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 $q_$ (for $i \neq j$) denotes the rate departing from $i$ and arriving in state $j$. The rates $q_ \geq 0$, and the diagonal elements $q_$ are defined such that
:$q_ = -\sum_ q_$,
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 $0 > \mathrm\ \geq 2 \min_i q_$.
* All eigenvectors $v$ with a non-zero eigenvalue fulfill $\sum_v_ = 0$.
* 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
:$Q=\begin -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &\mu & -(\mu+\lambda) & \lambda \\ &&\mu & -(\mu+\lambda) & \ddots &\\ &&&\ddots&\ddots \end.$

See also

* Stochastic matrix

References

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Markov processes

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Matrices (mathematics)