# Transition-rate matrix

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{{Use American English|date = January 2019}}
{{Short description|Matrix describing continuous-time Markov chains}}
In [probability theory](/source/probability_theory), a '''transition-rate matrix''' (also known as a '''Q-matrix''',{{sfn|Suhov|Kelbert|2008|loc=Definition 2.1.1}} '''intensity matrix''',<ref>{{Cite book | first = S. R. | last = Asmussen| doi = 10.1007/0-387-21525-5_2 | chapter = Markov Jump Processes | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 39–59 | year = 2003 | isbn = 978-0-387-00211-8 }}</ref> or '''[infinitesimal generator](/source/Infinitesimal_generator_(stochastic_processes)) matrix'''<ref>{{Cite book | last1 = Trivedi | first1 = K. S. | last2 = Kulkarni | first2 = V. G. | chapter = FSPNs: Fluid stochastic Petri nets | doi = 10.1007/3-540-56863-8_38 | title = Application and Theory of Petri Nets 1993 | series = Lecture Notes in Computer Science | volume = 691 | pages = 24 | year = 1993 | isbn = 978-3-540-56863-6 }}</ref>) is an array of numbers describing the instantaneous rate at which a [continuous-time Markov chain](/source/continuous-time_Markov_chain) transitions between states.

In a transition-rate matrix <math>Q</math> (sometimes written <math>A</math><ref>{{cite journal | last1 = Rubino | first1 = Gerardo | last2 =Sericola | first2 = Bruno | year = 1989 | title = Sojourn Times in Finite Markov Processes | journal = Journal of Applied Probability | volume = 26 | issue = 4 | pages = 744–756 | publisher = Applied Probability Trust | doi = 10.2307/3214379 | jstor = 3214379 | s2cid = 54623773 | url = https://hal.inria.fr/inria-00075739/file/RR-0812.pdf }}</ref>), element <math>q_{ij}</math> (for <math>i \neq j</math>) denotes the rate departing from <math>i</math> and arriving in state <math>j</math>. The rates <math>q_{ij} \geq 0</math>, and the diagonal elements <math>q_{ii}</math> are defined such that 
:<math>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](/source/Laplacian_matrix). The vertices of the graph correspond to the Markov chain's states.

==Properties==
The transition-rate matrix has following properties:<ref>{{Cite journal|last=Keizer|first=Joel|date=1972-11-01|title=On the solutions and the steady states of a master equation|url=https://doi.org/10.1007/BF01023679|journal=Journal of Statistical Physics|language=en|volume=6|issue=2|pages=67–72|doi=10.1007/BF01023679|bibcode=1972JSP.....6...67K |s2cid=120377514 |issn=1572-9613|url-access=subscription}}</ref> 
* There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of <math>Q</math> is strongly connected. 
* All other eigenvalues <math>\lambda</math> fulfill <math> 0 > \mathrm{Re}\{\lambda\} \geq 2 \min_i q_{ii}</math>.
* All eigenvectors <math>v</math> with a non-zero eigenvalue fulfill <math>\sum_{i}v_{i} = 0</math>.
* The Transition-rate matrix satisfies the relation <math>Q=P'(0)</math> where P(t) is the continuous [stochastic matrix](/source/stochastic_matrix).

==Example==
An [M/M/1 queue](/source/M%2FM%2F1_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>Q=\begin{pmatrix}
-\lambda & \lambda \\
\mu & -(\mu+\lambda) & \lambda \\
&\mu & -(\mu+\lambda) & \lambda \\
&&\mu & -(\mu+\lambda) & \ddots &\\
&&&\ddots&\ddots
\end{pmatrix}.</math>

== See also ==

* [Stochastic matrix](/source/Stochastic_matrix)

==References==
{{Reflist}}
* {{cite book|title=Markov Chains|year=1997|isbn=9780511810633|doi=10.1017/CBO9780511810633.005|last1=Norris|first1=J. R.|author-link1=James R. Norris}}
* {{cite book |last1=Suhov |first1=Yuri |last2=Kelbert |first2=Mark |title=Markov chains: a primer in random processes and their applications |date=2008 |publisher=Cambridge University Press}}
* {{cite book | last1 = Syski | first1 = R.  | year = 1992 | title = Passage Times for Markov Chains | publisher =  IOS Press | isbn = 90-5199-060-X | url = https://www.iospress.com/catalog/books/passage-times-for-markov-chains }}

Category:Markov processes

{{probability-stub}}
Category:Matrices (mathematics)

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