Transition rate matrix

A transition rate matrix ${\displaystyle Q}$ satisfies the following conditions[5]

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

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.

The transition rate matrix has following properties:[6]

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of ${\displaystyle Q}$ is strongly connected.
• All other eigenvalues ${\displaystyle \lambda }$ fulfill ${\displaystyle 0>\mathrm {Re} \{\lambda \}\geq 2\min _{i}q_{ii}}$.
• All eigenvectors ${\displaystyle v}$ with a non-zero eigenvalue fulfill ${\displaystyle \sum _{i}v_{i}=0}$.

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

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

• Stochastic matrix