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:


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