$1)(42$ pts$)$ In a hypothetical system with $1024$ bytes of main memory, $128$ bytes of cache, blocks of $32-$byte

size, with direct$-$map $($S $=1)$ placement policy and LRU replacement policy, answer each of the following

questions:

a$)(3$ pts$)$ Determine the address fields for index, tag and block offset in the memory address.

b$)(3$ pts$)$ How many sets are there in the cache?

c$)(3$ pts$)$ How many blocks in the main memory are mapped to the same set of blocks in cache?

d$)(3$ pts$)$ If associative $($parallel$)$ comparison is to be performed with all tags for identification, how

many XOR gates are required?

e$)$ Determine the hit ratio in running each of the following program segments $($each instruction occupies

a word$)$: $($briefly show your work$)$

i$)(10$ pts$)$ a sequence of $100$ consecutive instructions starting from address x$148$ with each

instruction executed once in order

ii$)(10$ pts$)$ a loop of $32$ consecutive instructions starting at location x$148$ and is iterated $20$ times

iii$)(10$ pts$)$ a loop of $64$ consecutive instructions starting at location x$148$ and is iterated $20$ times

$2)(42$ pts$)$ Redo problem #$1$ if it is with a $2-$way set associative placement policy $($S $=2).$