Matrix geometric method

In probability theory, the matrix geometric method is a method for the analysis of quasi-birth–death processes, continuous-time Markov chain whose transition rate matrices with a repetitive block structure. The method was developed "largely by Marcel F. Neuts and his students starting around 1975."

Method description
The method requires a transition rate matrix with tridiagonal block structure as follows


 * $$Q=\begin{pmatrix}

B_{00} & B_{01} \\ B_{10} & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& A_0 & A_1 & A_2 \\ &&&& \ddots & \ddots & \ddots \end{pmatrix}$$

where each of B00, B01, B10, A0, A1 and A2 are matrices. To compute the stationary distribution π writing π Q = 0 the balance equations are considered for sub-vectors πi


 * $$\begin{align}

\pi_0 B_{00} + \pi_1 B_{10} &= 0\\ \pi_0 B_{01} + \pi_1 A_1 + \pi_2 A_0 &= 0\\ \pi_1 A_2 + \pi_2 A_1 + \pi_3 A_0 &= 0 \\ & \vdots \\ \pi_{i-1} A_2 + \pi_i A_1 + \pi_{i+1} A_0 &= 0\\ & \vdots \\ \end{align}$$

Observe that the relationship


 * $$\pi_i = \pi_1 R^{i-1}$$

holds where R is the Neut's rate matrix, which can be computed numerically. Using this we write


 * $$\begin{align}

\begin{pmatrix}\pi_0 & \pi_1 \end{pmatrix} \begin{pmatrix}B_{00} & B_{01} \\ B_{10} & A_1 + RA_0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \end{pmatrix} \end{align}$$

which can be solve to find π0 and π1 and therefore iteratively all the πi.

Computation of R
The matrix R can be computed using cyclic reduction or logarithmic reduction.

Matrix analytic method
The matrix analytic method is a more complicated version of the matrix geometric solution method used to analyse models with block M/G/1 matrices. Such models are harder because no relationship like πi = π1 Ri – 1 used above holds.