My doubt on TLB and page fault

Question

First read this whole thing what I am writing below:

Case 1: If we have to access unit address in memory using TLB and we assume that no page fault occurs then,

EMAT=p( T+M )+( 1-p ) (T+M+M)

T=TLB access time, M=memory access time(page table access time is included), p= TLB hit. EMAT : estimated memory access time.

case 2: As case 1 , But here page fault occurs.

QUESTION: Now how will we calculate EMAT?

Does the following calculated EMAT affect or contributes to the solution of my question?

EMAT= X(S+M)+(1-X)(M)

X=page fault occurring ratio, S=page fault service time

Note: Here we are not using any TLB. This is for normal virtual memory concept without TLB

Now my main question is: when we calculate EMAT using TLB and if there page fault occurs then how does the last calculated EMAT here affects the first Estimated memory access time which we have calculated using TLB?

Answer 1

EMAT= X(S+M)+(1-X)(M)

just add one more M for page table access at the beginning and it will be correct:

EMAT= M + X(S+M)+ (1-X)(M)

In paging and cache concepts always go step by step, i.e., hierarchically. EMAT with TLB and Page Fault both in picture:

If TLB is updated asap(use this for numericals):

EMAT = t ( tat + m ) + (1-t) ( tat + m + pf(pfst+m) + (1-pf)m ) 

If TLB hasn't been update when we looked up for a page in it(helpful in interviews just to give a point):

EMAT = t ( tat + pf(pfst+m) + (1-pf)m ) + (1-t) ( tat + m + pf(pfst+m) + (1-pf)m ) 

where t=TLB hit probability, tat=TLB access time, pf=Page Fault probability, pfst=Page Fault service time and m=memory access time.

Pseudo-code/Procedure:

check with TLB
if found in TLB
    {
        go to memory
     }
else not in TLB
    {
        go to page table in memory 
        {
            go to memory
            {
                if page fault occurs
                    service page fault then acess memory
                else
                    access memory
            }
        }
    }

Comments

@ Akhilesh Singla I'm not able to understand this part of equation  t ( tat + pf(pfst+m) + (1-pf)m ) . 

if there's a tlb hit means that is page is already loaded in memory than why are you taking page fault rate and page fault service time in this case.

shouldn't it be like t(tat+m) ?

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Hmm, you are right. I included it for cases when TLB hasn't been updated about a page which has left the memory. I should have mentioned that. And the standard approach doesn't include my assumption. I will edit it now. Thanks for pointing it out.

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EMAT = t + m + (1-Ht)( p * PS  + (1-p) * m )

t= tlb access time

m= memory access time

Ht= hit rate of tlb

p= page fault rate

PS= pagefault service time

Can't we use this formula as t + m is access whether is TLB hit or TLB miss. Please help on this topic I'm confused. How tlb and page fault rate formula is changed when cache is also introduced.