Page Replacement.

Consider the following reference string of length $24$ generated by a process for which $3$ memory frames are allocated for a process with $9$ pages, $0,1,2,3,4,5,6,7,8.$ Indicate the content of the frames after each page indicated by the reference string is accessed, including the page fault. How many page faults are generated from the reference string? Answer the question based on $3$ different page replacement algorithms: $($a$)$ FIFO, $($b$)$ Optimal, and $($c$)$ LRU based on stack implementation. Show the content of the stack for $($c$).$ For Optimal, there could be multiple possible answers.

$012345243204810245124320$

Repeat the question with $4$ memory frames.

$($d$)$ Consider the case with $3$ memory frames. One would expect that when a reference string is made longer through inserting an item in between $($like Question $4$ in Tutorial $10),$ the performance could remain or get worsen. Is it possible to insert one item in the reference string that will instead reduce the number of page faults for FIFO? If possible, indicate your reference string. If not possible, say so$.$ Is it possible to insert one item in the reference string that will instead reduce the number of page faults for LRU? If possible, indicate your reference string. Is it possible to insert one item in the reference string that will instead reduce the number of page faults for both FIFO for LRU? If possible, indicate your reference string. If not possible, say so$.$