On some limit theorems following from Smith's Theorem

TL;DR

This paper generalizes Smith's ergodic theorem to semi-Markov processes with non-arithmetic support and provides uniform estimates for forward renewal times, aiding queueing theory research.

Contribution

It extends Smith's key renewal theorem to a broader class of semi-Markov processes with non-arithmetic support and offers new uniform expectation estimates.

Findings

Generalized ergodic theorem for semi-Markov processes

Provided uniform estimates of forward renewal times

Enhanced tools for queueing theory analysis

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.