Problem $4$: $(10$ points each, total: $30$ pts$)$ Cache Performance

A processor uses a two$-$level cache hierarchy $($L$1$ and L$2)$ and has the following properties:

$$ L$1$ Cache: Cache size: $32$ KB$,$ Block size: $64$ bytes, Associativity: $4-$way set associative, Hit time: $1$ cycle, and Miss rate: $5\%.$

$$ L$2$ Cache: Cache size: $256$ KB$,$ Block size: $64$ bytes, Associativity: $8-$way set associative, Hit time: $5$ cycles, and Miss rate: $10\%.$

$$ Main Memory: Access time: $100$ cycles

$$ Average memory instruction counts: $20\%$ of the total instructions

$$ Total number of instructions executed: $1$ billion

Questions:

a$.$ Calculate the average memory access time $($AMAT$)$ for the three$-$level cache hierarchy.

b$.$ Estimate the total cache misses at each cache level $($L$1$ and L$2)$ for the given instruction count.

c$.$ Determine the CPI of the processor, assuming a base CPI of $2$ without cache miss penalties.