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Sufficient conditions for a geometric tail in a QBD process with many countable levels and phases. (English) Zbl 1065.60133
Summary: We present sufficient conditions, under which the stationary probability vector of a QBD process with both infinite levels and phases decays geometrically, characterized by the convergence norm $$\eta$$ and the $$1/\eta$$-left-invariant vector $$x$$ of the rate matrix $$R$$. We also present a method to compute $$\eta$$ and $$x$$ based on spectral properties of the censored matrix of a matrix function constructed with the repeating blocks of the transition matrix of the QBD process. What makes this method attractive is its simplicity; finding $$\eta$$ reduces to determining the zeros of a polynomial. We demonstrate the application of our method through a few interesting examples.

##### MSC:
 60K25 Queueing theory (aspects of probability theory)
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