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."
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 REdit
Matrix analytic methodEdit
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.
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