[Paper Review] FIFO anomaly is unbounded

Péter Fornai, Antal Iványi|arXiv (Cornell University)|Mar 5, 2010

Optimization and Search Problems29 references3 citations

TL;DR

This paper proves that the FIFO page replacement algorithm can suffer from an unbounded anomaly: increasing memory size can drastically increase page faults, defying intuition. The authors construct reference strings where the anomaly ratio (page faults with larger memory divided by smaller memory) exceeds any finite bound, disproving Belady's conjecture that the ratio is at most 2.

ABSTRACT

Virtual memory of computers is usually implemented by demand paging. For some page replacement algorithms the number of page faults may increase as the number of page frames increases. Belady, Nelson and Shedler constructed reference strings for which page replacement algorithm FIFO produces near twice more page faults in a larger memory than in a smaller one. They formulated the conjecture that 2 is a general bound. We prove that this ratio can be arbitrarily large.

Motivation & Objective

• To disprove Belady's conjecture that the FIFO anomaly ratio is bounded by 2.
• To demonstrate that the anomaly ratio can be arbitrarily large for the FIFO page replacement algorithm.
• To construct explicit reference strings that produce arbitrarily high anomaly ratios.
• To provide a general algorithm for generating such reference strings for any desired anomaly ratio.

Proposed method

• Constructs a reference string R using a pseudocode algorithm ANOMALY that generates a sequence of page references.
• Uses a cyclic reference string of length n (odd, n ≥ 5) with n-1 frames and m = n-2 frames in the small memory.
• Employs a control state mechanism where the small memory is initialized to (w3, w4, ..., wn) and the large memory to (2, 3, ..., n).
• Analyzes the page fault behavior under FIFO, showing n faults in the small memory and n(n-1)/2 in the large memory.
• Demonstrates that the anomaly ratio fFIFO(R, M)/fFIFO(R, m) approaches (n-1)/2 as n increases.
• Uses number theory lemmas to prove the periodic and modular behavior of page replacement sequences under FIFO.

Experimental results

Research questions

• RQ1 Can the FIFO anomaly ratio exceed any finite bound, contradicting Belady's conjecture?
• RQ2 What reference string construction produces an arbitrarily large anomaly ratio for FIFO?
• RQ3 How does the anomaly ratio scale with memory size and reference string structure?
• RQ4 What is the maximum possible anomaly ratio for FIFO under cyclical reference strings?

Key findings

• The FIFO anomaly ratio can be made arbitrarily large, disproving Belady's conjecture that it is bounded by 2.
• For any large number L, there exist parameters m, M, and a reference string R such that fFIFO(R, M)/fFIFO(R, m) > L.
• The anomaly ratio approaches (n-1)/2 for odd n, where n is the number of distinct pages.
• The construction achieves an anomaly ratio greater than 2 for n = 15 (ratio > 7), and grows without bound as n increases.
• The ANOMALY algorithm generates a reference string that forces the small memory to reach a specific control state while maximizing page faults in the large memory.
• The paper proves that the anomaly is unbounded even under cyclical reference strings, which are often considered worst-case scenarios.