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 instantaneous rate at which a continuous time Markov chain transitions 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 (see condition 3 in the definition section).

Definition

A transition rate matrix ${\displaystyle Q}$ satisfies the following conditions[5]

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

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:[6]

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of ${\displaystyle Q}$ is strongly connected.
• All other eigenvalues ${\displaystyle \lambda }$ fulfill ${\displaystyle 0>\mathrm {Re} \{\lambda \}\geq 2\min _{i}q_{ii}}$.
• All eigenvectors ${\displaystyle v}$ with a non-zero eigenvalue fulfill ${\displaystyle \sum _{i}v_{i}=0}$.

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. 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. Vol. 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. Vol. 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" (PDF). Journal of Applied Probability. Applied Probability Trust. 26 (4): 744–756. doi:10.2307/3214379. JSTOR 3214379. S2CID 54623773.
5. ^ Norris, J. R. (1997). Markov Chains. doi:10.1017/CBO9780511810633. ISBN 9780511810633.
6. ^ Keizer, Joel (1972-11-01). "On the solutions and the steady states of a master equation". Journal of Statistical Physics. 6 (2): 67–72. Bibcode:1972JSP.....6...67K. doi:10.1007/BF01023679. ISSN 1572-9613. S2CID 120377514.