Asymmetric rendezvous on the line is a double linear search problem

We apply a new method of analysis to the asymmetric rendezvous search probl em on the line (ARSPL). This problem, previously studied in a paper of Alpe rn and Gal (1995), asks how two blind, speed one players placed a distance d apart on the line, can find each other in minimum expected time. The dist ance d is drawn from a known cumulative probability distribution G, and the players are faced in random directions. We show that the ARSPL is strategically equivalent to a new problem we call the double linear search problem (DLSP), where an object is placed equipro bably on one of two lines, and equiprobably at positions +/-d. A searcher i s placed at the origin of each of these lines. The two searchers move with a combined speed of one, to minimize the expected time before one of them f inds the object. Using results from a concurrent paper of the first author and J. V. Howard (1998), we solve the DLSP (and hence the ARSPL) for the case where G is con vex on its support, and show that the solution is that conjectured in a pap er of Baston and Gal (1998).