  This paper studies the asymptotic behavior of the steady-state waiting time,
W_infty, of the M/G/1 queue with subexponenential processing times for
different combinations of traffic intensities and overflow levels. In
particular, we provide insights into the regions of large deviations where the
so-called heavy traffic approximation and heavy tail asymptotic hold. For
queues whose service time distribution decays slower than e^{-sqrt{t}} we
identify a third region of asymptotics where neither the heavy traffic nor the
heavy tailed approximations are valid. These results are obtained by deriving
approximations for P(W_infty > x) that are either uniform in the traffic
intensity as the tail value goes to infinity or uniform on the positive axis as
the traffic intensity converges to one. Our approach makes clear the connection
between the asymptotic behavior of the steady-state waiting time distribution
and that of an associated random walk.
