Distributions of numbers of runs and scans on directed acyclic graphs with generation

In this paper, we introduce a class of a directed acyclic graph on the assumption that the collection of random variables indexed by the vertices has a Markov property. We present a flexible approach for the study of the exact distributions of runs and scans on the directed acyclic graph by extending the method of conditional probability generating functions. The results presented here provide a wide framework for developing the exact distribution theory of runs and scans on the graphical models. We also show that our theoretical results can easily be carried out through some computer algebra systems and give some numerical examples in order to demonstrate the feasibility of our theoretical results. As applications, two special reliability systems are considered, which are closely related to our general results. Finally, we address the parameter estimation in the distributions of runs and scans.