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Waiting time problems for a two-dimensional pattern

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Abstract

We consider waiting time problems for a two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. We deal with a two-dimensional pattern with a general shape in the two-dimensional lattice which is generated by the above sequence of random vectors. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given.

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References

  • Balakrishnan, N. and Koutras, M. V. (2002).Runs and Scans with Applications, Wiley, New York.

    MATH  Google Scholar 

  • Blom, G. and Thorburn, D. (1982). How many random digits are required until given sequences are obtained?,Journal of Applied Probability,19, 518–531.

    Article  MathSciNet  Google Scholar 

  • Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials,Statistica Sinica,6, 957–974.

    MathSciNet  Google Scholar 

  • Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach,Journal of the American Statistical Association,89, 1050–1058.

    Article  MathSciNet  Google Scholar 

  • Koutras, M. V. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables,Advances in Combinatorial Methods and Applications to Probability and Statistics (ed. N. Balakrishman), 363–384, Birkhäuser, Boston.

    Google Scholar 

  • Salvia, A. A. and Lasher, W. C. (1990). 2-dimensional consecutive-k-out-of-n:F models,IEEE Transactions on Reliability,R-39, 382–385.

    Article  Google Scholar 

  • Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables,Annals of the Institute of Statistical Mathematics,50, 655–671.

    Article  MathSciNet  Google Scholar 

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Author information

Author notes

  1. Sigeo Aki

    Present address: Department of Mathematics, Faculty of Engineering, Kansai University, 564-8680, Suita, Osaka, Japan

Authors and Affiliations

  1. Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, 5608531, Toyonaka, Japan

    Sigeo Aki

  2. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106-8569, Tokyo, Japan

    Katuomi Hirano

Authors
  1. Sigeo Aki
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  2. Katuomi Hirano
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Additional information

Department of Statistical Science, School of Mathematical and Physical Science, The Graduate University for Advanced Studies

This research was partially supported by the ISM Cooperative Research Program (2002-ISM-CRP-2007).

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Aki, S., Hirano, K. Waiting time problems for a two-dimensional pattern. Ann Inst Stat Math 56, 169–182 (2004). https://doi.org/10.1007/BF02530530

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  • DOI: https://doi.org/10.1007/BF02530530

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