Transition rate matrix

In probability theory, a transition rate matrix (also known as an intensity matrix^[1][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 Q (sometimes written A^[4]) 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

${\displaystyle q_{ii}=-\sum _{j\neq i}q_{ij}.}$

and therefore the rows of the matrix sum to zero (see condition 3 in the definition section).

Definition[]

A Q matrix (q_ij) satisfies the following conditions^[5]

1. ${\displaystyle 0\leq -q_{ii}<\infty }$
2. ${\displaystyle 0\leq q_{ij}:\mathrm {for} \;i\neq j}$
3. ${\displaystyle \sum _{j}q_{ij}=0:\mathrm {for} \;\mathrm {all} \;i}$

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

${\displaystyle Q={\begin{pmatrix}-\lambda &\lambda \\\mu &-(\mu +\lambda )&\lambda \\&\mu &-(\mu +\lambda )&\lambda \\&&\mu &-(\mu +\lambda )&\lambda &\\&&&&\ddots \end{pmatrix}}.}$

References[]

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