Heavy-traffic asymptotics for stationary GI/G/1-type Markov chains

Asmussen (1987) proved a heavy-traffic limit theorem for the GI/G/1-type Markov chain. As a corollary, the theorem shows that in the heavy-traffic limit, the stationary distribution of the properly scaled level variable is geometric and independent of the phase variable. In this paper, the heavy-traffic asymptotics of the stationary distribution is proved under a weaker condition than Asmussen’s, by the characteristic function approach.