### Sample Questions

```CMPT 300
Introduction to Operating
Systems
Virtual Memory Sample Questions
1
0. Page Table
 Q: Consider a system with 32-bit virtual address. The
virtual memory is implemented by paging, and the
page size is 4KB. A user process generates the
virtual address 0x11123456. What is the page table
 A: Since the page size is 2^12, the page table size is
2^20. The virtual address in binary form is 0001 0001
0001 0010 0011 0100 0101 0110. The low-order 12
bits 0100 0101 0110 are used as the offset into the
page, while the remaining 20 bits 0001 0001 0001
0010 0011 are used as the index into the page table,
or 0x11123.
2
1.1 Inverted Page Table
 Consider a system with 32-bit virtual address space, and 4 MB of
physical memory. The page size is 8KB (8192-byte). The system
uses an Inverted Page Table (IPT).
 Q: How many bits is each VPN? How many bits is each PPN?
Describe what PTE (page table entry) looks like. How many PTEs
(page table entries) does the IPT contain? (Remember: PT is
indexed by VPN and contains PPN; IPT is indexed by PPN and
contains VPN.)
 A: The virtual address consists of VPN: 19 bits; Offset: 13 bits

2^13=8KB, hence offset is 13 bits; the total virtual address length is 32
bits, hence VPN is 32-3=19 bits.
 There are a total of 4MB/8KB=512 physical pages, hence the PPN
is 9 bits (2^9=512). The IPT contains 512 PTEs.
 IPT maps from PPN to VPN. It contains 512 PTEs. Each PTE
contains a 19-bit field for VPN, plus another field for Process ID,
plus additional bits for protection etc. (Offset is not stored in page
tables)
3
1.2 Page Table
 Consider a system with 32-bit virtual address space,






and 4 MB of physical memory. The page size is 8KB
(8192-byte). The system uses a regular Page Table.
Q: How many bits is each VPN? How many bits is
each PPN? Describe what PTE (page table entry)
looks like. How many PTEs (page table entries) does
the PT contain?
A: Same as before, VPN: 19 bits; PPN: 9 bits.
Each PTE contains a 9-bit field for PPN, plus
additional protection bits for protection etc.
There are a total of 2^32/2^13=2^19 virtual pages,
hence the PT contains 2^19 PTEs.
Q: Why is there not a field for Process ID?
A: Each process has its own separate page table.
4
2. 1 Page Table
10
P1 index
10
P2 index
1K
PTEs
1K
PTEs
4KB
12
page offset
 Consider a system with 32-bit virtual address
using two-level page tables for address
translation. The format of the virtual address,
physical address, and PTE (page table entry)
are as shown. Assume each PTE in both 1st
and 2nd level PTs is 2 bytes, including all
permission bits. Size of physical memory is
4MB.
 Q: How large is each page? How many physical
pages in total? How many bits in PPN?
 A: Page size: 2^12 bytes =4 KB; Total number
of physical pages is 4MB/4KB=1K=2^10  #
bits in PPN is 10.
5
2. 2 Page Table
10
P1 index
10
P2 index
1K
PTEs
1K
PTEs
4KB
12
page offset
 Q: What is the largest possible memory
needed for storing all page tables?
 A: There is one page table at the first
level, and up to 2^10 = 1K page tables at
the 2nd level. Each PTE is 2 bytes. Size of
the 1st level page table is 2*(2^10) bytes =
2KB. Size of each 2nd level table is also
2*(2^10) bytes = 2KB. All in all, the page
tables use 2KB + 1024 * 2KB = 2050 KB
of memory.
6
2. 3 Page Table
10
P1 index



10
P2 index
1K
PTEs
1K
PTEs
4KB
12
page offset
Q: Consider a process that is using 512KB of physical
memory. What is the minimum number of page tables
used by this process? What is the maximum number of
page tables this process might use?
A: The process uses 512KB / 4 KB = 128 physical pages.
Since each 2nd level page can hold up to 1K PTEs, in the
best case scenario the process is covered by a single 2nd
level PT. we use only 2 page tables: 1st level PT + a 2nd
level PT.
In the worst case, the process may use a little bit of every
physical page (e.g., 0.5 KB of each physical page), and all
page tables will be populated. Thus, the process ends up
using 1 + 1024 = 1025 page tables.
7
2. 4 Page Table
 Q: Assume that instead of two-level PTs, we
use an inverted PT for address translation.
How many entries are used in the inverted
table by a process using 512KB of physical
memory?
 A: The inverted table contains one entry per
physical page, hence it has 2^10=1K
entries. In the worst case, the process uses
all physical pages, which yields 1K entries.
In the best case, the process fully uses each
physical page, which yields 128 entries.
3. TLB
 Consider a system with the page tables stored in
memory. If a memory reference takes 200
nanoseconds, how long does a paged memory
reference take? If we add a TLB, and 75 percent
of all page-table references are found in the TLB,
what is the effective memory reference time?
(Assume that finding a page-table entry in the TLB
takes zero time, if TLB hit.)
 A: 400 ns; 200 ns to access the page table and
200 ns to access the word in memory.
 Effective access time = 0.75 (200 ns) + 0.25 (400
ns) = 250 ns.
9
```