---
tag: 1WT57ED272
kind: definition
name: Queueing theory
msc: [60K25, 90B22, 60K30]
defines: P64V0HTE78
uses: [00MRN0FARV, 0DVDCMA2DK, 0WQNRCNY3M]
sources:
  - url: https://en.wikipedia.org/wiki/Queueing_theory
---

## Definition

Let $A$ and $B$ be [[probability distributions|0DVDCMA2DK]] on $(0, \infty)$ and let $c \ge 1$ be an integer.
A *single-station queue* $A/B/c$, in Kendall's notation, is the following model:

1. customers arrive at times $0 < \tau_1 < \tau_2 < \cdots$ whose gaps $\tau_{n+1} - \tau_n$ and first time $\tau_1$ are [[independent and identically distributed|00MRN0FARV]] with law $A$;
2. customer $n$ needs a service time $\sigma_n$, where $\sigma_1, \sigma_2, \ldots$ are independent with law $B$ and independent of the arrivals;
3. there are $c$ servers and an unlimited waiting room, and customers are served in order of arrival by the first free server.

The queue length $Q(t)$ is the number of customers in the system at time $t$, waiting or in service.
Queueing theory studies such models and their networks.

## Remarks

The letter $M$ stands for an exponential law, which makes the model Markov, and $D$ for a deterministic law.
With mean gap $1/\lambda$ and mean service time $1/\mu$, the traffic intensity is $\rho = \lambda / (c \mu)$.
Example: the [[M/M/1 queue|GTFRPF5J2K]] takes $A$ exponential of rate $\lambda$, $B$ exponential of rate $\mu$ and $c = 1$; then $Q$ is a birth–death process.
Non-example: if customers who find the server busy leave at once, the model is a loss system, not an $A/B/c$ queue, since clause 3 asks for an unlimited waiting room.
For the $A/B/1$ queue with finite means, $\rho < 1$ ensures that the waiting times converge in law to a finite limit, and $\rho > 1$ makes them tend to infinity.