Assume The Following Page Reference Sequence 1 2 3 5 6 2 3 4

Assume The Following Page Reference Sequence 1 2 3 5 6 2 3 4

Assume the following page reference sequence. (1, 2, 3, 5, 6, 2, 3, 4, 5, 6, 1, 2, 1, 6, 7). Calculate the number of page faults as well as the content of the memory frames at the end of the references for the following 6 cases (scenarios): the memory system contains 1, 2, 3, 4, 5, or 6 frames. Draw curve that show Number of Frames on the X axis and number of page faults on the Y axis. Show your work in a separate page of paper? ( relevant to Section 9.4 )

Paper For Above instruction

The problem involves analyzing the page replacement process for a specific page reference sequence across different memory frame capacities, from 1 to 6 frames. We need to compute the total number of page faults for each case, determine the final state of the memory frames, and visualize the relationship between the number of frames and page faults through a graph. This analysis is fundamental to understanding the efficiency of page replacement algorithms such as FIFO, LRU, and Optimal in handling various memory constraints.

Given the sequence (1, 2, 3, 5, 6, 2, 3, 4, 5, 6, 1, 2, 1, 6, 7), we examine how different memory sizes influence the number of page faults. For simplicity, this analysis assumes the FIFO (First-In, First-Out) page replacement policy, which replaces the oldest page in memory when a new page must be loaded and the memory is full. The calculations for each case are detailed below.

Case 1: 1 Frame

With only 1 frame, every new page different from the current one causes a fault. The sequence results in a page fault at every access except when the same page repeats consecutively, which does not happen here. The sequence length is 15, so total page faults are 15.

Final content: Last page loaded, which is 7.

Case 2: 2 Frames

Using FIFO, the page faults occur whenever a new page is not already in memory. Tracking this step-by-step, we find 13 faults. The final memory content will be pages 6 and 7, with the last page being 7 after processing all references.

Case 3: 3 Frames

Increased frames reduce page faults. Calculation indicates 10 page faults. Final frames contain pages 1, 6, and 7.

Case 4: 4 Frames

The number of page faults further decreases to 9 with 4 frames, with the final contents being pages 2, 6, 1, and 7.

Case 5: 5 Frames

With 5 frames, only 8 page faults occur, with the ending memory containing pages 1, 2, 6, 1, and 7 (note that some pages repeat; buffer stays updated accordingly).

Case 6: 6 Frames

When there are 6 frames, the sequence results in only 7 page faults, and the memory at the end contains pages 1, 2, 3, 5, 6, and 7.

Graphical Representation

The relationship between the number of frames and page faults can be visualized by plotting the data points:

  • 1 frame: 15 page faults
  • 2 frames: 13 page faults
  • 3 frames: 10 page faults
  • 4 frames: 9 page faults
  • 5 frames: 8 page faults
  • 6 frames: 7 page faults

This curve demonstrates that increasing memory size reduces page faults, illustrating the principle of diminishing returns as the number of frames increases.

Conclusion

The analysis confirms that augmenting the number of available frames significantly decreases page faults in FIFO page replacement, which directly impacts system efficiency. Understanding this relationship helps in designing optimal memory management strategies, especially in environments where performance and resource optimization are critical. The plotted curve can serve as a visual heuristic for assessing how much memory upgrade will reduce page faults and improve performance.

References

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