Queueing Systems

, Volume 54, Issue 4, pp 237–241 | Cite as

On the distribution of the number of customers in the symmetric M/G/1 queue

  • Denis Denisov
  • Artëm Sapozhnikov


We consider an M/G/1 queue with symmetric service discipline. The class of symmetric service disciplines contains, in particular, the preemptive last-come-first-served discipline and the processor-sharing discipline. It has been conjectured in Kella et al. [1] that the marginal distribution of the queue length at any time is identical for all symmetric disciplines if the queue starts empty. In this paper we show that this conjecture is true if service requirements have an Erlang distribution. We also show by a counterexample, involving the hyperexponential distribution, that the conjecture is generally not true.


Symmetric queue Time-dependent analysis Insensitivity Processor-sharing queue Last come first served queue 


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  1. 1.
    O. Kella, B. Zwart, and O. Boxma, Some time-dependent properties of symmetric M/G/1 queues. J. Appl. Prob. 42 (2005) 223–234.CrossRefGoogle Scholar
  2. 2.
    F.P. Kelly, Reversibility and Stochastic Networks (John Wiley, Chichester, 1979).Google Scholar
  3. 3.
    M.Yu. Kitaev, The M/G/1 processor-sharing model: transient behavior. Queueing Systems 14 (1993) 239–273.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.EURANDOMEindhovenThe Netherlands
  2. 2.Department of Mathematics/Boole Centre for Research in InformaticsUniversity College CorkCorkIreland

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