Asymmetric rendezvous search on the line

The classical asymmetric rendezvous search problem on the line is faced by two people who, facing in random directions, are placed somewhere on the real line. Their initial distance is drawn from a known cumulative distribution function F. Neither knows the direction of the other and by using different trajectories they aim to meet up as quickly as possible. Mathematically the aim is to find the asymmetric rendezvous value which is the least expected meeting time achievable by players moving at maximum unit velocity.We introduce a restriction of the classical asymmetric rendezvous search problem on the line, a restriction in which the players' strategy space PL comprises of speed-one piecewise linear continuous trajectories. We show that, in general, the asymmetric rendezvous values of the line in these two problems coincide. However we prove that, in problems involving distributions with right upper derivative infinite at zero, the asymmetric rendezvous value of the line can never be attained with strategies in PL.We demonstrate general necessary conditions on a strategy pair for it to be optimal. New upper bounds for the asymmetric rendezvous value of the line are given for certain problems of bounded distributions.We establish a necessary and sufficient condition on a bounded distribution function for the existence of a terminating pair of strategies with minimality properties. We prove that an optimal pair of trajectories is terminating if and only if F is fat. These results qualitatively characterize optimal strategies for bounded distribution problems.Our research also involves numerical treatment of issues connected to asymmetric rendezvous search on the line. A tool for the numerical computation of the players' expected meeting time is developed and simulation analysis is used to gain a better understanding of the problem.