Nearly optimal strategies for special cases of on-line capital investment

Suppose that some job must be done for a period of unspecified duration. The market offers a selection of devices that can do this job, each characterized by purchase and running costs. Which of them should we buy at what times, in order to minimize the total costs? As usual in competitive analysis, the cost of an on-line solution is compared to the optimum costs paid by a clearvoyant buyer. This problem which generalizes the basic rent-to-buy problem has been introduced by Y. Azar et al. In the so-called convex case where lower running costs always imply higher prices, a strategy with competitive ratio 6.83 has been proposed. Here we consider two natural subcases of the convex case in a continuous-time model where new devices can be bought at any time. For the static case where all devices are available at the beginning, we give a simple 4-competitive deterministic algorithm, and we show that 3.618 is a lower bound. (This is also the first non-trivial lower bound for the convex case, both for discrete and continuous time.) Furthermore we give a 2.88-competitive randomized algorithm. In the case that all devices have equal prices but are not all available at the beginning, we show that a very simple algorithm is 2-competitive, and we derive a 1.618 lower bound