Equilibrium analysis of skip free markov chains: nonlinear matrix equations

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Abstract
Nonlinear matrix equations of the form where Fi, i=0, 1,2 ... ,m are known nxn nonnegative sub-matrices of a state transition matrix arise ubiquitously in the analysis of various stochastic models utilized in queueing, inventory and communication theories. Computation of the minimal nonnegative solution of (1), called the rate matrix R′, is essential for the equilibrium analysis of these models. Previously, this matrix has been computed by iterative techniques. In this paper, an analytical method for the computation of the rate matrix is proposed. Specifically, the eigenvalues of the rate matrix are determined through a Localization Theorem. The eigenvectors and generalized left eigenvectors are then determined to arrive at the Jordan canonical form representation of the rate matrix. Also, the problem of computation of the rate matrix is formulated as a linear programming problem. This method can be adapted in a natural manner for the equilibrium analysis of M/G/l type Markov chains.

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