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Increasing the number of page frames can increase misses under a given page replacement policy on the same page-reference trace. For FIFO page replacement on the displayed twelve-reference sequence, an empty start with three page frames gives nine misses, while an empty start with four gives ten. The larger run hits only at indices 5 and 6; it misses at 1–4 and 7–12. These are miss counts, not observed running times.^[ref]

The explanation is visible before the final count. After access 6, the three-page frame queue is $(4,1,2)$ and the four-page frame queue is $(1,2,3,4)$, both oldest first. Loading memory page 5 at access 7 removes memory page 4 in the smaller run but memory page 1 in the larger. Their resulting resident sets are $\{1,2,5\}$ and $\{2,3,4,5\}$. The smaller resident set is not a subset of the larger: its next access to memory page 1 hits while the larger run misses. Different earlier evictions create different histories; an extra page frame is not simply an extra unchanged slot. FIFO page replacement can improve on other sequences; the anomaly is an existence claim, not a claim that more page frames usually hurt.

Evict the least recently accessed eligible item. Exact recency tracking on every memory reference can be expensive, whereas accesses to a buffer cache already pass through software. Recency captures temporal locality but a long scan can evict frequently reused items before their next use.

For page replacement, update recency on hits as well as misses; evict only when a miss requires a page frame and all eligible page frames are occupied. The four-page cycle in memory thrashing shows how exact LRU replacement behaves at capacities three and four.^[ref]

The LRU stack property compares different capacities on one fixed page-reference trace; it does not say that LRU replacement always beats FIFO page replacement. On the displayed sequence, three-page frame LRU replacement has ten misses, versus nine for FIFO page replacement, while four-page frame LRU replacement has only eight.

A memory access trace reduced to the ordered sequence of accessed memory page identities, retaining repetitions. An address-only reduction loses read/write information, so annotate writes when studying dirty memory pages. Use the following classic Belady anomaly sequence, where the numbers label memory pages, not physical addresses:Start with three empty, equal-sized page frames allocated to one operating-system process. Every memory page can occupy any page frame, and every resident memory page is eligible for eviction. Use demand paging: a hit finds the requested memory page in the resident set; a miss causes exactly one nonresident page fault and loads it before the next access. Fill the lowest-numbered empty page frame, then reuse the victim's page frame. Assume only reads, no fetching ahead of requests, no competing operating-system processes, and no external removal of memory pages. CLOCK page replacement uses its declared hand rule, which fills the same initial page frames here. The memory page labels alone do not determine the result without these assumptions.

Each column shows the state after servicing that access. Rows keep physical page frame positions fixed, rather than silently sorting the resident set. The CLOCK page replacement entries also show reference bits and the next hand position. The independently computed totals are:

The first three misses fill unused page frames; the fourth miss needs eviction. At access 4, FIFO page replacement and LRU replacement evict memory page 1. Optimal page replacement instead evicts memory page 3: the next uses of resident memory pages 1, 2 and 3 are at indices 5, 6 and 10. At access 10, FIFO page replacement evicts memory page 1 while LRU replacement evicts memory page 5. Consequently the last access to memory page 5 hits under FIFO page replacement but misses under LRU replacement. Matching totals for FIFO page replacement and CLOCK page replacement on this sequence do not make the policies equivalent. The generators compute every displayed state; a separate search over all legal victims confirms the seven-miss optimal page replacement bound for three page frames.

Choosing a resident memory page to evict when another memory page needs an occupied page frame. An unused page frame needs no eviction. A dirty memory page requires preservation before reuse, accounted for separately as page writeback. A page replacement policy balances likely future reuse against the overhead of tracking past memory references.

The opportunity comes from temporal locality: retaining a recently accessed memory page can spare a later nonresident page fault. Spatial locality helps consecutive accesses reuse the same memory page until they cross its boundary. A working set describes recent demand, while the resident set describes what is actually present. A small active footprint can fit in the available page frames even when the whole address space cannot. These are useful workload tendencies, not guarantees about future memory references.^[ref]

The page-reference trace below uses the counterexample discussed in Operating Systems: Three Easy Pieces by Remzi H. Arpaci-Dusseau and others (chapter 22, p. 5) and makes the decisions explicit. FIFO page replacement ranks arrival time, LRU replacement ranks last access, CLOCK page replacement samples recent use, and optimal page replacement knows the remaining sequence in advance. Count misses first; page-replacement cost accounting separately counts page writeback and explains why fewer misses need not mean less total service time.

// Destination: computer-science.bigb

Let $n$ be the number of accesses, $M$ the miss count and $D$ the number of dirty-victim page writebacks. Assume every requested memory page has a backing copy, every miss reads it, writes modify only part of its contents, transfers succeed and occur serially, and there is no background or final flushing. With common access cost $h$, additional load cost $r$ per miss, and additional outgoing write cost $w$ per dirty victim, an illustrative total service time isBookkeeping and other operating system costs are omitted. The two transfer directions are separate: a miss can require a read with no write, or both a victim write and a requested-page read. A retained dirty memory page contributes no outgoing write to this finite-page-reference trace count until evicted. Future flushing would need its own accounting boundary.

For three-page frame FIFO page replacement on the displayed page-reference trace, all reads give $(n,M,D)=(12,9,0)$. Change only access 1 to a partial write of memory page 1. Its initial miss still loads the old contents, the write marks it dirty, and eviction at access 4 adds exactly one page writeback. Later loads of memory page 1 are clean because the remaining accesses read. The result is $(12,9,1)$: identical residency and nine misses, but one extra outgoing write. With illustrative $h=100$ nanoseconds, $r=1$ millisecond and $w=2$ milliseconds, total service time changes from $9.0012$ to $11.0012$ milliseconds. These are model costs, not hardware measurements.

Preferring a clean victim can save an immediate page writeback, but evicting a soon-needed clean memory page can add later misses. This tradeoff is why a miss-minimizing optimal page replacement benchmark and a minimum-cost policy are different questions. Real page faults can also initialize a new memory page or perform copy-on-write without reading backing storage; those cases require different load costs.

// Destination: computer-science.bigb