In probability theory, a transition-rate matrix (also known as a Q-matrix, 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 ${\displaystyle Q}$ (sometimes written ${\displaystyle A}$), element ${\displaystyle q_{ij}}$ (for ${\displaystyle i\neq j}$) denotes the rate departing from ${\displaystyle i}$ and arriving in state ${\displaystyle j}$. The rates ${\displaystyle q_{ij}\geq 0}$, and the diagonal elements ${\displaystyle q_{ii}}$ are defined such that

and therefore the rows of the matrix sum to zero.

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.

The transition-rate matrix has following properties:

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