# Bounds on the bias terms for the Markov reward approach

## TL;DR

This paper develops theoretical conditions and a linear programming framework to bound bias terms in Markov reward models, enabling improved error bounds for stationary performance measures in certain Markov chains.

## Contribution

It introduces sufficient conditions for using quadratic and geometric functions to bound bias terms, filling a gap where affine bounds are insufficient.

## Key findings

- Quadratic and geometric bounds can be used under certain conditions.
- A linear programming framework is proposed for computing these bounds.
- The approach improves error bounds for specific Markov chain models.

## Abstract

An important step in the Markov reward approach to error bounds on stationary performance measures of Markov chains is to bound the bias terms. Affine functions have been successfully used for these bounds for various models, but there are also models for which it has not been possible to establish such bounds. So far, no theoretical results have been available that guarantee bounds on the bias terms. We consider random walks in the positive orthant and provide sufficient conditions under which quadratic and/or geometric functions can be used to bound the bias terms. In addition, we provide a linear programming framework that establishes the quadratic bounds as well as the resulting bound on the stationary performance.