  The main contribution of this paper is to prove the subexponential tail
equivalence of the stationary queue length distributions in the BMAP/GI/1
queues with and without retrials. We first present a
stochastic-decomposition-like result of the stationary queue length in the
BMAP/GI/1 retrial queue, which is an extension of the stochastic decomposition
of the stationary queue length in the M${}^X$/GI/1 retrial queue. The
stochastic-decomposition-like result shows that the stationary queue length
distribution in the BMAP/GI/1 retrial queue is decomposed into two parts: the
stationary conditional queue length distribution given that the server is idle;
and a certain matrix sequence associated with the stationary queue length
distribution in the corresponding standard BMAP/GI/1 queue (without retrials).
Using the stochastic-decomposition-like result and matrix analytic methods, we
prove the subexponential tail equivalence of the stationary queue length
distributions in the BMAP/GI/1 queues with and without retrials. This tail
equivalence result does not necessarily require that the size of an arriving
batch is light-tailed, unlike Yamamuro's result for the M${}^X$/GI/1 retrial
queue (Queueing Syst. 70:187--205, 2012). As a by-product, the key lemma to the
roof of the main theorem presents a subexponential asymptotic formula for the
stationary distribution of a level-dependent M/G/1-type Markov chain, which is
the first reported result on the subexponential asymptotics of level-dependent
block-structured Markov chains.
