infinitesimal generator matrix

Shanshan yuan · 收录于 2023-12-06 19:59:18 · source URL

Transition rate matrix (Redirected from Infinitesimal generator matrix)

In probability theory, a transition rate matrix (also known as an 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 (sometimes written A) element q i j q_{ij} qij (for i ≠ j) denotes the rate departing from i and arriving in state j. Diagonal elements q i i q_{ii} qii are defined such that

q i i = − ∑ j ≠ i q i j . q i i = − ∑ j ≠ i q i j {\displaystyle q_{ii}=-\sum _{j\neq i}q_{ij}.}q_{ii} = -\sum_{j\neq i} q_{ij} qii=j=iqij.qii=j=iqij.
and therefore the rows of the matrix sum to zero (see condition 3 in the definition section).

Definition
A Q matrix (qij) satisfies the following conditions[5]
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This definition can be interpreted as the Laplacian of a directed, weighted graph whose vertices correspond to the Markov chain’s states.

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 ) λ ⋱ ) . {\displaystyle Q={\begin{pmatrix}-\lambda &\lambda \\\mu &-(\mu +\lambda )&\lambda \\&\mu &-(\mu +\lambda )&\lambda \\&&\mu &-(\mu +\lambda )&\lambda &\\&&&&\ddots \end{pmatrix}}.} Q=λμλ(μ+λ)μλ(μ+λ)μλ(μ+λ)λ.