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 rate a continuous time Markov chain moves between states.

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

$q_{ii} = -\sum_{j\neq i} q_{ij}.$

and therefore the rows of the matrix sum to zero.

Definition

A Q matrix (q_ij) satisfies the following conditions

1. $0 \leq -q_{ii} < \infty$
2. $0 \leq q_{ij} : \mathrm{for}\; i \neq j$
3. $\sum_j q_{ij} = 0 : \mathrm{for}\;\mathrm{all}\; i$

This definition can be interpreted as the Laplacian of a directed, weighted graph whose vertices correspond to the Markov chain's states.

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

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