Upper bound on the rate of convergence and truncation bound for nonhomogeneous birth and death processes on $\mathbb{Z}$

TL;DR

This paper derives upper bounds on the convergence rate and truncation errors for nonhomogeneous birth-death processes on integers, providing new inequalities and conditions for limiting distributions, with applications in queueing theory.

Contribution

It introduces the first known upper bounds on convergence rates and truncation errors for nonhomogeneous birth-death processes, along with a new concentration inequality.

Findings

Derived upper bounds on convergence rates.

Provided truncation error bounds.

Illustrated with queueing theory examples.

Abstract

We consider the well-known problem of the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time birth and death processes on $\mathbb{Z}$ with time varying and possible state-dependent intensities. First in the literature upper bounds on the rate of convergence along with one new concentration inequality are provided. Upper bounds for the error of truncation are also given. Condition under which a limiting (time-dependent) distribution exists is formulated but relies on the quantities that need to be guessed in each use-case. The developed theory is illustrated by two numerical examples within the queueing theory context.