Problem $1$: $(10$ points each, total: $20$ pts$)$ Cache Performance Analysis

A computer system has a $2-$level cache with the following parameters:

L$1$ Cache:

$$ Size: $32$ KB

$$ Block Size: $64$ bytes

$$ Associativity: $4-$way set associative

$$ Hit Time: $2$ cycles

$$ Miss Rate: $10\%$

L$2$ Cache:

$$ Size: $256$ KB

$$ Block Size: $64$ bytes

$$ Associativity: $8-$way set associative

$$ Hit Time: $10$ cycles

$$ Miss Rate: $5\%($for accesses that missed in the L$1$ cache$)$

Main Memory Access Time: $100$ cycles.

Questions:

$($a$)(10$ points$)$ Calculate the Average Memory Access Time $($AMAT$)$ for this system. Show your calculations step by step.

$($b$)(10$ points$)$ If the L$1$ miss rate is reduced to $7\%$ due to improvements in cache management techniques, calculate the new AMAT. Determine the percentage improvement in AMAT over the original AMAT due to the reduction in the L$1$ miss rate.