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May 2004 On the minimal travel time needed to collect n items on a circle
Nelly Litvak, Willem R. van Zwet
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Ann. Appl. Probab. 14(2): 881-902 (May 2004). DOI: 10.1214/105051604000000152

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.

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Nelly Litvak. Willem R. van Zwet. "On the minimal travel time needed to collect n items on a circle." Ann. Appl. Probab. 14 (2) 881 - 902, May 2004. https://doi.org/10.1214/105051604000000152

Information

Published: May 2004
First available in Project Euclid: 23 April 2004

zbMATH: 1121.90012
MathSciNet: MR2052907
Digital Object Identifier: 10.1214/105051604000000152

Subjects:
Primary: 90B05
Secondary: 60F05 , 60G51 , 62E15

Keywords: asymptotics , carousel systems , exact distributions , Exponential functionals , Uniform spacings

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.14 • No. 2 • May 2004
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