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.
Contents
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
and therefore the rows of the matrix sum to zero.
Definition
A Q matrix (qij) satisfies the following conditions
-
0 ≤ − q i i < ∞ -
0 ≤ q i j : f o r i ≠ j -
∑ j q i j = 0 : f o r a l l 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