Inprobability theory, atransition-rate matrix (also known as aQ-matrix,^[1]intensity matrix,^[2] orinfinitesimal generator matrix^[3]) is an array of numbers describing the instantaneous rate at which acontinuous-time Markov chain transitions between states.

In a transition-rate matrix${\displaystyle Q}$ (sometimes written${\displaystyle A}$^[4]), element${\displaystyle q_{ij}}$ (for${\displaystyle i\ne j}$) denotes the rate departing from${\displaystyle i}$ and arriving in state${\displaystyle j}$. The rates${\displaystyle q_{ij}\ge 0}$, and the diagonal elements${\displaystyle q_{ii}}$ are defined such that

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

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 theLaplacian 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]

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of${\displaystyle Q}$ is strongly connected.
• All other eigenvalues${\displaystyle \lambda }$ fulfill${\displaystyle 0>\mathrm{R}\mathrm{e}\{\lambda \}\ge 2\underset{i}{min}{q}_{ii}}$.
• All eigenvectors${\displaystyle v}$ with a non-zero eigenvalue fulfill${\displaystyle \sum _i{v}_i=0}$.
• The Transition-rate matrix satisfies the relation${\displaystyle Q=P^{\prime }(0)}$ where P(t) is the continuousstochastic matrix.

AnM/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

• Stochastic matrix

• Norris, J. R. (1997).Markov Chains.doi:10.1017/CBO9780511810633.005.ISBN 9780511810633.
• Suhov, Yuri; Kelbert, Mark (2008).Markov chains: a primer in random processes and their applications. Cambridge University Press.
• Syski, R. (1992).Passage Times for Markov Chains. IOS Press.ISBN 90-5199-060-X.

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