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We consider a model for transitory queues in which only a finite number ofcustomers can join. The queue thus operates over a finite time horizon. In thissystem, also known as the $\Delta_{(i)}/G/1$ queue, the customers decideindependently when to join the queue by sampling their arrival time from acommon distribution. We prove that, when the queue satisfies a certainheavy-traffic condition and under the additional assumption that the secondmoment of the service time is finite, the rescaled queue length processconverges to a reflected Brownian motion with parabolic drift. Our result holdsfor general arrival times, thus improving on an earlier result which assumesexponential arrival times.

An alternative approach to heavy-traffic limits for finite-pool queues