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
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
Observe that the relationship
holds where R is the Neut's rate matrix, which can be computed numerically. Using this we write
which can be solve to find π₀ and π₁ and therefore iteratively all the π_i.

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 π_i = π₁ R^{i - 1} used above holds.