  We consider a queue to which only a finite pool of $n$ customers can arrive,
at times depending on their service requirement. A customer with stochastic
service requirement $S$ arrives to the queue after an exponentially distributed
time with mean $S^{-\alpha}$ for some $\alpha\in[0,1]$; so larger service
requirements trigger customers to join earlier. This finite-pool queue
interpolates between two previously studied cases: $\alpha = 0$ gives the
so-called $\Delta_{(i)}/G/1$ queue and $\alpha = 1$ is closely related to the
exploration process for inhomogeneous random graphs. We consider the asymptotic
regime in which the pool size $n$ grows to infinity and establish that the
scaled queue-length process converges to a diffusion process with a negative
quadratic drift. We leverage this asymptotic result to characterize the head
start that is needed to create a long period of activity. We also describe how
this first busy period of the queue gives rise to a critically connected random
forest.
