Renewal processes with costs and rewards

TL;DR

This paper reviews renewal reward processes, introduces a simplified proof of the renewal reward theorem, discusses extensions to partial rewards, and demonstrates applications like the inspection paradox and Little's law.

Contribution

It provides a new simplified proof of the renewal reward theorem and explores extensions to partial rewards with necessary regularity conditions.

Findings

Simplified proof of the renewal reward theorem

Counterexample for partial rewards extension

Applications to inspection paradox and Little's law

Abstract

We review the theory of renewal reward processes, which describes renewal processes that have some cost or reward associated with each cycle. We present a new simplified proof of the renewal reward theorem that mimics the proof of the elementary renewal theorem and avoids the technicalities in the proof that is presented in most textbooks. Moreover, we mention briefly the extension of the theory to partial rewards, where it is assumed that rewards are not accrued only at renewal epochs but also during the renewal cycle. For this case, we present a counterexample which indicates that the standard conditions for the renewal reward theorem are not sufficient; additional regularity assumptions are necessary. We present a few examples to indicate the usefulness of this theory, where we prove the inspection paradox and Little's law through the renewal reward theorem.