  We consider online algorithms as a request-answer game. An adversary that
generates input requests, and an online algorithm answers. We consider a
generalized version of the game that has a buffer of limited size. The
adversary loads data to the buffer, and the algorithm has random access to
elements of the buffer. We consider quantum and classical (deterministic or
randomized) algorithms for the model.
  In the paper, we provide a specific problem (The Most Frequent Keyword
Problem) and a quantum algorithm that works better than any classical
(deterministic or randomized) algorithm in terms of competitive ratio. At the
same time, for the problem, classical online algorithms in the standard model
are equivalent to the classical algorithms in the request-answer game with
buffer model.
