Problem $2.(20$ points$)$ Consider a database buffer manager that uses the algorithm explained in

Section $13\mathrm{.}5\mathrm{.}1$ of the textbook for a hard drive which takes $10$ milliseconds per disk seek and can

read $80$ MB per second.

$($a$)(10$ points$)$ Assume that $($i$)$ the size of each disk block is $8$ KB $,($ii$)$ the probability that the

buffer manager can handle a data request without accessing the disk $($i$.$e$.,$ the probability that

a requested block is already in the buffer$)$ is $90\%,($iii$)$ when the requested disk block is not in

the main memory $($with $10\%$ probability$),$ that block needs to be read from the disk following

a disk seek, $($iv$)$ no disk blocks are updated $($i$.$e$.,$ no need to write dirty blocks back to the

hard drive$),$ and $($ v $)$ the time spent for reading$/$writing data in the main memory is negligible

$($considered $0$ milliseconds$).$ Under these assumptions, calculate the expected data access time

$($i$.$e$.,$ how much time it would on average take to obtain a disk block from the buffer manager$).$

$($b$)(10$ points$)$ In contrast to the above scenario, assume that $($i$)$ the size of each disk block is

$16$ KB $,($ii$)$ the probability that the buffer manager can handle a data request without accessing

the disk $($i$.$e$.,$ the probability that a requested block is already in the buffer$)$ is $70\%,$ and $($iii$)$

with $5\%$ probability, the requested disk block needs to be read from disk following a disk seek.

Calculate the expected data access time as in $($a$).$ Also, based on the above calculations,

explain whether it would be more advantageous to use $8$ KB disk blocks or $16$ KB disk blocks.