---
tag: 45V0W838BQ
kind: concept
name: Renewal theory
msc: [60K05, 60K15]
sources:
  - url: https://en.wikipedia.org/wiki/Renewal_theory
same_as:
  wikidata: Q1356856
  wikipedia: Renewal theory
---

## Summary

Renewal theory studies processes that start afresh at random times, with independent and identically distributed waiting times between two restarts.
Its central object is the renewal process, which counts the restarts up to time $t$.
Its main results, such as the renewal theorems, describe the long-run rate of restarts.
It applies to the replacement of machine parts, to queues and to returns of a Markov chain to a state.

## Explanations

### L1 human

A light bulb burns out at a random time and is replaced at once by a new bulb of the same kind.
Each replacement is a renewal: the system starts over as if new.
Renewal theory asks how many bulbs are used by time $t$.
If a bulb lasts $\mu$ hours on average, then in the long run one uses about $t / \mu$ bulbs in $t$ hours.
The Poisson process is the renewal process with exponential waiting times.