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

• 成果类型: Article
• 署名作者: Litvak, N; van Zwet, WR
• 署名单位: University of Twente; Leiden University - Excl LUMC; Leiden University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000152
• 发表日期: 2004
• 页码: 881-902
• 关键词: similar markov-processes; strategies

摘要

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.