# American Institute of Mathematical Sciences

October  2017, 13(4): 2093-2146. doi: 10.3934/jimo.2017033

## Light-tailed asymptotics of GI/G/1-type Markov chains

 1 NTT Network Technology Laboratories, NTT Corporation, Tokyo 180–8585, Japan 2 Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606–8501, Japan

Received  October 2015 Revised  September 2016 Published  April 2017

This paper studies the light-tailed asymptotics of the stationary distribution of the GI/G/1-type Markov chain. We consider three cases:(ⅰ) the tail decay rate is determined by a certain parameter $\theta$ associated with the transition block matrices $\{\boldsymbol{A}(k);k=0,\pm1,\pm2,\dots\}$ in the non-boundary levels; (ⅱ) by the convergence radius of the generating function of the transition block matrices $\{\boldsymbol{B}(k);k=1,2,\dots\}$ in the boundary level; and (ⅲ) by the convergence radius of $\sum_{k=1}^{\infty}z^k \boldsymbol{A}(k)$. In the case (ⅰ), we extend the existing asymptotic formula for the M/G/1-type Markov chain to the GI/G/1-type one. In the case (ⅱ), we present general asymptotic formulas that include, as special cases, the existing results in the literature. In the case (ⅲ), we derive new asymptotic formulas. As far as we know, such formulas have not been reported in the literature.

Citation: Tatsuaki Kimura, Hiroyuki Masuyama, Yutaka Takahashi. Light-tailed asymptotics of GI/G/1-type Markov chains. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2093-2146. doi: 10.3934/jimo.2017033
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