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 qij (for i ≠ j) denotes the rate departing from i and arriving in state j. Diagonal elements qii are defined such that

${\displaystyle q_{ii}=-\sum _{j\neq i}q_{ij}.}$

and therefore the rows of the matrix sum to zero.

Definition

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

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

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

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

References

1. ^ Syski, R. (1992). Passage Times for Markov Chains. IOS Press. ISBN 90-5199-060-X. doi:10.3233/978-1-60750-950-9-i.
2. ^ Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 39–59. ISBN 978-0-387-00211-8. doi:10.1007/0-387-21525-5_2.
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. ISBN 978-3-540-56863-6. doi:10.1007/3-540-56863-8_38.
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". ISBN 9780511810633. doi:10.1017/CBO9780511810633.