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The Working Set Page Replacement Algorithm

In the purest form of paging, processes are started up with none of their pages in memory. As soon as the CPU tries to fetch the first instruction, it gets a page fault, causing the operating system to bring in the page containing the first instruction. Other page faults for global variables and the stack usually follow quickly. After a while, the process has most of the pages it needs and settles down to run with relatively few page faults. This strategy is called demand paging because pages are loaded only on demand, not in advance.

Of course, it is easy enough to write a test program that systematically reads all the pages in a large address space, causing so many page faults that there is not enough memory to hold them all. Fortunately, most processes do not work this way. They exhibit a locality of reference, meaning that during any phase of execution, the process references only a relatively small fraction of its pages. Each pass of a multipass compiler, for example, references only a fraction of all the pages, and a different fraction at that.

The set of pages that a process is currently using is called its working set (Denning, 1968a; Denning, 1980). If the entire working set is in memory, the process will run without causing many faults until it moves into another execution phase (e.g., the next pass of the compiler). If the available memory is too small to hold the entire working set, the process will cause many page faults and run slowly since executing an instruction takes a few nanoseconds and reading in a page from the disk typically takes 10 milliseconds. At a rate of one or two instructions per 10 milliseconds, it will take ages to finish. A program causing page faults every few instructions is said to be thrashing (Denning, 1968b).

In a multiprogramming system, processes are frequently moved to disk (i.e., all their pages are removed from memory) to let other processes have a turn at the CPU. The question arises of what to do when a process is brought back in again. Technically, nothing need be done. The process will just cause page faults until its working set has been loaded. The problem is that having 20, 100, or even 1000 page faults every time a process is loaded is slow, and it also wastes considerable CPU time, since it takes the operating system a few milliseconds of CPU time to process a page fault.

Therefore, many paging systems try to keep track of each process' working set and make sure that it is in memory before letting the process run. This approach is called the working set model (Denning, 1970). It is designed to greatly reduce the page fault rate. Loading the pages before letting processes run is also called prepaging. Note that the working set changes over time.

It has been long known that most programs do not reference their address space uniformly, but that the references tend to cluster on a small number of pages. A memory reference may fetch an instruction, it may fetch data, or it may store data. At any instant of time, t, there exists a set consisting of all the pages used by the k most recent memory references. This set, w(k, t), is the working set. Because the k = 1 most recent references must have used all the pages used by the k > 1 most recent references, and possibly others, w(k, t) is a monotonically nondecreasing function of k. The limit of w(k, t) as k becomes large is finite because a program cannot reference more pages than its address space contains, and few programs will use every single page. Figure 4-5 depicts the size of the working set as a function of k.

Figure 4-5. The working set is the set of pages used by the k most recent memory references. The function w(k, t) is the size of the working set at time t.

The fact that most programs randomly access a small number of pages, but that this set changes slowly in time explains the initial rapid rise of the curve and then the slow rise for large k. For example, a program that is executing a loop occupying two pages using data on four pages, may reference all six pages every 1000 instructions, but the most recent reference to some other page may be a million instructions earlier, during the initialization phase. Due to this asymptotic behavior, the contents of the working set is not sensitive to the value of k chosen. To put it differently, there exists a wide range of k values for which the working set is unchanged. Because the working set varies slowly with time, it is possible to make a reasonable guess as to which pages will be needed when the program is restarted on the basis of its working set when it was last stopped. Prepaging consists of loading these pages before the process is allowed to run again.

To implement the working set model, it is necessary for the operating system to keep track of which pages are in the working set. Having this information also immediately leads to a possible page replacement algorithm: when a page fault occurs, find a page not in the working set and evict it. To implement such an algorithm, we need a precise way of determining which pages are in the working set and which are not at any given moment in time.

As we mentioned above, the working set is the set of pages used in the k most recent memory references (some authors use the k most recent page references, but the choice is arbitrary). To implement any working set algorithm, some value of k must be chosen in advance. Once some value has been selected, after every memory reference, the set of pages used by the previous k memory references is uniquely determined.

Of course, having an operational definition of the working set does not mean that there is an efficient way to monitor it in real time, during program execution. One could imagine a shift register of length k, with every memory reference shifting the register left one position and inserting the most recently referenced page number on the right. The set of all k page numbers in the shift register would be the working set. In theory, at a page fault, the contents of the shift register could be read out and sorted. Duplicate pages could then be removed. The result would be the working set. However, maintaining the shift register and processing it at a page fault would both be prohibitively expensive, so this technique is never used.

Instead, various approximations are used. One commonly used approximation is to drop the idea of counting back k memory references and use execution time instead. For example, instead of defining the working set as those pages used during the previous 10 million memory references, we can define it as the set of pages used during the past 100 msec of execution time. In practice, such a definition is just as good and much easier to use. Note that for each process, only its own execution time counts. Thus if a process starts running at time T and has had 40 msec of CPU time at real time T + 100 msec, for working set purposes, its time is 40 msec. The amount of CPU time a process has actually used has since it started is often called its current virtual time. With this approximation, the working set of a process is the set of pages it has referenced during the past t seconds of virtual time.

Now let us look at a page replacement algorithm based on the working set. The basic idea is to find a page that is not in the working set and evict it. In Fig. 4-6 we see a portion of a page table for some machine. Because only pages that are in memory are considered as candidates for eviction, pages that are absent from memory are ignored by this algorithm. Each entry contains (at least) two items of information: the approximate time the page was last used and the R (Referenced) bit. The empty white rectangle symbolizes the other fields not needed for this algorithm, such as the page frame number, the protection bits, and the M (Modified) bit.

Figure 4-6. The working set algorithm.

The algorithm works as follows. The hardware is assumed to set the R and M bits, as we have discussed before. Similarly, a periodic clock interrupt is assumed to cause software to run that clears the Referenced bit on every clock tick. On every page fault, the page table is scanned to look for a suitable page to evict.

As each entry is processed, the R bit is examined. If it is 1, the current virtual time is written into the Time of last use field in the page table, indicating that the page was in use at the time the fault occurred. Since the page has been referenced during the current clock tick, it is clearly in the working set and is not a candidate for removal (t is assumed to span multiple clock ticks).

If R is 0, the page has not been referenced during the current clock tick and may be a candidate for removal. To see whether or not it should be removed, its age, that is, the current virtual time minus its Time of last use is computed and compared to t. If the age is greater than t, the page is no longer in the working set. It is reclaimed and the new page loaded here. The scan continues updating the remaining entries, however.

However, if R is 0 but the age is less than or equal to t, the page is still in the working set. The page is temporarily spared, but the page with the greatest age (smallest value of Time of last use) is noted. If the entire table is scanned without finding a candidate to evict, that means that all pages are in the working set. In that case, if one or more pages with R = 0 were found, the one with the greatest age is evicted. In the worst case, all pages have been referenced during the current clock tick (and thus all have R = 1), so one is chosen at random for removal, preferably a clean page, if one exists.

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