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Light-tailed asymptotics of the stationary probability vectors of Markov chains of GI/G/1 type. (English) Zbl 1099.60061
The paper deals with the light-tailed asymptotic behavior of stationary probability vectors of block-structured Markov chains. It presents a novel approach to evaluating the light-tailed asymptotics by means of the RG-factorization of both the repeating blocks and the Wiener-Hopf equations for the boundary blocks of the transition probability matrix, the RG-factorization plays a role similar to that played by the Wiener-Hopf factorization in analyzing waiting times. The stationary probability vector is partitioned into vectors $\left({\pi }_{0},{\pi }_{1},{\pi }_{2},...\right)$, $\left\{{\pi }_{k}\right\}$ are expressed in terms of the R-measure and, finally, in terms of the blocks in the transition probability matrix of GI/G/1 type. This expression can be used to show that $\left\{{\pi }_{k}\right\}$ is light-tailed under certain condition. The paper explicitly presents the tail-asymptotics of $\left\{{\pi }_{k}\right\}$. There are defined three classes of sequences of nonnegative matrices, two of them exhibit light-tailed asymptotics, the third heavy-tailed one. The classification of $\left\{{\pi }_{k}\right\}$ is discussed in terms of the classification of the repeating row and the boundary row.
##### MSC:
 60K25 Queueing theory 60K15 Markov renewal processes 60J22 Computational methods in Markov chains 90B22 Queues and service (optimization)
##### Keywords:
asymptotic analysis