Question

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? 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?

Answer 1

The page-replacement algorithm manages page frames with counters, initiating at zero and adjusting with page replacements. Calculation of page faults using this model for a given reference string requires step-by-step tracking. The minimum for an optimal algorithm needs predictive analysis.

Step-by-step explanation:

Page-Replacement Algorithm

According to the concept described, the page-replacement algorithm aims to minimize page faults by managing the distribution of page frames based on counters. Here's a defined algorithm addressing the specified issues:

• Initial value of counters: Each page frame counter is initialized to zero, indicating that no pages are currently associated with the frame.
• Counters increased: A counter for a page frame is increased when a page is placed into it, representing an additional page now associated with that frame.
• Counters decreased: A counter is decreased when a page associated with that frame is removed, whether it's due to being replaced or removed.
• Page selection for replacement: The algorithm selects the page for replacement by searching for the page frame with the smallest counter, implying it's the least frequently used frame.

Page Faults for Reference String

For the given reference string with four page frames, the number of page faults would need to be computed step by step, following the algorithm specified.

Optimal Page Faults

To determine the minimum number of page faults for an optimal page replacement strategy, one would typically use an algorithm like Belady's optimal algorithm, which replaces pages that will not be used for the longest period of time in the future. The calculation would be made against the same reference string with the same number of page frames.