Second Order Properties of Thinned Counts in Finite Birth--Death Processes

TL;DR

This paper analyzes the second order properties of thinned counting processes derived from finite birth-death processes, providing formulas for asymptotic variance and exploring applications in queueing models.

Contribution

It derives a formula for the asymptotic variance rate of thinned counts in finite birth-death processes, extending understanding of their second order properties.

Findings

Derived a formula for asymptotic variance rate.

Illustrated results with queueing model examples.

Conjectured and tested an analogous formula for infinite state spaces.

Abstract

The paper studies the counting process arising as a subset of births and deaths in a birth--death process on a finite state space. Whenever a birth or death occurs, the process is incremented or not depending on the outcome of an independent Bernoulli experiment whose probability is a state-dependent function of the birth and death and also depends on whether it is a birth or death that has occurred. We establish a formula for the asymptotic variance rate of this process, also presented as the ratio of the asymptotic variance and the asymptotic mean. Several examples including queueing models illustrate the scope of applicability of the results. An analogous formula for the countably infinite state space is conjectured and tested.