Stationary distribution convergence of the offered waiting processes for GI/ GI/ 1 + GI queues in heavy traffic

A result of Ward and Glynn (Queueing Syst 50(4):371–400, 2005) asserts that the sequence of scaled offered waiting time processes of the GI/ GI/ 1 + GI queue converges weakly to a reflected Ornstein–Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a GI/ GI/ 1 + GI queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in 2005. In comparison with Kingman’s classical result (Kingman in Proc Camb Philos Soc 57:902–904, 1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a GI / GI / 1 queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue’s stationary behavior.