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The k-server problem is that of planning the motion of k mobile servers on the vertices of a graph under a sequence of requests for service. Each request consists of the name of a vertex, and is satisfied by placing a server at the requested vertex. The requests must be satisfied in their order of occurrence. The cost of satisfying a sequence of requests is the distance moved by the servers. In this paper we study on-line algorithms for this problem from the competitive point of view. That is, we seek to develop on-line algorithms whose performance on any sequence of requests is as close as possible to the performance of the optimum off-line algorithm. We obtain optimally competitive algorithms for several important cases. Because of the flexibility in choosing the distances in the graph and the number of servers, the k-server problem can be used to model a number of important paging and caching problems. It can also be used as a building block for solving more general problems. We show how server algorithms can be used to solve a seemingly more general class of problems known as tusk systems. Q 1990 A~&C h, Inc