Newton's iteration for non-linear equations in Markov chains

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Abstract
A large number of queueing systems may be modelled as infinite Markov chains for which the transition matrix has a repetitive structure. In order to determine the stationary distribution for these Markov chains, it is necessary to find a particular solution of a non-linear matrix equation. Various iterative algorithms have been proposed to determine the matrix of interest. We consider here one particular algorithm and transform it by Newton's method. We show that Newton's algorithm is well defined and converges quadratically in the domain of interest.

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