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

Nelli 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 languageUndefined
Title of host publicationSelected Works of Willem van Zwet
EditorsSarah van der Geer, Marten Wegkamp
Place of PublicationNew York
PublisherSpringer
Pages371-392
Number of pages22
ISBN (Print)978-1-4614-1313-4
DOIs
Publication statusPublished - 2012

Publication series

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

Keywords

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

Cite this

Litvak, N., & van Zwet, W. R. (2012). On the minimal travel time needed to collect $n$ items on a circle. In S. van der Geer, & M. Wegkamp (Eds.), Selected Works of Willem van Zwet (pp. 371-392). (Selected Works in Probability and Statistics). New York: Springer. https://doi.org/10.1007/978-1-4614-1314-1_22