Transition-rate matrix

Matrix describing continuous-time Markov chains

In probability theory, a transition-rate matrix (also known as a Q-matrix,[1] intensity matrix,[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 {\displaystyle Q} (sometimes written A {\displaystyle A} [4]), 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:[5]

  • 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

  1. ^ Suhov & Kelbert 2008, Definition 2.1.1.
  2. ^ Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 39–59. doi:10.1007/0-387-21525-5_2. ISBN 978-0-387-00211-8.
  3. ^ Trivedi, K. S.; Kulkarni, V. G. (1993). "FSPNs: Fluid stochastic Petri nets". Application and Theory of Petri Nets 1993. Lecture Notes in Computer Science. Vol. 691. p. 24. doi:10.1007/3-540-56863-8_38. ISBN 978-3-540-56863-6.
  4. ^ Rubino, Gerardo; Sericola, Bruno (1989). "Sojourn Times in Finite Markov Processes" (PDF). Journal of Applied Probability. 26 (4). Applied Probability Trust: 744–756. doi:10.2307/3214379. JSTOR 3214379. S2CID 54623773.
  5. ^ Keizer, Joel (1972-11-01). "On the solutions and the steady states of a master equation". Journal of Statistical Physics. 6 (2): 67–72. Bibcode:1972JSP.....6...67K. doi:10.1007/BF01023679. ISSN 1572-9613. S2CID 120377514.
  • 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.


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