---
tag: E8DAD7052E
kind: concept
name: Birth–death process
msc: [60J27, 60J80, 60K25]
specializes: [TWQFE0FZZS]
sources:
  - url: https://en.wikipedia.org/wiki/Birth–death_process
same_as:
  wikidata: Q1456275
  wikipedia: Birth–death process
---

## Summary

A birth–death process is a [[continuous-time Markov chain|TWQFE0FZZS]] on the integers $0, 1, 2, \ldots$ that moves one step up or one step down at each jump.
The state is read as the size of a population: a step up is a birth and a step down is a death.
When it has a stationary distribution, that distribution has an explicit product formula.
It is the basic model for queues, population growth and chemical reactions.

## Explanations

### L1 human

Think of the number of people waiting in a shop.
A new customer arrives, so the number goes up by one, or a customer leaves, so it goes down by one.
Two things never happen at the same instant.
The speed of arrivals and of departures may depend on how many people are already there.
For example, the [[M/M/1 queue|GTFRPF5J2K]] has arrivals at a constant rate and departures at a constant rate whenever the shop is not empty.