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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.

60K25 Queueing theory (aspects of probability theory)
Full Text: DOI
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