  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 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 neighborhood of zero. The limiting random variable 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.
