  We consider the Erlang A model, or $M/M/m+M$ queue, with Poisson arrivals,
exponential service times, and $m$ parallel servers, and the property that
waiting customers abandon the queue after an exponential time. The queue length
process is in this case a birth-death process, for which we obtain explicit
expressions for the Laplace transforms of the time-dependent distribution and
the first passage time. These two transient characteristics were generally
presumed to be intractable. Solving for the Laplace transforms involves using
Green's functions and contour integrals related to hypergeometric functions.
Our results are specialized to the $M/M/\infty$ queue, the $M/M/m$ queue, and
the $M/M/m/m$ loss model. We also obtain some corresponding results for
diffusion approximations to these models.
