**Paging in OS-**

Before you go through this article, make sure that you have gone through the previous articles on **Paging in OS**.

We have discussed-

- Paging is a non-contiguous memory allocation technique.
**Page Table**is a table that maps a page number to the frame number containing that page.**Multilevel Paging**is a paging scheme where there exists a hierarchy of page tables.

**Also Read-****Important Formulas of Paging**

In this article, we will discuss practice problems based on multilevel paging.

**PRACTICE PROBLEMS BASED ON MULTILEVEL PAGING IN OS-**

**Problem-01:**

Consider a single level paging scheme. The page size is 4 KB and page table entry size is 4 bytes. The size of page table is 4 KB. Give the division of virtual address space.

**Solution-**

Given-

- Page size = 4 KB
- Page table entry size = 4 bytes
- Page table size = 4 KB

Let the number of bits in virtual address = n bits

**Number of Bits in Page Offset-**

We have,

Page size

= 4 KB

= 2^{12} B

Thus, Number of bits in page offset = 12 bits

**Process Size-**

Number of bits in virtual address = n bits

Thus,

Process size = 2^{n} bytes

**Number of Pages of Process-**

Number of pages the process is divided

= Process size / Page size

= 2^{n} B / 4 KB

= 2^{n} B / 2^{12} B

= 2^{n-12} pages

**Page Table Size-**

Page table keeps track of the frames storing the pages of process.

Page table size

= Number of entries in page table x Page table entry size

= Number of pages the process is divided x Page table entry size

= 2^{n-12} x 4 bytes

= 2^{n-12+2} bytes

= 2^{n-10} bytes

But we are given page table size = 4 KB

Thus,

2^{n-10} bytes = 4 KB

2^{n-10} = 2^{12}

n – 10 = 12

∴ n = 22

Thus, number of bits in virtual address = 22 bits

**Number of Bits Required to Search an Entry in Page Table-**

**Method-01:**

Number of bits required to search a particular entry in page table

= Number of bits in virtual address – Number of bits in page offset

= 22 bits – 12 bits

= 10 bits

**Method-02:**

Number of entries in page table

= Number of pages the process is divided

= 2^{n-12}

= 2^{22-12}

= 2^{10}

Thus, Number of bits required to search a particular entry in page table = 10 bits

**Problem-02:**

Consider a two level paging scheme. The page size is 4 KB and page table entry size is 4 bytes. The size of outer page table is 4 KB. Give the division of virtual address space.

**Solution-**

Given-

- Page size = 4 KB
- Page table entry size = 4 bytes
- Page table size = 4 KB

Let the number of bits in virtual address = n bits

**Number of Bits in Page Offset-**

We have,

Page size

= 4 KB

= 2^{12} B

Thus, Number of bits in page offset = 12 bits

**Process Size-**

Number of bits in virtual address = n bits

Thus,

Process size = 2^{n} bytes

**Number of Pages of Process-**

Number of pages the process is divided

= Process size / Page size

= 2^{n} B / 4 KB

= 2^{n} B / 2^{12} B

= 2^{n-12} pages

**Inner Page Table Size-**

Inner page table keeps track of the frames storing the pages of process.

Inner page table size

= Number of entries in inner page table x Page table entry size

= Number of pages the process is divided x Page table entry size

= 2^{n-12} x 4 bytes

= 2^{n-12+2} bytes

= 2^{n-10} bytes

**Number of Pages of Inner Page Table-**

Number of pages the inner page table is divided

= Inner page table size / Page size

= 2^{n-10} B / 4 KB

= 2^{n-10} B / 2^{12} B

= 2^{n-10-12}

= 2^{n-22} pages

Now, these 2^{n-22} pages of inner page table are stored in different frames of the main memory.

**Number of Page Table Entries in One Page of Inner Page Table-**

Number of page table entries in one page of inner page table

= Page size / Page table entry size

= 4 KB / 4 bytes

= 1 K

= 2^{10} entries

**Number of Bits Required to Search an Entry in One Page of Inner Page Table-**

One page of inner page table contains 2^{10} entries.

Thus,

Number of bits required to search a particular entry in one page of inner page table = 10 bits

**Outer Page Table Size-**

Outer page table is required to keep track of the frames storing the pages of inner page table.

Outer page table size

= Number of entries in outer page table x Page table entry size

= Number of pages the inner page table is divided x Page table entry size

= 2^{n-22} x 4 bytes

= 2^{n-22+2} bytes

= 2^{n-20} bytes

But we are given outer page table size = 4 KB

Thus,

2^{n-20} bytes = 4 KB

2^{n-20} = 2^{12}

n – 20 = 12

∴ n = 32

Thus, number of bits in virtual address = 32 bits

**Number of Bits Required to Search an Entry in Outer Page Table-**

**Method-01:**

Number of bits required to search a particular entry in outer page table

= Number of bits in virtual address – (Number of bits required to search an entry in inner page table + Number of bits in page offset)

= 32 bits – (10 bits + 12 bits)

= 32 bits – 22 bits

= 10 bits

**Method-02:**

Number of entries in outer page table

= Number of pages the inner page table is divided

= 2^{n-22}

= 2^{32-22}

= 2^{10}

Thus, Number of bits required to search a particular entry in outer page table = 10 bits

**Problem-03:**

Consider a three level paging scheme. The page size is 4 KB and page table entry size is 4 bytes. The size of outermost page table is 4 KB. Give the division of virtual address space.

