1. §12.4 Static Paging Algorithms
Douglas Wilhelm Harder, M.Math. LEL
Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
ece.uwaterloo.ca
© 2012 by Douglas Wilhelm Harder. Some rights reserved.
Based in part on Gary Nutt, Operating Systems, 3rd ed.

2. Outline This topic will cover address translation: TBW
Static Paging Algorithms
Outline
This topic will cover address translation:
TBW

3. Definitions Recall that
Static Paging Algorithms
Definitions
Recall that
The virtual address space of a process is divided into equally sized pages (for example, 4 KiB)
Main memory is divided into equally sized frames, each of which can hold one page
The virtual address space may be located on a hard drive and pages are copied into frames only as the processor needs them

4. Paging There are two classes of paging algorithms:
Static Paging Algorithms
Paging
There are two classes of paging algorithms:
Static paging algorithm allocate each process a fixed number of frames and this number does not change over the lifetime of the process
Dynamic paging algorithms allocate a variable number of frames, as necessary, for each process

5. Static Paging Algorithms
To implement paging, there are many processes requesting memory locations and all these must be loaded into memory for processes to execute
Operations we must be concerned with include:
A fetch policy which decides when a page should be loaded into main memory
A replacement policy which decides which pages should be removed
A placement policy which decides where new pages should be placed if there is room

6. Static Paging Algorithms
To model this, we require a means of describing how page references work:
A page-reference stream is a sequence of page numbers in the order in which references appear to them in an executing program
To make this reasonable:
We will consider accesses to the text, data, and stack segments separately, otherwise most page-reference streams will be rapid accesses between these segments
We will consider any continuous sequence of references to a single page to a single reference
Our interest is the order in which the pages are accessed—duplication in the list does not assist us here

7. Static Paging Algorithms
For example, within the text segment, a possible sequence during initialization may be
0, 1, 0, 1, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 4, 3, 4, 5, 6, 7
Recall that each page is 4 KiB—the time spent in each block may by perhaps milliseconds depending on any looping
In others, it may be just a single instruction—but it must still be loaded

8. Static Paging Algorithms
When a program is executed, it is allocated memory within the virtual address space for text, data, and the stack
Initially, none of these are in main memory
At the very least, at least one page from both the text and data segments must be loaded into main memory

9. Static Paging Algorithms
Fetch Policy
The fetch policy determines when a page should be copied from the virtual address space to main memory
Prefetching is the process where the memory manager attempts to determine which pages may be loaded into main memory at some point in the near future
Exceptionally difficult, as this requires knowledge of the future
With 32-bit instructions, that is 128 instructions per page, so there is no guarantee that loading one page will require the loading of the next
Normally, demand fetching is used: a page is loaded into a frame only if an instruction accesses a memory location from that page and it is not currently loaded

10. Demand Paging Algorithms
Static Paging Algorithms
Demand Paging Algorithms
Given a page-reference stream, we require an algorithm that specifies which frame a given page should be copied to
We will consider a number of such algorithms:
Random-replacement algorithms
Belady’s optimal algorithm
Least Recently Used (LRU) algorithm
Least Frequently Used (LFU) algorithm
First-in—First-out (FIFO) algorithm
Stack algorithms

11. Demand Paging Algorithms
Static Paging Algorithms
Demand Paging Algorithms
We will test these algorithms with the following page-reference stream: 0, 1, 0, 1, 2, 1, 2, 3, 1, 2, 4, 1, 3, 0, 2, 4, 3, 4, 5, 6, 7

12. Random-replacement Algorithms
Static Paging Algorithms
Random-replacement Algorithms
A random replacement algorithm
When a missing page fault occurs and there are no available frames, simply choose one frame at random and replace its contents
This algorithm, while easy to implement, has been found to be reasonably sub-optimal in comparison to other algorithms
We may use randomness if there is more than one page to choose from, however...

