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SOONER AND LATER WAITING TIME PROBLEMS

IN A TWO-STATE MARKOV CHAIN

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MASAYUKI UCHIDA AND SIGEO
AKI

*Department of Mathematical Science, Faculty of Engineering
Science, Osaka University,*

Machikaneyama-cho, Toyonaka, Osaka 560, Japan
(Received August 31, 1994; revised February 8, 1995)

**Abstract.**
Let *X*_{1}, *X*_{2}, ... be a time-homogeneous
{0,1}-valued Markov
chain. Let *F*_{0} be the event that *l* runs
of ``0'' of length *r* occur and let *F*_{1} be the event that *m*
runs of ``1'' of length *k* occur in the sequence *X*_{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 between *F*_{0} and *F*_{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 length *r* or
more and ``1''-runs of length *k* or more.

*Key words and phrases*:
Sooner and later problems,
discrete distribution, Markov chain, recurrence relations,
overlapping, non-overlapping, non-overlapping ``0''-runs of length
*r* or more, generalized probability generating function.

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