  The main contribution of this paper is to present a new sufficient condition
for the subexponential asymptotics of the stationary distribution of a
GI/GI/1-type Markov chain without jumps from level "infinity" to level zero.
For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains
without disasters} because they are often used to analyze semi-Markovian queues
without "disasters", which are negative customers who remove all the customers
in the system (including themselves) on their arrivals. In this paper, we
demonstrate the application of our main result to the stationary queue length
distribution in the standard BMAP/GI/1 queue. Thus we obtain new asymptotic
formulas and prove the existing formulas under weaker conditions than those in
the literature. In addition, applying our main result to a single-server queue
with Markovian arrivals and the $(a,b)$-bulk-service rule (i.e., MAP/${\rm
GI}^{(a,b)}$/1 queue), we obatin a subexponential asymptotic formula for the
stationary queue length distribution.
