Transition-rate matrix
In probability theory, a transition-rate matrix (also known as a Q-matrix,[1] intensity matrix,[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 [math]\displaystyle{ Q }[/math] (sometimes written [math]\displaystyle{ A }[/math][4]), element [math]\displaystyle{ q_{ij} }[/math] (for [math]\displaystyle{ i \neq j }[/math]) denotes the rate departing from [math]\displaystyle{ i }[/math] and arriving in state [math]\displaystyle{ j }[/math]. The rates [math]\displaystyle{ q_{ij} \geq 0 }[/math], and the diagonal elements [math]\displaystyle{ q_{ii} }[/math] are defined such that
- [math]\displaystyle{ q_{ii} = -\sum_{j\neq i} q_{ij} }[/math],
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:[5]
- There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of [math]\displaystyle{ Q }[/math] is strongly connected.
- All other eigenvalues [math]\displaystyle{ \lambda }[/math] fulfill [math]\displaystyle{ 0 \gt \mathrm{Re}\{\lambda\} \geq 2 \min_i q_{ii} }[/math].
- All eigenvectors [math]\displaystyle{ v }[/math] with a non-zero eigenvalue fulfill [math]\displaystyle{ \sum_{i}v_{i} = 0 }[/math].
- The Transition-rate matrix satisfies the relation [math]\displaystyle{ Q=P'(0) }[/math] 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
- [math]\displaystyle{ Q=\begin{pmatrix} -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &\mu & -(\mu+\lambda) & \lambda \\ &&\mu & -(\mu+\lambda) & \ddots &\\ &&&\ddots&\ddots \end{pmatrix}. }[/math]
See also
References
- ↑ Suhov & Kelbert 2008, Definition 2.1.1.
- ↑ 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.
- ↑ 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. pp. 24. doi:10.1007/3-540-56863-8_38. ISBN 978-3-540-56863-6.
- ↑ Rubino, Gerardo; Sericola, Bruno (1989). "Sojourn Times in Finite Markov Processes". Journal of Applied Probability (Applied Probability Trust) 26 (4): 744–756. doi:10.2307/3214379. https://hal.inria.fr/inria-00075739/file/RR-0812.pdf.
- ↑ Keizer, Joel (1972-11-01). "On the solutions and the steady states of a master equation" (in en). Journal of Statistical Physics 6 (2): 67–72. doi:10.1007/BF01023679. ISSN 1572-9613. Bibcode: 1972JSP.....6...67K. https://doi.org/10.1007/BF01023679.
- Norris, J. R. (1997). Markov Chains. doi:10.1017/CBO9780511810633.005. ISBN 9780511810633.
- Suhov, Yuri; Kelbert, Mark (2008). Markov chains: a primer in random processes and their applications. Cambridge University Press.
- Syski, R. (1992). Passage Times for Markov Chains. IOS Press. ISBN 90-5199-060-X. https://www.iospress.com/catalog/books/passage-times-for-markov-chains.
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