Scheduling for on-line taxi problem on a real line and competitive algorithms

The problem of k-taxi is a generalization of the famous k-server problem, where we must choose how k mobile servers serve a sequence of requests occurring in the metric space. In our case we consider how k taxies on a real line to serve a sequence of requests must be determined in an on-line manner. Our goal is to minimize the transportation cost of all taxies. With the help of position maintaining strategy (PMS for short), a competitive algorithm that has a ratio of k+2 is given. Besides, another deterministic on-line strategy is also illustrated, namely, the partial double coverage strategy (PDCS for short), a competitive algorithm with competitive ratio of k is given. In this paper, a particular case concerning elevator is also analyzed and some results have been obtained. Finally, some open problems for the on-line k-taxi problem on a real line are proposed.