13. Bélády’s Optimal Replacement
Static Paging Algorithms
Bélády’s Optimal Replacement
The ideal replacement, described by Bélády, is to choose that frame that will be referenced most distantly into the future
This is, of course, impossible to achieve in almost all but the most trivial of systems
This algorithm is guaranteed to minimize the number of missing page faults

14. Bélády’s Optimal Replacement
Static Paging Algorithms
Bélády’s Optimal Replacement
Here is Bélády’s optimal replacement algorithm in action
The number of missing-page faults is10

15. Static Paging Algorithms
Least Recently Used
We would like to somehow guess which page will be accessed most recently into the future
The Least Recently Used (LRU) algorithm chooses that frame that has been accessed most distantly in the past
We must stamp each frame every time the page it contains is accessed
We must keep an ordered list of frames
This essentially requires a modifiable heap data structure to be efficient...
It uses the principle of spatial locality

16. Least Recently Used Here is the LRU algorithm in action
Static Paging Algorithms
Least Recently Used
Here is the LRU algorithm in action
The number of missing-page faults is13

17. Static Paging Algorithms
Least Frequently Used
Another algorithm is the Least Frequently Used (LFU) algorithm
Remove that frame that has been least frequently accessed
This requires a modifiable heap together with a counter for each frame
Unfortunately, this algorithm reacts very slowly to changes in locality, as we will see...

18. Least Frequently Used Here is the LFU algorithm in action
Static Paging Algorithms
Least Frequently Used
Here is the LFU algorithm in action
The number of missing-page faults is13
Note, however, that pages 1 and 2 are still loaded...

19. First-in—First-out Another is the first-in—first-out (FIFO) algorithm
Static Paging Algorithms
First-in—First-out
Another is the first-in—first-out (FIFO) algorithm
Remove that frame that was loaded most distantly in the past
This would appear to require significantly more work, but it is quite easy: the frames are loaded in a round-robin fashion
It ignores the principle of locality, but

20. First-in—First-out Here is the FIFO algorithm in action
Static Paging Algorithms
First-in—First-out
Here is the FIFO algorithm in action
The number of missing-page faults is13

21. Comparison In this example:
Static Paging Algorithms
Comparison
In this example:
Bélády’s algorithm issued 10 page missing faults
All other algorithms required more
That they all required 13 is irrelevant—examples this small cannot be used to suggest which is better...
FIFO appears to be the easiest to implement—can we use it?

22. Bélády’s Anomaly Here is an interesting question:
Static Paging Algorithms
Bélády’s Anomaly
Here is an interesting question:
Suppose a process is given a fixed number of frames
Will all of these algorithms always reduce the number of missing-page faults if the amount of frames allocated is increased?
Oddly enough, the answer with respect to the FIFO algorithm is no!

23. Bélády’s Anomaly From their original paper:
Static Paging Algorithms
Bélády’s Anomaly
From their original paper:
0, 1, 2, 3, 0, 1, 4, 0, 1, 2, 3, 4
If the process has three frames assigned to it, there are nine missing-page faults

24. Bélády’s Anomaly From their original paper:
Static Paging Algorithms
Bélády’s Anomaly
From their original paper:
0, 1, 2, 3, 0, 1, 4, 0, 1, 2, 3, 4
If the process has four frames assigned to it, there are ten missing-page faults

25. Static Paging Algorithms
Inclusion Property
Consequently, while FIFO is the easiest to implement, it is also susceptible to this anomaly—we can’t use it
The inclusion property may be held by a frame-replacement algorithm if the set of pages loaded when n frames are available is always a subset of the pages loaded when n + 1 frames are loaded
This clearly true with the LRU algorithm:
At a particular instance, if n pages have been loaded, then those are the n pages that have been most recently used
If there were n + 1 pages available, then the n most recently used pages would still be loaded

26. Static Paging Algorithms
Inclusion Property
Nutt [1992] showed that any frame-replacement algorithm that satisfies the inclusion property is not susceptible to Bélády’s Anomaly
This includes LRU and LFU
LFU has issues, so we will try to implement LRU...

27. Static Paging Algorithms
Implementing LRU
One means of approximating FRU seems almost trivial but it works well in practice with sufficiently many pages of sufficient size
Associate each frame with a reference bit
These bits are periodically all set to 0
When a page in a frame is referenced, set that bit to 1
When a page-missing fault occurs, randomly replace a frame that has the referenced bit set to 0
In the very unlikely case that all referenced bits are set, pick any frame at random

28. Static Paging Algorithms
Dirty Bits
Another issue with frame replacement is copying out the old contents
If data within a frame has only been read but not written, when that frame is copied out, there is no need to write that page out to secondary memory
Associate each frame with a dirty bit
When a page is loaded into the frame, set the dirty bit to 0
If the page in the frame is written to, set the dirty bit to 1
When the page in the frame is being replaced, you only need to copy that frame out to secondary memory if the dirty bit was set to 1

29. Summary This topic covered an introduction to virtual memory: TBW
Static Paging Algorithms
Summary
This topic covered an introduction to virtual memory:
TBW