Renewal Reward Rate

The renewal reward theorem connects the long-run average reward per unit time to the expected reward and expected length of a single renewal cycle. Specifically, if a system goes through independent and identically distributed (IID) cycles, each with a random length L and a random reward R, the long-run rate at which reward is accumulated is E[R] / E[L].

Interactive Experiment

Open the Renewal Reward Rate experiment to observe the difference between these rates in real time.

1. Ensure all weights (Weight A, Weight B, Weight C) are set to 1.
2. Observe that the E[R]/E[L] result shows 4/7 (0.571), while the E[R/L] result shows 5/12 (0.417).
3. Change Weight A to 0 and Weight B to 0, leaving Weight C at 1.
4. Notice that both E[R]/E[L] and E[R/L] now equal 3/4 (0.75) because only a single cycle length is active.