Matrixgeometric

Matrixgeometric refers to a class of matrix-analytic methods and the associated stationary distributions used for certain Markov chains with a repeating level structure, commonly known as quasi-birth-death (QBD) processes. In these models, the state space is divided into levels, each containing a finite set of phases. Transitions typically move between neighboring levels or within the same level, producing a block structure in the generator that enables level-wise analysis.

A key feature is the matrix-geometric form of the stationary distribution. If pi_n denotes the stationary probability vector for level n, then for all levels n ≥ 1 the relation pi_n = pi_0 R^n holds, where R is the rate matrix (a nonnegative square matrix). The matrix R encodes the geometric decay across levels. R is the minimal nonnegative solution of a matrix quadratic equation of the form R^2 A2 + R A1 + A0 = 0 (the exact arrangement of A0, A1, A2 depends on the block structure of the generator). Once R is found, the boundary vector pi_0 is determined from the level-0 equations and normalization.

Matrixgeometric methods provide efficient computation for the stationary distribution of QBD-type processes and related M/G/1-type and GI/M/1-type models. They exploit the repetitive level structure to reduce a large problem to solving for the rate matrix R and a boundary vector, enabling the analysis of queues, reliability systems, inventory models, and telecommunications networks. The approach is a cornerstone of matrix-analytic methods in stochastic processes and has been developed by researchers in the field of applied probability and operations research.