  We consider a model for transitory queues in which only a finite number of
customers can join. The queue thus operates over a finite time horizon. In this
system, also known as the $\Delta_{(i)}/G/1$ queue, the customers decide
independently when to join the queue by sampling their arrival time from a
common distribution. We prove that, when the queue satisfies a certain
heavy-traffic condition and under the additional assumption that the second
moment of the service time is finite, the rescaled queue length process
converges to a reflected Brownian motion with parabolic drift. Our result holds
for general arrival times, thus improving on an earlier result which assumes
exponential arrival times.
