  We calculate asymptotics of the distribution of the number of customers in
orbit in a two-class priority retrial $M/G/1$-type queueing model. In this
model, priority customers wait in line while non-priority customers join an
orbit and retry later. Although the generating function and moments of the
number of customers in orbit has been analyzed before, asymptotics of the
distribution have not been thoroughly investigated. We use singularity analysis
of the probability generating function to do just that. Our results show that
different regimes exist for these asymptotics in case of light-tailed service
times: in what we call the `priority regime', the tail asymptotics have the
same decay ($\sim cn^{-3/2}R^{-n}$) as in the priority non-retrial queue and
the retrial rate only influences the constant $c$. In the `retrial regime', the
retrial rate also influences the sub-exponential factor of the asymptotics. In
this regime, asymptotics are very similar to asymptotics in retrial queues
without (priority) waiting line. Finally, we also analyze the case that the
service time distribution is power law (with or without exponential cut-off)
using the same technique.
