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

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 \begin{document}$\theta$\end{document} associated with the transition block matrices \begin{document}$\{\boldsymbol{A}(k);k=0,\pm1,\pm2,\dots\}$\end{document} in the non-boundary levels; (ⅱ) by the convergence radius of the generating function of the transition block matrices \begin{document}$\{\boldsymbol{B}(k);k=1,2,\dots\}$\end{document} in the boundary level; and (ⅲ) by the convergence radius of \begin{document}$\sum_{k=1}^{\infty}z^k \boldsymbol{A}(k)$\end{document} . 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.