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

Abstract

Let N(t) be a counting process associated with a sequence of non-negative random variables (Xj)1∞ where the distribution of Xn+1 depends only on the value of the partial sum Sn = Σj=1nXj. 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.