Compound geometric approximation under a failure rate constraint

TL;DR

This paper develops bounds for compound geometric approximation of nonnegative integer-valued variables using failure rate constraints, with applications to queuing systems, birth-death, and Poisson processes.

Contribution

It introduces a simple bound based on failure rate lower bounds for compound geometric approximation, applicable to various stochastic models.

Findings

Explicit bounds for M/G/1 queue metrics

Approximation bounds for birth-death processes

Bounds for Poisson process applications

Abstract

We consider compound geometric approximation for a nonnegative, integer-valued random variable $W$. The bound we give is straightforward but relies on having a lower bound on the failure rate of $W$. Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth-death processes and Poisson processes.