A large deviation analysis of a queueing system with general retrieval time

We consider logarithmic asymptotics of the large deviation probability in a single-server queue with Poisson input, where server, after completion of service, seeks a customer from an orbit (retrial customer) for the next service, unless new arrival captures server. This system is described by a regenerative process, and under stability assumption, the asymptotic of the probability that the number of customers in the system (indeed in the orbit) reaches a high threshold within a regeneration cycle is found. Some examples are given for particular retrieval time distributions.