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Heavy-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type. (English) Zbl 1076.60081
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.
60K25Queueing theory
60K15Markov renewal processes
60J22Computational methods in Markov chains
90B22Queues and service (optimization)