# Economical Caching with Stochastic Prices

## Abstract

In the *economical caching* problem, an online algorithm is given a sequence of prices for a certain commodity. The algorithm has to manage a buffer of fixed capacity over time. We assume that time proceeds in discrete steps. In step *i*, the commodity is available at price *c* _{ i } ∈ [*α*,*β*], where *β* > *α* ≥ 0 and *c* _{ i } ∈ ℕ. One unit of the commodity is consumed per step. The algorithm can buy this unit at the current price *c* _{ i }, can take a previously bought unit from the storage, or can buy more than one unit at price *c* _{ i } and put the remaining units into the storage.

In this paper, we study the economical caching problem in a probabilistic analysis, that is, we assume that the prices are generated by a random walk with reflecting boundaries *α* and *β*. We are able to identify the optimal online algorithm in this probabilistic model and analyze its expected cost and its expected *savings*, i.e., the cost that it saves in comparison to the cost that would arise without having a buffer. In particular, we compare the savings of the optimal online algorithm with the savings of the optimal offline algorithm in a probabilistic competitive analysis and obtain tight bounds (up to constant factors) on the ratio between the expected savings of these two algorithms.

### Keywords

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