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Zbl 1099.60061
Li, Quan-Lin; Zhao, Yiqiang Q.
Light-tailed asymptotics of the stationary probability vectors of Markov chains of GI/G/1 type.
(English)
[J] Adv. Appl. Probab. 37, No. 4, 1075-1093 (2005). ISSN 0001-8678

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 $(\pi _0,\pi _1,\pi _2,\ldots )$, $\{\pi _k\}$ 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 $\{\pi _k\}$ is light-tailed under certain condition. The paper explicitly presents the tail-asymptotics of $\{\pi _k\}$. 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 $\{\pi _k\}$ is discussed in terms of the classification of the repeating row and the boundary row.
[Laszlo Lakatos (Budapest)]
MSC 2000:
*60K25 Queueing theory
60K15 Markov renewal processes
60J22 Computational methods in Markov chains
90B22 Queues and service

Keywords: asymptotic analysis

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