- Source: Transition-rate matrix
In probability theory, a transition-rate matrix (also known as a Q-matrix, intensity matrix, or infinitesimal generator matrix) 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
{\displaystyle Q}
(sometimes written
A
{\displaystyle A}
), element
q
i
j
{\displaystyle q_{ij}}
(for
i
≠
j
{\displaystyle i\neq j}
) denotes the rate departing from
i
{\displaystyle i}
and arriving in state
j
{\displaystyle j}
. The rates
q
i
j
≥
0
{\displaystyle q_{ij}\geq 0}
, and the diagonal elements
q
i
i
{\displaystyle q_{ii}}
are defined such that
q
i
i
=
−
∑
j
≠
i
q
i
j
{\displaystyle q_{ii}=-\sum _{j\neq i}q_{ij}}
,
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:
There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of
Q
{\displaystyle Q}
is strongly connected.
All other eigenvalues
λ
{\displaystyle \lambda }
fulfill
0
>
R
e
{
λ
}
≥
2
min
i
q
i
i
{\displaystyle 0>\mathrm {Re} \{\lambda \}\geq 2\min _{i}q_{ii}}
.
All eigenvectors
v
{\displaystyle v}
with a non-zero eigenvalue fulfill
∑
i
v
i
=
0
{\displaystyle \sum _{i}v_{i}=0}
.
The Transition-rate matrix satisfies the relation
Q
=
P
′
(
0
)
{\displaystyle Q=P'(0)}
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
Q
=
(
−
λ
λ
μ
−
(
μ
+
λ
)
λ
μ
−
(
μ
+
λ
)
λ
μ
−
(
μ
+
λ
)
⋱
⋱
⋱
)
.
{\displaystyle Q={\begin{pmatrix}-\lambda &\lambda \\\mu &-(\mu +\lambda )&\lambda \\&\mu &-(\mu +\lambda )&\lambda \\&&\mu &-(\mu +\lambda )&\ddots &\\&&&\ddots &\ddots \end{pmatrix}}.}
See also
Stochastic matrix
References
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.
Kata Kunci Pencarian:
- Fosforilasi oksidatif
- Karbon
- Transition-rate matrix
- Stochastic matrix
- Markov chain
- Continuous-time Markov chain
- Master equation
- Laplacian matrix
- M/M/c queue
- Markovian arrival process
- M/M/1 queue
- List of statistics articles
