Transition Rate Matrices
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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, 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 A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a ...
transitions between states. In a transition-rate matrix Q (sometimes written A), element q_ (for i \neq j) denotes the rate departing from i and arriving in state j. The rates q_ \geq 0, and the diagonal elements q_ are defined such that :q_ = -\sum_ q_, 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 is strongly connected. * All other eigenvalues \lambda fulfill 0 > \mathrm\ \geq 2 \min_i q_. * All eigenvectors v with a non-zero eigenvalue fulfill \sum_v_ = 0. * The Transition-rate matrix satisfies the relation Q=P'(0) where P(t) is the continuous
stochastic matrix In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ''s ...
.


Example

An
M/M/1 queue In queueing theory, a discipline within the mathematical probability theory, theory of probability, an M/M/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service ...
, 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=\begin -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &\mu & -(\mu+\lambda) & \lambda \\ &&\mu & -(\mu+\lambda) & \ddots &\\ &&&\ddots&\ddots \end.


See also

*
Stochastic matrix In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ''s ...


References

* * * Markov processes {{probability-stub Matrices (mathematics)