  A single queueing system with time-dependent exponentially distributed
arrival processes and exponential machine processes (Kendall notation
$M_t/M_t/1$) is analyzed. Modeling the time evolution for the discrete
queue-length distribution by a continuous drift-diffusion process a
Smoluchowski equation on the half space is derived approximating the forward
Kolmogorov equations. The approximate model is analyzed and validated, showing
excellent agreement for the probabilities of all queue lengths and for all
queuing utilizations, including ones that are very small and some that are
significantly larger than one. Having an excellent approximation for the
probability of an empty queue generates an approximation of the expected
outflow of the queueing system. Comparisons to several well-established
approximation from the literature show significant improvements in several
numerical examples.
