Queues with Poisson Arrivals

  • Philippe Robert
Part of the Applications of Mathematics book series (SMAP, volume 52)


Throughout this chapter, the arrival process is assumed to be a marked Poisson point process. See Proposition 1.11 page 11 and Section 1.3.2 page 18 for the definition and the main properties of Poisson marked point processes. In this setting, four queueing models are analyzed: The queue with an infinite number of servers (the M/G/∞ queue) and the single server queue with the following service disciplines: FIFO, LIFO and Processor-Sharing. The Processor-Sharing queue receives a detailed treatment because of the central role played by an interesting branching process in the derivation of the distribution of the sojourn time. It is also an important discipline in modern stochastic models of communication networks. The last section is devoted to a common, important property of queues having a Poisson input.


Markov Process Point Process Sojourn Time Busy Period Infinitesimal Generator 
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    S.F. Yashkov, A derivation of response time distribution for an MICR processor-sharing queue, Problems of Control and Information Theory 12 (1983), no. 2, 133–148.Google Scholar
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    Op. cit. page 182.Google Scholar
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    D.Y. Burman, Insensitivity in queueing systems, Advances in Applied Probability 13 (1981), 846–859.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Philippe Robert
    • 1
  1. 1.Domaine de VoluceauINRIALe ChesnayFrance

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