On the minimal travel time needed to collect n items on a circle

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. In this paper we study the minimal travel time of the picker. We obtain upper bounds and analyze the exact travel time distribution. Further, we derive closed-form limiting results when n tends
to infinity. 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.