  In the literature, retrial queues with batch arrivals and heavy service times
have been studied and the so-called equivalence theorem has been established
under the condition that the service time is heavier than the batch size. The
equivalence theorem provides the distribution (or tail) equivalence between the
total number of customers in the system for the retrial queue and the total
number of customers in the corresponding standard (non-retrial) queue. In this
paper, under the assumption of regularly varying tails, we eliminate this
condition by allowing that the service time can be either heavier or lighter
than the batch size. The main contribution made in this paper is an asymptotic
characterization of the difference between two tail probabilities: the
probability of the total number of customers in the system for the $M^X/G/1$
retrial queue and the probability of the total number of customers in the
corresponding standard (non-retrial) queue. The equivalence theorem by allowing
a heavier batch size is another contribution in this paper.
