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 $i\ne j$) denotes the rate departing from $i$ and arriving in state $j$. The rates $q_{ij}\ge 0$, and the diagonal elements $q_{ii}$ are defined such that

$q_{ii}=-\sum _{j\ne i}{q}_{ij}$,

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 $0>\mathrm{R}\mathrm{e}\{\lambda \}\ge 2\underset{i}{\min }{q}_{ii}$.
• All eigenvectors $v$ with a non-zero eigenvalue fulfill $\sum _i{v}_i=0$.
• The Transition-rate matrix satisfies the relation $Q=P^{\prime }(0)$ 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

$Q=\left(\begin{array}{cc}-\lambda & \lambda \\ \mu & -(\mu +\lambda )& \lambda \\ & \mu & -(\mu +\lambda )& \lambda \\ & & \mu & -(\mu +\lambda )& \ddots & \\ & & & \ddots & \ddots \end{array}\right).$

See also

• Stochastic matrix