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

Nelli Litvak, Willem R. van Zwet

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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.
Original languageUndefined
Title of host publicationSelected Works of Willem van Zwet
EditorsSarah van der Geer, Marten Wegkamp
Place of PublicationNew York
Number of pages22
ISBN (Print)978-1-4614-1313-4
Publication statusPublished - 2012

Publication series

NameSelected Works in Probability and Statistics
ISSN (Print)1050-5164
ISSN (Electronic)2168-8737


  • EWI-22566
  • Uniform spacings
  • IR-83428
  • METIS-289794
  • Exponential functionals
  • Exact distributions
  • Asymptotics
  • Carousel systems

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