Exact time-dependent solutions for the M/D/1 queue

Jung Woo Baek, Ho Woo Lee, Soohan Ahn, Yun Han Bae

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Time-dependent solutions to queuing models are very useful for evaluating the performance of real-world systems. However, because of their mathematical complexity, few available results exist. In this paper, we derive the time-dependent performance measures for an M/D/1 queue starting with a positive number of initial customers. Using the limiting property of an Erlang distribution, we obtain closed-form time-dependent formulas for the queue length and the waiting time. Furthermore, the time-dependent queue length probability in a busy period is derived.

Original languageEnglish
Pages (from-to)692-695
Number of pages4
JournalOperations Research Letters
Issue number5
StatePublished - 1 Sep 2016


  • Closed-form solution
  • M/D/1 queue
  • Time-dependent queue length probability
  • Time-dependent waiting time distribution


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