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Zbl 1076.60081
Li, Quan-Lin; Zhao, Yiqiang Q.
Heavy-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type. (English)
[J] Adv. Appl. Probab. 37, No. 2, 482-509 (2005). ISSN 0001-8678

The paper presents a novel approach to analyze the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type. The use of the R-measure, the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation are the keys to this approach. The main contributions are threefold. First, some properties of heavy-tailed sequences of nonnegative scalars are extended to matrix form for sequences of nonnegative matrices. Second, the paper provides a necessary and sufficient condition under which the stationary probability vector is heavy-tailed. Third, the long-tailed asymptotics of the R-measure is derived in terms of the RG-factorization of the repeating matrix sequence and a Wiener-Hopf equation for the boundary matrix sequence.
[Oleg K. Zakusilo (Ky{\"\i}v)]

MSC 2000:: *60K25 Queueing theory
60K15 Markov renewal processes
60J22 Computational methods in Markov chains
90B22 Queues and service

Keywords: heavy tail; subexponentiality

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