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

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1082737116

Digital Object Identifier
doi:10.1214/105051604000000152

Mathematical Reviews number (MathSciNet)
MR2052907

Zentralblatt MATH identifier
1121.90012

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

Keywords
Uniform spacings carousel systems exact distributions asymptotics exponential functionals

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


Export citation

References

  • Ali, M. M. and Obaidullah, M. (1982). Distribution of linear combination of exponential variates. Comm. Statist. Theory Methods 11 1453–1463.
  • Askey, R. (1980). Ramanujan's extensions of the gamma and beta functions. Amer. Math. Monthly 87 346–359.
  • Bartholdi, J. J. III and Platzman, L. K. (1986). Retrieval strategies for a carousel conveyor. IIE Transactions 18 166–173.
  • Bertoin, J. and Yor, M. (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Comm. Probab. 6 95–106.
  • Bertoin, J. and Yor, M. (2002a). On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math. (6) 11 33–45.
  • Bertoin, J. and Yor, M. (2002b). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 389–400.
  • Bertoin, J., Biane, P. and Yor, M. (2002). Poissonian exponential functionals, $q$-series, $q$-integrals, and the moment problem for log-normal distributions. Technical Report PMA-705, Univ. Paris 6, Laboratoire de Probabilités.
  • Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, ed.) 73–130. Iberoamericana, Madrid.
  • Davis, R. A. and Resnick, S. I. (1991). Extremes of moving averages of random variables with finite endpoint. Ann. Probab. 19 312–328.
  • Dumas, V., Guillemin, F. and Robert, Ph. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab. 34 85–111.
  • Feller, W. (1970). An Introduction to Probability Theory and Its Applications $\mathbf{II}$. Wiley, London.
  • Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge Univ. Press.
  • Guillemin, F., Robert, Ph. and Zwart, B. (2002). AIMD algorithms and exponential functionals. Technical Report 4447, INRIA.
  • Litvak, N. and Adan, I. (2001). The travel time in carousel systems under the nearest item heuristic. J. Appl. Probab. 38 45–54.
  • Litvak, N. and Adan, I. (2002). On a class of order pick strategies in paternosters. Oper. Res. Lett. 30 377–386.
  • Litvak, N. (2001). Some peculiarities of exponential random variables. J. Appl. Probab. 38 787–792.
  • Pyke, R. (1965). Spacings. J. R. Stat. Soc. Ser. B Stat. Methodol. 27 395–449.
  • Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.