The Annals of Applied Probability

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

Nelly Litvak and 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.

Article information

Ann. Appl. Probab., Volume 14, Number 2 (2004), 881-902.

First available in Project Euclid: 23 April 2004

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Zentralblatt MATH identifier

Primary: 90B05: Inventory, storage, reservoirs
Secondary: 62E15: Exact distribution theory 60F05: Central limit and other weak theorems 60G51: Processes with independent increments; Lévy processes

Uniform spacings carousel systems exact distributions asymptotics exponential functionals


Litvak, Nelly; van Zwet, Willem R. On the minimal travel time needed to collect n items on a circle. Ann. Appl. Probab. 14 (2004), no. 2, 881--902. doi:10.1214/105051604000000152.

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