  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.
