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This paper considers the subexponential asymptotics of the stationarydistributions of GI/G/1-type Markov chains in two cases: (i) the phasetransition matrix in non-boundary levels is stochastic; and (ii) it is strictlysubstochastic. For the case (i), we present a weaker sufficient condition forthe sub exponential asymptotics than those given in the literature. As for thecase (ii), the subexponential asymptotics has not been studied, as far as weknow. We show that the subexponential asymptotics in the case (ii) is differentfrom that in the case (i). We also study the locally subexponential asymptoticsof the stationary distributions in both cases (i) and (ii).

Subexponential Asymptotics of the Stationary Distributions of GI/G/1-Type Markov Chains