---
tag: P64V0HTE78
kind: concept
name: Queueing theory
msc: [60K25, 90B22, 60K30]
sources:
  - url: https://en.wikipedia.org/wiki/Queueing_theory
same_as:
  wikidata: Q847526
  wikipedia: Queueing theory
---

## Summary

Queueing theory is the mathematical study of waiting lines.
Customers arrive at random times, wait, are served by one or more servers, and leave.
The theory computes the laws of the queue length and of the waiting time, and it asks when the queue stays stable.
It is used to design telephone networks, computer systems, call centres and traffic systems.

## Explanations

### L1 human

Think of a post office with one clerk.
People arrive at random and each needs a random amount of time at the desk.
If people arrive faster than the clerk can serve them, the line grows without bound.
If they arrive more slowly, the line goes up and down but stays of moderate size, and one can compute its average length.
The [[M/M/1 queue|GTFRPF5J2K]] is the simplest model of this kind, and [[Little's law|JN2YME6F4A]] links the average length with the average wait.