On some limit theorems following from Smith's Theorem

Abstract

We give a generalization of the ergodic theorem for semi-Markov linear-type processes. This generalization is proved for the case when a common support of distributions defining this process is not arithmetic. Also we give an uniform estimate of expectations of the forward renewal time. These facts are based on the key renewal (Walter Smith's) theorem, and they are very useful in research in the queueing theory.