Subexponential Asymptotics of the Stationary Distributions of   GI/G/1-Type Markov Chains
  This paper considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For the case (i), we present a weaker sufficient condition for the sub exponential asymptotics than those given in the literature. As for the case (ii), the subexponential asymptotics has not been studied, as far as we know. We show that the subexponential asymptotics in the case (ii) is different from that in the case (i). We also study the locally subexponential asymptotics of the stationary distributions in both cases (i) and (ii). 
Tatsuaki Kimura, Hiroyuki Masuyama, Yutaka Takahashi
math.PR