Waiting time problems for a two-dimensional pattern

  • Sigeo Aki
  • Katuomi Hirano


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.

Key words and phrases

Waiting time problem two-dimensional pattern probability generating function discrete distribution conditional distribution reliability consecutive system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Balakrishnan, N. and Koutras, M. V. (2002).Runs and Scans with Applications, Wiley, New York.MATHGoogle Scholar
  2. Blom, G. and Thorburn, D. (1982). How many random digits are required until given sequences are obtained?,Journal of Applied Probability,19, 518–531.CrossRefMathSciNetGoogle Scholar
  3. Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials,Statistica Sinica,6, 957–974.MathSciNetGoogle Scholar
  4. 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.CrossRefMathSciNetGoogle Scholar
  5. 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
  6. 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.CrossRefGoogle Scholar
  7. 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.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics 2004

Authors and Affiliations

  • Sigeo Aki
    • 1
  • Katuomi Hirano
    • 2
  1. 1.Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering ScienceOsaka UniversityToyonakaJapan
  2. 2.The Institute of Statistical MathematicsTokyoJapan

Personalised recommendations