We give an O(log k)-competitive randomized algorithm for weighted paging with delay. This improves upon the O(log k log n)-competitive algorithm of Gupta, Kumar, and Panigrahi (STOC 2020, SICOMP 2022) by removing the dependence on the number n of pages. Without delay, the optimal competitive ratio of randomized algorithms is Theta(log k), and since the lower bound carries over, our bound is tight. Our algorithm is fractional and combines mirror descent, used by Bubeck et al. (STOC 2018) for weighted paging without delay, with preemptive service, introduced by Azar et al. (STOC 2017) for online service with delay: when a request is served, an increase in the mirror descent potential is offset by preemptive service to other pending requests, with a budget derived from the mirror descent path. To obtain a randomized algorithm, we use the online rounding of Bansal, Buchbinder, and Naor (FOCS 2007) from classical paging.

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