  We consider the $\Delta_{(i)}/G/1$ queue, in which a a total of $n$ customers
independently demand service after an exponential time. We focus on the case of
heavy-tailed service times, and assume that the tail of the service time
distribution decays like $x^{-\alpha}$, with $\alpha \in (1,2)$. We consider
the asymptotic regime in which the population size grows to infinity and
establish that the scaled queue length process converges to an \alpha-stable
process with a negative quadratic drift. We leverage this asymptotic result to
characterize the headstart that is needed to create a long period of activity.
This result should be contrasted with the case of light-tailed service times,
which was shown to have a similar scaling limit, but then with a Brownian
motion instead of an \alpha-stable process.
