1
0 0 0 0
7 7 7 2 2
* * * * Hit

11. First in First Out (FIFO)..
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11
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1
0 0 0 0 3
7 7 7 2 2 2
* * * * Hit
*

12. First in First Out (FIFO)..
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12
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0
0 0 0 0 3 3
7 7 7 2 2 2 2
* * * * Hit
* *

13. First in First Out (FIFO)..
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13
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0
0 0 0 0 3 3 3
7 7 7 2 2 2 2 4
* * * * Hit
* * *

14. First in First Out (FIFO)..
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14
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0
0 0 0 0 3 3 3 2
7 7 7 2 2 2 2 4 4
* * * * Hit
* * * *

15. First in First Out (FIFO)..
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15
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *

16. First in First Out (FIFO)..
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16
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3
2
0
*

17. First in First Out (FIFO)..
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17
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3
2 2
0 0
* Hit

18. First in First Out (FIFO)..
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18
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3
2 2 2
0 0 0
* Hit Hit

19. First in First Out (FIFO)..
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19
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3
2 2 2 1
0 0 0 0
* Hit Hit
*

20. First in First Out (FIFO)..
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20
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2
2 2 2 1 1
0 0 0 0 0
* Hit Hit
* *

21. First in First Out (FIFO)..
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21
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2 2
2 2 2 1 1 1
0 0 0 0 0 0
* Hit Hit
* * Hit

22. First in First Out (FIFO)..
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22
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2 2 2
2 2 2 1 1 1 1
0 0 0 0 0 0 0
* Hit Hit
* * Hit Hit

23. First in First Out (FIFO)..
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23
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2 2 2 2
2 2 2 1 1 1 1 1
0 0 0 0 0 0 0 7
* Hit Hit
* * Hit Hit
*

24. First in First Out (FIFO)..
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24
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2 2 2 2 2
2 2 2 1 1 1 1 1 0
0 0 0 0 0 0 0 7 7
* Hit Hit
* * Hit Hit
* *

25. First in First Out (FIFO)..
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25
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2 2 2 2 2 1
2 2 2 1 1 1 1 1 0 0
0 0 0 0 0 0 0 7 7 7
* Hit Hit
* * Hit Hit
* * *

26. First in First Out (FIFO)..
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26
7 0 1 2 0 3 0 4 2 3 0 3 2 1 2 0 1 7 0 1
1 1 1 1 0 0 0 3
0 0 0 0 3 3 3 2 2
7 7 7 2 2 2 2 4 4 4
* * * * Hit
* * * * *
3 3 3 3 2 2 2 2 2 1
2 2 2 1 1 1 1 1 0 0
0 0 0 0 0 0 0 7 7 7
* Hit Hit
* * Hit Hit
* * *
HIT = 5 Miss = 15 Hit Ratio = 25% (5/20)

27. Belady's Anomaly
• As we know, the number of BLOCK misses decreases with the
increase in number of available blocks, but for some block-
replacement algorithms, the block- misses may increase as the
number of allocated blocks increases.
This most unexpected result is known as Belady's
Anomaly.
• The FIFO Algorithm suffers from this Anomaly.
Consider the following reference string:
1, 2, 3, 4, 1, 2, 5, 1, 2, 3, 4, 5
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28. BELADY’S ANOMALY
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28
1 2 3 4 1 2 5 1 2 3 4 5
USING 3 BLOCKS

29. BELADY’S ANOMALY
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29
1
*
1 2 3 4 1 2 5 1 2 3 4 5

30. BELADY’S ANOMALY
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30
1 2 3 4 1 2 5 1 2 3 4 5
2
1 1
* *

31. BELADY’S ANOMALY
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31
1 2 3 4 1 2 5 1 2 3 4 5
3
2 2
1 1 1
* * *

32. BELADY’S ANOMALY
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32
1 2 3 4 1 2 5 1 2 3 4 5
3 3
2 2 2
1 1 1 4
* * * *

33. BELADY’S ANOMALY
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33
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3
2 2 2 1
1 1 1 4 4
* * * * *

34. BELADY’S ANOMALY
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34
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2
2 2 2 1 1
1 1 1 4 4 4
* * * * * *

35. BELADY’S ANOMALY
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35
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2 2
2 2 2 1 1 1
1 1 1 4 4 4 5
* * * * * * *

