This paper investigates the destination and desirability of anticipatory moves by an idle server aimed at reducing its response time for future requests. It is motivated by certain types of automated storage and retrieval systems, computer disks and elevators. Requests' timing and location are assumed to be uncertain. If a request arrives during an anticipatory move, the server will travel to the request upon the completion of the noninterruptible move. Otherwise, another anticipatory move can be considered in order to be closer to the statie optimum. A stochastic dynamic programming model is developed for finding whether and where to move from each starting location. Structural properties of the optimal solutions are analyzed. Numerical examples are explored to provide insight into the nature and benefit of such moves. These numerical examples show the benefits depend heavily on requests' arrival rate and their location distribution, as well as on the server's setup time and speed.