On the minimal travel time needed to collect n items on a circle
Nelly Litvak and Willem R. van Zwet
Source: Ann. Appl. Probab. Volume 14, Number 2
(2004), 881-902.
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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Keywords: Uniform spacings; carousel systems; exact distributions; asymptotics; exponential functionals
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1082737116
Digital Object Identifier: doi:10.1214/105051604000000152
Mathematical Reviews number (MathSciNet): MR2052907
Zentralblatt MATH identifier: 02100759
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