Sooner and later waiting time problems in a two-state Markov chain |
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Authors: | Masayuki Uchida Sigeo Aki |
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Institution: | (1) Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Machikaneyama-cho, 560 Toyonaka, Osaka, Japan;(2) Present address: the Institute of Statistical Mathematics, Japan |
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Abstract: | LetX
1,X
2,... be a time-homogeneous {0, 1}-valued Markov chain. LetF
0 be the event thatl runs of 0 of lengthr occur and letF
1 be the event thatm runs of 1 of lengthk occur in the sequenceX
1,X
2, ... We obtained the recurrence relations of the probability generating functions of the distributions of the waiting time for the sooner and later occurring events betweenF
0 andF
1 by the non-overlapping way of counting and overlapping way of counting. We also obtained the recurrence relations of the probability generating functions of the distributions of the sooner and later waiting time by the non-overlapping way of counting of 0 -runs of lengthr or more and 1 -runs of lengthk or more. |
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Keywords: | Sooner and later problems discrete distribution Markov chain recurrence relations overlapping non-overlapping non-over-lapping 0 gif" alt="ldquo" align="MIDDLE" BORDER="0">0 -runs of lengthr or more" target="_blank">gif" alt="rdquo" align="MIDDLE" BORDER="0">-runs of lengthr or more generalized probability generating function |
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