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 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 (see condition 3 in the definition section).

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

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