Refined tail asymptotic properties for the $$M^X/G/1$$ retrial queue

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