# Transition rate matrix

In probability theory, a transition rate matrix (also known as an intensity matrix or infinitesimal generator matrix) is an array of numbers describing the instantaneous rate at which a continuous time Markov chain transitions between states.

In a transition rate matrix Q (sometimes written A) element qij (for i ≠ j) denotes the rate departing from i and arriving in state j. Diagonal elements qii are defined such that

$q_{ii}=-\sum _{j\neq i}q_{ij}.$ and therefore the rows of the matrix sum to zero (see condition 3 in the definition section).

## Definition

A transition rate matrix $Q$ satisfies the following conditions

1. $0\leq -q_{ii}<\infty$ 2. $0\leq q_{ij}:\mathrm {for} \;i\neq j$ 3. $\sum _{j}q_{ij}=0:\mathrm {for} \;\mathrm {all} \;i$ Up to a global sign, a large class of examples of such matrices is provided by the Laplacian of a directed, weighted graph. The vertices of the graph correspond to the Markov chain's states.

## Properties

The transition rate matrix has following properties:

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of $Q$ is strongly connected.
• All other eigenvalues $\lambda$ fulfill $0>\mathrm {Re} \{\lambda \}\geq 2\min _{i}q_{ii}$ .
• All eigenvectors $v$ with a non-zero eigenvalue fulfill $\sum _{i}v_{i}=0$ .

## Example

An M/M/1 queue, a model which counts the number of jobs in a queueing system with arrivals at rate λ and services at rate μ, has transition rate matrix

$Q={\begin{pmatrix}-\lambda &\lambda \\\mu &-(\mu +\lambda )&\lambda \\&\mu &-(\mu +\lambda )&\lambda \\&&\mu &-(\mu +\lambda )&\lambda &\\&&&&\ddots \end{pmatrix}}.$ 