Transition-rate matrix
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  (sometimes written ), 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. Properties
The transition-rate matrix has following properties: 
 There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of  is strongly connected. All other eigenvalues  fulfill . All eigenvectors  with a non-zero eigenvalue fulfill . 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

See also 

 Stochastic matrix

References

 
 
 

Markov processes

Matrices