Distributed Patrolling with Two-Speed Robots (and an Application to Transportation).

We initiate the study of patrolling a unit interval with primitive two-speed, autonomous robots, i.e. robots without memory, no communication capabilities and no computation power. Robots have only two moving-states, one for patrolling and one for walking, each associated with a direction and speed. The robots are moving perpetually, and their moving-states and moving directions change only when they collide. Such a dynamic system induces the so-called idleness for patrolling a unit interval, i.e. the smallest time interval within which every point of the domain is patrolled by some robot. Our main technical contribution is an analytic study of the induced dynamic system of robots, which allows us to decide efficiently whether or not the system converges to a stable configuration that is also shown to be optimal. As a warm-up for our main result, we show how robots can be centrally coordinated, carefully choosing initial locations, so that the induced idleness is optimal. Our main result pertaining to the idleness of primitive robots follows by a technical analysis of their collision locations, which we show, under some conditions, converge to the optimal initial locations for non-distributed robots. Our result finds an application to a transportation problem concerning Scheduling with Regular Delivery. In this optimization problem, an infinite quantity of a commodity, residing at an endpoint of an interval, needs to be transported to the other endpoint. To that end, we show that the already established patrolling schedules of an interval correspond to optimal strategies that guarantee that the flow of the commodity is the largest possible.