Ergodicity and uniform in time truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin

TL;DR

This paper extends the analysis of inhomogeneous birth-death processes with special origin transitions, providing new ergodicity and truncation bounds applicable even when transition decay is slower than exponential.

Contribution

It introduces improved bounds for ergodicity and truncation of these processes, covering cases with slower decay rates than previously addressed.

Findings

Bounds are valid for non-exponential decay rates.

Numerical results demonstrate the bounds' effectiveness.

Extended analysis includes transitions from and to the origin.

Abstract

In this paper one presents the extension of the transient analysis of the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes which were known only for the case when transitions from the origin decay exponentially (other intensities must have unique uniform upper bound). We show how the bounds can be obtained the decay rate is slower than exponential. Numerical results are also provided.