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

Nelly Litvak, Willem R. van Zwet

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Abstract

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 languageEnglish
Pages (from-to)881-902
Number of pages22
JournalAnnals of applied probability
Volume14
Issue number2
DOIs
Publication statusPublished - 2004

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Keywords

  • METIS-223770
  • EWI-17676
  • Uniform spacings
  • Asymptotics
  • Exponential functionals
  • Exact distributions
  • Carousel systems
  • IR-70339

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