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Matrix geometric method

Method of analysis in probability theory

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Source updated March 10, 2026

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 matrix has a repetitive block structure. The method was developed "largely by Marcel F. Neuts and his students starting around 1975."

01Method 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 π_i

${\begin{aligned}\pi _{0}B_{00}+\pi _{1}B_{10}&=0\\\pi _{0}B_{01}+\pi _{1}A_{1}+\pi _{2}A_{0}&=0\\\pi _{1}A_{2}+\pi _{2}A_{1}+\pi _{3}A_{0}&=0\\&\vdots \\\pi _{i-1}A_{2}+\pi _{i}A_{1}+\pi _{i+1}A_{0}&=0\\&\vdots \\\end{aligned}}$

Observe that the relationship

$\pi _{i}=\pi _{1}R^{i-1}$

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

${\begin{aligned}{\begin{pmatrix}\pi _{0}&\pi _{1}\end{pmatrix}}{\begin{pmatrix}B_{00}&B_{01}\\B_{10}&A_{1}+RA_{0}\end{pmatrix}}={\begin{pmatrix}0&0\end{pmatrix}}\end{aligned}}$

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

02Computation of R

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

03Matrix 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 π_i = π₁ R^{i , 1} used above holds.

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Sources and credits

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