The Waiting Time Distribution of Competing Patterns in Markov-Dependent Bernoulli Trials

Competing patterns are compound patterns that compete to be the first to occur a pattern-specific number of times, known as a stopping rule. In this paper, we study a higher-order Markovian dependent Bernoulli trials model with competing patterns. The waiting time distribution refers to the distribution of the number of trials required until the stopping rule is met. Based on a finite Markov chain, a hierarchical algorithm is proposed to derive the conditional probability generating function (pgf) of the waiting time of the competing patterns model. By applying the law of total expectation, the final pgf is then obtained. Using examples, we further demonstrate that the proposed algorithm is an effective and easy-to-implement tool.