Transition rate matrix

In probability theory, a transition rate matrix (also known as an intensity matrix^[1][2] or infinitesimal generator matrix^[3]) is an array of numbers describing the rate a continuous time Markov chain moves between states.

In a transition rate matrix Q (sometimes written A^[4]) element q_ij (for i ≠ j) denotes the rate departing from i and arriving in state j. Diagonal elements q_ii are defined such that

$q_{ii} = -\sum_{j\neq i} q_{ij}.$

and therefore the rows of the matrix sum to zero.

Definition

A Q matrix (q_ij) satisfies the following conditions^[5]

1. $0 \leq -q_{ii} < \infty$
2. $0 \leq q_{ij} : \mathrm{for}\; i \neq j$
3. $\sum_j q_{ij} = 0 : \mathrm{for}\;\mathrm{all}\; i$

This definition can be interpreted as the Laplacian of a directed, weighted graph whose vertices correspond to the Markov chain's states.

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

$Q=\begin{pmatrix} -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &\mu & -(\mu+\lambda) & \lambda \\ &&\mu & -(\mu+\lambda) & \lambda &\\ &&&&\ddots \end{pmatrix}^T.$

References

1. Syski, R. (1992). Passage Times for Markov Chains. IOS Press. doi:10.3233/978-1-60750-950-9-i. ISBN 90-5199-060-X.
2. Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability 51. pp. 39–59. doi:10.1007/0-387-21525-5_2. ISBN 978-0-387-00211-8.
3. Trivedi, K. S.; Kulkarni, V. G. (1993). "FSPNs: Fluid stochastic Petri nets". Application and Theory of Petri Nets 1993. Lecture Notes in Computer Science 691. p. 24. doi:10.1007/3-540-56863-8_38. ISBN 978-3-540-56863-6.
4. Rubino, Gerardo; Sericola, Bruno (1989). "Sojourn Times in Finite Markov Processes". Journal of Applied Probability (Applied Probability Trust) 26 (4): 744–756. JSTOR 3214379.
5. Norris, J. R. (1997). "Markov Chains". doi:10.1017/CBO9780511810633. ISBN 9780511810633.