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 $$Q$$ (sometimes written $$A$$ ), element $$q_{ij}$$ (for $$i \neq j$$) denotes the rate departing from $$i$$ and arriving in state $$j$$. The rates $$q_{ij} \geq 0$$, and the 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.

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 $$Q$$ is strongly connected.
 * All other eigenvalues $$\lambda$$ fulfill $$ 0 > \mathrm{Re}\{\lambda\} \geq 2 \min_i q_{ii}$$.
 * All eigenvectors $$v$$ with a non-zero eigenvalue fulfill $$\sum_{i}v_{i} = 0$$.
 * The Transition-rate matrix satisfies the relation $$Q=P'(0)$$ where P(t) is the continuous stochastic matrix.

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) & \ddots &\\ &&&\ddots&\ddots \end{pmatrix}.$$