  In this paper we study the maximum queue length $M$ (in terms of the number
of customers present) in a busy cycle in the M/G/1 queue. Assume that the
service times have a logconvex density. For such (heavy-tailed) service-time
distributions the Foreground Background service discipline is optimal. This
discipline gives service to the customer(s) that have received the least amount
of service so far. It is shown that under this discipline $M$ has an
exponentially decreasing tail. From the behaviour of $M$ we obtain asymptotics
of the maximum queue length $M(t)$ over the interval $(0,t)$ for $t\to\infty$.
These are applied to calculate the time to overflow of a buffer, both in stable
and unstable queues.
