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A single queueing system with time-dependent exponentially distributedarrival processes and exponential machine processes (Kendall notation$M_t/M_t/1$) is analyzed. Modeling the time evolution for the discretequeue-length distribution by a continuous drift-diffusion process aSmoluchowski equation on the half space is derived approximating the forwardKolmogorov equations. The approximate model is analyzed and validated, showingexcellent agreement for the probabilities of all queue lengths and for allqueuing utilizations, including ones that are very small and some that aresignificantly larger than one. Having an excellent approximation for theprobability of an empty queue generates an approximation of the expectedoutflow of the queueing system. Comparisons to several well-establishedapproximation from the literature show significant improvements in severalnumerical examples.

Continuous approximation of $M_t/M_t/1$ distributions with application to production