SOONER AND LATER WAITING TIME PROBLEMS FOR SUCCESS AND FAILURE RUNS IN HIGHER ORDER

MARKOV DEPENDENT TRIALS

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

1Department of Mathematical Science, Osaka University, Toyonaka, Osaka 560, Japan

2Department 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 mth 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.