Consider a demand paging system where an integer is stored in $4$ bytes, page size is $512$ bytes, uses LRU page replacement algorithm and each process can be assigned $4$ frames. A process executes the following code where the code occupies $80\%$ of page $0$ and integers x and y are stored in general purpose registers of the processor.

int x $=0,$ y$,$ a$[200][200]$;

e$($x $+200)($

$[$CO$3]$

y $=0$

while$($y$++<200)$

a$[$x$][$y$]=0$

$\}$

$($a$)$ How many pages are needed for the array if all the elements of the array are stored contiguously.

$($b$)$ How many page faults will this program generate? Jerstify the answer.

$($c$)$ Calculate the amount of internal fragmentation in bytes.

$($d$)$ If the program makes the following page references: $0,1,2,3,0,4,0,6,0,4,8,9,0,11,0,13,10,0,15,16,17,0$ What is the working set W$($t$,$ Delta$)$ with t equal to time at $11$th reference assuming window size equal to $6$ references?