Transition-rate matrix explained

In probability theory, a transition-rate matrix (also known as a Q-matrix, intensity matrix,^[1] or infinitesimal generator matrix^[2]) 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 ^[3]), element q _{ij} (for i ≠ 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 ≠} 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:^[4]

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of

Q is strongly connected.

• All other eigenvalues

λ fulfill 0 > Re \{ λ \} \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} -λ & λ \\ \mu & - (\mu +λ ) & λ \\ &\mu & - (\mu +λ ) & λ \\ &&\mu & - (\mu +λ ) & \ddots &\\ &&&\ddots&\ddots \end{pmatrix}.

See also

• Stochastic matrix

References

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

Notes and References

1. Book: Asmussen, S. R. . 10.1007/0-387-21525-5_2 . Markov Jump Processes . Applied Probability and Queues . Stochastic Modelling and Applied Probability . 51 . 39–59 . 2003 . 978-0-387-00211-8 .
2. Book: Trivedi . K. S. . Kulkarni . V. G. . FSPNs: Fluid stochastic Petri nets . 10.1007/3-540-56863-8_38 . Application and Theory of Petri Nets 1993 . Lecture Notes in Computer Science . 691 . 24 . 1993 . 978-3-540-56863-6 .
3. Rubino . Gerardo . Sericola . Bruno . 1989 . Sojourn Times in Finite Markov Processes . Journal of Applied Probability . 26 . 4 . 744–756 . Applied Probability Trust . 10.2307/3214379 . 3214379 . 54623773 .
4. Keizer. Joel. 1972-11-01. On the solutions and the steady states of a master equation. Journal of Statistical Physics. en. 6. 2. 67–72. 10.1007/BF01023679. 1972JSP.....6...67K . 120377514 . 1572-9613.