This capstone develops the single most practically useful result to come out of renewal theory — the renewal-reward theorem — and applies it to a genuine optimisation problem (when to replace an aging asset) and a genuine two-state system (an alternating renewal process). The theorem's slogan, long-run average reward per unit time equals expected reward per cycle divided by expected cycle length, converts a hard question about a whole infinite trajectory into an easy question about a single typical cycle — exactly the same "compute one representative unit, then divide" logic already used for the cycle-trick stationary distribution and the inspection paradox.
Learning objectives. By the end of this note you should be able to:
- state and apply the renewal-reward theorem;
- set up and solve a long-run-cost optimisation problem using it;
- state the alternating-renewal-process theorem and apply it to a two-state system;
- recognise the "reward per cycle over mean cycle length" pattern across different applications.
In this lesson
- The renewal-reward theorem
- Worked example: optimal replacement policy
- Alternating renewal processes
The full lesson, with worked examples and exercise solutions, is for members.
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