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

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]

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

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

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

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References

Syski, R. (1992). Passage Times for Markov Chains. IOS Press. doi:10.3233/978-1-60750-950-9-i. ISBN 90-5199-060-X.
Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 39–59. doi:10.1007/0-387-21525-5_2. ISBN 978-0-387-00211-8.
Trivedi, K. S.; Kulkarni, V. G. (1993). "FSPNs: Fluid stochastic Petri nets". Application and Theory of Petri Nets 1993. Lecture Notes in Computer Science. 691. p. 24. doi:10.1007/3-540-56863-8_38. ISBN 978-3-540-56863-6.
Rubino, Gerardo; Sericola, Bruno (1989). "Sojourn Times in Finite Markov Processes". Journal of Applied Probability. Applied Probability Trust. 26 (4): 744–756. JSTOR 3214379.
Norris, J. R. (1997). "Markov Chains". doi:10.1017/CBO9780511810633. ISBN 9780511810633. Cite journal requires |journal= (help)

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[1] Syski, R. (1992). Passage Times for Markov Chains. IOS Press. doi:10.3233/978-1-60750-950-9-i. ISBN 90-5199-060-X.

[2] Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 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. 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". Journal of Applied Probability. Applied Probability Trust. 26 (4): 744–756. JSTOR 3214379.

[5] Norris, J. R. (1997). "Markov Chains". doi:10.1017/CBO9780511810633. ISBN 9780511810633. Cite journal requires |journal= (help)