In probability theory, a transition-rate matrix (also known as a Q-matrix,[1] intensity matrix,[2] or infinitesimal generator matrix[3]) is an array of numbers describing the instantaneous rate at which a continuous-time Markov chain transitions between states.
In a transition-rate matrix (sometimes written [4]), element (for ) denotes the rate departing from and arriving in state . The rates , and the diagonal elements 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:[5]
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