A page$-$replacement algorithm should minimize the number of page faults. We can achieve this minimization by distributing heavily used pages evenly over all of memory, rather than having them compete for a small number of page frames. We can associate with each page frame a counter of the number of pages associated with that frame. Then, to replace a page, we can search for the page frame with the smallest counter.

a$.$

Define a page$-$replacement algorithm using this basic idea. Specifically address these problems:

i$.$

What is the initial value of the counters?

ii$.$

When are counters increased?

iii.

When are counters decreased?

iv$.$

How is the page to be replaced selected?

b$.$

How many page faults occur for your algorithm for the following reference string with four page frames? Please calculate the number of page faults, and clearly write out the whole process. Don't use the Random method

$1,2,3,4,5,3,4,1,6,7,8,7,8,9,7,8,9,5,4,5,4,2.$

c$.$

What is the minimum number of page faults for an optimal page$-$replacement strategy for the reference string in part b with four page frames? Please calculate the number of page faults, and clearly write out the whole process.