The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 14, Number 2 (2004), 881-902.
On the minimal travel time needed to collect n items on a circle
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.
Ann. Appl. Probab., Volume 14, Number 2 (2004), 881-902.
First available in Project Euclid: 23 April 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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