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."
Contents
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 πi
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 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.