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SOONER AND LATER WAITING TIME PROBLEMS FOR

SUCCESS AND FAILURE RUNS IN HIGHER ORDER

MARKOV DEPENDENT TRIALS

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S. AKI^{ 1}, N. BALAKRISHNAN^{ 2} AND S. G. MOHANTY^{ 2}

^{1} *Department of Mathematical Science, Osaka University,
Toyonaka, Osaka 560, Japan*

^{2} *Department of Mathematics and Statistics, McMaster
University,*

Hamilton, Ontario, Canada L8S 4K1
(Received May 19, 1995; revised November 13, 1995)

**Abstract.**
The probability generating functions of the
waiting times for the first success run of length *k* and for the
sooner run and the later run between a success run of length *k* and
a failure run of length *r* in the second order Markov dependent
trials are derived using the probability generating function method
and the combinatorial method. Further, the systems of equations of
2^{m} conditional probability generating functions of the waiting
times in the *m*-th order Markov dependent trials are given. Since
the systems of equations are linear with respect to the conditional
probability generating functions, they can be solved exactly, and
hence the probability generating functions of the waiting time
distributions are obtained. If *m* is large, some computer algebra
systems are available to solve the linear systems of equations.

*Key words and phrases*:
Probability generating function,
discrete distributions, sooner and later problem, success and
failure runs, Markov chain, waiting time.

**Source**
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