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1 results

  • Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

  • Part of
  • Hans Daduna, Christian Malchin, Ryszard Szekli
  • Journal:
    Journal of Applied Probability / Volume 45 / Issue 2 / June 2008
    Published online by Cambridge University Press:
    14 July 2016, pp. 333-346
    Print publication:
    June 2008
    • Article
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  • We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.