**Solution-**

Given-

- Page size = 4 KB
- Page table entry size = 4 bytes
- Page table size = 4 KB

Let the number of bits in virtual address = n bits

**Number of Bits in Page Offset-**

We have,

Page size

= 4 KB

= 2^{12} B

Thus, Number of bits in page offset = 12 bits

**Process Size-**

Number of bits in virtual address = n bits

Thus,

Process size = 2^{n} bytes

**Number of Pages of Process-**

Number of pages the process is divided

= Process size / Page size

= 2^{n} B / 4 KB

= 2^{n} B / 2^{12} B

= 2^{n-12} pages

**Inner Page Table Size-**

Inner page table keeps track of the frames storing the pages of process.

Inner page table size

= Number of entries in inner page table x Page table entry size

= Number of pages the process is divided x Page table entry size

= 2^{n-12} x 4 bytes

= 2^{n-12+2} bytes

= 2^{n-10} bytes

**Number of Pages of Inner Page Table-**

Number of pages the inner page table is divided

= Inner page table size / Page size

= 2^{n-10} B / 4 KB

= 2^{n-10} B / 2^{12} B

= 2^{n-10-12}

= 2^{n-22} pages

Now, these 2^{n-22} pages of inner page table are stored in different frames of the main memory.

**Number of Page Table Entries in One Page of Inner Page Table-**

Number of page table entries in one page of inner page table

= Page size / Page table entry size

= 4 KB / 4 bytes

= 1 K

= 2^{10} entries

**Number of Bits Required to Search an Entry in One Page of Inner Page Table-**

One page of inner page table contains 2^{10} entries.

Thus,

Number of bits required to search a particular entry in one page of inner page table = 10 bits

**Outer Page Table-1 Size-**

Outer page table-1 is required to keep track of the frames storing the pages of inner page table.

Outer page table-1 size

= Number of entries in outer page table-1 x Page table entry size

= Number of pages the inner page table is divided x Page table entry size

= 2^{n-22} x 4 bytes

= 2^{n-22+2} bytes

= 2^{n-20} bytes

**Number of Pages of Outer Page Table-1**

Number of pages the outer page table-1 is divided

= Outer page table-1 size / Page size

= 2^{n-20} B / 4 KB

= 2^{n-20} B / 2^{12} B

= 2^{n-20-12}

= 2^{n-32} pages

Now, these 2^{n-32} pages of outer page table-1 are stored in different frames of the main memory.

**Number of Page Table Entries in One Page of Outer Page Table-1**

Number of page table entries in one page of outer page table-1

= Page size / Page table entry size

= 4 KB / 4 bytes

= 1 K

= 2^{10} entries

**Number of Bits Required to Search an Entry in One Page of Outer Page Table-1**

One page of outer page table-1 contains 2^{10} entries.

Thus,

Number of bits required to search a particular entry in one page of outer page table-1 = 10 bits

**Outer Page Table-2 Size-**

Outer page table-2 is required to keep track of the frames storing the pages of outer page table-1.

Outer page table-2 size

= Number of entries in outer page table-2 x Page table entry size

= Number of pages the outer page table-1 is divided x Page table entry size

= 2^{n-32} x 4 bytes

= 2^{n-32+2} bytes

= 2^{n-30} bytes

But we are given outermost page table size = 4 KB

Thus,

2^{n-30} bytes = 4 KB

2^{n-30} = 2^{12}

n – 30 = 12

∴ n = 42

Thus, number of bits in virtual address = 42 bits

**Number of Bits Required to Search an Entry in Outer Page Table-2**

**Method-01:**

Number of bits required to search a particular entry in outer page table-2

= Number of bits in virtual address – (Number of bits required to search an entry in outer page table-1 + Number of bits required to search an entry in inner page table + Number of bits in page offset)

= 42 bits – (10 bits + 10 bits + 12 bits)

= 42 bits – 32 bits

= 10 bits

**Method-02:**

Number of entries in outer page table-2

= Number of pages the outer page table-1 is divided

= 2^{n-32}

= 2^{42-32}

= 2^{10}

Thus, Number of bits required to search a particular entry in outer page table-2 = 10 bits

**Problem-04:**

Complete the following table-

Page Size | Page Table Entry Size | Outermost Page Table Size | Levels of Paging | Virtual Address Space Division |

4 KB | 4 bytes | 256 bytes | 1 | ? |

4 KB | 4 bytes | 256 bytes | 2 | ? |

4 KB | 4 bytes | 256 bytes | 3 | ? |

**Solution-**

- We have to solve these problems in exactly the same manner as we have solved above.
- So, try yourself.

The answers to these problems are-

- 6 bits, 12 bits
- 6 bits, 10 bits, 12 bits
- 6 bits, 10 bits, 10 bits, 12 bits

**Next Article-****Practice Problems On Multilevel Paging Using TLB | Set-03**

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