We consider the $Δ_{(i)}/G/1$ 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 $x^{-α}$, with $α\in (1,2)$. We consider the asymptotic regime in which the population size grows to infinity and establish that the scaled queue length process converges to an α-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 α-stable process.