Steady-State Distribution Convergence for GI/GI/1 + GI Queues in Heavy Traffic

We establish the validity of the heavy traffic steady-state approximation for a single server queue, operating under the FIFO service discipline, in which each customer abandons the system if his waiting time exceeds his generally-distributed patience time. This follows from early results of Kingman when the loading factor approaches one from below, but has not been shown in more generality. We prove the convergence of the steady-state distributions of the offered waiting time process and their moments both under the assumption that the hazard rate of the abandonment distribution is scaled and that it is not scaled. As a consequence, we establish the limit behavior of the steady-state abandonment probability and mean queue-length.