  We consider a single server queue that serves a finite population of $n$
customers that will enter the queue (require service) only once, also known as
the $\Delta_{(i)}/G/1$ queue. This paper presents a method for analyzing
heavy-traffic behavior by using uniform acceleration, which simultaneously lets
$n$ and the service rate grow large, while the initial resource utilization
approaches one. A key feature of the model is that, as time progresses, more
customers have joined the queue, and fewer customers can potentially join. This
diminishing population gives rise to a class of reflected stochastic processes
that vanish over time, and hence do not have a stationary distribution. We
establish that, when the arrival times are exponentially distributed, by
suitably rescaling space and time, the queue length process converges to a
Brownian motion with parabolic drift, a stochastic-process limit that captures
the effect of a diminishing population by a negative quadratic drift. When the
arrival times are generally distributed, our techniques provide information on
the typical queue length and the first busy period.
