Continuous approximation of mt=mt=1 distributions with application to production

Dieter Armbruster, Simone Göttlich, Stephan Knapp

Research output: Contribution to journalArticlepeer-review


A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation Mt=Mt=1) is analyzed. Modeling the time evolution for the discrete queuelength 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 approximations from the literature show significant improvements in several numerical examples.

Original languageEnglish (US)
Pages (from-to)243-269
Number of pages27
JournalJournal of Computational Dynamics
Issue number2
StatePublished - 2020


  • Approximate model
  • Production
  • Queueing theory

ASJC Scopus subject areas

  • Computational Mechanics
  • Computational Mathematics


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