System theoretic approach to teletraffic problems. A unifying framework

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Abstract
A new theory that is suitable for efficient and reliable computation for a rich class of teletraffic problems based on Markov chains of M/G/1 and G/M/1 type has been reported by Akar et al. (see Queueing Systems, 1996 and Commun. Stat.-Stochastic Models, 1996) and the computation of bases for stable invariant subspaces of real matrices plays a key role in this approach. We provide a unifying framework based on state space representations for a set of teletraffic models some of which cannot be analyzed via the M/G/1 or G/M/1 paradigms and for which the concept of invariant subspaces is essential. Once the dynamical state equations are obtained, the problem naturally reduces to the following open-loop control problem: bring the dynamical system with some unstable modes to an initial state so that all the states remain bounded. From a system theory point of view, this problem is equivalent to posing that the initial state of the representation should lie in the stable subspace of the state matrix. An efficient solution to this problem is proposed through the matrix sign function iterations with quadratic convergence rates without the need for computing the individual eigenvalues and eigenvectors.

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