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 {\displaystyle Q} (sometimes written A {\displaystyle A}), element q i j {\displaystyle q_{ij}} (for i ≠ j {\displaystyle i\neq j}) denotes the rate departing from i {\displaystyle i} and arriving in state j {\displaystyle j}. The rates q i j ≥ 0 {\displaystyle q_{ij}\geq 0}, and the diagonal elements q i i {\displaystyle q_{ii}} are defined such that

q i i = − ∑ j ≠ i q i j {\displaystyle 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 {\displaystyle Q} is strongly connected.
  • All other eigenvalues λ {\displaystyle \lambda } fulfill 0 > R e { λ } ≥ 2 min i q i i {\displaystyle 0>\mathrm {Re} \{\lambda \}\geq 2\min _{i}q_{ii}}.
  • All eigenvectors v {\displaystyle v} with a non-zero eigenvalue fulfill ∑ i v i = 0 {\displaystyle \sum _{i}v_{i}=0}.
  • The Transition-rate matrix satisfies the relation Q = P ′ ( 0 ) {\displaystyle 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 = ( − λ λ μ − ( μ + λ ) λ μ − ( μ + λ ) λ μ − ( μ + λ ) ⋱ ⋱ ⋱ ) . {\displaystyle Q={\begin{pmatrix}-\lambda &\lambda \\\mu &-(\mu +\lambda )&\lambda \\&\mu &-(\mu +\lambda )&\lambda \\&&\mu &-(\mu +\lambda )&\ddots &\\&&&\ddots &\ddots \end{pmatrix}}.}

See also

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