Estimates of the probability of a regenerative process reaching a high level
Kateryna Akbash, Ivan Matsak, Oleg Zakusylo
TL;DR
This paper addresses estimating the probability that a regenerative process reaches a high level. It develops two-sided bounds for the hitting probability by reducing the problem to geometric sums with delay and exploiting auxiliary renewal- and moment-based quantities, notably $q^{*}$, $m_1^{-}$, and $\hat{m}_1^{+}$. The main contribution is Theorem 1.1, which bounds the deviation $\Delta_X(x)$ between the actual hitting distribution and an exponential benchmark, with refinements via $\mathbb{C}_0$, $\mathbb{C}_1$, and $\mathbb{C}_2$, under small failure probability $q$ and $0<x<1$, complemented by two technical lemmas on geometric sums with delay. The results are illustrated through queue-length processes, including explicit M/G/1 and M/M/1 examples, providing practically usable bounds for reliability and queueing-performance analyses where regenerative modeling is appropriate.
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.