{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "FV0E9GEPCD"
    },
    {
      "field": "instance_of",
      "target": "GTFRPF5J2K"
    }
  ],
  "body": "\n## Summary\n\nA 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.\nThe state is read as the size of a population: a step up is a birth and a step down is a death.\nWhen it has a stationary distribution, that distribution has an explicit product formula.\nIt is the basic model for queues, population growth and chemical reactions.\n\n## Explanations\n\n### L1 human\n\nThink of the number of people waiting in a shop.\nA new customer arrives, so the number goes up by one, or a customer leaves, so it goes down by one.\nTwo things never happen at the same instant.\nThe speed of arrivals and of departures may depend on how many people are already there.\nFor 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.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "60J27",
      "60J80",
      "60K25"
    ],
    "name": "Birth–death process",
    "same_as": {
      "wikidata": "Q1456275",
      "wikipedia": "Birth–death process"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Birth–death_process"
      }
    ],
    "specializes": [
      "TWQFE0FZZS"
    ],
    "tag": "E8DAD7052E"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/E8DAD7052E/",
  "links": [
    {
      "field": "specializes",
      "target": "TWQFE0FZZS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/E8DAD7052E.md",
  "msc": [
    "60J27",
    "60J80",
    "60K25"
  ],
  "name": "Birth–death process",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E8D/E8DAD7052E.md",
  "status": null,
  "tag": "E8DAD7052E",
  "uri": "m2://E8DAD7052E",
  "url": "https://math2.tamnd.com/wiki/birth-death-process/",
  "writing": {
    "tag": "E8DAD7052E",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}