Optimizing Page Size

Question

Given that the logical address is "d" bits.Page table entry size is 4  bytes,what must be the optimal page size P by minimizing  the page table size overhead and internal fragmentation in paging ?

a) (4d)^1/2                   b) (4*d)²

C) (8*2^(2d))^1/2        d) (2^(d+3))^1/2

Comments

it will be D...

Answer 1

Let X=page table size overhead + internal fragmentation overhead

We need to minimize X.

If we take very large P i.e. large page size then chances of internal fragmentation will be high.

If we take very small P then no. of pages required for a process will be high which means no. of page table entry will increase which in turn means the page table size will increase.

We need to manage both.

If P is the page size then log(P) is the page offset bits. Since logical address bits is "d" so no. of bits for pages= (d-logP). 

So no. of pages= 2^(d-logP) = 2^d/2^logP = 2^d/P

Size of page table= (2^d/P)*4 B

internal fragmentation in page table:

Min internal frag=0 i.e. when the page table is full

Max internal frag=P-4 i.e. when only one entry(i.e. of size 4B) is present in page table (it can be approximated to P)

So average internal frag=0+P/2=P/2.

X= (2^d/P)*4 + P/2

To minimize X(i.e. overall overhead) we need to find dX/dP

dX/dP= (-4*2^d)/P² + 1/2

dX/dP=0 then P= (2^(d+3))^1/2

Since d²X/dP² > 0 at P =(2^(d+3))^1/2 so we know that X is minimum of that value of P.

Option D

 

Answer 2

Overhead=page table size+p/2

F(o)=(V.A/P)*e +P/2

where V.A=virtual address

P=page size

e=page table entry

Differentiating with respect to P

F'(o)=-K*e/$p^{2}$+1/2

F'(o)=0

P=$(2*k*e)^{1/2}$         (1)

Where K=virtual address=$(2)^{d}$

Putting this value in eq (1)

Option (D) will satisfied.