36. BELADY’S ANOMALY
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36
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2 2 2
2 2 2 1 1 1 1
1 1 1 4 4 4 5 5
* * * * * * * Hit

37. BELADY’S ANOMALY
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37
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2 2 2 2
2 2 2 1 1 1 1 1
1 1 1 4 4 4 5 5 5
* * * * * * * Hit Hit

38. BELADY’S ANOMALY
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38
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2 2 2 2 2
2 2 2 1 1 1 1 1 3
1 1 1 4 4 4 5 5 5 5
* * * * * * * Hit Hit
*

39. BELADY’S ANOMALY
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39
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2 2 2 2 2
2 2 2 1 1 1 1 1 3
1 1 1 4 4 4 5 5 5 5
* * * * * * * Hit Hit
*
4
3
5
*

40. BELADY’S ANOMALY
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40
1 2 3 4 1 2 5 1 2 3 4 5
3 3 3 2 2 2 2 2
2 2 2 1 1 1 1 1 3
1 1 1 4 4 4 5 5 5 5
* * * * * * * Hit Hit
*
4 4
3 3
5 5
* Hit
Total no of misses = 9

41. BELADY’S ANOMALY
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41
1 2 3 4 1 2 5 1 2 3 4 5
USING 4 BLOCKS

42. BELADY’S ANOMALY
Tuesday, March
42
1
*
1 2 3 4 1 2 5 1 2 3 4 5

43. BELADY’S ANOMALY
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43
1 2 3 4 1 2 5 1 2 3 4 5
2
1 1
* *

44. BELADY’S ANOMALY
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44
1 2 3 4 1 2 5 1 2 3 4 5
3
2 2
1 1 1
* * *

45. BELADY’S ANOMALY
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45
1 2 3 4 1 2 5 1 2 3 4 5
* * * *
4
3 3
2 2 2
1 1 1 1

46. BELADY’S ANOMALY
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46
1 2 3 4 1 2 5 1 2 3 4 5
* * * * Hit
4 4
3 3 3
2 2 2 2
1 1 1 1 1

47. BELADY’S ANOMALY
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47
1 2 3 4 1 2 5 1 2 3 4 5
* * * * Hit
4 4 4
3 3 3 3
2 2 2 2 2
1 1 1 1 1 1
Hit

48. BELADY’S ANOMALY
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48
1 2 3 4 1 2 5 1 2 3 4 5
* * * * *
Hit
4 4 4 4
3 3 3 3 3
2 2 2 2 2 2
1 1 1 1 1 1 5
Hit

49. BELADY’S ANOMALY
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49
1 2 3 4 1 2 5 1 2 3 4 5
* * * * *
Hit
4 4 4 4 4
3 3 3 3 3 3
2 2 2 2 2 2 1
1 1 1 1 1 1 5 5
Hit
*

50. BELADY’S ANOMALY
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50
1 2 3 4 1 2 5 1 2 3 4 5
* * * * * *
*
Hit
4 4 4 4 4 4
3 3 3 3 3 3 2
2 2 2 2 2 2 1 1
1 1 1 1 1 1 5 5 5
Hit

51. BELADY’S ANOMALY
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51
1 2 3 4 1 2 5 1 2 3 4 5
* * * * *
* *
* Hit
4 4 4 4 4 4 3
3 3 3 3 3 3 2 2
2 2 2 2 2 2 1 1 1
1 1 1 1 1 1 5 5 5 5
Hit

52. BELADY’S ANOMALY
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52
1 2 3 4 1 2 5 1 2 3 4 5
3
2
1
4
*
* * * * *
* *
* Hit
4 4 4 4 4 4 3
3 3 3 3 3 3 2 2
2 2 2 2 2 2 1 1 1
1 1 1 1 1 1 5 5 5 5
Hit

53. BELADY’S ANOMALY
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53
1 2 3 4 1 2 5 1 2 3 4 5
3 3
2 2
1 5
4 4
*
* * * * *
* *
* Hit
4 4 4 4 4 4 3
3 3 3 3 3 3 2 2
2 2 2 2 2 2 1 1 1
1 1 1 1 1 1 5 5 5 5
Hit
*
No of Misses=10

54. Tuesday, March 16,
54
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