LRU replacement

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 finite-capacity cache using LRU replacement: accessing an item makes it most recent, and a miss evicts the least recently used item when the cache is full.

For exact LRU replacement, let $R_k(t)$ be the resident set after the first $t$ accesses of a fixed page-reference trace with capacity $k$. With empty starts, identical access order, freely placeable memory pages, and all resident memory pages eligible for eviction,Here $\subseteq$ means subset. At every time, form one capacity-independent ordered list of distinct memory pages, most recently accessed first. An access moves its memory page to the front, preserving the order of the others. Exact LRU replacement with $k$ page frames retains the first $k$ entries, or all entries if fewer than $k$ have appeared. The first $k$ entries are contained in the first $k+1$, proving the property. This recency list is not the push/pop interface of a LIFO stack.^[ref]

Apply inclusion immediately before an access: a hit with $k$ page frames must also hit with $k+1$. Thus cumulative misses with $k+1$ cannot exceed those with $k$, at any prefix of the same page-reference trace. On the displayed sequence, the counts decrease from ten to eight when capacity increases from three to four. Arbitrary initial contents, different access orders, protected victims, or clean-victim preference fall outside the proof. Nor does it prove monotonic page writeback costs, or apply automatically to CLOCK page replacement just because that policy approximates LRU replacement.

// Destination: computer-science.bigb