Transition rate matrix
In probability theory, a transition rate 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 element qij denotes the rate departing from i and arriving in state j. Diagonal elements qii are defined such that
and therefore the rows of the matrix sum to zero.Definition
A Q matrix satisfies the following conditions
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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
