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An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. (English) Zbl 0896.60061

Summary: Let \(A_k\), \(k\geq 0\), be a sequence of \(m\times m\) nonnegative matrices and let \(A(z)= \sum^\infty_{k=0} A_kz^k\) be such that \(A(1)\) is an irreducible stochastic matrix. The unique power-bounded solution of the nonlinear matrix equation \(G= \sum^\infty_{k=0} A_kG^k\) has been shown to play a key role in the analysis of Markov chains of M/G/1 type. Assuming that the matrix \(A(z)\) is rational, we show that the solution of this matrix equation reduces to finding an invariant subspace of a certain matrix. We present an iterative method for computing this subspace which is globally convergent. Moreover, the method can be implemented with quadratic or higher convergence rate matrix sign function iterations, which brings in a new dimension to the analysis of M/G/1 type Markov chains for which the existing algorithms may suffer from low linear convergence rates. The method can be viewed as a “bridge” between the matrix analytic methods and transform techniques whereas it circumvents the requirements for a large number of iterations which may be encountered in the methods of the former type and the root finding problem of the techniques of the latter type. Similar results are obtained for computing the unique power-summable solution of the matrix equation \(R= \sum^\infty_{k=0} R^kA_k\), which appears in the analysis of G/M/1 type Markov chains.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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