Renewal-Reward Process

1. Definition

• let $N(t)$ be a renewal process
• Let $R_n$ = reward earned at $n$-th renewal
• Assume $R_n$ are i.i.d, but can depend on $X_n$ (length of the $n$-th cycle)

Then $R(t) = \sum^{N(t)}_{n=1} R_n$ is a renewal reward process.

Intuitive explanation: $R(t)$ = cumulative reward earned up to time t

2. Renewal Reward Theorem 

Proposition 7.3 

• $lim_{t \to \infty} \frac{R(t)}{t} = \frac{E[R_n]}{E[X_n]}$
• $lim_{t \to \infty} \frac{E[R(t)]}{t} = \frac{E[R_n]}{E[X_n]} $

Provided $E[R_n] < \infty, E[X_n] < \infty$

3. Example: Inventory Problem

Problem: 

Solution: