Exact and asymptotic solutions to a PDE that arises in time-dependent queues

We consider a diffusing particle in one dimension that is subject to a time -dependent drift or potential field. A reflecting barrier constrains the pa rticle's position to the half-line X greater than or equal to 0. Such model s arise naturally in the study of queues with time-dependent arrival rates, as well as in advection-diffusion problems of mathematical physics. We sol ve for the probability distribution of the particle as a function of space and time. Then we do a detailed study of the asymptotic properties of the s olution, for various ranges of space and time. We also relate our asymptoti c results to those obtained by probabilistic approaches, such as central li mit theorems and large deviations. We consider drifts that are either piece wise constant or Linear functions of time.