Paging with Per-Replacement Maximum Delay

Abstract

Classical paging serves every miss immediately. We study paging with per-replacement maximum delay, where loading a pending page costs one movement plus the age of its oldest outstanding request and clears that page's entire episode. Equivalently, the holding rate is the number of pending pages. For cache size $k$, threshold LRU is strictly $(5k+3)$-competitive, while a randomized algorithm is strictly $5H_k$-competitive against an oblivious adversary; classical constructions give matching $Ω(k)$ and $Ω(H_k)$ orders. Offline, we obtain an exact $O(nk)$ dynamic program with one hole, an exact configuration algorithm for any fixed number of holes, and a nonproactive polynomial-time $5$-approximation in general. Physical farthest-next-use can nevertheless fail with only three pages. For page-dependent fetch costs with spread $ρ=w_{\max}/w_{\min}$, the exact fixed-number-of-holes algorithms persist. We obtain a $(3ρ+2)$-approximation and online guarantees with multiplicative factors $O(ρk)$ deterministically and $O(ρH_k)$ randomly against an oblivious adversary, plus any additive term inherited from the corresponding classical guarantee. A strict $Ω(\sqrtρ)$ randomized lower bound already holds for one cache slot. Thus maximum delay preserves the unit-cost competitive hierarchy but disrupts classical offline structure and makes weighted timing spread-sensitive.