We consider a single-server queue that is initially empty and operates under the first-in–first-out service discipline. In this system, delays (waiting times in queue) experienced by subsequent arriving customers form a transient process. We investigate its transient behavior by constructing a sample-path coupling of the transient and a general (delayed) processes. From the coupling, we obtain an identity that relates the sample paths of these two processes. This identity helps us to better understand the queue's approach to the stationary limit and to derive upper and lower bounds on the expected transient delay. In addition, we use a Brownian-motion model to approximate the identity. This produces an approximation of the expected transient delay. The approximation turns out to be identical to the corresponding first moment of a reflected Brownian motion. Thus, it is easy to compute and its accuracy is supported by numerical experiments.