en>SchreiberBike: You can help. Repaired link to the disambiguation page, Convergence.

2013-12-07T00:23:41Z

You can help. Repaired link to the disambiguation page, Convergence.

New page

In [[probability theory]], a '''transition rate matrix''' (also known as an '''intensity matrix'''<ref>{{cite doi|10.3233/978-1-60750-950-9-i}}</ref><ref>{{cite doi|10.1007/0-387-21525-5_2}}</ref> or '''[[Infinitesimal generator (stochastic processes)|infinitesimal generator]] matrix'''<ref>{{cite doi|10.1007/3-540-56863-8_38}}</ref>) 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''<ref>{{cite jstor|3214379}}</ref>) 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
:<math>q_{ii} = -\sum_{j\neq i} q_{ij}.</math>
and therefore the rows of the matrix sum to zero.

==Definition==

A ''Q'' matrix (''q''''ij'') satisfies the following conditions<ref>{{cite doi|10.1017/CBO9780511810633}}</ref>

# 0 ≤ -''q''''ii'' ≤ ∞
# 0 ≤ ''q''''ij'' for all ''i'' ≠ ''j''
# <math style="vertical-align:-120%;">\sum_j q_{ij} = 0</math> for all ''i''.

==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>Q=\begin{pmatrix}
-\lambda & \lambda \\
\mu & -(\mu+\lambda) & \lambda \\
&\mu & -(\mu+\lambda) & \lambda \\
&&\mu & -(\mu+\lambda) & \lambda &\\
&&&&\ddots
\end{pmatrix}.</math>

==References==

{{Reflist}}

{{probability-stub}}

[[Category:Stochastic processes]]

False diffusion - Revision history

en>SchreiberBike: You can help. Repaired link to the disambiguation page, Convergence.