Consider n items located randomly on a circle of length 1. The locations of the items are assumed to be independent and uniformly distributed on [0,1). A picker starts at point 0, and he has to collect all n items by moving along the circle at unit speed in either direction. We study the minimal travel time of the picker.
We analyze the exact travel time distribution. Further, we derive closed-form limiting results when n tends to infinity. These results can be extended to non-uniform i..i.d. items' locations, for example, if their density is strictly positive on [0,1]. We determine the behavior of the limiting distribution in a positive neighbourhood of zero. The limiting distribution is closely related to exponential functionals associated with a Poisson process. These functionals occur in many areas and have been intensively studied in recent literature.
This is a joint work with Willem van Zwet