AISM 52, 778-789

## Waiting time problems in a two-state Markov chain

### Qing Han and Sigeo Aki

Department of Informatics and Mathematical Science,
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

(Received May 18, 1998; revised August 6, 1999)

Abstract.
Let $F_0$ be the event that $l_0$ $0$-runs of length $k_0$ occur and $F_1$ be the event that $l_1$ $1$-runs of length $k_1$ occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between $F_0$ and $F_1$ by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the
distributions of the sooner and later waiting time problems.

Key words and phrases:
Waiting time problems,
discrete distributions of order $k$, Markov chain,
Markov chain imbedding method, probability generating function.

**Source**
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