  A result of Ward and Glynn (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 Ward and Glynn (2005). In comparison to
Kingman's classical result in Kingman (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
stationary behavior.
