A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments

Let N ( t ) be a counting process associated with a sequence of non-negative random variables ( X j ) 1 ∞ where the distribution of X n +1 depends only on the value of the partial sum S n = Σ j=1 n X j . In this paper, we study the structure of the function H ( t ) = E N ( t ), extending the ordinary renewal theory. It is shown under certain conditions that h ( t ) = ( d/dt ) H ( t ) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h ( t ) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h ( t ) and H ( t ) as t → ∞ is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.