In probability theory, the matrix geometric method is a method for the analysis of quasi-birthâÂÂdeath 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."
The method requires a transition rate matrix with tridiagonal block structure as follows
where each of B<sub>00</sub>, B<sub>01</sub>, B<sub>10</sub>, A<sub>0</sub>, A<sub>1</sub> and A<sub>2</sub> are matrices. To compute the stationary distribution ÃÂ writing ÃÂ Q = 0 the balance equations are considered for sub-vectors ÃÂ<sub>i</sub>
Observe that the relationship
holds where R is the Neuts' rate matrix, which can be computed numerically. Using this we write
which can be solve to find ÃÂ<sub>0</sub> and ÃÂ<sub>1</sub> and therefore iteratively all the ÃÂ<sub>i</sub>.
The matrix R can be computed using cyclic reduction or logarithmic reduction.
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 ÃÂ<sub>i</sub> = ÃÂ<sub>1</sub> R<sup>i – 1</sup> used above holds.