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

$q_{ii}=-\sum _{j\neq i}q_{ji}.$

and therefore the rows of the matrix sum to zero.

Definition

A Q matrix (q_ij) satisfies the following conditions

1. 0 ≤ -q_ii < ∞
2. 0 ≤ q_ij for all i ≠ j
3. $\sum _{j}q_{ji}=0$ for all i.

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

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