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Waiting time problems for a two-dimensional pattern |
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| Authors: | Aki Sigeo Hirano Katuomi |
| Affiliation: | (1) Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, 5608531 Toyonaka, Japan;(2) The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106-8569 Tokyo, Japan;(3) Present address: Department of Mathematics, Faculty of Engineering, Kansai University, 564-8680 Suita, Osaka, Japan |
| 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. 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). |
| Keywords: | Waiting time problem two-dimensional pattern probability generating function discrete distribution conditional distribution reliability consecutive system |
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