  We consider a single-server GI/GI/1 queueing system with feedback. We assume
the service times distribution to be (intermediate) regularly varying. We find
the tail asymptotics for a customer's sojourn time in two regimes: the customer
arrives in an empty system, and the customer arrives in the system in the
stationary regime. In particular, in the case of Poisson input we use the
branching processes structure and provide more precise formulae. As auxiliary
results, we find the tail asymptotics for the busy period distribution in a
single-server queue with an intermediate varying service times distribution and
establish the principle-of-a-single-big-jump equivalences that characterise the
asymptotics.
