Assume Computer XY has a $4$ GB main memory and a direct$-$mapped $($L$1)$ cache $($there are no other cache levels and no secondary or tertiary storage$)$ with a $32$B line size. On XY$,$ cache hits take $2$ clock cycles, and cache misses take $90$ clock cycles. Also on XY$,$ if the CPU had disabled the cache and went directly to main memory, the main memory access time would be $100$ cycles.

If Program A was run on XY when XY had a $32$ KB L$1$ cache$$ and A is a program that has a $2$ GB array of $(32$ bit$)$ integers $($and essentially no other use of memory$)$ that is accessed in a sequential way and thus has high spatial locality of reference, what would the overall average memory access time be in terms of CPU clock cycles? If the cache was instead a fully associative cache, what effect would you expect that to have on memory access time for this scenario? $($N$.$B$.$: in cache memory discussions K is not $1000,$ instead it denotes $2^10.)$

To calculate the hit rate, we need to determine the number of cache hits and cache misses.

Given:

$-$ Cache Line Size $=32$B

$-$ L$1$ Cache Size $=32$ KB $=\backslash (32\backslash$times $2^\{10\}\backslash )$ bytes

$-$ Array Size $=2$ GB of $32-$bit integers $=\backslash (2\backslash$times $2^\{30\}\backslash )$ bytes

We have previously calculated:

$-$ Number of cache lines $=1024$

$-$ Total number of integers in the array $=\backslash (2^\{29\}\backslash )$

$-$ Number of cache lines needed to accommodate the array $=\backslash (2^\{26\}\backslash )$

Since the cache has $1024$ cache lines and the number of cache lines needed to accommodate the array is $\backslash (2^\{26\}\backslash ),$ all the memory blocks needed for the array can fit into the cache at some point.

Now, let's calculate the hit rate:

Number of cache hits $=$ Total number of accesses $-$ Number of cache misses

Total number of accesses $=$ Total number of integers in the array $=\backslash (2^\{29\}\backslash )$

Number of cache misses $=$ Number of cache lines needed to accommodate the array $=\backslash (2^\{26\}\backslash )$

So$,$

Number of cache hits $=????$

Given:

$-$ Cache Line Size $=32$B

$-$ L$1$ Cache Size $=32$ KB $=\backslash (32\backslash$times $2^\{10\}\backslash )$ bytes

$-$ Array Size $=2$ GB of $32-$bit integers $=\backslash (2\backslash$times $2^\{30\}\backslash )$ bytes

Since the integers are $32$ bits $(4$ bytes$)$ each, the array contains $\backslash (\backslash$frac$\{2\backslash$times $2^\{30\}\}\{4\}\backslash )$ integers. With sequential access and high spatial locality, most accesses will hit in the cache after the initial miss that loads a line. Each line can hold $\backslash (\backslash$frac$\{32\}\{4\}=8\backslash )$ integers, so the miss rate is effectively the fraction of accesses that require a new line to be loaded.

For a direct$-$mapped cache, each new line access after every $8$ integer accesses will likely result in a cache miss $($assuming no conflict or capacity misses, which is optimistic$).$ Thus, the miss rate can be approximated as $\backslash (\backslash$frac$\{1\}\{8\}\backslash ).$

What is the overall hit rate here $?????$ As said above for one line the hit rate was $7/8,$ for computing $($or$)$ accessing the overall array elements what is the hit rate????