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a second-level page table. There can be 1024 second-level page tables, each with 1024 entries, so all one million pages in the logical address space can still be mapped. However, almost no program requires the full million-page logical address space. The beauty of the multilevel page table is that the OS does not need to devote space to page table entries that will never be used. Each second-level page table can represent 1024 pages, and if each page is 4K, then each second-level page table can represent 4 Mbytes of address space. For example, as I write, Microsoft Word is using almost 11 Mbytes of memory. That much memory could be mapped with only three second-level page tables. The sum of entries from the top-level page table and three second-level page tables is 4096. If each entry is 32 bits (4 bytes), then the page table space required for Word is 16K, not the 4 Mbytes required to represent the entire logical address space. Depending on the actual locations being referenced, the OS may in fact require more than the minimum number of second-level page tables to manage the mappings, but the improvement in memory utilization by means of the nested page table is still apt to be two orders of magnitude. With the introduction of 64-bit computers, a new solution to the problem of large page tables is being implemented. This approach is called the inverted page table. The page table as we have discussed it so far has one entry for each page of logical memory. Flipping this idea on its head, the inverted page table has one entry for each frame of physical memory. And, instead of storing a physical frame number, the inverted page table stores a logical page number. Since the number of frames of physical memory is likely to be much smaller than the number of pages of logical memory in a 64-bit computer (with 4K pages, 252 = way too many logical pages to contemplate millions of billions), the page table size remains reasonable. Each entry in the inverted page table contains a process identifier and a logical page number. When a process accesses memory, the system scans the inverted page table looking for a match of process ID and logical page number. Since each entry in the inverted page table corresponds to a physical frame, when the system finds a match, the index into the inverted page table becomes the frame number for the actual memory reference. For instance, if a match occurs in the 53rd entry (entry 52, since the first entry is entry 0) of the inverted page table, then the frame for physical memory is frame 52. The system adds the offset within the page to the frame number of 52, and the physical address is complete. The inverted page table keeps the size of the page table reasonable. Note, too, that there is only one page table for the whole system not one page table per process. The only problem is that searching through the page table for a match on process ID and logical page number can require many memory references itself. Once again, the TLB delivers a workable solution. Programs still show strong locality, so the TLB continues to resolve most address lookups at high speed. TLB misses do require a lengthy search, but engineering means making sensible tradeoffs. Finally on the subject of page tables, page table entries have additional status bits associated with them. Earlier we mentioned the valid bit, used for signaling whether the page was in memory or not. In addition, read and write permission bits allow the system to provide protection to the contents of memory, page by page. There are also bits to indicate whether the page has been accessed and whether it has been modified. When it comes time to replace the mapping of one page with another, the accessed bit allows the OS to replace a page that isn t being accessed very often instead of one which is. The modified bit also helps during page replacement decisions. Other things being equal, it makes more sense to replace a page that has not been modified, because the contents of memory don t need to be written to disk as the page is replaced; nothing has been changed. On the other hand, a modified, or dirty, page will need to be written to disk to store the changes before its memory can be reallocated. Page Replacement Algorithms When a process needs a page that is not yet mapped into memory, and another page must be replaced to accommodate that, the OS must decide which page to sacrifice. Much research and experimentation have gone into creating efficient algorithms for replacing pages in memory. As an example, we will discuss the clock or second chance algorithm. The clock algorithm uses the Referenced bits in the page table, and a pointer into the page table. Every read or write to a page of memory sets the Referenced bit for the page to 1. On every clock tick interrupt to the
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