## The Annals of Applied Probability

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 23 April 2004

https://projecteuclid.org/euclid.aoap/1082737116

Digital Object Identifier
doi:10.1214/105051604000000152

Mathematical Reviews number (MathSciNet)
MR2052907

Zentralblatt MATH identifier
1121.90012

#### Citation

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. https://projecteuclid.org/euclid.aoap/1082737116

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