Estimates of the probability of a regenerative process reaching a high level

TL;DR

This paper develops two-sided estimates for the probability that a regenerative process reaches a high level, with applications to queue length processes in queueing theory.

Contribution

It introduces new two-sided bounds for high-level reaching probabilities of regenerative processes, utilizing auxiliary results for geometric sums with delay.

Findings

Established two-sided estimates for high-level probabilities.

Applied results to queue length processes in queueing systems.

Provided examples demonstrating practical applications.

Abstract

The problem of estimating the probability of a random process reaching a certain level is well known. In this article, two-sided estimates are established for the probability that a regenerative process reaches a high level. Two auxiliary results for geometric sums with delay will play an important role. Examples of application to random processes describing queue lengths in queueing theory are also given.